""" This code is automatically generated. Never edit it manually. For details of generating the code see `rubi_parsing_guide.md` in `parsetools`. """ from sympy.external import import_module matchpy = import_module("matchpy") if matchpy: from matchpy import Pattern, ReplacementRule, CustomConstraint, is_match from sympy.integrals.rubi.utility_function import ( Int, Sum, Set, With, Module, Scan, MapAnd, FalseQ, ZeroQ, NegativeQ, NonzeroQ, FreeQ, NFreeQ, List, Log, PositiveQ, PositiveIntegerQ, NegativeIntegerQ, IntegerQ, IntegersQ, ComplexNumberQ, PureComplexNumberQ, RealNumericQ, PositiveOrZeroQ, NegativeOrZeroQ, FractionOrNegativeQ, NegQ, Equal, Unequal, IntPart, FracPart, RationalQ, ProductQ, SumQ, NonsumQ, Subst, First, Rest, SqrtNumberQ, SqrtNumberSumQ, LinearQ, Sqrt, ArcCosh, Coefficient, Denominator, Hypergeometric2F1, Not, Simplify, FractionalPart, IntegerPart, AppellF1, EllipticPi, EllipticE, EllipticF, ArcTan, ArcCot, ArcCoth, ArcTanh, ArcSin, ArcSinh, ArcCos, ArcCsc, ArcSec, ArcCsch, ArcSech, Sinh, Tanh, Cosh, Sech, Csch, Coth, LessEqual, Less, Greater, GreaterEqual, FractionQ, IntLinearcQ, Expand, IndependentQ, PowerQ, IntegerPowerQ, PositiveIntegerPowerQ, FractionalPowerQ, AtomQ, ExpQ, LogQ, Head, MemberQ, TrigQ, SinQ, CosQ, TanQ, CotQ, SecQ, CscQ, Sin, Cos, Tan, Cot, Sec, Csc, HyperbolicQ, SinhQ, CoshQ, TanhQ, CothQ, SechQ, CschQ, InverseTrigQ, SinCosQ, SinhCoshQ, LeafCount, Numerator, NumberQ, NumericQ, Length, ListQ, Im, Re, InverseHyperbolicQ, InverseFunctionQ, TrigHyperbolicFreeQ, InverseFunctionFreeQ, RealQ, EqQ, FractionalPowerFreeQ, ComplexFreeQ, PolynomialQ, FactorSquareFree, PowerOfLinearQ, Exponent, QuadraticQ, LinearPairQ, BinomialParts, TrinomialParts, PolyQ, EvenQ, OddQ, PerfectSquareQ, NiceSqrtAuxQ, NiceSqrtQ, Together, PosAux, PosQ, CoefficientList, ReplaceAll, ExpandLinearProduct, GCD, ContentFactor, NumericFactor, NonnumericFactors, MakeAssocList, GensymSubst, KernelSubst, ExpandExpression, Apart, SmartApart, MatchQ, PolynomialQuotientRemainder, FreeFactors, NonfreeFactors, RemoveContentAux, RemoveContent, FreeTerms, NonfreeTerms, ExpandAlgebraicFunction, CollectReciprocals, ExpandCleanup, AlgebraicFunctionQ, Coeff, LeadTerm, RemainingTerms, LeadFactor, RemainingFactors, LeadBase, LeadDegree, Numer, Denom, hypergeom, Expon, MergeMonomials, PolynomialDivide, BinomialQ, TrinomialQ, GeneralizedBinomialQ, GeneralizedTrinomialQ, FactorSquareFreeList, PerfectPowerTest, SquareFreeFactorTest, RationalFunctionQ, RationalFunctionFactors, NonrationalFunctionFactors, Reverse, RationalFunctionExponents, RationalFunctionExpand, ExpandIntegrand, SimplerQ, SimplerSqrtQ, SumSimplerQ, BinomialDegree, TrinomialDegree, CancelCommonFactors, SimplerIntegrandQ, GeneralizedBinomialDegree, GeneralizedBinomialParts, GeneralizedTrinomialDegree, GeneralizedTrinomialParts, MonomialQ, MonomialSumQ, MinimumMonomialExponent, MonomialExponent, LinearMatchQ, PowerOfLinearMatchQ, QuadraticMatchQ, CubicMatchQ, BinomialMatchQ, TrinomialMatchQ, GeneralizedBinomialMatchQ, GeneralizedTrinomialMatchQ, QuotientOfLinearsMatchQ, PolynomialTermQ, PolynomialTerms, NonpolynomialTerms, PseudoBinomialParts, NormalizePseudoBinomial, PseudoBinomialPairQ, PseudoBinomialQ, PolynomialGCD, PolyGCD, AlgebraicFunctionFactors, NonalgebraicFunctionFactors, QuotientOfLinearsP, QuotientOfLinearsParts, QuotientOfLinearsQ, Flatten, Sort, AbsurdNumberQ, AbsurdNumberFactors, NonabsurdNumberFactors, SumSimplerAuxQ, Prepend, Drop, CombineExponents, FactorInteger, FactorAbsurdNumber, SubstForInverseFunction, SubstForFractionalPower, SubstForFractionalPowerOfQuotientOfLinears, FractionalPowerOfQuotientOfLinears, SubstForFractionalPowerQ, SubstForFractionalPowerAuxQ, FractionalPowerOfSquareQ, FractionalPowerSubexpressionQ, Apply, FactorNumericGcd, MergeableFactorQ, MergeFactor, MergeFactors, TrigSimplifyQ, TrigSimplify, TrigSimplifyRecur, Order, FactorOrder, Smallest, OrderedQ, MinimumDegree, PositiveFactors, Sign, NonpositiveFactors, PolynomialInAuxQ, PolynomialInQ, ExponentInAux, ExponentIn, PolynomialInSubstAux, PolynomialInSubst, Distrib, DistributeDegree, FunctionOfPower, DivideDegreesOfFactors, MonomialFactor, FullSimplify, FunctionOfLinearSubst, FunctionOfLinear, NormalizeIntegrand, NormalizeIntegrandAux, NormalizeIntegrandFactor, NormalizeIntegrandFactorBase, NormalizeTogether, NormalizeLeadTermSigns, AbsorbMinusSign, NormalizeSumFactors, SignOfFactor, NormalizePowerOfLinear, SimplifyIntegrand, SimplifyTerm, TogetherSimplify, SmartSimplify, SubstForExpn, ExpandToSum, UnifySum, UnifyTerms, UnifyTerm, CalculusQ, FunctionOfInverseLinear, PureFunctionOfSinhQ, PureFunctionOfTanhQ, PureFunctionOfCoshQ, IntegerQuotientQ, OddQuotientQ, EvenQuotientQ, FindTrigFactor, FunctionOfSinhQ, FunctionOfCoshQ, OddHyperbolicPowerQ, FunctionOfTanhQ, FunctionOfTanhWeight, FunctionOfHyperbolicQ, SmartNumerator, SmartDenominator, SubstForAux, ActivateTrig, ExpandTrig, TrigExpand, SubstForTrig, SubstForHyperbolic, InertTrigFreeQ, LCM, SubstForFractionalPowerOfLinear, FractionalPowerOfLinear, InverseFunctionOfLinear, InertTrigQ, InertReciprocalQ, DeactivateTrig, FixInertTrigFunction, DeactivateTrigAux, PowerOfInertTrigSumQ, PiecewiseLinearQ, KnownTrigIntegrandQ, KnownSineIntegrandQ, KnownTangentIntegrandQ, KnownCotangentIntegrandQ, KnownSecantIntegrandQ, TryPureTanSubst, TryTanhSubst, TryPureTanhSubst, AbsurdNumberGCD, AbsurdNumberGCDList, ExpandTrigExpand, ExpandTrigReduce, ExpandTrigReduceAux, NormalizeTrig, TrigToExp, ExpandTrigToExp, TrigReduce, FunctionOfTrig, AlgebraicTrigFunctionQ, FunctionOfHyperbolic, FunctionOfQ, FunctionOfExpnQ, PureFunctionOfSinQ, PureFunctionOfCosQ, PureFunctionOfTanQ, PureFunctionOfCotQ, FunctionOfCosQ, FunctionOfSinQ, OddTrigPowerQ, FunctionOfTanQ, FunctionOfTanWeight, FunctionOfTrigQ, FunctionOfDensePolynomialsQ, FunctionOfLog, PowerVariableExpn, PowerVariableDegree, PowerVariableSubst, EulerIntegrandQ, FunctionOfSquareRootOfQuadratic, SquareRootOfQuadraticSubst, Divides, EasyDQ, ProductOfLinearPowersQ, Rt, NthRoot, AtomBaseQ, SumBaseQ, NegSumBaseQ, AllNegTermQ, SomeNegTermQ, TrigSquareQ, RtAux, TrigSquare, IntSum, IntTerm, Map2, ConstantFactor, SameQ, ReplacePart, CommonFactors, MostMainFactorPosition, FunctionOfExponentialQ, FunctionOfExponential, FunctionOfExponentialFunction, FunctionOfExponentialFunctionAux, FunctionOfExponentialTest, FunctionOfExponentialTestAux, stdev, rubi_test, If, IntQuadraticQ, IntBinomialQ, RectifyTangent, RectifyCotangent, Inequality, Condition, Simp, SimpHelp, SplitProduct, SplitSum, SubstFor, SubstForAux, FresnelS, FresnelC, Erfc, Erfi, Gamma, FunctionOfTrigOfLinearQ, ElementaryFunctionQ, Complex, UnsameQ, _SimpFixFactor, SimpFixFactor, _FixSimplify, FixSimplify, _SimplifyAntiderivativeSum, SimplifyAntiderivativeSum, _SimplifyAntiderivative, SimplifyAntiderivative, _TrigSimplifyAux, TrigSimplifyAux, Cancel, Part, PolyLog, D, Dist, Sum_doit, PolynomialQuotient, Floor, PolynomialRemainder, Factor, PolyLog, CosIntegral, SinIntegral, LogIntegral, SinhIntegral, CoshIntegral, Rule, Erf, PolyGamma, ExpIntegralEi, ExpIntegralE, LogGamma , UtilityOperator, Factorial, Zeta, ProductLog, DerivativeDivides, HypergeometricPFQ, IntHide, OneQ, Null, rubi_exp as exp, rubi_log as log, Discriminant, Negative, Quotient ) from sympy.core.add import Add from sympy.core.mod import Mod from sympy.core.mul import Mul from sympy.core import EulerGamma from sympy.core.numbers import (Float, I, Integer) from sympy.core.power import Pow from sympy.core.singleton import S from sympy.functions.elementary.complexes import (Abs, sign) from sympy.functions.elementary.miscellaneous import sqrt from sympy.integrals.integrals import Integral from sympy.logic.boolalg import (And, Or) from sympy.simplify.simplify import simplify from sympy.integrals.rubi.symbol import WC from sympy.core.symbol import symbols, Symbol from sympy.functions import (sin, cos, tan, cot, csc, sec, sqrt, erf) from sympy.functions.elementary.hyperbolic import (acosh, asinh, atanh, acoth, acsch, asech, cosh, sinh, tanh, coth, sech, csch) from sympy.functions.elementary.trigonometric import (atan, acsc, asin, acot, acos, asec, atan2) from sympy.core.numbers import pi as Pi A_, B_, C_, F_, G_, H_, a_, b_, c_, d_, e_, f_, g_, h_, i_, j_, k_, l_, m_, n_, p_, q_, r_, t_, u_, v_, s_, w_, x_, y_, z_ = [WC(i) for i in 'ABCFGHabcdefghijklmnpqrtuvswxyz'] a1_, a2_, b1_, b2_, c1_, c2_, d1_, d2_, n1_, n2_, e1_, e2_, f1_, f2_, g1_, g2_, n1_, n2_, n3_, Pq_, Pm_, Px_, Qm_, Qr_, Qx_, jn_, mn_, non2_, RFx_, RGx_ = [WC(i) for i in ['a1', 'a2', 'b1', 'b2', 'c1', 'c2', 'd1', 'd2', 'n1', 'n2', 'e1', 'e2', 'f1', 'f2', 'g1', 'g2', 'n1', 'n2', 'n3', 'Pq', 'Pm', 'Px', 'Qm', 'Qr', 'Qx', 'jn', 'mn', 'non2', 'RFx', 'RGx']] i, ii, Pqq, Q, R, r, C, k, u = symbols('i ii Pqq Q R r C k u') _UseGamma = False ShowSteps = False StepCounter = None def sine(): from sympy.integrals.rubi.constraints import cons1251, cons74, cons68, cons2, cons3, cons50, cons127, cons19, cons4, cons1252, cons1253, cons1254, cons95, cons168, cons91, cons1255, cons33, cons1172, cons96, cons1256, cons1257, cons1258, cons1259, cons1260, cons1261, cons167, cons1262, cons21, cons25, cons523, cons8, cons29, cons810, cons1263, cons1264, cons1265, cons545, cons1266, cons1267, cons150, cons1268, cons89, cons45, cons450, cons1269, cons1270, cons1271, cons1272, cons1273, cons483, cons484, cons1274, cons1275, cons1276, cons1277, cons1278, cons210, cons5, cons20, cons13, cons139, cons1279, cons1280, cons1281, cons1282, cons1283, cons145, cons1284, cons1285, cons1286, cons1287, cons246, cons170, cons1288, cons1289, cons1290, cons148, cons1291, cons1292, cons1293, cons248, cons1294, cons1295, cons1296, cons1297, cons1298, cons86, cons1299, cons149, cons56, cons1300, cons1301, cons1302, cons1303, cons1304, cons64, cons269, cons1305, cons1306, cons1307, cons1308, cons517, cons274, cons1309, cons1310, cons73, cons72, cons1311, cons1312, cons1313, cons1314, cons348, cons1315, cons157, cons1316, cons1317, cons116, cons1318, cons1319, cons1320, cons1321, cons1322, cons1323, cons79, cons1324, cons1325, cons1326, cons1327, cons1328, cons1329, cons1330, cons465, cons90, cons1331, cons1332, cons87, cons1333, cons1334, cons1335, cons1336, cons1337, cons1338, cons1339, cons1340, cons1341, cons1342, cons1343, cons1344, cons1345, cons1346, cons1347, cons1348, cons1349, cons1350, cons1351, cons1352, cons1353, cons1354, cons1355, cons1356, cons1357, cons1358, cons1359, cons1360, cons1361, cons1362, cons1363, cons1364, cons1365, cons1366, cons144, cons337, cons1367, cons1368, cons1369, cons1370, cons1371, cons1372, cons172, cons1373, cons1374, cons1375, cons1376, cons1377, cons255, cons1378, cons1379, cons1380, cons1381, cons1382, cons360, cons1383, cons1384, cons1385, cons1386, cons1387, cons1388, cons1389, cons1390, cons1391, cons1392, cons1393, cons1394, cons1395, cons1396, cons215, cons586, cons1397, cons1398, cons1399, cons1400, cons1401, cons1402, cons1403, cons1404, cons1405, cons1406, cons1407, cons1408, cons1409, cons1410, cons1411, cons1412, cons1413, cons107, cons1414, cons1415, cons1416, cons1417, cons1418, cons1419, cons40, cons1420, cons36, cons37, cons1421, cons1422, cons1423, cons685, cons1424, cons1425, cons1426, cons1427, cons1428, cons1429, cons1430, cons1431, cons1432, cons1433, cons38, cons1230, cons1434, cons35, cons1435, cons1436, cons1437, cons1438, cons216, cons1439, cons1440, cons1441, cons1442, cons1443, cons1444, cons1445, cons1446, cons1447, cons1448, cons1154, cons1449, cons198, cons130, cons65, cons152, cons377, cons324, cons1450, cons1451, cons1452, cons1453, cons1454, cons1455, cons1456, cons78, cons1457, cons1458, cons1459, cons1460, cons1461, cons1462, cons1463, cons1464, cons1465, cons1466, cons1467, cons1468, cons1469, cons1470, cons1471, cons1472, cons1473, cons1247, cons1474, cons1475, cons1476, cons1477, cons1478, cons1479, cons1045, cons1480, cons1481, cons1482, cons1483, cons1484, cons1485, cons1486, cons1487, cons1488, cons1489, cons48, cons47, cons228, cons378, cons1118, cons178, cons1490, cons1491, cons247, cons249, cons1492, cons1493, cons1494, cons1495, cons812, cons813, cons746, cons1496, cons1497, cons55, cons598, cons1498, cons1499, cons491, cons1500, cons70, cons71, cons825, cons826, cons1501, cons1502, cons1503, cons1504, cons58, cons1505, cons1506, cons369, cons1507, cons358, cons856, cons1508, cons820, cons1133, cons49, cons241, cons1134, cons1135, cons1509, cons821 pattern2167 = Pattern(Integral(u_, x_), cons1251) rule2167 = ReplacementRule(pattern2167, replacement2167) pattern2168 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons50, cons127, cons19, cons4, cons74, cons68) rule2168 = ReplacementRule(pattern2168, replacement2168) pattern2169 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**WC('n', S(1)), x_), cons2, cons50, cons127, cons19, cons1252, cons1253) rule2169 = ReplacementRule(pattern2169, replacement2169) pattern2170 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**WC('n', S(1)), x_), cons2, cons50, cons127, cons19, cons1252, cons1254) rule2170 = ReplacementRule(pattern2170, replacement2170) pattern2171 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons95, cons168, cons91, cons1255) rule2171 = ReplacementRule(pattern2171, replacement2171) pattern2172 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons95, cons168, cons91, cons1255) rule2172 = ReplacementRule(pattern2172, replacement2172) pattern2173 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons4, cons33, cons168, cons1172) rule2173 = ReplacementRule(pattern2173, replacement2173) pattern2174 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons4, cons33, cons168, cons1172) rule2174 = ReplacementRule(pattern2174, replacement2174) pattern2175 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons4, cons33, cons96, cons1172) rule2175 = ReplacementRule(pattern2175, replacement2175) pattern2176 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons4, cons33, cons96, cons1172) rule2176 = ReplacementRule(pattern2176, replacement2176) pattern2177 = Pattern(Integral(sqrt(WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons1256) rule2177 = ReplacementRule(pattern2177, replacement2177) pattern2178 = Pattern(Integral(S(1)/(sqrt(WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons50, cons127, cons1256) rule2178 = ReplacementRule(pattern2178, replacement2178) pattern2179 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons1257, cons33, cons1258) rule2179 = ReplacementRule(pattern2179, With2179) pattern2180 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons1257, cons33, cons1258) rule2180 = ReplacementRule(pattern2180, With2180) pattern2181 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons19, cons4, cons1259) rule2181 = ReplacementRule(pattern2181, replacement2181) pattern2182 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons19, cons4, cons1260) rule2182 = ReplacementRule(pattern2182, replacement2182) pattern2183 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons50, cons127, cons19, cons4, cons1261, cons68) rule2183 = ReplacementRule(pattern2183, replacement2183) pattern2184 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons50, cons127, cons19, cons4, cons1261, cons68) rule2184 = ReplacementRule(pattern2184, replacement2184) pattern2185 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons95, cons168, cons167, cons1172) rule2185 = ReplacementRule(pattern2185, replacement2185) pattern2186 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons95, cons168, cons167, cons1172) rule2186 = ReplacementRule(pattern2186, replacement2186) pattern2187 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons4, cons33, cons168, cons1262, cons1172) rule2187 = ReplacementRule(pattern2187, replacement2187) pattern2188 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons4, cons33, cons168, cons1262, cons1172) rule2188 = ReplacementRule(pattern2188, replacement2188) pattern2189 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons4, cons33, cons96, cons1172) rule2189 = ReplacementRule(pattern2189, replacement2189) pattern2190 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons4, cons33, cons96, cons1172) rule2190 = ReplacementRule(pattern2190, replacement2190) pattern2191 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons19, cons4, cons21, cons25) rule2191 = ReplacementRule(pattern2191, replacement2191) pattern2192 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons19, cons4, cons21, cons25) rule2192 = ReplacementRule(pattern2192, replacement2192) pattern2193 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons19, cons4, cons21, cons25) rule2193 = ReplacementRule(pattern2193, replacement2193) pattern2194 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons19, cons4, cons21, cons25) rule2194 = ReplacementRule(pattern2194, replacement2194) pattern2195 = Pattern(Integral(sin(x_*WC('d', S(1)) + WC('c', S(0)))**n_, x_), cons8, cons29, cons523) rule2195 = ReplacementRule(pattern2195, replacement2195) pattern2196 = Pattern(Integral(cos(x_*WC('d', S(1)) + WC('c', S(0)))**n_, x_), cons8, cons29, cons523) rule2196 = ReplacementRule(pattern2196, replacement2196) pattern2197 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons3, cons8, cons29, cons810, cons167) rule2197 = ReplacementRule(pattern2197, replacement2197) pattern2198 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons3, cons8, cons29, cons810, cons167) rule2198 = ReplacementRule(pattern2198, replacement2198) pattern2199 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons3, cons8, cons29, cons810, cons91) rule2199 = ReplacementRule(pattern2199, replacement2199) pattern2200 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons3, cons8, cons29, cons810, cons91) rule2200 = ReplacementRule(pattern2200, replacement2200) pattern2201 = Pattern(Integral(sin(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons8, cons29, cons1263) rule2201 = ReplacementRule(pattern2201, replacement2201) pattern2202 = Pattern(Integral(cos(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons8, cons8, cons1264) rule2202 = ReplacementRule(pattern2202, replacement2202) pattern2203 = Pattern(Integral(sqrt(sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons8, cons29, cons1263) rule2203 = ReplacementRule(pattern2203, replacement2203) pattern2204 = Pattern(Integral(sqrt(cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons8, cons29, cons1263) rule2204 = ReplacementRule(pattern2204, replacement2204) pattern2205 = Pattern(Integral(sqrt(b_*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons3, cons8, cons29, cons1265) rule2205 = ReplacementRule(pattern2205, replacement2205) pattern2206 = Pattern(Integral(sqrt(b_*cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons3, cons8, cons29, cons1265) rule2206 = ReplacementRule(pattern2206, replacement2206) pattern2207 = Pattern(Integral(S(1)/sqrt(sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons8, cons29, cons1263) rule2207 = ReplacementRule(pattern2207, replacement2207) pattern2208 = Pattern(Integral(S(1)/sqrt(cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons8, cons29, cons1263) rule2208 = ReplacementRule(pattern2208, replacement2208) pattern2209 = Pattern(Integral(S(1)/sqrt(b_*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons3, cons8, cons29, cons1265) rule2209 = ReplacementRule(pattern2209, replacement2209) pattern2210 = Pattern(Integral(S(1)/sqrt(b_*cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons3, cons8, cons29, cons1265) rule2210 = ReplacementRule(pattern2210, replacement2210) pattern2211 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons3, cons8, cons29, cons4, cons545) rule2211 = ReplacementRule(pattern2211, replacement2211) pattern2212 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons3, cons8, cons29, cons4, cons545) rule2212 = ReplacementRule(pattern2212, replacement2212) pattern2213 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**S(2), x_), cons2, cons3, cons8, cons29, cons1266) rule2213 = ReplacementRule(pattern2213, replacement2213) pattern2214 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))))**S(2), x_), cons2, cons3, cons8, cons29, cons1266) rule2214 = ReplacementRule(pattern2214, replacement2214) pattern2215 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons4, cons1267, cons150) rule2215 = ReplacementRule(pattern2215, replacement2215) pattern2216 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons4, cons1267, cons150) rule2216 = ReplacementRule(pattern2216, replacement2216) pattern2217 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1267) rule2217 = ReplacementRule(pattern2217, replacement2217) pattern2218 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1267) rule2218 = ReplacementRule(pattern2218, replacement2218) pattern2219 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons1267, cons1268) rule2219 = ReplacementRule(pattern2219, replacement2219) pattern2220 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons1267, cons1268) rule2220 = ReplacementRule(pattern2220, replacement2220) pattern2221 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1267) rule2221 = ReplacementRule(pattern2221, replacement2221) pattern2222 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1267) rule2222 = ReplacementRule(pattern2222, replacement2222) pattern2223 = Pattern(Integral(S(1)/sqrt(a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1267) rule2223 = ReplacementRule(pattern2223, replacement2223) pattern2224 = Pattern(Integral(S(1)/sqrt(a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1267) rule2224 = ReplacementRule(pattern2224, replacement2224) pattern2225 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons1267, cons89, cons91, cons810) rule2225 = ReplacementRule(pattern2225, replacement2225) pattern2226 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons1267, cons89, cons91, cons810) rule2226 = ReplacementRule(pattern2226, replacement2226) pattern2227 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons4, cons1267, cons545, cons45) rule2227 = ReplacementRule(pattern2227, replacement2227) pattern2228 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons4, cons1267, cons545, cons45) rule2228 = ReplacementRule(pattern2228, replacement2228) pattern2229 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons4, cons1267, cons545, cons450) rule2229 = ReplacementRule(pattern2229, replacement2229) pattern2230 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons4, cons1267, cons545, cons450) rule2230 = ReplacementRule(pattern2230, replacement2230) pattern2231 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1269, cons1270) rule2231 = ReplacementRule(pattern2231, replacement2231) pattern2232 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1269, cons1270) rule2232 = ReplacementRule(pattern2232, replacement2232) pattern2233 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1269, cons1271) rule2233 = ReplacementRule(pattern2233, replacement2233) pattern2234 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1269, cons1271) rule2234 = ReplacementRule(pattern2234, replacement2234) pattern2235 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1269, cons1272) rule2235 = ReplacementRule(pattern2235, replacement2235) pattern2236 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1269, cons1272) rule2236 = ReplacementRule(pattern2236, replacement2236) pattern2237 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons1269, cons89, cons167, cons810) rule2237 = ReplacementRule(pattern2237, replacement2237) pattern2238 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons1269, cons89, cons167, cons810) rule2238 = ReplacementRule(pattern2238, replacement2238) pattern2239 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1273, cons483) rule2239 = ReplacementRule(pattern2239, With2239) pattern2240 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1273, cons483) rule2240 = ReplacementRule(pattern2240, With2240) pattern2241 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1273, cons484) rule2241 = ReplacementRule(pattern2241, With2241) pattern2242 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1273, cons484) rule2242 = ReplacementRule(pattern2242, With2242) pattern2243 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1269, cons1274) rule2243 = ReplacementRule(pattern2243, With2243) pattern2244 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1269) rule2244 = ReplacementRule(pattern2244, With2244) pattern2245 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1269) rule2245 = ReplacementRule(pattern2245, With2245) pattern2246 = Pattern(Integral(S(1)/sqrt(a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1269, cons1270) rule2246 = ReplacementRule(pattern2246, replacement2246) pattern2247 = Pattern(Integral(S(1)/sqrt(a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1269, cons1270) rule2247 = ReplacementRule(pattern2247, replacement2247) pattern2248 = Pattern(Integral(S(1)/sqrt(a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1269, cons1271) rule2248 = ReplacementRule(pattern2248, replacement2248) pattern2249 = Pattern(Integral(S(1)/sqrt(a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1269, cons1271) rule2249 = ReplacementRule(pattern2249, replacement2249) pattern2250 = Pattern(Integral(S(1)/sqrt(a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1269, cons1272) rule2250 = ReplacementRule(pattern2250, replacement2250) pattern2251 = Pattern(Integral(S(1)/sqrt(a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1269, cons1272) rule2251 = ReplacementRule(pattern2251, replacement2251) pattern2252 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons1269, cons89, cons91, cons810) rule2252 = ReplacementRule(pattern2252, replacement2252) pattern2253 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons1269, cons89, cons91, cons810) rule2253 = ReplacementRule(pattern2253, replacement2253) pattern2254 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons4, cons1269, cons545) rule2254 = ReplacementRule(pattern2254, replacement2254) pattern2255 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons4, cons1269, cons545) rule2255 = ReplacementRule(pattern2255, replacement2255) pattern2256 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))*cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons4, cons1275) rule2256 = ReplacementRule(pattern2256, replacement2256) pattern2257 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons50, cons127, cons19, cons1276, cons1267, cons1277) rule2257 = ReplacementRule(pattern2257, replacement2257) pattern2258 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons50, cons127, cons19, cons1276, cons1267, cons1277) rule2258 = ReplacementRule(pattern2258, replacement2258) pattern2259 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons50, cons127, cons19, cons1276, cons1269) rule2259 = ReplacementRule(pattern2259, replacement2259) pattern2260 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons50, cons127, cons19, cons1276, cons1269) rule2260 = ReplacementRule(pattern2260, replacement2260) pattern2261 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons210, cons5, cons1278) rule2261 = ReplacementRule(pattern2261, replacement2261) pattern2262 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons210, cons5, cons1278) rule2262 = ReplacementRule(pattern2262, replacement2262) pattern2263 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons1267, cons20, cons13, cons139, cons1279) rule2263 = ReplacementRule(pattern2263, replacement2263) pattern2264 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons1267, cons20, cons13, cons139, cons1279) rule2264 = ReplacementRule(pattern2264, replacement2264) pattern2265 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons19, cons5, cons1267, cons1280, cons1281) rule2265 = ReplacementRule(pattern2265, replacement2265) pattern2266 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons19, cons5, cons1267, cons1280, cons1281) rule2266 = ReplacementRule(pattern2266, replacement2266) pattern2267 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons19, cons5, cons1267, cons1282, cons1283, cons145) rule2267 = ReplacementRule(pattern2267, replacement2267) pattern2268 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons19, cons5, cons1267, cons1282, cons1283, cons145) rule2268 = ReplacementRule(pattern2268, replacement2268) pattern2269 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons19, cons5, cons1267, cons1284, cons1285) rule2269 = ReplacementRule(pattern2269, replacement2269) pattern2270 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons19, cons5, cons1267, cons1284, cons1285) rule2270 = ReplacementRule(pattern2270, replacement2270) pattern2271 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons19, cons5, cons1267, cons1286, cons1287) rule2271 = ReplacementRule(pattern2271, replacement2271) pattern2272 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons19, cons5, cons1267, cons1286, cons1287) rule2272 = ReplacementRule(pattern2272, replacement2272) pattern2273 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons1267, cons246, cons170, cons1288, cons1289) rule2273 = ReplacementRule(pattern2273, replacement2273) pattern2274 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons1267, cons246, cons170, cons1288, cons1289) rule2274 = ReplacementRule(pattern2274, replacement2274) pattern2275 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons1267, cons246, cons168, cons139, cons1290) rule2275 = ReplacementRule(pattern2275, replacement2275) pattern2276 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons1267, cons246, cons168, cons139, cons1290) rule2276 = ReplacementRule(pattern2276, replacement2276) pattern2277 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons210, cons1267) rule2277 = ReplacementRule(pattern2277, replacement2277) pattern2278 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons210, cons1267) rule2278 = ReplacementRule(pattern2278, replacement2278) pattern2279 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons19, cons5, cons1267, cons33, cons170, cons1287, cons1290) rule2279 = ReplacementRule(pattern2279, replacement2279) pattern2280 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons19, cons5, cons1267, cons33, cons170, cons1287, cons1290) rule2280 = ReplacementRule(pattern2280, replacement2280) pattern2281 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons1267, cons246, cons96, cons148, cons1291, cons1287, cons1290) rule2281 = ReplacementRule(pattern2281, replacement2281) pattern2282 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons1267, cons246, cons96, cons148, cons1291, cons1287, cons1290) rule2282 = ReplacementRule(pattern2282, replacement2282) pattern2283 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons1267, cons246, cons1292, cons148, cons1283, cons1293, cons1290) rule2283 = ReplacementRule(pattern2283, replacement2283) pattern2284 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons1267, cons246, cons1292, cons148, cons1283, cons1293, cons1290) rule2284 = ReplacementRule(pattern2284, replacement2284) pattern2285 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons19, cons5, cons1267, cons33, cons96, cons1283, cons1290) rule2285 = ReplacementRule(pattern2285, replacement2285) pattern2286 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons19, cons5, cons1267, cons33, cons96, cons1283, cons1290) rule2286 = ReplacementRule(pattern2286, replacement2286) pattern2287 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons210, cons1267, cons13, cons148, cons248) rule2287 = ReplacementRule(pattern2287, replacement2287) pattern2288 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons210, cons1267, cons13, cons148, cons248) rule2288 = ReplacementRule(pattern2288, replacement2288) pattern2289 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons210, cons5, cons1267, cons1294, cons248) rule2289 = ReplacementRule(pattern2289, replacement2289) pattern2290 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons210, cons5, cons1267, cons1294, cons248) rule2290 = ReplacementRule(pattern2290, replacement2290) pattern2291 = Pattern(Integral(sqrt(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons210, cons1267) rule2291 = ReplacementRule(pattern2291, replacement2291) pattern2292 = Pattern(Integral(sqrt(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons210, cons1267) rule2292 = ReplacementRule(pattern2292, replacement2292) pattern2293 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons210, cons1267) rule2293 = ReplacementRule(pattern2293, replacement2293) pattern2294 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons210, cons1267) rule2294 = ReplacementRule(pattern2294, replacement2294) pattern2295 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_/sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons210, cons1267, cons13, cons1295, cons248) rule2295 = ReplacementRule(pattern2295, replacement2295) pattern2296 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_/sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons210, cons1267, cons13, cons1295, cons248) rule2296 = ReplacementRule(pattern2296, replacement2296) pattern2297 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_/sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons210, cons1267, cons13, cons139, cons248) rule2297 = ReplacementRule(pattern2297, replacement2297) pattern2298 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_/sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons210, cons1267, cons13, cons139, cons248) rule2298 = ReplacementRule(pattern2298, replacement2298) pattern2299 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons50, cons127, cons210, cons5, cons1267, cons20) rule2299 = ReplacementRule(pattern2299, replacement2299) pattern2300 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons50, cons127, cons210, cons5, cons1267, cons20) rule2300 = ReplacementRule(pattern2300, replacement2300) pattern2301 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons50, cons127, cons210, cons19, cons5, cons1267, cons21) rule2301 = ReplacementRule(pattern2301, replacement2301) pattern2302 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons50, cons127, cons210, cons19, cons5, cons1267, cons21) rule2302 = ReplacementRule(pattern2302, replacement2302) pattern2303 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons1269, cons246, cons1258, cons139, cons1296) rule2303 = ReplacementRule(pattern2303, replacement2303) pattern2304 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons1269, cons246, cons1258, cons139, cons1296) rule2304 = ReplacementRule(pattern2304, replacement2304) pattern2305 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons1269, cons246, cons168, cons139, cons1296) rule2305 = ReplacementRule(pattern2305, replacement2305) pattern2306 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons1269, cons246, cons168, cons139, cons1296) rule2306 = ReplacementRule(pattern2306, replacement2306) pattern2307 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons5, cons1269, cons33, cons168, cons1287, cons1296) rule2307 = ReplacementRule(pattern2307, replacement2307) pattern2308 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons5, cons1269, cons33, cons168, cons1287, cons1296) rule2308 = ReplacementRule(pattern2308, replacement2308) pattern2309 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons1269, cons246, cons96, cons148, cons1290) rule2309 = ReplacementRule(pattern2309, replacement2309) pattern2310 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons1269, cons246, cons96, cons148, cons1290) rule2310 = ReplacementRule(pattern2310, replacement2310) pattern2311 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons5, cons1269, cons33, cons96, cons1290) rule2311 = ReplacementRule(pattern2311, replacement2311) pattern2312 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons5, cons1269, cons33, cons96, cons1290) rule2312 = ReplacementRule(pattern2312, replacement2312) pattern2313 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons19, cons1269, cons13, cons148, cons1287, cons1290) rule2313 = ReplacementRule(pattern2313, replacement2313) pattern2314 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons19, cons1269, cons13, cons148, cons1287, cons1290) rule2314 = ReplacementRule(pattern2314, replacement2314) pattern2315 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons19, cons1269, cons13, cons139, cons1290) rule2315 = ReplacementRule(pattern2315, replacement2315) pattern2316 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons19, cons1269, cons13, cons139, cons1290) rule2316 = ReplacementRule(pattern2316, replacement2316) pattern2317 = Pattern(Integral(S(1)/(sqrt(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons50, cons127, cons210, cons1269) rule2317 = ReplacementRule(pattern2317, replacement2317) pattern2318 = Pattern(Integral(S(1)/(sqrt(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons50, cons127, cons210, cons1269) rule2318 = ReplacementRule(pattern2318, replacement2318) pattern2319 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons19, cons5, cons1269, cons1280) rule2319 = ReplacementRule(pattern2319, replacement2319) pattern2320 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons19, cons5, cons1269, cons1280) rule2320 = ReplacementRule(pattern2320, replacement2320) pattern2321 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons19, cons5, cons1269, cons1297) rule2321 = ReplacementRule(pattern2321, replacement2321) pattern2322 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons19, cons5, cons1269, cons1297) rule2322 = ReplacementRule(pattern2322, replacement2322) pattern2323 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons19, cons5, cons1269, cons1298) rule2323 = ReplacementRule(pattern2323, replacement2323) pattern2324 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons19, cons5, cons1269, cons1298) rule2324 = ReplacementRule(pattern2324, replacement2324) pattern2325 = Pattern(Integral(sqrt(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons210, cons1269) rule2325 = ReplacementRule(pattern2325, With2325) pattern2326 = Pattern(Integral(sqrt(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons210, cons1269) rule2326 = ReplacementRule(pattern2326, With2326) pattern2327 = Pattern(Integral(S(1)/(sqrt(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons50, cons127, cons210, cons1269) rule2327 = ReplacementRule(pattern2327, With2327) pattern2328 = Pattern(Integral(S(1)/(sqrt(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons50, cons127, cons210, cons1269) rule2328 = ReplacementRule(pattern2328, With2328) pattern2329 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons5, cons1269, cons86, cons1299) rule2329 = ReplacementRule(pattern2329, replacement2329) pattern2330 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons5, cons1269, cons86, cons1299) rule2330 = ReplacementRule(pattern2330, replacement2330) pattern2331 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons19, cons5, cons1269, cons145) rule2331 = ReplacementRule(pattern2331, replacement2331) pattern2332 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons19, cons5, cons1269, cons145) rule2332 = ReplacementRule(pattern2332, replacement2332) pattern2333 = Pattern(Integral((WC('g', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons50, cons127, cons210, cons19, cons5, cons149) rule2333 = ReplacementRule(pattern2333, replacement2333) pattern2334 = Pattern(Integral((WC('g', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons50, cons127, cons210, cons19, cons5, cons149) rule2334 = ReplacementRule(pattern2334, replacement2334) pattern2335 = Pattern(Integral((WC('g', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons210, cons5, cons1267, cons56) rule2335 = ReplacementRule(pattern2335, replacement2335) pattern2336 = Pattern(Integral((WC('g', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons210, cons5, cons1267, cons56) rule2336 = ReplacementRule(pattern2336, replacement2336) pattern2337 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons50, cons127, cons19, cons1267, cons1300) rule2337 = ReplacementRule(pattern2337, replacement2337) pattern2338 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(S(1)/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons50, cons127, cons19, cons1267, cons1300) rule2338 = ReplacementRule(pattern2338, replacement2338) pattern2339 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons50, cons127, cons1267, cons1301, cons1302) rule2339 = ReplacementRule(pattern2339, replacement2339) pattern2340 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(S(1)/tan(x_*WC('f', S(1)) + WC('e', S(0))))**p_, x_), cons2, cons3, cons50, cons127, cons1267, cons1301, cons1302) rule2340 = ReplacementRule(pattern2340, replacement2340) pattern2341 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*tan(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons50, cons127, cons1267, cons1303, cons1304) rule2341 = ReplacementRule(pattern2341, replacement2341) pattern2342 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(S(1)/tan(x_*WC('f', S(1)) + WC('e', S(0))))**p_, x_), cons2, cons3, cons50, cons127, cons1267, cons1303, cons1304) rule2342 = ReplacementRule(pattern2342, replacement2342) pattern2343 = Pattern(Integral((WC('g', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons50, cons127, cons210, cons5, cons1267, cons64) rule2343 = ReplacementRule(pattern2343, replacement2343) pattern2344 = Pattern(Integral((WC('g', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons50, cons127, cons210, cons5, cons1267, cons64) rule2344 = ReplacementRule(pattern2344, replacement2344) pattern2345 = Pattern(Integral((WC('g', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons5, cons1267, cons86) rule2345 = ReplacementRule(pattern2345, replacement2345) pattern2346 = Pattern(Integral((WC('g', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons5, cons1267, cons86) rule2346 = ReplacementRule(pattern2346, replacement2346) pattern2347 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons50, cons127, cons1267, cons21, cons33, cons269) rule2347 = ReplacementRule(pattern2347, replacement2347) pattern2348 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons50, cons127, cons1267, cons21, cons33, cons269) rule2348 = ReplacementRule(pattern2348, replacement2348) pattern2349 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons50, cons127, cons19, cons1267, cons21, cons1305) rule2349 = ReplacementRule(pattern2349, replacement2349) pattern2350 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons50, cons127, cons19, cons1267, cons21, cons1305) rule2350 = ReplacementRule(pattern2350, replacement2350) pattern2351 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons50, cons127, cons19, cons1267, cons1306) rule2351 = ReplacementRule(pattern2351, replacement2351) pattern2352 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons50, cons127, cons19, cons1267, cons1306) rule2352 = ReplacementRule(pattern2352, replacement2352) pattern2353 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons50, cons127, cons1267, cons1306, cons96) rule2353 = ReplacementRule(pattern2353, replacement2353) pattern2354 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons50, cons127, cons1267, cons1306, cons96) rule2354 = ReplacementRule(pattern2354, replacement2354) pattern2355 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons50, cons127, cons19, cons1267, cons1306, cons1307) rule2355 = ReplacementRule(pattern2355, replacement2355) pattern2356 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons50, cons127, cons19, cons1267, cons1306, cons1307) rule2356 = ReplacementRule(pattern2356, replacement2356) pattern2357 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons50, cons127, cons1267, cons1306, cons96) rule2357 = ReplacementRule(pattern2357, replacement2357) pattern2358 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons50, cons127, cons1267, cons1306, cons96) rule2358 = ReplacementRule(pattern2358, replacement2358) pattern2359 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons50, cons127, cons19, cons1267, cons1306, cons1307) rule2359 = ReplacementRule(pattern2359, replacement2359) pattern2360 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons50, cons127, cons19, cons1267, cons1306, cons1307) rule2360 = ReplacementRule(pattern2360, replacement2360) pattern2361 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*tan(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons50, cons127, cons19, cons1267, cons21, cons1308) rule2361 = ReplacementRule(pattern2361, replacement2361) pattern2362 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(S(1)/tan(x_*WC('f', S(1)) + WC('e', S(0))))**p_, x_), cons2, cons3, cons50, cons127, cons19, cons1267, cons21, cons1308) rule2362 = ReplacementRule(pattern2362, replacement2362) pattern2363 = Pattern(Integral((WC('g', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons19, cons5, cons1267, cons21, cons149) rule2363 = ReplacementRule(pattern2363, replacement2363) pattern2364 = Pattern(Integral((WC('g', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons210, cons19, cons5, cons1267, cons21, cons149) rule2364 = ReplacementRule(pattern2364, replacement2364) pattern2365 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons50, cons127, cons19, cons1269, cons1300) rule2365 = ReplacementRule(pattern2365, replacement2365) pattern2366 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(S(1)/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons50, cons127, cons19, cons1269, cons1300) rule2366 = ReplacementRule(pattern2366, replacement2366) pattern2367 = Pattern(Integral((WC('g', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons50, cons127, cons210, cons5, cons1269, cons64) rule2367 = ReplacementRule(pattern2367, replacement2367) pattern2368 = Pattern(Integral((WC('g', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons50, cons127, cons210, cons5, cons1269, cons64) rule2368 = ReplacementRule(pattern2368, replacement2368) pattern2369 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons50, cons127, cons19, cons1269) rule2369 = ReplacementRule(pattern2369, replacement2369) pattern2370 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons50, cons127, cons19, cons1269) rule2370 = ReplacementRule(pattern2370, replacement2370) pattern2371 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons50, cons127, cons1269, cons33, cons96, cons517) rule2371 = ReplacementRule(pattern2371, replacement2371) pattern2372 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons50, cons127, cons1269, cons33, cons96, cons517) rule2372 = ReplacementRule(pattern2372, replacement2372) pattern2373 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons50, cons127, cons19, cons1269, cons274, cons517) rule2373 = ReplacementRule(pattern2373, replacement2373) pattern2374 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons50, cons127, cons19, cons1269, cons274, cons517) rule2374 = ReplacementRule(pattern2374, replacement2374) pattern2375 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(6), x_), cons2, cons3, cons50, cons127, cons19, cons1269, cons1285, cons517) rule2375 = ReplacementRule(pattern2375, replacement2375) pattern2376 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(6), x_), cons2, cons3, cons50, cons127, cons19, cons1269, cons1285, cons517) rule2376 = ReplacementRule(pattern2376, replacement2376) pattern2377 = Pattern(Integral((WC('g', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons210, cons1269, cons1309, cons148) rule2377 = ReplacementRule(pattern2377, replacement2377) pattern2378 = Pattern(Integral((WC('g', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons210, cons1269, cons1309, cons148) rule2378 = ReplacementRule(pattern2378, replacement2378) pattern2379 = Pattern(Integral((WC('g', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons210, cons1269, cons1309, cons139) rule2379 = ReplacementRule(pattern2379, replacement2379) pattern2380 = Pattern(Integral((WC('g', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons210, cons1269, cons1309, cons139) rule2380 = ReplacementRule(pattern2380, replacement2380) pattern2381 = Pattern(Integral(sqrt(WC('g', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons210, cons1269) rule2381 = ReplacementRule(pattern2381, replacement2381) pattern2382 = Pattern(Integral(sqrt(WC('g', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons210, cons1269) rule2382 = ReplacementRule(pattern2382, replacement2382) pattern2383 = Pattern(Integral(S(1)/(sqrt(g_*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons50, cons127, cons210, cons1269) rule2383 = ReplacementRule(pattern2383, replacement2383) pattern2384 = Pattern(Integral(S(1)/(sqrt(g_/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons50, cons127, cons210, cons1269) rule2384 = ReplacementRule(pattern2384, replacement2384) pattern2385 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*tan(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons50, cons127, cons1269, cons1303) rule2385 = ReplacementRule(pattern2385, replacement2385) pattern2386 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(S(1)/tan(x_*WC('f', S(1)) + WC('e', S(0))))**p_, x_), cons2, cons3, cons50, cons127, cons1269, cons1303) rule2386 = ReplacementRule(pattern2386, replacement2386) pattern2387 = Pattern(Integral((WC('g', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons50, cons127, cons210, cons19, cons5, cons1310) rule2387 = ReplacementRule(pattern2387, replacement2387) pattern2388 = Pattern(Integral((WC('g', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons50, cons127, cons210, cons19, cons5, cons1310) rule2388 = ReplacementRule(pattern2388, replacement2388) pattern2389 = Pattern(Integral((WC('g', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons50, cons127, cons210, cons19, cons5, cons149) rule2389 = ReplacementRule(pattern2389, replacement2389) pattern2390 = Pattern(Integral((WC('g', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons50, cons127, cons210, cons19, cons5, cons149) rule2390 = ReplacementRule(pattern2390, replacement2390) pattern2391 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73) rule2391 = ReplacementRule(pattern2391, replacement2391) pattern2392 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73) rule2392 = ReplacementRule(pattern2392, replacement2392) pattern2393 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73) rule2393 = ReplacementRule(pattern2393, replacement2393) pattern2394 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73) rule2394 = ReplacementRule(pattern2394, replacement2394) pattern2395 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons72, cons1267, cons20, cons1311) rule2395 = ReplacementRule(pattern2395, replacement2395) pattern2396 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons72, cons1267, cons20, cons1311) rule2396 = ReplacementRule(pattern2396, replacement2396) pattern2397 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons72, cons1267) rule2397 = ReplacementRule(pattern2397, replacement2397) pattern2398 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons72, cons1267) rule2398 = ReplacementRule(pattern2398, replacement2398) pattern2399 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons72, cons1267, cons1312) rule2399 = ReplacementRule(pattern2399, replacement2399) pattern2400 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons72, cons1267, cons1312) rule2400 = ReplacementRule(pattern2400, replacement2400) pattern2401 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons72, cons1267, cons1313, cons89, cons91, cons1314) rule2401 = ReplacementRule(pattern2401, replacement2401) pattern2402 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons72, cons1267, cons1313, cons89, cons91, cons1314) rule2402 = ReplacementRule(pattern2402, replacement2402) pattern2403 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons72, cons1267, cons1313, cons348, cons1315, cons1314) rule2403 = ReplacementRule(pattern2403, replacement2403) pattern2404 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons72, cons1267, cons1313, cons348, cons1315, cons1314) rule2404 = ReplacementRule(pattern2404, replacement2404) pattern2405 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons72, cons1267) rule2405 = ReplacementRule(pattern2405, replacement2405) pattern2406 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons72, cons1267) rule2406 = ReplacementRule(pattern2406, replacement2406) pattern2407 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons72, cons1267, cons157, cons1316) rule2407 = ReplacementRule(pattern2407, replacement2407) pattern2408 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons72, cons1267, cons157, cons1316) rule2408 = ReplacementRule(pattern2408, replacement2408) pattern2409 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons72, cons1267, cons1317, cons1316, cons116) rule2409 = ReplacementRule(pattern2409, replacement2409) pattern2410 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons72, cons1267, cons1317, cons1316, cons116) rule2410 = ReplacementRule(pattern2410, replacement2410) pattern2411 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons72, cons1267, cons33, cons96, cons1318, cons1172) rule2411 = ReplacementRule(pattern2411, replacement2411) pattern2412 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons72, cons1267, cons33, cons96, cons1318, cons1172) rule2412 = ReplacementRule(pattern2412, replacement2412) pattern2413 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons72, cons1267, cons1319) rule2413 = ReplacementRule(pattern2413, replacement2413) pattern2414 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons72, cons1267, cons1319) rule2414 = ReplacementRule(pattern2414, replacement2414) pattern2415 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**S(2)/(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73) rule2415 = ReplacementRule(pattern2415, replacement2415) pattern2416 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**S(2)/(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73) rule2416 = ReplacementRule(pattern2416, replacement2416) pattern2417 = Pattern(Integral(S(1)/((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73) rule2417 = ReplacementRule(pattern2417, replacement2417) pattern2418 = Pattern(Integral(S(1)/((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73) rule2418 = ReplacementRule(pattern2418, replacement2418) pattern2419 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons3, cons8, cons29, cons50, cons127, cons19, cons1320) rule2419 = ReplacementRule(pattern2419, replacement2419) pattern2420 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons3, cons8, cons29, cons50, cons127, cons19, cons1320) rule2420 = ReplacementRule(pattern2420, replacement2420) pattern2421 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons73, cons1267, cons1321) rule2421 = ReplacementRule(pattern2421, replacement2421) pattern2422 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons73, cons1267, cons1321) rule2422 = ReplacementRule(pattern2422, replacement2422) pattern2423 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons33, cons1322) rule2423 = ReplacementRule(pattern2423, replacement2423) pattern2424 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons33, cons1322) rule2424 = ReplacementRule(pattern2424, replacement2424) pattern2425 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons73, cons1267, cons1323) rule2425 = ReplacementRule(pattern2425, replacement2425) pattern2426 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons73, cons1267, cons1323) rule2426 = ReplacementRule(pattern2426, replacement2426) pattern2427 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269) rule2427 = ReplacementRule(pattern2427, replacement2427) pattern2428 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269) rule2428 = ReplacementRule(pattern2428, replacement2428) pattern2429 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons33, cons170, cons517) rule2429 = ReplacementRule(pattern2429, replacement2429) pattern2430 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons33, cons170, cons517) rule2430 = ReplacementRule(pattern2430, replacement2430) pattern2431 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons33, cons96, cons517) rule2431 = ReplacementRule(pattern2431, replacement2431) pattern2432 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons33, cons96, cons517) rule2432 = ReplacementRule(pattern2432, replacement2432) pattern2433 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons73, cons1269, cons79, cons1324) rule2433 = ReplacementRule(pattern2433, replacement2433) pattern2434 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons73, cons1269, cons79, cons1324) rule2434 = ReplacementRule(pattern2434, replacement2434) pattern2435 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons73, cons1269) rule2435 = ReplacementRule(pattern2435, replacement2435) pattern2436 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons73, cons1269) rule2436 = ReplacementRule(pattern2436, replacement2436) pattern2437 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons29, cons50, cons127, cons4, cons1267, cons64, cons89) rule2437 = ReplacementRule(pattern2437, replacement2437) pattern2438 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons29, cons50, cons127, cons4, cons1267, cons64, cons89) rule2438 = ReplacementRule(pattern2438, replacement2438) pattern2439 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons50, cons127, cons1267, cons33, cons1322) rule2439 = ReplacementRule(pattern2439, replacement2439) pattern2440 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons50, cons127, cons1267, cons33, cons1322) rule2440 = ReplacementRule(pattern2440, replacement2440) pattern2441 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons50, cons127, cons19, cons1267, cons1323) rule2441 = ReplacementRule(pattern2441, replacement2441) pattern2442 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons50, cons127, cons19, cons1267, cons1323) rule2442 = ReplacementRule(pattern2442, replacement2442) pattern2443 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons33, cons96) rule2443 = ReplacementRule(pattern2443, replacement2443) pattern2444 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons33, cons96) rule2444 = ReplacementRule(pattern2444, replacement2444) pattern2445 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons73, cons1267, cons274) rule2445 = ReplacementRule(pattern2445, replacement2445) pattern2446 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons73, cons1267, cons274) rule2446 = ReplacementRule(pattern2446, replacement2446) pattern2447 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons1325, cons95, cons168, cons91, cons1326) rule2447 = ReplacementRule(pattern2447, replacement2447) pattern2448 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons1325, cons95, cons168, cons91, cons1326) rule2448 = ReplacementRule(pattern2448, replacement2448) pattern2449 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons73, cons1267, cons1325, cons33, cons168, cons348, cons1326) rule2449 = ReplacementRule(pattern2449, replacement2449) pattern2450 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons73, cons1267, cons1325, cons33, cons168, cons348, cons1326) rule2450 = ReplacementRule(pattern2450, replacement2450) pattern2451 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons1325, cons95, cons96, cons1327, cons1328) rule2451 = ReplacementRule(pattern2451, replacement2451) pattern2452 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons1325, cons95, cons96, cons1327, cons1328) rule2452 = ReplacementRule(pattern2452, replacement2452) pattern2453 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons1325, cons95, cons96, cons167, cons1328) rule2453 = ReplacementRule(pattern2453, replacement2453) pattern2454 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons1325, cons95, cons96, cons167, cons1328) rule2454 = ReplacementRule(pattern2454, replacement2454) pattern2455 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons73, cons1267, cons1325, cons33, cons96, cons1329, cons1328) rule2455 = ReplacementRule(pattern2455, replacement2455) pattern2456 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons73, cons1267, cons1325, cons33, cons96, cons1329, cons1328) rule2456 = ReplacementRule(pattern2456, replacement2456) pattern2457 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons1325, cons89, cons167, cons1330) rule2457 = ReplacementRule(pattern2457, replacement2457) pattern2458 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons1325, cons89, cons167, cons1330) rule2458 = ReplacementRule(pattern2458, replacement2458) pattern2459 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons1325, cons89, cons465, cons1330) rule2459 = ReplacementRule(pattern2459, replacement2459) pattern2460 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons1325, cons89, cons465, cons1330) rule2460 = ReplacementRule(pattern2460, replacement2460) pattern2461 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons73, cons1267, cons1325, cons1330) rule2461 = ReplacementRule(pattern2461, replacement2461) pattern2462 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons73, cons1267, cons1325, cons1330) rule2462 = ReplacementRule(pattern2462, replacement2462) pattern2463 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons1325, cons89, cons90, cons810) rule2463 = ReplacementRule(pattern2463, replacement2463) pattern2464 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons1325, cons89, cons90, cons810) rule2464 = ReplacementRule(pattern2464, replacement2464) pattern2465 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons1325) rule2465 = ReplacementRule(pattern2465, replacement2465) pattern2466 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons1325) rule2466 = ReplacementRule(pattern2466, replacement2466) pattern2467 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons1325, cons89, cons91, cons1331, cons810) rule2467 = ReplacementRule(pattern2467, replacement2467) pattern2468 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons1325, cons89, cons91, cons810) rule2468 = ReplacementRule(pattern2468, replacement2468) pattern2469 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons1325) rule2469 = ReplacementRule(pattern2469, replacement2469) pattern2470 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons1325) rule2470 = ReplacementRule(pattern2470, replacement2470) pattern2471 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons1267, cons1332) rule2471 = ReplacementRule(pattern2471, replacement2471) pattern2472 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons1267, cons1332) rule2472 = ReplacementRule(pattern2472, replacement2472) pattern2473 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons1325) rule2473 = ReplacementRule(pattern2473, replacement2473) pattern2474 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons1325) rule2474 = ReplacementRule(pattern2474, replacement2474) pattern2475 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons73, cons1267, cons1325, cons545) rule2475 = ReplacementRule(pattern2475, replacement2475) pattern2476 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons73, cons1267, cons1325, cons545) rule2476 = ReplacementRule(pattern2476, replacement2476) pattern2477 = Pattern(Integral(sqrt(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons1325) rule2477 = ReplacementRule(pattern2477, replacement2477) pattern2478 = Pattern(Integral(sqrt(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons1325) rule2478 = ReplacementRule(pattern2478, replacement2478) pattern2479 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_/sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons1325, cons89, cons167, cons810) rule2479 = ReplacementRule(pattern2479, replacement2479) pattern2480 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_/sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons1325, cons89, cons167, cons810) rule2480 = ReplacementRule(pattern2480, replacement2480) pattern2481 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_/sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons1325, cons89, cons91, cons810) rule2481 = ReplacementRule(pattern2481, replacement2481) pattern2482 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_/sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons1325, cons89, cons91, cons810) rule2482 = ReplacementRule(pattern2482, replacement2482) pattern2483 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons1325) rule2483 = ReplacementRule(pattern2483, replacement2483) pattern2484 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons1325) rule2484 = ReplacementRule(pattern2484, replacement2484) pattern2485 = Pattern(Integral(S(1)/(sqrt(WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons29, cons50, cons127, cons1267, cons1332, cons45) rule2485 = ReplacementRule(pattern2485, replacement2485) pattern2486 = Pattern(Integral(S(1)/(sqrt(WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons29, cons50, cons127, cons1267, cons1332, cons45) rule2486 = ReplacementRule(pattern2486, replacement2486) pattern2487 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons1325) rule2487 = ReplacementRule(pattern2487, replacement2487) pattern2488 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons1325) rule2488 = ReplacementRule(pattern2488, replacement2488) pattern2489 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons73, cons1267, cons1325, cons89, cons167, cons87) rule2489 = ReplacementRule(pattern2489, replacement2489) pattern2490 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons73, cons1267, cons1325, cons89, cons167, cons87) rule2490 = ReplacementRule(pattern2490, replacement2490) pattern2491 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons73, cons1267, cons1325, cons20) rule2491 = ReplacementRule(pattern2491, replacement2491) pattern2492 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons73, cons1267, cons1325, cons20) rule2492 = ReplacementRule(pattern2492, replacement2492) pattern2493 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons1267, cons21, cons45, cons1333) rule2493 = ReplacementRule(pattern2493, replacement2493) pattern2494 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons1267, cons21, cons45, cons1333) rule2494 = ReplacementRule(pattern2494, replacement2494) pattern2495 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons1267, cons21, cons45, cons1334) rule2495 = ReplacementRule(pattern2495, replacement2495) pattern2496 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons1267, cons21, cons45, cons1334) rule2496 = ReplacementRule(pattern2496, replacement2496) pattern2497 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons1267, cons21, cons450) rule2497 = ReplacementRule(pattern2497, replacement2497) pattern2498 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons1267, cons21, cons450) rule2498 = ReplacementRule(pattern2498, replacement2498) pattern2499 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons73, cons1267, cons1325, cons21) rule2499 = ReplacementRule(pattern2499, replacement2499) pattern2500 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons73, cons1267, cons1325, cons21) rule2500 = ReplacementRule(pattern2500, replacement2500) pattern2501 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons3, cons8, cons29, cons50, cons127, cons19, cons1320) rule2501 = ReplacementRule(pattern2501, replacement2501) pattern2502 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons3, cons8, cons29, cons50, cons127, cons19, cons1320) rule2502 = ReplacementRule(pattern2502, replacement2502) pattern2503 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons33, cons96) rule2503 = ReplacementRule(pattern2503, replacement2503) pattern2504 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons33, cons96) rule2504 = ReplacementRule(pattern2504, replacement2504) pattern2505 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons73, cons1269, cons274) rule2505 = ReplacementRule(pattern2505, replacement2505) pattern2506 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons73, cons1269, cons274) rule2506 = ReplacementRule(pattern2506, replacement2506) pattern2507 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons1325, cons95, cons1335, cons91, cons1336) rule2507 = ReplacementRule(pattern2507, replacement2507) pattern2508 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons1325, cons95, cons1335, cons91, cons1336) rule2508 = ReplacementRule(pattern2508, replacement2508) pattern2509 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons73, cons1269, cons1325, cons33, cons1335, cons1336, cons1337) rule2509 = ReplacementRule(pattern2509, replacement2509) pattern2510 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons73, cons1269, cons1325, cons33, cons1335, cons1336, cons1337) rule2510 = ReplacementRule(pattern2510, replacement2510) pattern2511 = Pattern(Integral(sqrt(WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2), x_), cons2, cons3, cons29, cons50, cons127, cons1269) rule2511 = ReplacementRule(pattern2511, replacement2511) pattern2512 = Pattern(Integral(sqrt(WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2), x_), cons2, cons3, cons29, cons50, cons127, cons1269) rule2512 = ReplacementRule(pattern2512, replacement2512) pattern2513 = Pattern(Integral(sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons1325) rule2513 = ReplacementRule(pattern2513, replacement2513) pattern2514 = Pattern(Integral(sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons1325) rule2514 = ReplacementRule(pattern2514, replacement2514) pattern2515 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons1325, cons95, cons96, cons1327, cons1172) rule2515 = ReplacementRule(pattern2515, replacement2515) pattern2516 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons1325, cons95, cons96, cons1327, cons1172) rule2516 = ReplacementRule(pattern2516, replacement2516) pattern2517 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2), x_), cons2, cons3, cons29, cons50, cons127, cons1269) rule2517 = ReplacementRule(pattern2517, replacement2517) pattern2518 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2), x_), cons2, cons3, cons29, cons50, cons127, cons1269) rule2518 = ReplacementRule(pattern2518, replacement2518) pattern2519 = Pattern(Integral((c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons1325) rule2519 = ReplacementRule(pattern2519, replacement2519) pattern2520 = Pattern(Integral((c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons1325) rule2520 = ReplacementRule(pattern2520, replacement2520) pattern2521 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons1325, cons95, cons96, cons1338, cons1172) rule2521 = ReplacementRule(pattern2521, replacement2521) pattern2522 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons1325, cons95, cons96, cons1338, cons1172) rule2522 = ReplacementRule(pattern2522, replacement2522) pattern2523 = Pattern(Integral(S(1)/(sqrt(WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)), x_), cons2, cons3, cons29, cons50, cons127, cons1269) rule2523 = ReplacementRule(pattern2523, replacement2523) pattern2524 = Pattern(Integral(S(1)/(sqrt(WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)), x_), cons2, cons3, cons29, cons50, cons127, cons1269) rule2524 = ReplacementRule(pattern2524, replacement2524) pattern2525 = Pattern(Integral(S(1)/((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sqrt(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons1325) rule2525 = ReplacementRule(pattern2525, replacement2525) pattern2526 = Pattern(Integral(S(1)/((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sqrt(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons1325) rule2526 = ReplacementRule(pattern2526, replacement2526) pattern2527 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons73, cons1269, cons1325, cons33, cons96, cons1172, cons1339) rule2527 = ReplacementRule(pattern2527, replacement2527) pattern2528 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons73, cons1269, cons1325, cons33, cons96, cons1172, cons1339) rule2528 = ReplacementRule(pattern2528, replacement2528) pattern2529 = Pattern(Integral(sqrt(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons1325) rule2529 = ReplacementRule(pattern2529, replacement2529) pattern2530 = Pattern(Integral(sqrt(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons1325) rule2530 = ReplacementRule(pattern2530, replacement2530) pattern2531 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons1325) rule2531 = ReplacementRule(pattern2531, replacement2531) pattern2532 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons1325) rule2532 = ReplacementRule(pattern2532, replacement2532) pattern2533 = Pattern(Integral(S(1)/((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons1325, cons1340) rule2533 = ReplacementRule(pattern2533, replacement2533) pattern2534 = Pattern(Integral(S(1)/((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons1325, cons1340) rule2534 = ReplacementRule(pattern2534, replacement2534) pattern2535 = Pattern(Integral(S(1)/((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons1325, cons1341) rule2535 = ReplacementRule(pattern2535, replacement2535) pattern2536 = Pattern(Integral(S(1)/((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons1325, cons1341) rule2536 = ReplacementRule(pattern2536, replacement2536) pattern2537 = Pattern(Integral(S(1)/((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons1325, cons1342) rule2537 = ReplacementRule(pattern2537, replacement2537) pattern2538 = Pattern(Integral(S(1)/((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons1325, cons1342) rule2538 = ReplacementRule(pattern2538, replacement2538) pattern2539 = Pattern(Integral(sqrt(WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons3, cons8, cons29, cons50, cons127, cons1343, cons1344, cons1345) rule2539 = ReplacementRule(pattern2539, replacement2539) pattern2540 = Pattern(Integral(sqrt(WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons3, cons8, cons29, cons50, cons127, cons1343, cons1344, cons1345) rule2540 = ReplacementRule(pattern2540, replacement2540) pattern2541 = Pattern(Integral(sqrt(WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons3, cons8, cons29, cons50, cons127, cons1325, cons1344) rule2541 = ReplacementRule(pattern2541, replacement2541) pattern2542 = Pattern(Integral(sqrt(WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons3, cons8, cons29, cons50, cons127, cons1325, cons1344) rule2542 = ReplacementRule(pattern2542, replacement2542) pattern2543 = Pattern(Integral(sqrt(WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons3, cons8, cons29, cons50, cons127, cons1325, cons1346) rule2543 = ReplacementRule(pattern2543, replacement2543) pattern2544 = Pattern(Integral(sqrt(WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons3, cons8, cons29, cons50, cons127, cons1325, cons1346) rule2544 = ReplacementRule(pattern2544, replacement2544) pattern2545 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons1325, cons1347) rule2545 = ReplacementRule(pattern2545, replacement2545) pattern2546 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons1325, cons1347) rule2546 = ReplacementRule(pattern2546, replacement2546) pattern2547 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons1325, cons1348) rule2547 = ReplacementRule(pattern2547, replacement2547) pattern2548 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons1325, cons1348) rule2548 = ReplacementRule(pattern2548, replacement2548) pattern2549 = Pattern(Integral(S(1)/(sqrt(WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons29, cons50, cons127, cons1349, cons1350, cons1351) rule2549 = ReplacementRule(pattern2549, replacement2549) pattern2550 = Pattern(Integral(S(1)/(sqrt(WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons29, cons50, cons127, cons1349, cons1350, cons1351) rule2550 = ReplacementRule(pattern2550, replacement2550) pattern2551 = Pattern(Integral(S(1)/(sqrt(WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons29, cons50, cons127, cons1349, cons1352, cons1353) rule2551 = ReplacementRule(pattern2551, replacement2551) pattern2552 = Pattern(Integral(S(1)/(sqrt(WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons29, cons50, cons127, cons1349, cons1352, cons1353) rule2552 = ReplacementRule(pattern2552, replacement2552) pattern2553 = Pattern(Integral(S(1)/(sqrt(WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons29, cons50, cons127, cons1273, cons1354, cons1355) rule2553 = ReplacementRule(pattern2553, replacement2553) pattern2554 = Pattern(Integral(S(1)/(sqrt(WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons29, cons50, cons127, cons1273, cons1354, cons1355) rule2554 = ReplacementRule(pattern2554, replacement2554) pattern2555 = Pattern(Integral(S(1)/(sqrt(WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons29, cons50, cons127, cons1269, cons1354) rule2555 = ReplacementRule(pattern2555, replacement2555) pattern2556 = Pattern(Integral(S(1)/(sqrt(WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons29, cons50, cons127, cons1269, cons1354) rule2556 = ReplacementRule(pattern2556, replacement2556) pattern2557 = Pattern(Integral(S(1)/(sqrt(WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons29, cons50, cons127, cons1269, cons1356) rule2557 = ReplacementRule(pattern2557, replacement2557) pattern2558 = Pattern(Integral(S(1)/(sqrt(WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons29, cons50, cons127, cons1269, cons1356) rule2558 = ReplacementRule(pattern2558, replacement2558) pattern2559 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons1325, cons1357) rule2559 = ReplacementRule(pattern2559, replacement2559) pattern2560 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons1325, cons1357) rule2560 = ReplacementRule(pattern2560, replacement2560) pattern2561 = Pattern(Integral(S(1)/(sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons1325, cons1358) rule2561 = ReplacementRule(pattern2561, replacement2561) pattern2562 = Pattern(Integral(S(1)/(sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons1325, cons1358) rule2562 = ReplacementRule(pattern2562, replacement2562) pattern2563 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/sqrt(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons1269) rule2563 = ReplacementRule(pattern2563, replacement2563) pattern2564 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/sqrt(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons1269) rule2564 = ReplacementRule(pattern2564, replacement2564) pattern2565 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons1325, cons95, cons1359, cons1360, cons1361, cons1336) rule2565 = ReplacementRule(pattern2565, replacement2565) pattern2566 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons1325, cons95, cons1359, cons1360, cons1361, cons1336) rule2566 = ReplacementRule(pattern2566, replacement2566) pattern2567 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons73, cons64) rule2567 = ReplacementRule(pattern2567, replacement2567) pattern2568 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons73, cons64) rule2568 = ReplacementRule(pattern2568, replacement2568) pattern2569 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons73, cons1269, cons1325) rule2569 = ReplacementRule(pattern2569, replacement2569) pattern2570 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons73, cons1269, cons1325) rule2570 = ReplacementRule(pattern2570, replacement2570) pattern2571 = Pattern(Integral(((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*WC('c', S(1)))**n_*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons5, cons25) rule2571 = ReplacementRule(pattern2571, replacement2571) pattern2572 = Pattern(Integral(((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*WC('c', S(1)))**n_*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons5, cons25) rule2572 = ReplacementRule(pattern2572, replacement2572) pattern2573 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons87) rule2573 = ReplacementRule(pattern2573, replacement2573) pattern2574 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons87) rule2574 = ReplacementRule(pattern2574, replacement2574) pattern2575 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons25, cons20) rule2575 = ReplacementRule(pattern2575, replacement2575) pattern2576 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons25, cons20) rule2576 = ReplacementRule(pattern2576, replacement2576) pattern2577 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons25, cons21) rule2577 = ReplacementRule(pattern2577, replacement2577) pattern2578 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons25, cons21) rule2578 = ReplacementRule(pattern2578, replacement2578) pattern2579 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons1362) rule2579 = ReplacementRule(pattern2579, replacement2579) pattern2580 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons1362) rule2580 = ReplacementRule(pattern2580, replacement2580) pattern2581 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons29, cons50, cons127, cons4, cons5, cons1276, cons87, cons1363) rule2581 = ReplacementRule(pattern2581, replacement2581) pattern2582 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons29, cons50, cons127, cons4, cons5, cons1276, cons87, cons1363) rule2582 = ReplacementRule(pattern2582, replacement2582) pattern2583 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons4, cons5, cons1276, cons1267, cons87, cons1364) rule2583 = ReplacementRule(pattern2583, replacement2583) pattern2584 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons4, cons5, cons1276, cons1267, cons87, cons1364) rule2584 = ReplacementRule(pattern2584, replacement2584) pattern2585 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons50, cons127, cons8, cons29, cons19, cons4, cons1276, cons1267) rule2585 = ReplacementRule(pattern2585, replacement2585) pattern2586 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons50, cons127, cons8, cons29, cons19, cons4, cons1276, cons1267) rule2586 = ReplacementRule(pattern2586, replacement2586) pattern2587 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons1276, cons1269) rule2587 = ReplacementRule(pattern2587, replacement2587) pattern2588 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons1276, cons1269) rule2588 = ReplacementRule(pattern2588, replacement2588) pattern2589 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons210, cons4, cons5, cons1365) rule2589 = ReplacementRule(pattern2589, replacement2589) pattern2590 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons210, cons4, cons5, cons1365) rule2590 = ReplacementRule(pattern2590, replacement2590) pattern2591 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons210, cons4, cons5, cons1267) rule2591 = ReplacementRule(pattern2591, replacement2591) pattern2592 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons210, cons4, cons5, cons1267) rule2592 = ReplacementRule(pattern2592, replacement2592) pattern2593 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons4, cons5, cons72, cons1267, cons20, cons1366) rule2593 = ReplacementRule(pattern2593, replacement2593) pattern2594 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons4, cons5, cons72, cons1267, cons20, cons1366) rule2594 = ReplacementRule(pattern2594, replacement2594) pattern2595 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons5, cons72, cons1267, cons1308) rule2595 = ReplacementRule(pattern2595, replacement2595) pattern2596 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons5, cons72, cons1267, cons1308) rule2596 = ReplacementRule(pattern2596, replacement2596) pattern2597 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons5, cons72, cons1267) rule2597 = ReplacementRule(pattern2597, replacement2597) pattern2598 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons5, cons72, cons1267) rule2598 = ReplacementRule(pattern2598, replacement2598) pattern2599 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons72, cons1267, cons1284, cons144) rule2599 = ReplacementRule(pattern2599, replacement2599) pattern2600 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons72, cons1267, cons1284, cons144) rule2600 = ReplacementRule(pattern2600, replacement2600) pattern2601 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons72, cons1267, cons1284, cons337) rule2601 = ReplacementRule(pattern2601, replacement2601) pattern2602 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons72, cons1267, cons1284, cons337) rule2602 = ReplacementRule(pattern2602, replacement2602) pattern2603 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons5, cons72, cons1267, cons1286, cons89, cons91, cons1367, cons1368) rule2603 = ReplacementRule(pattern2603, replacement2603) pattern2604 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons5, cons72, cons1267, cons1286, cons89, cons91, cons1367, cons1368) rule2604 = ReplacementRule(pattern2604, replacement2604) pattern2605 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons4, cons5, cons72, cons1267, cons1286, cons348, cons1369, cons1368) rule2605 = ReplacementRule(pattern2605, replacement2605) pattern2606 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons4, cons5, cons72, cons1267, cons1286, cons348, cons1369, cons1368) rule2606 = ReplacementRule(pattern2606, replacement2606) pattern2607 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons5, cons72, cons1267, cons1370) rule2607 = ReplacementRule(pattern2607, replacement2607) pattern2608 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons5, cons72, cons1267, cons1370) rule2608 = ReplacementRule(pattern2608, replacement2608) pattern2609 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons72, cons1267, cons1371, cons1262) rule2609 = ReplacementRule(pattern2609, replacement2609) pattern2610 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons72, cons1267, cons1371, cons1262) rule2610 = ReplacementRule(pattern2610, replacement2610) pattern2611 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons72, cons1267, cons1372, cons1283, cons116) rule2611 = ReplacementRule(pattern2611, replacement2611) pattern2612 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons72, cons1267, cons1372, cons1283, cons116) rule2612 = ReplacementRule(pattern2612, replacement2612) pattern2613 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons5, cons72, cons1267, cons95, cons170, cons91, cons1367, cons172) rule2613 = ReplacementRule(pattern2613, replacement2613) pattern2614 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons5, cons72, cons1267, cons95, cons170, cons91, cons1367, cons172) rule2614 = ReplacementRule(pattern2614, replacement2614) pattern2615 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons4, cons5, cons72, cons1267, cons33, cons170, cons1373, cons1374, cons172) rule2615 = ReplacementRule(pattern2615, replacement2615) pattern2616 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons4, cons5, cons72, cons1267, cons33, cons170, cons1373, cons1374, cons172) rule2616 = ReplacementRule(pattern2616, replacement2616) pattern2617 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons4, cons5, cons72, cons1267, cons33, cons96, cons1283, cons1318, cons172) rule2617 = ReplacementRule(pattern2617, replacement2617) pattern2618 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons4, cons5, cons72, cons1267, cons33, cons96, cons1283, cons1318, cons172) rule2618 = ReplacementRule(pattern2618, replacement2618) pattern2619 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons72, cons1267, cons1319) rule2619 = ReplacementRule(pattern2619, replacement2619) pattern2620 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons72, cons1267, cons1319) rule2620 = ReplacementRule(pattern2620, replacement2620) pattern2621 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons5, cons1267, cons1375) rule2621 = ReplacementRule(pattern2621, replacement2621) pattern2622 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons5, cons1267, cons1375) rule2622 = ReplacementRule(pattern2622, replacement2622) pattern2623 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons1267, cons246, cons1376, cons139) rule2623 = ReplacementRule(pattern2623, replacement2623) pattern2624 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons1267, cons246, cons1376, cons139) rule2624 = ReplacementRule(pattern2624, replacement2624) pattern2625 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons5, cons1267, cons1377, cons255) rule2625 = ReplacementRule(pattern2625, replacement2625) pattern2626 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons5, cons1267, cons1377, cons255) rule2626 = ReplacementRule(pattern2626, replacement2626) pattern2627 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons1267, cons33, cons1378) rule2627 = ReplacementRule(pattern2627, replacement2627) pattern2628 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons1267, cons33, cons1378) rule2628 = ReplacementRule(pattern2628, replacement2628) pattern2629 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons1267, cons33, cons1379) rule2629 = ReplacementRule(pattern2629, replacement2629) pattern2630 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons1267, cons33, cons1379) rule2630 = ReplacementRule(pattern2630, replacement2630) pattern2631 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons5, cons1267, cons1380, cons1283) rule2631 = ReplacementRule(pattern2631, replacement2631) pattern2632 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons5, cons1267, cons1380, cons1283) rule2632 = ReplacementRule(pattern2632, replacement2632) pattern2633 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons5, cons1267, cons255) rule2633 = ReplacementRule(pattern2633, replacement2633) pattern2634 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons5, cons1267, cons255) rule2634 = ReplacementRule(pattern2634, replacement2634) pattern2635 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons1269, cons246, cons170, cons139, cons517) rule2635 = ReplacementRule(pattern2635, replacement2635) pattern2636 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons1269, cons246, cons170, cons139, cons517) rule2636 = ReplacementRule(pattern2636, replacement2636) pattern2637 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons5, cons1269, cons33, cons170, cons1381, cons517) rule2637 = ReplacementRule(pattern2637, replacement2637) pattern2638 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons5, cons1269, cons33, cons170, cons1381, cons517) rule2638 = ReplacementRule(pattern2638, replacement2638) pattern2639 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons1269, cons246, cons96, cons148, cons255, cons517) rule2639 = ReplacementRule(pattern2639, replacement2639) pattern2640 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons1269, cons246, cons96, cons148, cons255, cons517) rule2640 = ReplacementRule(pattern2640, replacement2640) pattern2641 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons5, cons1269, cons33, cons96, cons517) rule2641 = ReplacementRule(pattern2641, replacement2641) pattern2642 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons5, cons1269, cons33, cons96, cons517) rule2642 = ReplacementRule(pattern2642, replacement2642) pattern2643 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons1269, cons13, cons148, cons1287, cons255, cons517) rule2643 = ReplacementRule(pattern2643, replacement2643) pattern2644 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons1269, cons13, cons148, cons1287, cons255, cons517) rule2644 = ReplacementRule(pattern2644, replacement2644) pattern2645 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons1269, cons13, cons139, cons517) rule2645 = ReplacementRule(pattern2645, replacement2645) pattern2646 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons1269, cons13, cons139, cons517) rule2646 = ReplacementRule(pattern2646, replacement2646) pattern2647 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons1269) rule2647 = ReplacementRule(pattern2647, replacement2647) pattern2648 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons1269) rule2648 = ReplacementRule(pattern2648, replacement2648) pattern2649 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons5, cons1269, cons1324) rule2649 = ReplacementRule(pattern2649, replacement2649) pattern2650 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons5, cons1269, cons1324) rule2650 = ReplacementRule(pattern2650, replacement2650) pattern2651 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons29, cons50, cons127, cons4, cons1267, cons1301, cons1382) rule2651 = ReplacementRule(pattern2651, replacement2651) pattern2652 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons29, cons50, cons127, cons4, cons1267, cons1301, cons1382) rule2652 = ReplacementRule(pattern2652, replacement2652) pattern2653 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons50, cons127, cons210, cons19, cons5, cons1267, cons360) rule2653 = ReplacementRule(pattern2653, replacement2653) pattern2654 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons50, cons127, cons210, cons19, cons5, cons1267, cons360) rule2654 = ReplacementRule(pattern2654, replacement2654) pattern2655 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons50, cons127, cons210, cons19, cons5, cons1267, cons1280) rule2655 = ReplacementRule(pattern2655, replacement2655) pattern2656 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons50, cons127, cons210, cons19, cons5, cons1267, cons1280) rule2656 = ReplacementRule(pattern2656, replacement2656) pattern2657 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons29, cons50, cons127, cons1267, cons1383, cons1384) rule2657 = ReplacementRule(pattern2657, replacement2657) pattern2658 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons29, cons50, cons127, cons1267, cons1383, cons1384) rule2658 = ReplacementRule(pattern2658, replacement2658) pattern2659 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons29, cons50, cons127, cons210, cons4, cons5, cons1267, cons64) rule2659 = ReplacementRule(pattern2659, replacement2659) pattern2660 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons29, cons50, cons127, cons210, cons4, cons5, cons1267, cons64) rule2660 = ReplacementRule(pattern2660, replacement2660) pattern2661 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons1267, cons1385) rule2661 = ReplacementRule(pattern2661, replacement2661) pattern2662 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons1267, cons1385) rule2662 = ReplacementRule(pattern2662, replacement2662) pattern2663 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons29, cons50, cons127, cons210, cons4, cons5, cons1267, cons86) rule2663 = ReplacementRule(pattern2663, replacement2663) pattern2664 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons29, cons50, cons127, cons210, cons4, cons5, cons1267, cons86) rule2664 = ReplacementRule(pattern2664, replacement2664) pattern2665 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons29, cons50, cons127, cons210, cons4, cons1267, cons20, cons13, cons1386) rule2665 = ReplacementRule(pattern2665, replacement2665) pattern2666 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons29, cons50, cons127, cons210, cons4, cons1267, cons20, cons13, cons1386) rule2666 = ReplacementRule(pattern2666, replacement2666) pattern2667 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons50, cons127, cons210, cons5, cons1267, cons33, cons1387, cons1283) rule2667 = ReplacementRule(pattern2667, replacement2667) pattern2668 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons50, cons127, cons210, cons5, cons1267, cons33, cons1387, cons1283) rule2668 = ReplacementRule(pattern2668, replacement2668) pattern2669 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons50, cons127, cons210, cons19, cons5, cons1267, cons1388) rule2669 = ReplacementRule(pattern2669, replacement2669) pattern2670 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons50, cons127, cons210, cons19, cons5, cons1267, cons1388) rule2670 = ReplacementRule(pattern2670, replacement2670) pattern2671 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons1267, cons1172) rule2671 = ReplacementRule(pattern2671, replacement2671) pattern2672 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons1267, cons1172) rule2672 = ReplacementRule(pattern2672, replacement2672) pattern2673 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons29, cons50, cons127, cons4, cons1267, cons33, cons96) rule2673 = ReplacementRule(pattern2673, replacement2673) pattern2674 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons29, cons50, cons127, cons4, cons1267, cons33, cons96) rule2674 = ReplacementRule(pattern2674, replacement2674) pattern2675 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons1267, cons145) rule2675 = ReplacementRule(pattern2675, replacement2675) pattern2676 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons1267, cons145) rule2676 = ReplacementRule(pattern2676, replacement2676) pattern2677 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons29, cons50, cons127, cons4, cons1267, cons1308, cons20) rule2677 = ReplacementRule(pattern2677, replacement2677) pattern2678 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons29, cons50, cons127, cons4, cons1267, cons1308, cons20) rule2678 = ReplacementRule(pattern2678, replacement2678) pattern2679 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons1267, cons1308, cons21) rule2679 = ReplacementRule(pattern2679, replacement2679) pattern2680 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons1267, cons1308, cons21) rule2680 = ReplacementRule(pattern2680, replacement2680) pattern2681 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons29, cons50, cons127, cons210, cons4, cons5, cons1267, cons64, cons1389) rule2681 = ReplacementRule(pattern2681, replacement2681) pattern2682 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons29, cons50, cons127, cons210, cons4, cons5, cons1267, cons64, cons1389) rule2682 = ReplacementRule(pattern2682, replacement2682) pattern2683 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons29, cons50, cons127, cons4, cons5, cons1267, cons20) rule2683 = ReplacementRule(pattern2683, replacement2683) pattern2684 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons29, cons50, cons127, cons4, cons5, cons1267, cons20) rule2684 = ReplacementRule(pattern2684, replacement2684) pattern2685 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons5, cons1267, cons21) rule2685 = ReplacementRule(pattern2685, replacement2685) pattern2686 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons5, cons1267, cons21) rule2686 = ReplacementRule(pattern2686, replacement2686) pattern2687 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/sqrt(WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons210, cons1269, cons33, cons96, cons1390) rule2687 = ReplacementRule(pattern2687, replacement2687) pattern2688 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/sqrt(WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons210, cons1269, cons33, cons96, cons1390) rule2688 = ReplacementRule(pattern2688, replacement2688) pattern2689 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/sqrt(WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons210, cons1269, cons33, cons170, cons1391) rule2689 = ReplacementRule(pattern2689, replacement2689) pattern2690 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/sqrt(WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons210, cons1269, cons33, cons170, cons1391) rule2690 = ReplacementRule(pattern2690, replacement2690) pattern2691 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons1269, cons1392) rule2691 = ReplacementRule(pattern2691, replacement2691) pattern2692 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons1269, cons1392) rule2692 = ReplacementRule(pattern2692, replacement2692) pattern2693 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons29, cons50, cons127, cons1269, cons1172, cons96, cons91) rule2693 = ReplacementRule(pattern2693, replacement2693) pattern2694 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons29, cons50, cons127, cons1269, cons1172, cons96, cons91) rule2694 = ReplacementRule(pattern2694, replacement2694) pattern2695 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons29, cons50, cons127, cons4, cons1269, cons1172, cons96, cons1393, cons1394) rule2695 = ReplacementRule(pattern2695, replacement2695) pattern2696 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons29, cons50, cons127, cons4, cons1269, cons1172, cons96, cons1393, cons1394) rule2696 = ReplacementRule(pattern2696, replacement2696) pattern2697 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons29, cons50, cons127, cons4, cons1269, cons1172, cons96, cons1393, cons1395) rule2697 = ReplacementRule(pattern2697, replacement2697) pattern2698 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons29, cons50, cons127, cons4, cons1269, cons1172, cons96, cons1393, cons1395) rule2698 = ReplacementRule(pattern2698, replacement2698) pattern2699 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons29, cons50, cons127, cons19, cons1269, cons1392, cons1307, cons89, cons91, cons1396) rule2699 = ReplacementRule(pattern2699, replacement2699) pattern2700 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons29, cons50, cons127, cons19, cons1269, cons1392, cons1307, cons89, cons91, cons1396) rule2700 = ReplacementRule(pattern2700, replacement2700) pattern2701 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons29, cons50, cons127, cons19, cons1269, cons1392, cons1307, cons89, cons91, cons1395) rule2701 = ReplacementRule(pattern2701, replacement2701) pattern2702 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons29, cons50, cons127, cons19, cons1269, cons1392, cons1307, cons89, cons91, cons1395) rule2702 = ReplacementRule(pattern2702, replacement2702) pattern2703 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons1269, cons1392, cons1307, cons348, cons215, cons1395) rule2703 = ReplacementRule(pattern2703, replacement2703) pattern2704 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(4), x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons1269, cons1392, cons1307, cons348, cons215, cons1395) rule2704 = ReplacementRule(pattern2704, replacement2704) pattern2705 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(6), x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons1269, cons1172, cons586, cons1397, cons1398, cons1399, cons145) rule2705 = ReplacementRule(pattern2705, replacement2705) pattern2706 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(6), x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons1269, cons1172, cons586, cons1397, cons1398, cons1399, cons145) rule2706 = ReplacementRule(pattern2706, replacement2706) pattern2707 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons29, cons50, cons127, cons1269, cons1400, cons1401) rule2707 = ReplacementRule(pattern2707, replacement2707) pattern2708 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons29, cons50, cons127, cons1269, cons1400, cons1401) rule2708 = ReplacementRule(pattern2708, replacement2708) pattern2709 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**n_/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons210, cons5, cons1269, cons87, cons1402) rule2709 = ReplacementRule(pattern2709, replacement2709) pattern2710 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**n_/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons210, cons5, cons1269, cons87, cons1402) rule2710 = ReplacementRule(pattern2710, replacement2710) pattern2711 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons210, cons1269, cons1403, cons148, cons1404) rule2711 = ReplacementRule(pattern2711, replacement2711) pattern2712 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons210, cons1269, cons1403, cons148, cons1404) rule2712 = ReplacementRule(pattern2712, replacement2712) pattern2713 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons210, cons1269, cons1403, cons148, cons1405) rule2713 = ReplacementRule(pattern2713, replacement2713) pattern2714 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons210, cons1269, cons1403, cons148, cons1405) rule2714 = ReplacementRule(pattern2714, replacement2714) pattern2715 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons210, cons1269, cons1403, cons148) rule2715 = ReplacementRule(pattern2715, replacement2715) pattern2716 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons210, cons1269, cons1403, cons148) rule2716 = ReplacementRule(pattern2716, replacement2716) pattern2717 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons210, cons1269, cons1403, cons139, cons167) rule2717 = ReplacementRule(pattern2717, replacement2717) pattern2718 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons210, cons1269, cons1403, cons139, cons167) rule2718 = ReplacementRule(pattern2718, replacement2718) pattern2719 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons210, cons1269, cons1403, cons139, cons90) rule2719 = ReplacementRule(pattern2719, replacement2719) pattern2720 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons210, cons1269, cons1403, cons139, cons90) rule2720 = ReplacementRule(pattern2720, replacement2720) pattern2721 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons210, cons1269, cons1403, cons139) rule2721 = ReplacementRule(pattern2721, replacement2721) pattern2722 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons210, cons1269, cons1403, cons139) rule2722 = ReplacementRule(pattern2722, replacement2722) pattern2723 = Pattern(Integral(sqrt(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons50, cons127, cons210, cons1269) rule2723 = ReplacementRule(pattern2723, replacement2723) pattern2724 = Pattern(Integral(sqrt(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons50, cons127, cons210, cons1269) rule2724 = ReplacementRule(pattern2724, replacement2724) pattern2725 = Pattern(Integral(sqrt(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(d_*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons29, cons50, cons127, cons210, cons1269) rule2725 = ReplacementRule(pattern2725, replacement2725) pattern2726 = Pattern(Integral(sqrt(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(d_*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons29, cons50, cons127, cons210, cons1269) rule2726 = ReplacementRule(pattern2726, replacement2726) pattern2727 = Pattern(Integral(sqrt(WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons29, cons50, cons127, cons1269) rule2727 = ReplacementRule(pattern2727, With2727) pattern2728 = Pattern(Integral(sqrt(WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons29, cons50, cons127, cons1269) rule2728 = ReplacementRule(pattern2728, With2728) pattern2729 = Pattern(Integral(sqrt(WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons29, cons50, cons127, cons210, cons1269) rule2729 = ReplacementRule(pattern2729, replacement2729) pattern2730 = Pattern(Integral(sqrt(WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons29, cons50, cons127, cons210, cons1269) rule2730 = ReplacementRule(pattern2730, replacement2730) pattern2731 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons210, cons1269, cons1403, cons1406, cons90) rule2731 = ReplacementRule(pattern2731, replacement2731) pattern2732 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons210, cons1269, cons1403, cons1406, cons90) rule2732 = ReplacementRule(pattern2732, replacement2732) pattern2733 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons210, cons1269, cons1403, cons1406, cons465) rule2733 = ReplacementRule(pattern2733, replacement2733) pattern2734 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons210, cons1269, cons1403, cons1406, cons465) rule2734 = ReplacementRule(pattern2734, replacement2734) pattern2735 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons29, cons50, cons127, cons210, cons4, cons5, cons1269, cons20, cons1407) rule2735 = ReplacementRule(pattern2735, replacement2735) pattern2736 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons29, cons50, cons127, cons210, cons4, cons5, cons1269, cons20, cons1407) rule2736 = ReplacementRule(pattern2736, replacement2736) pattern2737 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons29, cons50, cons127, cons210, cons1269, cons1408, cons269, cons148, cons1409) rule2737 = ReplacementRule(pattern2737, replacement2737) pattern2738 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons29, cons50, cons127, cons210, cons1269, cons1408, cons269, cons148, cons1409) rule2738 = ReplacementRule(pattern2738, replacement2738) pattern2739 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons1267, cons1301, cons1382) rule2739 = ReplacementRule(pattern2739, replacement2739) pattern2740 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons1267, cons1301, cons1382) rule2740 = ReplacementRule(pattern2740, replacement2740) pattern2741 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons4, cons1267, cons20, cons13, cons1386) rule2741 = ReplacementRule(pattern2741, replacement2741) pattern2742 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons4, cons1267, cons20, cons13, cons1386) rule2742 = ReplacementRule(pattern2742, replacement2742) pattern2743 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons1267, cons1172) rule2743 = ReplacementRule(pattern2743, replacement2743) pattern2744 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons1267, cons1172) rule2744 = ReplacementRule(pattern2744, replacement2744) pattern2745 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons1267, cons1308, cons20) rule2745 = ReplacementRule(pattern2745, replacement2745) pattern2746 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons1267, cons1308, cons20) rule2746 = ReplacementRule(pattern2746, replacement2746) pattern2747 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons1267, cons1308, cons21) rule2747 = ReplacementRule(pattern2747, replacement2747) pattern2748 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons1267, cons1308, cons21) rule2748 = ReplacementRule(pattern2748, replacement2748) pattern2749 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons4, cons5, cons1267, cons64, cons1389) rule2749 = ReplacementRule(pattern2749, replacement2749) pattern2750 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons4, cons5, cons1267, cons64, cons1389) rule2750 = ReplacementRule(pattern2750, replacement2750) pattern2751 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons5, cons1267, cons20) rule2751 = ReplacementRule(pattern2751, replacement2751) pattern2752 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons5, cons1267, cons20) rule2752 = ReplacementRule(pattern2752, replacement2752) pattern2753 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons5, cons1267, cons21) rule2753 = ReplacementRule(pattern2753, replacement2753) pattern2754 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons5, cons1267, cons21) rule2754 = ReplacementRule(pattern2754, replacement2754) pattern2755 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons1269, cons1392) rule2755 = ReplacementRule(pattern2755, replacement2755) pattern2756 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons1269, cons1392) rule2756 = ReplacementRule(pattern2756, replacement2756) pattern2757 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons1269, cons1410, cons1392) rule2757 = ReplacementRule(pattern2757, replacement2757) pattern2758 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons1269, cons1410, cons1392) rule2758 = ReplacementRule(pattern2758, replacement2758) pattern2759 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons5, cons1269, cons1172) rule2759 = ReplacementRule(pattern2759, replacement2759) pattern2760 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons5, cons1269, cons1172) rule2760 = ReplacementRule(pattern2760, replacement2760) pattern2761 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons1269) rule2761 = ReplacementRule(pattern2761, replacement2761) pattern2762 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons1269) rule2762 = ReplacementRule(pattern2762, replacement2762) pattern2763 = Pattern(Integral((WC('g', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons149) rule2763 = ReplacementRule(pattern2763, replacement2763) pattern2764 = Pattern(Integral((WC('g', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons149) rule2764 = ReplacementRule(pattern2764, replacement2764) pattern2765 = Pattern(Integral(sqrt(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons73, cons1411) rule2765 = ReplacementRule(pattern2765, replacement2765) pattern2766 = Pattern(Integral(sqrt(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons73, cons1411) rule2766 = ReplacementRule(pattern2766, replacement2766) pattern2767 = Pattern(Integral(sqrt(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons73, cons1269, cons1325) rule2767 = ReplacementRule(pattern2767, replacement2767) pattern2768 = Pattern(Integral(sqrt(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons73, cons1269, cons1325) rule2768 = ReplacementRule(pattern2768, replacement2768) pattern2769 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons73, cons1267) rule2769 = ReplacementRule(pattern2769, replacement2769) pattern2770 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons73, cons1267) rule2770 = ReplacementRule(pattern2770, replacement2770) pattern2771 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/((c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons1412, cons1413, cons107) rule2771 = ReplacementRule(pattern2771, replacement2771) pattern2772 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/((c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons1412, cons1413, cons107) rule2772 = ReplacementRule(pattern2772, replacement2772) pattern2773 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons73, cons1269, cons1324) rule2773 = ReplacementRule(pattern2773, replacement2773) pattern2774 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons73, cons1269, cons1324) rule2774 = ReplacementRule(pattern2774, replacement2774) pattern2775 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons73, cons1269, cons1325) rule2775 = ReplacementRule(pattern2775, replacement2775) pattern2776 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons73, cons1269, cons1325) rule2776 = ReplacementRule(pattern2776, replacement2776) pattern2777 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/((c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267) rule2777 = ReplacementRule(pattern2777, replacement2777) pattern2778 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/((c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267) rule2778 = ReplacementRule(pattern2778, replacement2778) pattern2779 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/((c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269) rule2779 = ReplacementRule(pattern2779, replacement2779) pattern2780 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/((c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269) rule2780 = ReplacementRule(pattern2780, replacement2780) pattern2781 = Pattern(Integral(sqrt(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons73, cons1411) rule2781 = ReplacementRule(pattern2781, replacement2781) pattern2782 = Pattern(Integral(sqrt(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons73, cons1411) rule2782 = ReplacementRule(pattern2782, replacement2782) pattern2783 = Pattern(Integral(sqrt(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons73, cons1269, cons1325) rule2783 = ReplacementRule(pattern2783, replacement2783) pattern2784 = Pattern(Integral(sqrt(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons73, cons1269, cons1325) rule2784 = ReplacementRule(pattern2784, replacement2784) pattern2785 = Pattern(Integral(S(1)/(sqrt(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons73, cons1411) rule2785 = ReplacementRule(pattern2785, replacement2785) pattern2786 = Pattern(Integral(S(1)/(sqrt(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons73, cons1411) rule2786 = ReplacementRule(pattern2786, replacement2786) pattern2787 = Pattern(Integral(S(1)/(sqrt(WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons73, cons1269, cons1325) rule2787 = ReplacementRule(pattern2787, replacement2787) pattern2788 = Pattern(Integral(S(1)/(sqrt(WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons73, cons1269, cons1325) rule2788 = ReplacementRule(pattern2788, replacement2788) pattern2789 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267) rule2789 = ReplacementRule(pattern2789, replacement2789) pattern2790 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267) rule2790 = ReplacementRule(pattern2790, replacement2790) pattern2791 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269) rule2791 = ReplacementRule(pattern2791, replacement2791) pattern2792 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269) rule2792 = ReplacementRule(pattern2792, replacement2792) pattern2793 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons72) rule2793 = ReplacementRule(pattern2793, replacement2793) pattern2794 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons72) rule2794 = ReplacementRule(pattern2794, replacement2794) pattern2795 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons1414) rule2795 = ReplacementRule(pattern2795, replacement2795) pattern2796 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons1414) rule2796 = ReplacementRule(pattern2796, replacement2796) pattern2797 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons1324) rule2797 = ReplacementRule(pattern2797, replacement2797) pattern2798 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons1324) rule2798 = ReplacementRule(pattern2798, replacement2798) pattern2799 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons1325) rule2799 = ReplacementRule(pattern2799, replacement2799) pattern2800 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1269, cons1325) rule2800 = ReplacementRule(pattern2800, replacement2800) pattern2801 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons1324) rule2801 = ReplacementRule(pattern2801, replacement2801) pattern2802 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons1324) rule2802 = ReplacementRule(pattern2802, replacement2802) pattern2803 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1415) rule2803 = ReplacementRule(pattern2803, replacement2803) pattern2804 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1415) rule2804 = ReplacementRule(pattern2804, replacement2804) pattern2805 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons1324) rule2805 = ReplacementRule(pattern2805, replacement2805) pattern2806 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1267, cons1324) rule2806 = ReplacementRule(pattern2806, replacement2806) pattern2807 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1415) rule2807 = ReplacementRule(pattern2807, replacement2807) pattern2808 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1415) rule2808 = ReplacementRule(pattern2808, replacement2808) pattern2809 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons72, cons1267, cons1416, cons87) rule2809 = ReplacementRule(pattern2809, replacement2809) pattern2810 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons72, cons1267, cons1416, cons87) rule2810 = ReplacementRule(pattern2810, replacement2810) pattern2811 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons73, cons1267, cons1325, cons1306) rule2811 = ReplacementRule(pattern2811, replacement2811) pattern2812 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons73, cons1267, cons1325, cons1306) rule2812 = ReplacementRule(pattern2812, replacement2812) pattern2813 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons4, cons5, cons73, cons1417, cons1418) rule2813 = ReplacementRule(pattern2813, replacement2813) pattern2814 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons4, cons5, cons73, cons1417, cons1418) rule2814 = ReplacementRule(pattern2814, replacement2814) pattern2815 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons1418) rule2815 = ReplacementRule(pattern2815, replacement2815) pattern2816 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons1418) rule2816 = ReplacementRule(pattern2816, replacement2816) pattern2817 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('a', S(0)) + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons5, cons73, cons149, cons20, cons87) rule2817 = ReplacementRule(pattern2817, replacement2817) pattern2818 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('a', S(0)) + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons5, cons73, cons149, cons20, cons87) rule2818 = ReplacementRule(pattern2818, replacement2818) pattern2819 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('a', S(0)) + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons73, cons149, cons1419) rule2819 = ReplacementRule(pattern2819, replacement2819) pattern2820 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('a', S(0)) + WC('b', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons73, cons149, cons1419) rule2820 = ReplacementRule(pattern2820, replacement2820) pattern2821 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons5, cons87) rule2821 = ReplacementRule(pattern2821, replacement2821) pattern2822 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons5, cons87) rule2822 = ReplacementRule(pattern2822, replacement2822) pattern2823 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons25, cons20, cons40) rule2823 = ReplacementRule(pattern2823, replacement2823) pattern2824 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons25, cons20, cons40) rule2824 = ReplacementRule(pattern2824, replacement2824) pattern2825 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons4, cons5, cons25, cons20, cons149) rule2825 = ReplacementRule(pattern2825, replacement2825) pattern2826 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons4, cons5, cons25, cons20, cons149) rule2826 = ReplacementRule(pattern2826, replacement2826) pattern2827 = Pattern(Integral((WC('g', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons25, cons21) rule2827 = ReplacementRule(pattern2827, replacement2827) pattern2828 = Pattern(Integral((WC('g', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons25, cons21) rule2828 = ReplacementRule(pattern2828, replacement2828) pattern2829 = Pattern(Integral((WC('g', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons5, cons73, cons149, cons20, cons87) rule2829 = ReplacementRule(pattern2829, replacement2829) pattern2830 = Pattern(Integral((WC('g', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons5, cons73, cons149, cons20, cons87) rule2830 = ReplacementRule(pattern2830, replacement2830) pattern2831 = Pattern(Integral((WC('g', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons73, cons149, cons1419) rule2831 = ReplacementRule(pattern2831, replacement2831) pattern2832 = Pattern(Integral((WC('g', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons73, cons149, cons1419) rule2832 = ReplacementRule(pattern2832, replacement2832) pattern2833 = Pattern(Integral((WC('g', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons4, cons5, cons20) rule2833 = ReplacementRule(pattern2833, replacement2833) pattern2834 = Pattern(Integral((WC('g', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons4, cons5, cons20) rule2834 = ReplacementRule(pattern2834, replacement2834) pattern2835 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(S(1)/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons21, cons87, cons40) rule2835 = ReplacementRule(pattern2835, replacement2835) pattern2836 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(S(1)/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons21, cons87, cons40) rule2836 = ReplacementRule(pattern2836, replacement2836) pattern2837 = Pattern(Integral((WC('g', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons5, cons21, cons87, cons149) rule2837 = ReplacementRule(pattern2837, replacement2837) pattern2838 = Pattern(Integral((WC('g', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons5, cons21, cons87, cons149) rule2838 = ReplacementRule(pattern2838, replacement2838) pattern2839 = Pattern(Integral((WC('g', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons21, cons25) rule2839 = ReplacementRule(pattern2839, replacement2839) pattern2840 = Pattern(Integral((WC('g', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons21, cons25) rule2840 = ReplacementRule(pattern2840, replacement2840) pattern2841 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons50, cons127, cons36, cons37, cons1420, cons1267, cons20, cons87) rule2841 = ReplacementRule(pattern2841, replacement2841) pattern2842 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons50, cons127, cons36, cons37, cons1420, cons1267, cons20, cons87) rule2842 = ReplacementRule(pattern2842, replacement2842) pattern2843 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons4, cons72, cons1267, cons20, cons1311) rule2843 = ReplacementRule(pattern2843, replacement2843) pattern2844 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons4, cons72, cons1267, cons20, cons1311) rule2844 = ReplacementRule(pattern2844, replacement2844) pattern2845 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons73) rule2845 = ReplacementRule(pattern2845, replacement2845) pattern2846 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons73) rule2846 = ReplacementRule(pattern2846, replacement2846) pattern2847 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons72, cons1267) rule2847 = ReplacementRule(pattern2847, replacement2847) pattern2848 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons72, cons1267) rule2848 = ReplacementRule(pattern2848, replacement2848) pattern2849 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons4, cons72, cons1267, cons1421, cons1316) rule2849 = ReplacementRule(pattern2849, replacement2849) pattern2850 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons4, cons72, cons1267, cons1421, cons1316) rule2850 = ReplacementRule(pattern2850, replacement2850) pattern2851 = Pattern(Integral((c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons4, cons72, cons1267) rule2851 = ReplacementRule(pattern2851, replacement2851) pattern2852 = Pattern(Integral((c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons4, cons72, cons1267) rule2852 = ReplacementRule(pattern2852, replacement2852) pattern2853 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons4, cons72, cons1267, cons1422, cons1423) rule2853 = ReplacementRule(pattern2853, replacement2853) pattern2854 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons4, cons72, cons1267, cons1422, cons1423) rule2854 = ReplacementRule(pattern2854, replacement2854) pattern2855 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons4, cons72, cons1267, cons1323, cons685) rule2855 = ReplacementRule(pattern2855, replacement2855) pattern2856 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons4, cons72, cons1267, cons1323, cons685) rule2856 = ReplacementRule(pattern2856, replacement2856) pattern2857 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons4, cons73, cons1267, cons1325, cons74, cons1424) rule2857 = ReplacementRule(pattern2857, replacement2857) pattern2858 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons4, cons73, cons1267, cons1325, cons74, cons1424) rule2858 = ReplacementRule(pattern2858, replacement2858) pattern2859 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1267, cons1325, cons95, cons1425, cons91, cons517, cons1330) rule2859 = ReplacementRule(pattern2859, replacement2859) pattern2860 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1267, cons1325, cons95, cons1425, cons91, cons517, cons1330) rule2860 = ReplacementRule(pattern2860, replacement2860) pattern2861 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons4, cons73, cons1267, cons1325, cons33, cons1425, cons348, cons517, cons1330) rule2861 = ReplacementRule(pattern2861, replacement2861) pattern2862 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons4, cons73, cons1267, cons1325, cons33, cons1425, cons348, cons517, cons1330) rule2862 = ReplacementRule(pattern2862, replacement2862) pattern2863 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1267, cons1325, cons95, cons1322, cons90, cons517, cons1330) rule2863 = ReplacementRule(pattern2863, replacement2863) pattern2864 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1267, cons1325, cons95, cons1322, cons90, cons517, cons1330) rule2864 = ReplacementRule(pattern2864, replacement2864) pattern2865 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons4, cons73, cons1267, cons1325, cons33, cons1322, cons1329, cons517, cons1330) rule2865 = ReplacementRule(pattern2865, replacement2865) pattern2866 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons4, cons73, cons1267, cons1325, cons33, cons1322, cons1329, cons517, cons1330) rule2866 = ReplacementRule(pattern2866, replacement2866) pattern2867 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons4, cons73, cons1267, cons1325, cons1426) rule2867 = ReplacementRule(pattern2867, replacement2867) pattern2868 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons4, cons73, cons1267, cons1325, cons1426) rule2868 = ReplacementRule(pattern2868, replacement2868) pattern2869 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1267, cons1325, cons89, cons91) rule2869 = ReplacementRule(pattern2869, replacement2869) pattern2870 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1267, cons1325, cons89, cons91) rule2870 = ReplacementRule(pattern2870, replacement2870) pattern2871 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons4, cons73, cons1267, cons1325, cons348) rule2871 = ReplacementRule(pattern2871, replacement2871) pattern2872 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons4, cons73, cons1267, cons1325, cons348) rule2872 = ReplacementRule(pattern2872, replacement2872) pattern2873 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1267, cons1325) rule2873 = ReplacementRule(pattern2873, replacement2873) pattern2874 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1267, cons1325) rule2874 = ReplacementRule(pattern2874, replacement2874) pattern2875 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons73, cons1267, cons1325, cons89, cons90, cons1427) rule2875 = ReplacementRule(pattern2875, replacement2875) pattern2876 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons73, cons1267, cons1325, cons89, cons90, cons1427) rule2876 = ReplacementRule(pattern2876, replacement2876) pattern2877 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons73, cons1267, cons1325, cons89, cons91, cons1427) rule2877 = ReplacementRule(pattern2877, replacement2877) pattern2878 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons73, cons1267, cons1325, cons89, cons91, cons1427) rule2878 = ReplacementRule(pattern2878, replacement2878) pattern2879 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1267, cons1325) rule2879 = ReplacementRule(pattern2879, replacement2879) pattern2880 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1267, cons1325) rule2880 = ReplacementRule(pattern2880, replacement2880) pattern2881 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons73, cons1267, cons1325, cons1316) rule2881 = ReplacementRule(pattern2881, replacement2881) pattern2882 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons73, cons1267, cons1325, cons1316) rule2882 = ReplacementRule(pattern2882, replacement2882) pattern2883 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons4, cons73, cons1267, cons1325, cons1428) rule2883 = ReplacementRule(pattern2883, replacement2883) pattern2884 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons4, cons73, cons1267, cons1325, cons1428) rule2884 = ReplacementRule(pattern2884, replacement2884) pattern2885 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1269, cons1325, cons95, cons168, cons91) rule2885 = ReplacementRule(pattern2885, replacement2885) pattern2886 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1269, cons1325, cons95, cons168, cons91) rule2886 = ReplacementRule(pattern2886, replacement2886) pattern2887 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons4, cons73, cons1269, cons1325, cons33, cons168, cons1429) rule2887 = ReplacementRule(pattern2887, replacement2887) pattern2888 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons4, cons73, cons1269, cons1325, cons33, cons168, cons1429) rule2888 = ReplacementRule(pattern2888, replacement2888) pattern2889 = Pattern(Integral(sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2), x_), cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons1325) rule2889 = ReplacementRule(pattern2889, replacement2889) pattern2890 = Pattern(Integral(sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2), x_), cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons1325) rule2890 = ReplacementRule(pattern2890, replacement2890) pattern2891 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1269, cons1325) rule2891 = ReplacementRule(pattern2891, replacement2891) pattern2892 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1269, cons1325) rule2892 = ReplacementRule(pattern2892, replacement2892) pattern2893 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)), x_), cons2, cons3, cons29, cons50, cons127, cons36, cons37, cons1269) rule2893 = ReplacementRule(pattern2893, replacement2893) pattern2894 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)), x_), cons2, cons3, cons29, cons50, cons127, cons36, cons37, cons1269) rule2894 = ReplacementRule(pattern2894, replacement2894) pattern2895 = Pattern(Integral((A_ + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/((WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons1325, cons1430, cons1344) rule2895 = ReplacementRule(pattern2895, replacement2895) pattern2896 = Pattern(Integral((A_ + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/((WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons1325, cons1430, cons1344) rule2896 = ReplacementRule(pattern2896, replacement2896) pattern2897 = Pattern(Integral((A_ + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/((WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons1325, cons1430, cons1346) rule2897 = ReplacementRule(pattern2897, replacement2897) pattern2898 = Pattern(Integral((A_ + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/((WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons1325, cons1430, cons1346) rule2898 = ReplacementRule(pattern2898, replacement2898) pattern2899 = Pattern(Integral((A_ + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1269, cons1325, cons1430, cons1347) rule2899 = ReplacementRule(pattern2899, replacement2899) pattern2900 = Pattern(Integral((A_ + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1269, cons1325, cons1430, cons1347) rule2900 = ReplacementRule(pattern2900, replacement2900) pattern2901 = Pattern(Integral((A_ + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1269, cons1325, cons1430, cons1348) rule2901 = ReplacementRule(pattern2901, replacement2901) pattern2902 = Pattern(Integral((A_ + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1269, cons1325, cons1430, cons1348) rule2902 = ReplacementRule(pattern2902, replacement2902) pattern2903 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1269, cons1325, cons1431) rule2903 = ReplacementRule(pattern2903, replacement2903) pattern2904 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1269, cons1325, cons1431) rule2904 = ReplacementRule(pattern2904, replacement2904) pattern2905 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1269, cons1325, cons95, cons96, cons90) rule2905 = ReplacementRule(pattern2905, replacement2905) pattern2906 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1269, cons1325, cons95, cons96, cons90) rule2906 = ReplacementRule(pattern2906, replacement2906) pattern2907 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons4, cons73, cons1269, cons1325, cons33, cons96, cons1339) rule2907 = ReplacementRule(pattern2907, replacement2907) pattern2908 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons4, cons73, cons1269, cons1325, cons33, cons96, cons1339) rule2908 = ReplacementRule(pattern2908, replacement2908) pattern2909 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1269, cons1325) rule2909 = ReplacementRule(pattern2909, replacement2909) pattern2910 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1269, cons1325) rule2910 = ReplacementRule(pattern2910, replacement2910) pattern2911 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons73, cons1269, cons1325) rule2911 = ReplacementRule(pattern2911, replacement2911) pattern2912 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons73, cons1269, cons1325) rule2912 = ReplacementRule(pattern2912, replacement2912) pattern2913 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1269, cons1325, cons89, cons1432) rule2913 = ReplacementRule(pattern2913, replacement2913) pattern2914 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1269, cons1325, cons89, cons1432) rule2914 = ReplacementRule(pattern2914, replacement2914) pattern2915 = Pattern(Integral((A_ + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons50, cons127, cons36, cons37, cons107, cons1413, cons1430) rule2915 = ReplacementRule(pattern2915, replacement2915) pattern2916 = Pattern(Integral((A_ + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons50, cons127, cons36, cons37, cons107, cons1413, cons1430) rule2916 = ReplacementRule(pattern2916, replacement2916) pattern2917 = Pattern(Integral((A_ + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(d_*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons50, cons127, cons29, cons36, cons37, cons107, cons1413, cons1430) rule2917 = ReplacementRule(pattern2917, replacement2917) pattern2918 = Pattern(Integral((A_ + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(d_*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons50, cons127, cons29, cons36, cons37, cons107, cons1413, cons1430) rule2918 = ReplacementRule(pattern2918, replacement2918) pattern2919 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1269, cons1325) rule2919 = ReplacementRule(pattern2919, replacement2919) pattern2920 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))/(sqrt(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1269, cons1325) rule2920 = ReplacementRule(pattern2920, replacement2920) pattern2921 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons4, cons73, cons1269, cons1325) rule2921 = ReplacementRule(pattern2921, replacement2921) pattern2922 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons4, cons73, cons1269, cons1325) rule2922 = ReplacementRule(pattern2922, replacement2922) pattern2923 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons4, cons5, cons72, cons1267) rule2923 = ReplacementRule(pattern2923, replacement2923) pattern2924 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons4, cons5, cons72, cons1267) rule2924 = ReplacementRule(pattern2924, replacement2924) pattern2925 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons50, cons127, cons37, cons38, cons19, cons1433) rule2925 = ReplacementRule(pattern2925, replacement2925) pattern2926 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons50, cons127, cons37, cons38, cons19, cons1433) rule2926 = ReplacementRule(pattern2926, replacement2926) pattern2927 = Pattern(Integral((A_ + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1)), x_), cons50, cons127, cons36, cons38, cons1230) rule2927 = ReplacementRule(pattern2927, replacement2927) pattern2928 = Pattern(Integral((A_ + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1)), x_), cons50, cons127, cons36, cons38, cons1230) rule2928 = ReplacementRule(pattern2928, replacement2928) pattern2929 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(A_ + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons50, cons127, cons36, cons38, cons19, cons1434) rule2929 = ReplacementRule(pattern2929, replacement2929) pattern2930 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(A_ + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons50, cons127, cons36, cons38, cons19, cons1434) rule2930 = ReplacementRule(pattern2930, replacement2930) pattern2931 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(A_ + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons50, cons127, cons36, cons38, cons33, cons96) rule2931 = ReplacementRule(pattern2931, replacement2931) pattern2932 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(A_ + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons50, cons127, cons36, cons38, cons33, cons96) rule2932 = ReplacementRule(pattern2932, replacement2932) pattern2933 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(A_ + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons50, cons127, cons36, cons38, cons19, cons274) rule2933 = ReplacementRule(pattern2933, replacement2933) pattern2934 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(A_ + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons50, cons127, cons36, cons38, cons19, cons274) rule2934 = ReplacementRule(pattern2934, replacement2934) pattern2935 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons50, cons127, cons36, cons37, cons38, cons19, cons35) rule2935 = ReplacementRule(pattern2935, replacement2935) pattern2936 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons50, cons127, cons36, cons37, cons38, cons19, cons35) rule2936 = ReplacementRule(pattern2936, replacement2936) pattern2937 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons50, cons127, cons36, cons38, cons19, cons1435) rule2937 = ReplacementRule(pattern2937, replacement2937) pattern2938 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons50, cons127, cons36, cons38, cons19, cons1435) rule2938 = ReplacementRule(pattern2938, replacement2938) pattern2939 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons50, cons127, cons36, cons37, cons38, cons19, cons1436, cons79) rule2939 = ReplacementRule(pattern2939, replacement2939) pattern2940 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons50, cons127, cons36, cons37, cons38, cons19, cons1436, cons79) rule2940 = ReplacementRule(pattern2940, replacement2940) pattern2941 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons50, cons127, cons36, cons38, cons19, cons1437, cons79) rule2941 = ReplacementRule(pattern2941, replacement2941) pattern2942 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons50, cons127, cons36, cons38, cons19, cons1437, cons79) rule2942 = ReplacementRule(pattern2942, replacement2942) pattern2943 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons50, cons127, cons36, cons37, cons38, cons33, cons96, cons1267) rule2943 = ReplacementRule(pattern2943, replacement2943) pattern2944 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons50, cons127, cons36, cons37, cons38, cons33, cons96, cons1267) rule2944 = ReplacementRule(pattern2944, replacement2944) pattern2945 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons50, cons127, cons36, cons38, cons33, cons96, cons1267) rule2945 = ReplacementRule(pattern2945, replacement2945) pattern2946 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons50, cons127, cons36, cons38, cons33, cons96, cons1267) rule2946 = ReplacementRule(pattern2946, replacement2946) pattern2947 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons50, cons127, cons36, cons37, cons38, cons33, cons96, cons1269) rule2947 = ReplacementRule(pattern2947, replacement2947) pattern2948 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons50, cons127, cons36, cons37, cons38, cons33, cons96, cons1269) rule2948 = ReplacementRule(pattern2948, replacement2948) pattern2949 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons50, cons127, cons36, cons38, cons33, cons96, cons1269) rule2949 = ReplacementRule(pattern2949, replacement2949) pattern2950 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons50, cons127, cons36, cons38, cons33, cons96, cons1269) rule2950 = ReplacementRule(pattern2950, replacement2950) pattern2951 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons50, cons127, cons36, cons37, cons38, cons19, cons274) rule2951 = ReplacementRule(pattern2951, replacement2951) pattern2952 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons50, cons127, cons36, cons37, cons38, cons19, cons274) rule2952 = ReplacementRule(pattern2952, replacement2952) pattern2953 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons50, cons127, cons36, cons38, cons19, cons274) rule2953 = ReplacementRule(pattern2953, replacement2953) pattern2954 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons50, cons127, cons36, cons38, cons19, cons274) rule2954 = ReplacementRule(pattern2954, replacement2954) pattern2955 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_)**m_*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons50, cons127, cons36, cons37, cons38, cons19, cons5, cons21) rule2955 = ReplacementRule(pattern2955, replacement2955) pattern2956 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_)**m_*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons50, cons127, cons36, cons37, cons38, cons19, cons5, cons21) rule2956 = ReplacementRule(pattern2956, replacement2956) pattern2957 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_)**m_*(WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons50, cons127, cons36, cons38, cons19, cons5, cons21) rule2957 = ReplacementRule(pattern2957, replacement2957) pattern2958 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_)**m_*(WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons50, cons127, cons36, cons38, cons19, cons5, cons21) rule2958 = ReplacementRule(pattern2958, replacement2958) pattern2959 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons73, cons1269, cons33, cons96) rule2959 = ReplacementRule(pattern2959, replacement2959) pattern2960 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons73, cons1269, cons33, cons96) rule2960 = ReplacementRule(pattern2960, replacement2960) pattern2961 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons73, cons1269, cons33, cons96) rule2961 = ReplacementRule(pattern2961, replacement2961) pattern2962 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons73, cons1269, cons33, cons96) rule2962 = ReplacementRule(pattern2962, replacement2962) pattern2963 = Pattern(Integral((c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons19, cons73, cons1269, cons274) rule2963 = ReplacementRule(pattern2963, replacement2963) pattern2964 = Pattern(Integral((c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons19, cons73, cons1269, cons274) rule2964 = ReplacementRule(pattern2964, replacement2964) pattern2965 = Pattern(Integral((c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons19, cons73, cons1269, cons274) rule2965 = ReplacementRule(pattern2965, replacement2965) pattern2966 = Pattern(Integral((c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons19, cons73, cons1269, cons274) rule2966 = ReplacementRule(pattern2966, replacement2966) pattern2967 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons19, cons4, cons72, cons1267, cons1438) rule2967 = ReplacementRule(pattern2967, replacement2967) pattern2968 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons19, cons4, cons72, cons1267, cons1438) rule2968 = ReplacementRule(pattern2968, replacement2968) pattern2969 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons19, cons4, cons72, cons1267, cons1438) rule2969 = ReplacementRule(pattern2969, replacement2969) pattern2970 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons19, cons4, cons72, cons1267, cons1438) rule2970 = ReplacementRule(pattern2970, replacement2970) pattern2971 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/sqrt(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons19, cons72, cons1267, cons1323) rule2971 = ReplacementRule(pattern2971, replacement2971) pattern2972 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/sqrt(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons19, cons72, cons1267, cons1323) rule2972 = ReplacementRule(pattern2972, replacement2972) pattern2973 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))/sqrt(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons19, cons72, cons1267, cons1323) rule2973 = ReplacementRule(pattern2973, replacement2973) pattern2974 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))/sqrt(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons19, cons72, cons1267, cons1323) rule2974 = ReplacementRule(pattern2974, replacement2974) pattern2975 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons19, cons4, cons72, cons1267, cons1323, cons216) rule2975 = ReplacementRule(pattern2975, replacement2975) pattern2976 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons19, cons4, cons72, cons1267, cons1323, cons216) rule2976 = ReplacementRule(pattern2976, replacement2976) pattern2977 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons19, cons4, cons72, cons1267, cons1323, cons216) rule2977 = ReplacementRule(pattern2977, replacement2977) pattern2978 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons19, cons4, cons72, cons1267, cons1323, cons216) rule2978 = ReplacementRule(pattern2978, replacement2978) pattern2979 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons4, cons73, cons1267, cons1325, cons33, cons1322) rule2979 = ReplacementRule(pattern2979, replacement2979) pattern2980 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons4, cons73, cons1267, cons1325, cons33, cons1322) rule2980 = ReplacementRule(pattern2980, replacement2980) pattern2981 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons4, cons73, cons1267, cons1325, cons33, cons1322) rule2981 = ReplacementRule(pattern2981, replacement2981) pattern2982 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons4, cons73, cons1267, cons1325, cons33, cons1322) rule2982 = ReplacementRule(pattern2982, replacement2982) pattern2983 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons19, cons73, cons1267, cons1325, cons1323, cons1439) rule2983 = ReplacementRule(pattern2983, replacement2983) pattern2984 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons19, cons73, cons1267, cons1325, cons1323, cons1439) rule2984 = ReplacementRule(pattern2984, replacement2984) pattern2985 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons19, cons73, cons1267, cons1325, cons1323, cons1439) rule2985 = ReplacementRule(pattern2985, replacement2985) pattern2986 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons19, cons73, cons1267, cons1325, cons1323, cons1439) rule2986 = ReplacementRule(pattern2986, replacement2986) pattern2987 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons19, cons4, cons73, cons1267, cons1325, cons1323, cons216) rule2987 = ReplacementRule(pattern2987, replacement2987) pattern2988 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons19, cons4, cons73, cons1267, cons1325, cons1323, cons216) rule2988 = ReplacementRule(pattern2988, replacement2988) pattern2989 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons19, cons4, cons73, cons1267, cons1325, cons1323, cons216) rule2989 = ReplacementRule(pattern2989, replacement2989) pattern2990 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons19, cons4, cons73, cons1267, cons1325, cons1323, cons216) rule2990 = ReplacementRule(pattern2990, replacement2990) pattern2991 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons73, cons1269, cons1325, cons95, cons170, cons91) rule2991 = ReplacementRule(pattern2991, replacement2991) pattern2992 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons73, cons1269, cons1325, cons95, cons170, cons91) rule2992 = ReplacementRule(pattern2992, replacement2992) pattern2993 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons73, cons1269, cons1325, cons95, cons170, cons91) rule2993 = ReplacementRule(pattern2993, replacement2993) pattern2994 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons73, cons1269, cons1325, cons95, cons170, cons91) rule2994 = ReplacementRule(pattern2994, replacement2994) pattern2995 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons4, cons73, cons1269, cons1325, cons33, cons170, cons1440) rule2995 = ReplacementRule(pattern2995, replacement2995) pattern2996 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons4, cons73, cons1269, cons1325, cons33, cons170, cons1440) rule2996 = ReplacementRule(pattern2996, replacement2996) pattern2997 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons4, cons73, cons1269, cons1325, cons33, cons170, cons1440) rule2997 = ReplacementRule(pattern2997, replacement2997) pattern2998 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons4, cons73, cons1269, cons1325, cons33, cons170, cons1440) rule2998 = ReplacementRule(pattern2998, replacement2998) pattern2999 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(sqrt(WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)), x_), cons2, cons3, cons29, cons50, cons127, cons36, cons37, cons38, cons1269) rule2999 = ReplacementRule(pattern2999, replacement2999) pattern3000 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(sqrt(WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)), x_), cons2, cons3, cons29, cons50, cons127, cons36, cons37, cons38, cons1269) rule3000 = ReplacementRule(pattern3000, replacement3000) pattern3001 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(sqrt(WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)), x_), cons2, cons3, cons29, cons50, cons127, cons36, cons38, cons1269) rule3001 = ReplacementRule(pattern3001, replacement3001) pattern3002 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(sqrt(WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)), x_), cons2, cons3, cons29, cons50, cons127, cons36, cons38, cons1269) rule3002 = ReplacementRule(pattern3002, replacement3002) pattern3003 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sqrt(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons73, cons1269, cons1325) rule3003 = ReplacementRule(pattern3003, replacement3003) pattern3004 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sqrt(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons73, cons1269, cons1325) rule3004 = ReplacementRule(pattern3004, replacement3004) pattern3005 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sqrt(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons73, cons1269, cons1325) rule3005 = ReplacementRule(pattern3005, replacement3005) pattern3006 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sqrt(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons73, cons1269, cons1325) rule3006 = ReplacementRule(pattern3006, replacement3006) pattern3007 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons4, cons73, cons1269, cons1325, cons33, cons96, cons1339) rule3007 = ReplacementRule(pattern3007, replacement3007) pattern3008 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons4, cons73, cons1269, cons1325, cons33, cons96, cons1339) rule3008 = ReplacementRule(pattern3008, replacement3008) pattern3009 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons4, cons73, cons1269, cons1325, cons33, cons96, cons1339) rule3009 = ReplacementRule(pattern3009, replacement3009) pattern3010 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons4, cons73, cons1269, cons1325, cons33, cons96, cons1339) rule3010 = ReplacementRule(pattern3010, replacement3010) pattern3011 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons73, cons1269, cons1325) rule3011 = ReplacementRule(pattern3011, replacement3011) pattern3012 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons73, cons1269, cons1325) rule3012 = ReplacementRule(pattern3012, replacement3012) pattern3013 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons73, cons1269, cons1325) rule3013 = ReplacementRule(pattern3013, replacement3013) pattern3014 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons73, cons1269, cons1325) rule3014 = ReplacementRule(pattern3014, replacement3014) pattern3015 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(sqrt(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons73, cons1269, cons1325) rule3015 = ReplacementRule(pattern3015, replacement3015) pattern3016 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(sqrt(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons73, cons1269, cons1325) rule3016 = ReplacementRule(pattern3016, replacement3016) pattern3017 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(sqrt(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons73, cons1269, cons1325) rule3017 = ReplacementRule(pattern3017, replacement3017) pattern3018 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(sqrt(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons73, cons1269, cons1325) rule3018 = ReplacementRule(pattern3018, replacement3018) pattern3019 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons73, cons1269, cons1325) rule3019 = ReplacementRule(pattern3019, replacement3019) pattern3020 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons73, cons1269, cons1325) rule3020 = ReplacementRule(pattern3020, replacement3020) pattern3021 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(sqrt(c_ + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons73, cons1269, cons1325) rule3021 = ReplacementRule(pattern3021, replacement3021) pattern3022 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(sqrt(c_ + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons73, cons1269, cons1325) rule3022 = ReplacementRule(pattern3022, replacement3022) pattern3023 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons19, cons4, cons73, cons1269, cons1325) rule3023 = ReplacementRule(pattern3023, replacement3023) pattern3024 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons19, cons4, cons73, cons1269, cons1325) rule3024 = ReplacementRule(pattern3024, replacement3024) pattern3025 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons19, cons4, cons73, cons1269, cons1325) rule3025 = ReplacementRule(pattern3025, replacement3025) pattern3026 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons19, cons4, cons73, cons1269, cons1325) rule3026 = ReplacementRule(pattern3026, replacement3026) pattern3027 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_)**m_*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons19, cons4, cons5, cons21) rule3027 = ReplacementRule(pattern3027, replacement3027) pattern3028 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_)**m_*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons19, cons4, cons5, cons21) rule3028 = ReplacementRule(pattern3028, replacement3028) pattern3029 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**p_)**m_*(WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons19, cons4, cons5, cons21) rule3029 = ReplacementRule(pattern3029, replacement3029) pattern3030 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**p_)**m_*(WC('A', S(0)) + WC('C', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons19, cons4, cons5, cons21) rule3030 = ReplacementRule(pattern3030, replacement3030) pattern3031 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons4, cons1441) rule3031 = ReplacementRule(pattern3031, replacement3031) pattern3032 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons1442, cons523) rule3032 = ReplacementRule(pattern3032, replacement3032) pattern3033 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons1442, cons1443, cons89, cons167) rule3033 = ReplacementRule(pattern3033, replacement3033) pattern3034 = Pattern(Integral(S(1)/(WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1442) rule3034 = ReplacementRule(pattern3034, replacement3034) pattern3035 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**(S(-2)), x_), cons2, cons3, cons8, cons29, cons1442) rule3035 = ReplacementRule(pattern3035, replacement3035) pattern3036 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons1442, cons89, cons91, cons1444) rule3036 = ReplacementRule(pattern3036, replacement3036) pattern3037 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons4, cons1445, cons1446) rule3037 = ReplacementRule(pattern3037, replacement3037) pattern3038 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons4, cons1445, cons1447) rule3038 = ReplacementRule(pattern3038, replacement3038) pattern3039 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*sin(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons1257, cons1441, cons89, cons167) rule3039 = ReplacementRule(pattern3039, replacement3039) pattern3040 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*cos(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons1257, cons1441, cons89, cons167) rule3040 = ReplacementRule(pattern3040, replacement3040) pattern3041 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*sin(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons1257, cons1441, cons89, cons465) rule3041 = ReplacementRule(pattern3041, replacement3041) pattern3042 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*cos(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons1257, cons1441, cons89, cons465) rule3042 = ReplacementRule(pattern3042, replacement3042) pattern3043 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*sin(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons4, cons1257, cons1441, cons25) rule3043 = ReplacementRule(pattern3043, replacement3043) pattern3044 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*cos(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons4, cons1257, cons1441, cons25) rule3044 = ReplacementRule(pattern3044, replacement3044) pattern3045 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons1257, cons87, cons1442) rule3045 = ReplacementRule(pattern3045, replacement3045) pattern3046 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons1257, cons87, cons1442) rule3046 = ReplacementRule(pattern3046, replacement3046) pattern3047 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*sin(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons87, cons1448, cons1154, cons1449) rule3047 = ReplacementRule(pattern3047, replacement3047) pattern3048 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*cos(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons87, cons1448, cons1154, cons1449) rule3048 = ReplacementRule(pattern3048, replacement3048) pattern3049 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons20, cons150) rule3049 = ReplacementRule(pattern3049, replacement3049) pattern3050 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**WC('n', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons20, cons150) rule3050 = ReplacementRule(pattern3050, replacement3050) pattern3051 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*sin(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons19, cons1441, cons198) rule3051 = ReplacementRule(pattern3051, replacement3051) pattern3052 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*cos(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons19, cons1441, cons198) rule3052 = ReplacementRule(pattern3052, replacement3052) pattern3053 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_/sin(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons1442, cons89, cons91) rule3053 = ReplacementRule(pattern3053, replacement3053) pattern3054 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_/cos(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons1442, cons89, cons91) rule3054 = ReplacementRule(pattern3054, replacement3054) pattern3055 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*sin(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons1442, cons95, cons167, cons96) rule3055 = ReplacementRule(pattern3055, replacement3055) pattern3056 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*cos(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons1442, cons95, cons167, cons96) rule3056 = ReplacementRule(pattern3056, replacement3056) pattern3057 = Pattern(Integral(sin(x_*WC('d', S(1)) + WC('c', S(0)))/(WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1442) rule3057 = ReplacementRule(pattern3057, replacement3057) pattern3058 = Pattern(Integral(cos(x_*WC('d', S(1)) + WC('c', S(0)))/(WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1442) rule3058 = ReplacementRule(pattern3058, replacement3058) pattern3059 = Pattern(Integral(sin(x_*WC('d', S(1)) + WC('c', S(0)))**m_/(WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1442, cons33, cons168) rule3059 = ReplacementRule(pattern3059, replacement3059) pattern3060 = Pattern(Integral(cos(x_*WC('d', S(1)) + WC('c', S(0)))**m_/(WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1442, cons33, cons168) rule3060 = ReplacementRule(pattern3060, replacement3060) pattern3061 = Pattern(Integral(S(1)/((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1442) rule3061 = ReplacementRule(pattern3061, replacement3061) pattern3062 = Pattern(Integral(S(1)/((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))*cos(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1442) rule3062 = ReplacementRule(pattern3062, replacement3062) pattern3063 = Pattern(Integral(sin(x_*WC('d', S(1)) + WC('c', S(0)))**m_/(WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1442, cons33, cons96) rule3063 = ReplacementRule(pattern3063, replacement3063) pattern3064 = Pattern(Integral(cos(x_*WC('d', S(1)) + WC('c', S(0)))**m_/(WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1442, cons33, cons96) rule3064 = ReplacementRule(pattern3064, replacement3064) pattern3065 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*sin(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons1442, cons95, cons91, cons96) rule3065 = ReplacementRule(pattern3065, replacement3065) pattern3066 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*cos(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons1442, cons95, cons91, cons96) rule3066 = ReplacementRule(pattern3066, replacement3066) pattern3067 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons19, cons4, cons130) rule3067 = ReplacementRule(pattern3067, replacement3067) pattern3068 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_*sin(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons19, cons4, cons1441, cons65) rule3068 = ReplacementRule(pattern3068, replacement3068) pattern3069 = Pattern(Integral(sin(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/(WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1442, cons152, cons170, cons90) rule3069 = ReplacementRule(pattern3069, replacement3069) pattern3070 = Pattern(Integral(sin(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/(WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons19, cons4, cons152) rule3070 = ReplacementRule(pattern3070, replacement3070) pattern3071 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))) + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**p_*sin(x_*WC('d', S(1)) + WC('c', S(0)))**WC('n', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons1442, cons377, cons170, cons90, cons324) rule3071 = ReplacementRule(pattern3071, replacement3071) pattern3072 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons1450) rule3072 = ReplacementRule(pattern3072, replacement3072) pattern3073 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons1450, cons89, cons90) rule3073 = ReplacementRule(pattern3073, replacement3073) pattern3074 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons1450) rule3074 = ReplacementRule(pattern3074, replacement3074) pattern3075 = Pattern(Integral(S(1)/sqrt(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons1450) rule3075 = ReplacementRule(pattern3075, replacement3075) pattern3076 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons1450, cons89, cons91) rule3076 = ReplacementRule(pattern3076, replacement3076) pattern3077 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons1451) rule3077 = ReplacementRule(pattern3077, replacement3077) pattern3078 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons1452, cons1453) rule3078 = ReplacementRule(pattern3078, replacement3078) pattern3079 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons1454, cons1452, cons1455) rule3079 = ReplacementRule(pattern3079, replacement3079) pattern3080 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons1454, cons89, cons167) rule3080 = ReplacementRule(pattern3080, replacement3080) pattern3081 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons1456) rule3081 = ReplacementRule(pattern3081, With3081) pattern3082 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons78) rule3082 = ReplacementRule(pattern3082, With3082) pattern3083 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons1457, cons1458) rule3083 = ReplacementRule(pattern3083, With3083) pattern3084 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons1454) rule3084 = ReplacementRule(pattern3084, With3084) pattern3085 = Pattern(Integral(S(1)/sqrt(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons1451) rule3085 = ReplacementRule(pattern3085, replacement3085) pattern3086 = Pattern(Integral(S(1)/sqrt(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons1452, cons1453) rule3086 = ReplacementRule(pattern3086, replacement3086) pattern3087 = Pattern(Integral(S(1)/sqrt(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons1454, cons1452, cons1455) rule3087 = ReplacementRule(pattern3087, replacement3087) pattern3088 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**(S(-3)/2), x_), cons2, cons3, cons8, cons29, cons50, cons1454) rule3088 = ReplacementRule(pattern3088, replacement3088) pattern3089 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons1454, cons89, cons91, cons1459) rule3089 = ReplacementRule(pattern3089, replacement3089) pattern3090 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))/(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons38, cons1451) rule3090 = ReplacementRule(pattern3090, replacement3090) pattern3091 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))/(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons36, cons38, cons1451) rule3091 = ReplacementRule(pattern3091, replacement3091) pattern3092 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))/(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons1451) rule3092 = ReplacementRule(pattern3092, replacement3092) pattern3093 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))/(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons38, cons1452, cons1460) rule3093 = ReplacementRule(pattern3093, replacement3093) pattern3094 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))/(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons36, cons38, cons1452, cons1461) rule3094 = ReplacementRule(pattern3094, replacement3094) pattern3095 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))/(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons1452, cons1462) rule3095 = ReplacementRule(pattern3095, replacement3095) pattern3096 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))/(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons38, cons1452, cons1463) rule3096 = ReplacementRule(pattern3096, replacement3096) pattern3097 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))/(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons36, cons38, cons1452, cons1464) rule3097 = ReplacementRule(pattern3097, replacement3097) pattern3098 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))/(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons1452, cons1465) rule3098 = ReplacementRule(pattern3098, replacement3098) pattern3099 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons38, cons4, cons586, cons1450, cons1466) rule3099 = ReplacementRule(pattern3099, replacement3099) pattern3100 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))*(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons36, cons38, cons4, cons586, cons1450, cons1467) rule3100 = ReplacementRule(pattern3100, replacement3100) pattern3101 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))*(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons4, cons586, cons1450, cons1468) rule3101 = ReplacementRule(pattern3101, replacement3101) pattern3102 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons38, cons4, cons586, cons1450, cons1469) rule3102 = ReplacementRule(pattern3102, replacement3102) pattern3103 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))*(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons36, cons38, cons4, cons586, cons1450, cons1470) rule3103 = ReplacementRule(pattern3103, replacement3103) pattern3104 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))*(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons4, cons586, cons1450, cons1471) rule3104 = ReplacementRule(pattern3104, replacement3104) pattern3105 = Pattern(Integral((WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))*(WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons3, cons8, cons29, cons50, cons37, cons38, cons586, cons1452, cons1472) rule3105 = ReplacementRule(pattern3105, replacement3105) pattern3106 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons38, cons89, cons90, cons1454) rule3106 = ReplacementRule(pattern3106, replacement3106) pattern3107 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))*(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons36, cons38, cons89, cons90, cons1454) rule3107 = ReplacementRule(pattern3107, replacement3107) pattern3108 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))*(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons89, cons90, cons1454) rule3108 = ReplacementRule(pattern3108, replacement3108) pattern3109 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))/sqrt(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons38, cons1473, cons1247) rule3109 = ReplacementRule(pattern3109, replacement3109) pattern3110 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))/(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**S(2), x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons38, cons1454, cons1474) rule3110 = ReplacementRule(pattern3110, replacement3110) pattern3111 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))/(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**S(2), x_), cons2, cons3, cons8, cons29, cons50, cons36, cons38, cons1454, cons1475) rule3111 = ReplacementRule(pattern3111, replacement3111) pattern3112 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))/(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**S(2), x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons1454, cons1476) rule3112 = ReplacementRule(pattern3112, replacement3112) pattern3113 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))/(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**S(2), x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons38, cons1454, cons1477) rule3113 = ReplacementRule(pattern3113, replacement3113) pattern3114 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))/(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**S(2), x_), cons2, cons3, cons8, cons29, cons50, cons36, cons38, cons1454, cons1478) rule3114 = ReplacementRule(pattern3114, replacement3114) pattern3115 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))/(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**S(2), x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons1454, cons1479) rule3115 = ReplacementRule(pattern3115, replacement3115) pattern3116 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons38, cons89, cons91, cons1454, cons1444) rule3116 = ReplacementRule(pattern3116, replacement3116) pattern3117 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons36, cons38, cons89, cons91, cons1454, cons1444) rule3117 = ReplacementRule(pattern3117, replacement3117) pattern3118 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons89, cons91, cons1454, cons1444) rule3118 = ReplacementRule(pattern3118, replacement3118) pattern3119 = Pattern(Integral(S(1)/(WC('a', S(0)) + WC('b', S(1))/cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons1045) rule3119 = ReplacementRule(pattern3119, replacement3119) pattern3120 = Pattern(Integral(S(1)/(WC('a', S(0)) + WC('b', S(1))/sin(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons1045) rule3120 = ReplacementRule(pattern3120, replacement3120) pattern3121 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons87) rule3121 = ReplacementRule(pattern3121, replacement3121) pattern3122 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/sin(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons87) rule3122 = ReplacementRule(pattern3122, replacement3122) pattern3123 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0))))**n_*cos(x_*WC('e', S(1)) + WC('d', S(0)))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons25) rule3123 = ReplacementRule(pattern3123, replacement3123) pattern3124 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/sin(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0))))**n_*sin(x_*WC('e', S(1)) + WC('d', S(0)))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons25) rule3124 = ReplacementRule(pattern3124, replacement3124) pattern3125 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0))))**m_*(S(1)/cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons1257, cons87) rule3125 = ReplacementRule(pattern3125, replacement3125) pattern3126 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/sin(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0))))**m_*(S(1)/sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons1257, cons87) rule3126 = ReplacementRule(pattern3126, replacement3126) pattern3127 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0))))**m_*(S(1)/cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons1257, cons25) rule3127 = ReplacementRule(pattern3127, replacement3127) pattern3128 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/sin(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0))))**m_*(S(1)/sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons1257, cons25) rule3128 = ReplacementRule(pattern3128, replacement3128) pattern3129 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**p_, x_), cons3, cons8, cons29, cons13, cons148) rule3129 = ReplacementRule(pattern3129, replacement3129) pattern3130 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**p_, x_), cons3, cons8, cons29, cons13, cons148) rule3130 = ReplacementRule(pattern3130, replacement3130) pattern3131 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**p_, x_), cons3, cons8, cons29, cons13, cons139) rule3131 = ReplacementRule(pattern3131, replacement3131) pattern3132 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**p_, x_), cons3, cons8, cons29, cons13, cons139) rule3132 = ReplacementRule(pattern3132, replacement3132) pattern3133 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**p_, x_), cons2, cons3, cons8, cons29, cons1456, cons40) rule3133 = ReplacementRule(pattern3133, replacement3133) pattern3134 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**p_, x_), cons2, cons3, cons8, cons29, cons1456, cons40) rule3134 = ReplacementRule(pattern3134, replacement3134) pattern3135 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**p_, x_), cons2, cons3, cons8, cons29, cons5, cons1456, cons149) rule3135 = ReplacementRule(pattern3135, replacement3135) pattern3136 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**p_, x_), cons2, cons3, cons8, cons29, cons5, cons1456, cons149) rule3136 = ReplacementRule(pattern3136, replacement3136) pattern3137 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**p_, x_), cons2, cons3, cons8, cons29, cons40) rule3137 = ReplacementRule(pattern3137, With3137) pattern3138 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**p_, x_), cons2, cons3, cons8, cons29, cons40) rule3138 = ReplacementRule(pattern3138, With3138) pattern3139 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**p_, x_), cons2, cons3, cons8, cons29, cons5, cons1480, cons149) rule3139 = ReplacementRule(pattern3139, replacement3139) pattern3140 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**p_, x_), cons2, cons3, cons8, cons29, cons5, cons1480, cons149) rule3140 = ReplacementRule(pattern3140, replacement3140) pattern3141 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**p_, x_), cons2, cons3, cons8, cons29, cons87, cons130) rule3141 = ReplacementRule(pattern3141, replacement3141) pattern3142 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**p_, x_), cons2, cons3, cons8, cons29, cons87, cons130) rule3142 = ReplacementRule(pattern3142, replacement3142) pattern3143 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**p_, x_), cons2, cons3, cons8, cons29, cons1481, cons40, cons139) rule3143 = ReplacementRule(pattern3143, With3143) pattern3144 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**p_, x_), cons2, cons3, cons8, cons29, cons1481, cons40, cons139) rule3144 = ReplacementRule(pattern3144, With3144) pattern3145 = Pattern(Integral(u_*(a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**p_, x_), cons2, cons3, cons8, cons29, cons1456, cons40) rule3145 = ReplacementRule(pattern3145, replacement3145) pattern3146 = Pattern(Integral(u_*(a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**p_, x_), cons2, cons3, cons8, cons29, cons1456, cons40) rule3146 = ReplacementRule(pattern3146, replacement3146) pattern3147 = Pattern(Integral(u_*(a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**p_, x_), cons2, cons3, cons8, cons29, cons5, cons1456, cons149) rule3147 = ReplacementRule(pattern3147, replacement3147) pattern3148 = Pattern(Integral(u_*(a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**S(2))**p_, x_), cons2, cons3, cons8, cons29, cons5, cons1456, cons149) rule3148 = ReplacementRule(pattern3148, replacement3148) pattern3149 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**p_*sin(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons1481, cons1482, cons40) rule3149 = ReplacementRule(pattern3149, With3149) pattern3150 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**p_*cos(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons1481, cons1482, cons40) rule3150 = ReplacementRule(pattern3150, With3150) pattern3151 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**p_*sin(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons5, cons1481, cons1483) rule3151 = ReplacementRule(pattern3151, With3151) pattern3152 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**p_*cos(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons5, cons1481, cons1483) rule3152 = ReplacementRule(pattern3152, With3152) pattern3153 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**p_*sin(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons377) rule3153 = ReplacementRule(pattern3153, replacement3153) pattern3154 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**p_*cos(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons377) rule3154 = ReplacementRule(pattern3154, replacement3154) pattern3155 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**p_*cos(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons1482, cons1481, cons40) rule3155 = ReplacementRule(pattern3155, With3155) pattern3156 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**p_*sin(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons1482, cons1481, cons40) rule3156 = ReplacementRule(pattern3156, With3156) pattern3157 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**p_*cos(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons1482, cons1484, cons40, cons170) rule3157 = ReplacementRule(pattern3157, replacement3157) pattern3158 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**p_*sin(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons1482, cons1484, cons40, cons170) rule3158 = ReplacementRule(pattern3158, replacement3158) pattern3159 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**p_*cos(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons1482, cons1484, cons40, cons269, cons139) rule3159 = ReplacementRule(pattern3159, replacement3159) pattern3160 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**p_*sin(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons1482, cons1484, cons40, cons269, cons139) rule3160 = ReplacementRule(pattern3160, replacement3160) pattern3161 = Pattern(Integral((a_ + (WC('e', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**WC('p', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons4, cons5, cons1483) rule3161 = ReplacementRule(pattern3161, With3161) pattern3162 = Pattern(Integral((a_ + (WC('e', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**WC('p', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons4, cons5, cons1483) rule3162 = ReplacementRule(pattern3162, With3162) pattern3163 = Pattern(Integral((a_ + (WC('e', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**WC('p', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons4, cons1483, cons248) rule3163 = ReplacementRule(pattern3163, With3163) pattern3164 = Pattern(Integral((a_ + (WC('e', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**WC('p', S(1))*(S(1)/tan(x_*WC('d', S(1)) + WC('c', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons4, cons1483, cons248) rule3164 = ReplacementRule(pattern3164, With3164) pattern3165 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**WC('p', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons19, cons1485, cons1481, cons40) rule3165 = ReplacementRule(pattern3165, With3165) pattern3166 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**WC('p', S(1))*(S(1)/tan(x_*WC('d', S(1)) + WC('c', S(0))))**m_, x_), cons2, cons3, cons8, cons29, cons19, cons1485, cons1481, cons40) rule3166 = ReplacementRule(pattern3166, With3166) pattern3167 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**WC('p', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons19, cons5, cons1485, cons1481, cons149) rule3167 = ReplacementRule(pattern3167, With3167) pattern3168 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**WC('p', S(1))*(S(1)/tan(x_*WC('d', S(1)) + WC('c', S(0))))**m_, x_), cons2, cons3, cons8, cons29, cons19, cons5, cons1485, cons1481, cons149) rule3168 = ReplacementRule(pattern3168, With3168) pattern3169 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**p_ + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**q_)**n_*sin(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons1482, cons1486, cons1487, cons87, cons1488) rule3169 = ReplacementRule(pattern3169, With3169) pattern3170 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**p_ + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**q_)**n_*cos(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons1482, cons1486, cons1487, cons87, cons1488) rule3170 = ReplacementRule(pattern3170, With3170) pattern3171 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**p_ + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**q_)**n_*sin(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons1482, cons1486, cons1487, cons87, cons1489) rule3171 = ReplacementRule(pattern3171, With3171) pattern3172 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**p_ + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**q_)**n_*cos(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons1482, cons1486, cons1487, cons87, cons1489) rule3172 = ReplacementRule(pattern3172, With3172) pattern3173 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons4, cons48, cons47, cons40) rule3173 = ReplacementRule(pattern3173, replacement3173) pattern3174 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons4, cons48, cons47, cons40) rule3174 = ReplacementRule(pattern3174, replacement3174) pattern3175 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons4, cons5, cons48, cons47, cons149) rule3175 = ReplacementRule(pattern3175, replacement3175) pattern3176 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons4, cons5, cons48, cons47, cons149) rule3176 = ReplacementRule(pattern3176, replacement3176) pattern3177 = Pattern(Integral(S(1)/(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons4, cons48, cons228) rule3177 = ReplacementRule(pattern3177, With3177) pattern3178 = Pattern(Integral(S(1)/(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons4, cons48, cons228) rule3178 = ReplacementRule(pattern3178, With3178) pattern3179 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**p_*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons48, cons47, cons40) rule3179 = ReplacementRule(pattern3179, replacement3179) pattern3180 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**p_*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons48, cons47, cons40) rule3180 = ReplacementRule(pattern3180, replacement3180) pattern3181 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**p_*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons5, cons48, cons47, cons149) rule3181 = ReplacementRule(pattern3181, replacement3181) pattern3182 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**p_*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons5, cons48, cons47, cons149) rule3182 = ReplacementRule(pattern3182, replacement3182) pattern3183 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**n_ + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**n2_)**p_*sin(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons48, cons1482, cons228, cons1481, cons40) rule3183 = ReplacementRule(pattern3183, With3183) pattern3184 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**n_ + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**n2_)**p_*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons48, cons1482, cons228, cons1481, cons40) rule3184 = ReplacementRule(pattern3184, With3184) pattern3185 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**p_*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons48, cons228, cons377) rule3185 = ReplacementRule(pattern3185, replacement3185) pattern3186 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**p_*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons48, cons228, cons377) rule3186 = ReplacementRule(pattern3186, replacement3186) pattern3187 = Pattern(Integral(((WC('f', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (WC('f', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons5, cons48, cons1483) rule3187 = ReplacementRule(pattern3187, With3187) pattern3188 = Pattern(Integral(((WC('f', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (WC('f', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**WC('p', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons5, cons48, cons1483) rule3188 = ReplacementRule(pattern3188, With3188) pattern3189 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**WC('p', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons48, cons1485, cons47, cons40) rule3189 = ReplacementRule(pattern3189, replacement3189) pattern3190 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**WC('p', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons48, cons1485, cons47, cons40) rule3190 = ReplacementRule(pattern3190, replacement3190) pattern3191 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**p_*cos(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons5, cons48, cons1485, cons47, cons149) rule3191 = ReplacementRule(pattern3191, replacement3191) pattern3192 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**p_*sin(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons5, cons48, cons1485, cons47, cons149) rule3192 = ReplacementRule(pattern3192, replacement3192) pattern3193 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**n_ + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**n2_)**WC('p', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons48, cons1482, cons228, cons1481, cons40) rule3193 = ReplacementRule(pattern3193, With3193) pattern3194 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**n_ + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**n2_)**WC('p', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons48, cons1482, cons228, cons1481, cons40) rule3194 = ReplacementRule(pattern3194, With3194) pattern3195 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**WC('p', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons48, cons1482, cons228, cons378) rule3195 = ReplacementRule(pattern3195, replacement3195) pattern3196 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**WC('p', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons48, cons1482, cons228, cons378) rule3196 = ReplacementRule(pattern3196, replacement3196) pattern3197 = Pattern(Integral((a_ + (WC('f', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**n_*WC('b', S(1)) + (WC('f', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons1483, cons248) rule3197 = ReplacementRule(pattern3197, With3197) pattern3198 = Pattern(Integral((a_ + (WC('f', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))**n_*WC('b', S(1)) + (WC('f', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*(S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons1483, cons248) rule3198 = ReplacementRule(pattern3198, With3198) pattern3199 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**WC('p', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons48, cons1485, cons47, cons40) rule3199 = ReplacementRule(pattern3199, replacement3199) pattern3200 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**WC('p', S(1))*(S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons48, cons1485, cons47, cons40) rule3200 = ReplacementRule(pattern3200, replacement3200) pattern3201 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**p_*tan(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons5, cons48, cons1485, cons47, cons149) rule3201 = ReplacementRule(pattern3201, replacement3201) pattern3202 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**p_*(S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons5, cons48, cons1485, cons47, cons149) rule3202 = ReplacementRule(pattern3202, replacement3202) pattern3203 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**n_ + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**n2_)**WC('p', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons48, cons1485, cons228, cons1481, cons40) rule3203 = ReplacementRule(pattern3203, With3203) pattern3204 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**n_ + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**n2_)**WC('p', S(1))*(S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons48, cons1485, cons228, cons1481, cons40) rule3204 = ReplacementRule(pattern3204, With3204) pattern3205 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**WC('p', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons48, cons1482, cons228, cons378) rule3205 = ReplacementRule(pattern3205, replacement3205) pattern3206 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**WC('p', S(1))*(S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons48, cons1482, cons228, cons378) rule3206 = ReplacementRule(pattern3206, replacement3206) pattern3207 = Pattern(Integral((a_ + (WC('f', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**n_*WC('b', S(1)) + (WC('f', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*(S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons1483, cons248) rule3207 = ReplacementRule(pattern3207, With3207) pattern3208 = Pattern(Integral((a_ + (WC('f', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))**n_*WC('b', S(1)) + (WC('f', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons1483, cons248) rule3208 = ReplacementRule(pattern3208, With3208) pattern3209 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**WC('p', S(1))*(S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons48, cons1485, cons47, cons40) rule3209 = ReplacementRule(pattern3209, replacement3209) pattern3210 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**WC('p', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons48, cons1485, cons47, cons40) rule3210 = ReplacementRule(pattern3210, replacement3210) pattern3211 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**p_*(S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons5, cons48, cons1485, cons47, cons149) rule3211 = ReplacementRule(pattern3211, replacement3211) pattern3212 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**p_*tan(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons5, cons48, cons1485, cons47, cons149) rule3212 = ReplacementRule(pattern3212, replacement3212) pattern3213 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**n_ + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**n2_)**WC('p', S(1))*(S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons48, cons1481, cons40) rule3213 = ReplacementRule(pattern3213, With3213) pattern3214 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**n_ + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**n2_)**WC('p', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons48, cons1481, cons40) rule3214 = ReplacementRule(pattern3214, With3214) pattern3215 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**WC('p', S(1))*(S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons48, cons1482, cons228, cons378) rule3215 = ReplacementRule(pattern3215, replacement3215) pattern3216 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**WC('p', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons48, cons1482, cons228, cons378) rule3216 = ReplacementRule(pattern3216, replacement3216) pattern3217 = Pattern(Integral((A_ + WC('B', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))*(a_ + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons47, cons87) rule3217 = ReplacementRule(pattern3217, replacement3217) pattern3218 = Pattern(Integral((A_ + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))*(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons47, cons87) rule3218 = ReplacementRule(pattern3218, replacement3218) pattern3219 = Pattern(Integral((A_ + WC('B', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))*(a_ + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons47, cons25) rule3219 = ReplacementRule(pattern3219, replacement3219) pattern3220 = Pattern(Integral((A_ + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))*(a_ + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons47, cons25) rule3220 = ReplacementRule(pattern3220, replacement3220) pattern3221 = Pattern(Integral((A_ + WC('B', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))/(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons228) rule3221 = ReplacementRule(pattern3221, With3221) pattern3222 = Pattern(Integral((A_ + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))/(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons228) rule3222 = ReplacementRule(pattern3222, With3222) pattern3223 = Pattern(Integral((A_ + WC('B', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*sin(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons228, cons87) rule3223 = ReplacementRule(pattern3223, replacement3223) pattern3224 = Pattern(Integral((A_ + WC('B', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*cos(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons228, cons87) rule3224 = ReplacementRule(pattern3224, replacement3224) pattern3225 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons8, cons29, cons50, cons127, cons33, cons170) rule3225 = ReplacementRule(pattern3225, replacement3225) pattern3226 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons8, cons29, cons50, cons127, cons33, cons170) rule3226 = ReplacementRule(pattern3226, replacement3226) pattern3227 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons8, cons29, cons50, cons127, cons33, cons96) rule3227 = ReplacementRule(pattern3227, replacement3227) pattern3228 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons8, cons29, cons50, cons127, cons33, cons96) rule3228 = ReplacementRule(pattern3228, replacement3228) pattern3229 = Pattern(Integral(sin(x_*WC('f', S(1)) + WC('e', S(0)))/(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons8, cons29, cons50, cons127, cons1118) rule3229 = ReplacementRule(pattern3229, replacement3229) pattern3230 = Pattern(Integral(cos(x_*WC('f', S(1)) + WC('e', S(0)))/(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons8, cons29, cons50, cons127, cons1118) rule3230 = ReplacementRule(pattern3230, replacement3230) pattern3231 = Pattern(Integral(sin(x_*WC('f', S(1)) + WC('e', S(0)))/(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons8, cons29, cons50, cons127, cons178) rule3231 = ReplacementRule(pattern3231, replacement3231) pattern3232 = Pattern(Integral(cos(x_*WC('f', S(1)) + WC('e', S(0)))/(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons8, cons29, cons50, cons127, cons178) rule3232 = ReplacementRule(pattern3232, replacement3232) pattern3233 = Pattern(Integral(sin(x_*WC('f', S(1)) + WC('e', S(0)))/sqrt(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons8, cons29, cons50, cons127, cons1118) rule3233 = ReplacementRule(pattern3233, replacement3233) pattern3234 = Pattern(Integral(cos(x_*WC('f', S(1)) + WC('e', S(0)))/sqrt(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons8, cons29, cons50, cons127, cons1118) rule3234 = ReplacementRule(pattern3234, replacement3234) pattern3235 = Pattern(Integral(sin(x_*WC('f', S(1)) + WC('e', S(0)))/sqrt(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons8, cons29, cons50, cons127, cons178) rule3235 = ReplacementRule(pattern3235, replacement3235) pattern3236 = Pattern(Integral(cos(x_*WC('f', S(1)) + WC('e', S(0)))/sqrt(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons8, cons29, cons50, cons127, cons178) rule3236 = ReplacementRule(pattern3236, replacement3236) pattern3237 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons8, cons29, cons50, cons127, cons19, cons1490) rule3237 = ReplacementRule(pattern3237, replacement3237) pattern3238 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons8, cons29, cons50, cons127, cons19, cons1490) rule3238 = ReplacementRule(pattern3238, replacement3238) pattern3239 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons3, cons8, cons29, cons50, cons127, cons89, cons167) rule3239 = ReplacementRule(pattern3239, replacement3239) pattern3240 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons3, cons8, cons29, cons50, cons127, cons89, cons167) rule3240 = ReplacementRule(pattern3240, replacement3240) pattern3241 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons3, cons8, cons29, cons50, cons127, cons95, cons167, cons168) rule3241 = ReplacementRule(pattern3241, replacement3241) pattern3242 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons3, cons8, cons29, cons50, cons127, cons95, cons167, cons168) rule3242 = ReplacementRule(pattern3242, replacement3242) pattern3243 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**n_, x_), cons8, cons29, cons50, cons127, cons19, cons87, cons167, cons1491) rule3243 = ReplacementRule(pattern3243, replacement3243) pattern3244 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**n_, x_), cons8, cons29, cons50, cons127, cons19, cons87, cons167, cons1491) rule3244 = ReplacementRule(pattern3244, replacement3244) pattern3245 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**m_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**n_, x_), cons8, cons29, cons50, cons127, cons19, cons87, cons167, cons33, cons247) rule3245 = ReplacementRule(pattern3245, replacement3245) pattern3246 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**m_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**n_, x_), cons8, cons29, cons50, cons127, cons19, cons87, cons167, cons33, cons247) rule3246 = ReplacementRule(pattern3246, replacement3246) pattern3247 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons3, cons8, cons29, cons50, cons127, cons95, cons167, cons249) rule3247 = ReplacementRule(pattern3247, replacement3247) pattern3248 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons3, cons8, cons29, cons50, cons127, cons95, cons167, cons249) rule3248 = ReplacementRule(pattern3248, replacement3248) pattern3249 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons3, cons8, cons29, cons50, cons127, cons89, cons91, cons1444) rule3249 = ReplacementRule(pattern3249, replacement3249) pattern3250 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons3, cons8, cons29, cons50, cons127, cons89, cons91, cons1444) rule3250 = ReplacementRule(pattern3250, replacement3250) pattern3251 = Pattern(Integral((WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons3, cons8, cons29, cons50, cons127, cons95, cons91, cons1444, cons168) rule3251 = ReplacementRule(pattern3251, replacement3251) pattern3252 = Pattern(Integral((WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons3, cons8, cons29, cons50, cons127, cons95, cons91, cons1444, cons168) rule3252 = ReplacementRule(pattern3252, replacement3252) pattern3253 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons150, cons1492) rule3253 = ReplacementRule(pattern3253, replacement3253) pattern3254 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons150, cons1492) rule3254 = ReplacementRule(pattern3254, replacement3254) pattern3255 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons1267, cons87) rule3255 = ReplacementRule(pattern3255, replacement3255) pattern3256 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons1267, cons810, cons1493) rule3256 = ReplacementRule(pattern3256, replacement3256) pattern3257 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons1494, cons87) rule3257 = ReplacementRule(pattern3257, replacement3257) pattern3258 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons1456, cons87) rule3258 = ReplacementRule(pattern3258, replacement3258) pattern3259 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons1494, cons810, cons1493) rule3259 = ReplacementRule(pattern3259, replacement3259) pattern3260 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons1456, cons810, cons1493) rule3260 = ReplacementRule(pattern3260, replacement3260) pattern3261 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons1269, cons64) rule3261 = ReplacementRule(pattern3261, replacement3261) pattern3262 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons1269, cons64) rule3262 = ReplacementRule(pattern3262, replacement3262) pattern3263 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/(a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons1269, cons64) rule3263 = ReplacementRule(pattern3263, replacement3263) pattern3264 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/(a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons1269, cons64) rule3264 = ReplacementRule(pattern3264, replacement3264) pattern3265 = Pattern(Integral((a_ + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons1269, cons1495, cons64) rule3265 = ReplacementRule(pattern3265, replacement3265) pattern3266 = Pattern(Integral((a_ + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons1269, cons1495, cons64) rule3266 = ReplacementRule(pattern3266, replacement3266) pattern3267 = Pattern(Integral(u_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*sin(v_))**WC('n', S(1)), x_), cons2, cons3, cons19, cons4, cons812, cons813) rule3267 = ReplacementRule(pattern3267, replacement3267) pattern3268 = Pattern(Integral(u_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*cos(v_))**WC('n', S(1)), x_), cons2, cons3, cons19, cons4, cons812, cons813) rule3268 = ReplacementRule(pattern3268, replacement3268) pattern3269 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons1362) rule3269 = ReplacementRule(pattern3269, replacement3269) pattern3270 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons1362) rule3270 = ReplacementRule(pattern3270, replacement3270) pattern3271 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons4, cons130) rule3271 = ReplacementRule(pattern3271, replacement3271) pattern3272 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons4, cons130) rule3272 = ReplacementRule(pattern3272, replacement3272) pattern3273 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*sin(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons40, cons150, cons139, cons746) rule3273 = ReplacementRule(pattern3273, replacement3273) pattern3274 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*cos(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons40, cons150, cons139, cons746) rule3274 = ReplacementRule(pattern3274, replacement3274) pattern3275 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*sin(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons65, cons150, cons1496) rule3275 = ReplacementRule(pattern3275, replacement3275) pattern3276 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*cos(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons65, cons150, cons1496) rule3276 = ReplacementRule(pattern3276, replacement3276) pattern3277 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*sin(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons65, cons198) rule3277 = ReplacementRule(pattern3277, replacement3277) pattern3278 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*cos(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons65, cons198) rule3278 = ReplacementRule(pattern3278, replacement3278) pattern3279 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*sin(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons4, cons5, cons1497) rule3279 = ReplacementRule(pattern3279, replacement3279) pattern3280 = Pattern(Integral((a_ + x_**n_*WC('b', S(1)))**p_*cos(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons4, cons5, cons1497) rule3280 = ReplacementRule(pattern3280, replacement3280) pattern3281 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons130) rule3281 = ReplacementRule(pattern3281, replacement3281) pattern3282 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons130) rule3282 = ReplacementRule(pattern3282, replacement3282) pattern3283 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*sin(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons40, cons55, cons13, cons139, cons598) rule3283 = ReplacementRule(pattern3283, replacement3283) pattern3284 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*cos(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons40, cons55, cons13, cons139, cons598) rule3284 = ReplacementRule(pattern3284, replacement3284) pattern3285 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*sin(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons40, cons150, cons33, cons139, cons1498) rule3285 = ReplacementRule(pattern3285, replacement3285) pattern3286 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*cos(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons40, cons150, cons33, cons139, cons1498) rule3286 = ReplacementRule(pattern3286, replacement3286) pattern3287 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*sin(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons65, cons20, cons150, cons1496) rule3287 = ReplacementRule(pattern3287, replacement3287) pattern3288 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*cos(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons65, cons20, cons150, cons1496) rule3288 = ReplacementRule(pattern3288, replacement3288) pattern3289 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*sin(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons19, cons65, cons198) rule3289 = ReplacementRule(pattern3289, replacement3289) pattern3290 = Pattern(Integral(x_**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**p_*cos(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons19, cons65, cons198) rule3290 = ReplacementRule(pattern3290, replacement3290) pattern3291 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons5, cons1499) rule3291 = ReplacementRule(pattern3291, replacement3291) pattern3292 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(a_ + x_**n_*WC('b', S(1)))**WC('p', S(1))*cos(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons5, cons1499) rule3292 = ReplacementRule(pattern3292, replacement3292) pattern3293 = Pattern(Integral(sin(x_**S(2)*WC('d', S(1))), x_), cons29, cons29) rule3293 = ReplacementRule(pattern3293, replacement3293) pattern3294 = Pattern(Integral(cos(x_**S(2)*WC('d', S(1))), x_), cons29, cons29) rule3294 = ReplacementRule(pattern3294, replacement3294) pattern3295 = Pattern(Integral(sin(c_ + x_**S(2)*WC('d', S(1))), x_), cons8, cons29, cons1263) rule3295 = ReplacementRule(pattern3295, replacement3295) pattern3296 = Pattern(Integral(cos(c_ + x_**S(2)*WC('d', S(1))), x_), cons8, cons29, cons1263) rule3296 = ReplacementRule(pattern3296, replacement3296) pattern3297 = Pattern(Integral(sin(x_**n_*WC('d', S(1)) + WC('c', S(0))), x_), cons8, cons29, cons87, cons746) rule3297 = ReplacementRule(pattern3297, replacement3297) pattern3298 = Pattern(Integral(cos(x_**n_*WC('d', S(1)) + WC('c', S(0))), x_), cons8, cons29, cons87, cons746) rule3298 = ReplacementRule(pattern3298, replacement3298) pattern3299 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons8, cons29, cons378, cons167, cons148) rule3299 = ReplacementRule(pattern3299, replacement3299) pattern3300 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons8, cons29, cons378, cons167, cons148) rule3300 = ReplacementRule(pattern3300, replacement3300) pattern3301 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons40, cons198) rule3301 = ReplacementRule(pattern3301, replacement3301) pattern3302 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons40, cons198) rule3302 = ReplacementRule(pattern3302, replacement3302) pattern3303 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons40, cons491) rule3303 = ReplacementRule(pattern3303, With3303) pattern3304 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons40, cons491) rule3304 = ReplacementRule(pattern3304, With3304) pattern3305 = Pattern(Integral(sin(x_**n_*WC('d', S(1)) + WC('c', S(0))), x_), cons8, cons29, cons4, cons1500) rule3305 = ReplacementRule(pattern3305, replacement3305) pattern3306 = Pattern(Integral(cos(x_**n_*WC('d', S(1)) + WC('c', S(0))), x_), cons8, cons29, cons4, cons1500) rule3306 = ReplacementRule(pattern3306, replacement3306) pattern3307 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons8, cons29, cons4, cons130) rule3307 = ReplacementRule(pattern3307, replacement3307) pattern3308 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons8, cons29, cons4, cons130) rule3308 = ReplacementRule(pattern3308, replacement3308) pattern3309 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(u_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons4, cons40, cons70, cons71) rule3309 = ReplacementRule(pattern3309, replacement3309) pattern3310 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(u_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons4, cons40, cons70, cons71) rule3310 = ReplacementRule(pattern3310, replacement3310) pattern3311 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(u_**n_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons8, cons29, cons4, cons5, cons70) rule3311 = ReplacementRule(pattern3311, replacement3311) pattern3312 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(u_**n_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons8, cons29, cons4, cons5, cons70) rule3312 = ReplacementRule(pattern3312, replacement3312) pattern3313 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*sin(u_))**WC('p', S(1)), x_), cons2, cons3, cons5, cons825, cons826) rule3313 = ReplacementRule(pattern3313, replacement3313) pattern3314 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*cos(u_))**WC('p', S(1)), x_), cons2, cons3, cons5, cons825, cons826) rule3314 = ReplacementRule(pattern3314, replacement3314) pattern3315 = Pattern(Integral(sin(x_**n_*WC('d', S(1)))/x_, x_), cons29, cons4, cons1501) rule3315 = ReplacementRule(pattern3315, replacement3315) pattern3316 = Pattern(Integral(cos(x_**n_*WC('d', S(1)))/x_, x_), cons29, cons4, cons1501) rule3316 = ReplacementRule(pattern3316, replacement3316) pattern3317 = Pattern(Integral(sin(c_ + x_**n_*WC('d', S(1)))/x_, x_), cons8, cons29, cons4, cons1500) rule3317 = ReplacementRule(pattern3317, replacement3317) pattern3318 = Pattern(Integral(cos(c_ + x_**n_*WC('d', S(1)))/x_, x_), cons8, cons29, cons4, cons1500) rule3318 = ReplacementRule(pattern3318, replacement3318) pattern3319 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons19, cons4, cons40, CustomConstraint(With3319)) rule3319 = ReplacementRule(pattern3319, replacement3319) pattern3320 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons19, cons4, cons40, CustomConstraint(With3320)) rule3320 = ReplacementRule(pattern3320, replacement3320) pattern3321 = Pattern(Integral((e_*x_)**m_*(WC('a', S(0)) + WC('b', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons40, CustomConstraint(With3321)) rule3321 = ReplacementRule(pattern3321, replacement3321) pattern3322 = Pattern(Integral((e_*x_)**m_*(WC('a', S(0)) + WC('b', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons40, CustomConstraint(With3322)) rule3322 = ReplacementRule(pattern3322, replacement3322) pattern3323 = Pattern(Integral(x_**WC('m', S(1))*sin(x_**n_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons19, cons4, cons1502) rule3323 = ReplacementRule(pattern3323, replacement3323) pattern3324 = Pattern(Integral(x_**WC('m', S(1))*cos(x_**n_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons19, cons4, cons1502) rule3324 = ReplacementRule(pattern3324, replacement3324) pattern3325 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))), x_), cons8, cons29, cons50, cons150, cons33, cons1503) rule3325 = ReplacementRule(pattern3325, replacement3325) pattern3326 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))), x_), cons8, cons29, cons50, cons150, cons33, cons1503) rule3326 = ReplacementRule(pattern3326, replacement3326) pattern3327 = Pattern(Integral((x_*WC('e', S(1)))**m_*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))), x_), cons8, cons29, cons50, cons150, cons33, cons96) rule3327 = ReplacementRule(pattern3327, replacement3327) pattern3328 = Pattern(Integral((x_*WC('e', S(1)))**m_*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))), x_), cons8, cons29, cons50, cons150, cons33, cons96) rule3328 = ReplacementRule(pattern3328, replacement3328) pattern3329 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))), x_), cons8, cons29, cons50, cons19, cons150) rule3329 = ReplacementRule(pattern3329, replacement3329) pattern3330 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))), x_), cons8, cons29, cons50, cons19, cons150) rule3330 = ReplacementRule(pattern3330, replacement3330) pattern3331 = Pattern(Integral(x_**WC('m', S(1))*sin(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons378, cons1257, cons148, cons1504) rule3331 = ReplacementRule(pattern3331, replacement3331) pattern3332 = Pattern(Integral(x_**WC('m', S(1))*cos(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons378, cons1257, cons148, cons1504) rule3332 = ReplacementRule(pattern3332, replacement3332) pattern3333 = Pattern(Integral(x_**WC('m', S(1))*sin(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons19, cons4, cons58, cons13, cons148) rule3333 = ReplacementRule(pattern3333, replacement3333) pattern3334 = Pattern(Integral(x_**WC('m', S(1))*cos(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons19, cons4, cons58, cons13, cons148) rule3334 = ReplacementRule(pattern3334, replacement3334) pattern3335 = Pattern(Integral(x_**WC('m', S(1))*sin(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons152, cons13, cons148, cons1505) rule3335 = ReplacementRule(pattern3335, replacement3335) pattern3336 = Pattern(Integral(x_**WC('m', S(1))*cos(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons152, cons13, cons148, cons1505) rule3336 = ReplacementRule(pattern3336, replacement3336) pattern3337 = Pattern(Integral(x_**WC('m', S(1))*sin(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons152, cons13, cons148, cons1506, cons685) rule3337 = ReplacementRule(pattern3337, replacement3337) pattern3338 = Pattern(Integral(x_**WC('m', S(1))*cos(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons152, cons13, cons148, cons1506, cons685) rule3338 = ReplacementRule(pattern3338, replacement3338) pattern3339 = Pattern(Integral((x_*WC('e', S(1)))**m_*(WC('a', S(0)) + WC('b', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons40, cons150, cons369) rule3339 = ReplacementRule(pattern3339, With3339) pattern3340 = Pattern(Integral((x_*WC('e', S(1)))**m_*(WC('a', S(0)) + WC('b', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons40, cons150, cons369) rule3340 = ReplacementRule(pattern3340, With3340) pattern3341 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons19, cons40, cons150, cons148) rule3341 = ReplacementRule(pattern3341, replacement3341) pattern3342 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons19, cons40, cons150, cons148) rule3342 = ReplacementRule(pattern3342, replacement3342) pattern3343 = Pattern(Integral(x_**WC('m', S(1))*sin(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons19, cons4, cons58, cons13, cons139, cons1507) rule3343 = ReplacementRule(pattern3343, replacement3343) pattern3344 = Pattern(Integral(x_**WC('m', S(1))*cos(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons19, cons4, cons58, cons13, cons139, cons1507) rule3344 = ReplacementRule(pattern3344, replacement3344) pattern3345 = Pattern(Integral(x_**WC('m', S(1))*sin(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons152, cons13, cons139, cons1507, cons1505) rule3345 = ReplacementRule(pattern3345, replacement3345) pattern3346 = Pattern(Integral(x_**WC('m', S(1))*cos(x_**n_*WC('b', S(1)) + WC('a', S(0)))**p_, x_), cons2, cons3, cons152, cons13, cons139, cons1507, cons1505) rule3346 = ReplacementRule(pattern3346, replacement3346) pattern3347 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons40, cons198, cons20) rule3347 = ReplacementRule(pattern3347, replacement3347) pattern3348 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons40, cons198, cons20) rule3348 = ReplacementRule(pattern3348, replacement3348) pattern3349 = Pattern(Integral((x_*WC('e', S(1)))**m_*(WC('a', S(0)) + WC('b', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons40, cons198, cons369) rule3349 = ReplacementRule(pattern3349, With3349) pattern3350 = Pattern(Integral((x_*WC('e', S(1)))**m_*(WC('a', S(0)) + WC('b', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons40, cons198, cons369) rule3350 = ReplacementRule(pattern3350, With3350) pattern3351 = Pattern(Integral((x_*WC('e', S(1)))**m_*(WC('a', S(0)) + WC('b', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons40, cons198, cons358) rule3351 = ReplacementRule(pattern3351, replacement3351) pattern3352 = Pattern(Integral((x_*WC('e', S(1)))**m_*(WC('a', S(0)) + WC('b', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons40, cons198, cons358) rule3352 = ReplacementRule(pattern3352, replacement3352) pattern3353 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons19, cons40, cons491) rule3353 = ReplacementRule(pattern3353, With3353) pattern3354 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons19, cons40, cons491) rule3354 = ReplacementRule(pattern3354, With3354) pattern3355 = Pattern(Integral((e_*x_)**m_*(WC('a', S(0)) + WC('b', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons40, cons491) rule3355 = ReplacementRule(pattern3355, replacement3355) pattern3356 = Pattern(Integral((e_*x_)**m_*(WC('a', S(0)) + WC('b', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons40, cons491) rule3356 = ReplacementRule(pattern3356, replacement3356) pattern3357 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons19, cons4, cons40, cons68, cons856, cons25) rule3357 = ReplacementRule(pattern3357, replacement3357) pattern3358 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons19, cons4, cons40, cons68, cons856, cons25) rule3358 = ReplacementRule(pattern3358, replacement3358) pattern3359 = Pattern(Integral((e_*x_)**m_*(WC('a', S(0)) + WC('b', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons40, cons68, cons856, cons25) rule3359 = ReplacementRule(pattern3359, replacement3359) pattern3360 = Pattern(Integral((e_*x_)**m_*(WC('a', S(0)) + WC('b', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons40, cons68, cons856, cons25) rule3360 = ReplacementRule(pattern3360, replacement3360) pattern3361 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))), x_), cons8, cons29, cons50, cons19, cons4, cons1508) rule3361 = ReplacementRule(pattern3361, replacement3361) pattern3362 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))), x_), cons8, cons29, cons50, cons19, cons4, cons1508) rule3362 = ReplacementRule(pattern3362, replacement3362) pattern3363 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*sin(x_**n_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons130) rule3363 = ReplacementRule(pattern3363, replacement3363) pattern3364 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*cos(x_**n_*WC('d', S(1)) + WC('c', S(0))))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons130) rule3364 = ReplacementRule(pattern3364, replacement3364) pattern3365 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*sin(u_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons4, cons5, cons70, cons71, cons20) rule3365 = ReplacementRule(pattern3365, replacement3365) pattern3366 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*cos(u_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons4, cons5, cons70, cons71, cons20) rule3366 = ReplacementRule(pattern3366, replacement3366) pattern3367 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*sin(u_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons5, cons70) rule3367 = ReplacementRule(pattern3367, replacement3367) pattern3368 = Pattern(Integral((x_*WC('e', S(1)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*cos(u_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons5, cons70) rule3368 = ReplacementRule(pattern3368, replacement3368) pattern3369 = Pattern(Integral((e_*x_)**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*sin(u_))**WC('p', S(1)), x_), cons2, cons3, cons50, cons19, cons5, cons825, cons826) rule3369 = ReplacementRule(pattern3369, replacement3369) pattern3370 = Pattern(Integral((e_*x_)**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*cos(u_))**WC('p', S(1)), x_), cons2, cons3, cons50, cons19, cons5, cons825, cons826) rule3370 = ReplacementRule(pattern3370, replacement3370) pattern3371 = Pattern(Integral(x_**WC('m', S(1))*sin(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*cos(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons19, cons4, cons5, cons55, cons56) rule3371 = ReplacementRule(pattern3371, replacement3371) pattern3372 = Pattern(Integral(x_**WC('m', S(1))*sin(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))*cos(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons19, cons4, cons5, cons55, cons56) rule3372 = ReplacementRule(pattern3372, replacement3372) pattern3373 = Pattern(Integral(x_**WC('m', S(1))*sin(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*cos(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons5, cons95, cons1503, cons56) rule3373 = ReplacementRule(pattern3373, replacement3373) pattern3374 = Pattern(Integral(x_**WC('m', S(1))*sin(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))*cos(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons5, cons95, cons1503, cons56) rule3374 = ReplacementRule(pattern3374, replacement3374) pattern3375 = Pattern(Integral(sin(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons47) rule3375 = ReplacementRule(pattern3375, replacement3375) pattern3376 = Pattern(Integral(cos(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons47) rule3376 = ReplacementRule(pattern3376, replacement3376) pattern3377 = Pattern(Integral(sin(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons228) rule3377 = ReplacementRule(pattern3377, replacement3377) pattern3378 = Pattern(Integral(cos(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons228) rule3378 = ReplacementRule(pattern3378, replacement3378) pattern3379 = Pattern(Integral(sin(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**n_, x_), cons2, cons3, cons8, cons87, cons167) rule3379 = ReplacementRule(pattern3379, replacement3379) pattern3380 = Pattern(Integral(cos(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**n_, x_), cons2, cons3, cons8, cons87, cons167) rule3380 = ReplacementRule(pattern3380, replacement3380) pattern3381 = Pattern(Integral(sin(v_)**WC('n', S(1)), x_), cons150, cons820, cons1133) rule3381 = ReplacementRule(pattern3381, replacement3381) pattern3382 = Pattern(Integral(cos(v_)**WC('n', S(1)), x_), cons150, cons820, cons1133) rule3382 = ReplacementRule(pattern3382, replacement3382) pattern3383 = Pattern(Integral((d_ + x_*WC('e', S(1)))*sin(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons50, cons49) rule3383 = ReplacementRule(pattern3383, replacement3383) pattern3384 = Pattern(Integral((d_ + x_*WC('e', S(1)))*cos(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons50, cons49) rule3384 = ReplacementRule(pattern3384, replacement3384) pattern3385 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))*sin(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons50, cons241) rule3385 = ReplacementRule(pattern3385, replacement3385) pattern3386 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))*cos(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons50, cons241) rule3386 = ReplacementRule(pattern3386, replacement3386) pattern3387 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*sin(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons50, cons33, cons168, cons1134) rule3387 = ReplacementRule(pattern3387, replacement3387) pattern3388 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*cos(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons50, cons33, cons168, cons1134) rule3388 = ReplacementRule(pattern3388, replacement3388) pattern3389 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*sin(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons50, cons33, cons168, cons1135) rule3389 = ReplacementRule(pattern3389, replacement3389) pattern3390 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*cos(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons50, cons33, cons168, cons1135) rule3390 = ReplacementRule(pattern3390, replacement3390) pattern3391 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*sin(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons50, cons33, cons96, cons1134) rule3391 = ReplacementRule(pattern3391, replacement3391) pattern3392 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*cos(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons50, cons33, cons96, cons1134) rule3392 = ReplacementRule(pattern3392, replacement3392) pattern3393 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*sin(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons50, cons33, cons96, cons1135) rule3393 = ReplacementRule(pattern3393, replacement3393) pattern3394 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**m_*cos(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons50, cons33, cons96, cons1135) rule3394 = ReplacementRule(pattern3394, replacement3394) pattern3395 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*sin(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons1509) rule3395 = ReplacementRule(pattern3395, replacement3395) pattern3396 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*cos(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons1509) rule3396 = ReplacementRule(pattern3396, replacement3396) pattern3397 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*sin(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons19, cons87, cons167) rule3397 = ReplacementRule(pattern3397, replacement3397) pattern3398 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*cos(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons19, cons87, cons167) rule3398 = ReplacementRule(pattern3398, replacement3398) pattern3399 = Pattern(Integral(u_**WC('m', S(1))*sin(v_)**WC('n', S(1)), x_), cons19, cons150, cons70, cons820, cons821) rule3399 = ReplacementRule(pattern3399, replacement3399) pattern3400 = Pattern(Integral(u_**WC('m', S(1))*cos(v_)**WC('n', S(1)), x_), cons19, cons150, cons70, cons820, cons821) rule3400 = ReplacementRule(pattern3400, replacement3400) return [rule2167, rule2168, rule2169, rule2170, rule2171, rule2172, rule2173, rule2174, rule2175, rule2176, rule2177, rule2178, rule2179, rule2180, rule2181, rule2182, rule2183, rule2184, rule2185, rule2186, rule2187, rule2188, rule2189, rule2190, rule2191, rule2192, rule2193, rule2194, rule2195, rule2196, rule2197, rule2198, rule2199, rule2200, rule2201, rule2202, rule2203, rule2204, rule2205, rule2206, rule2207, rule2208, rule2209, rule2210, rule2211, rule2212, rule2213, rule2214, rule2215, rule2216, rule2217, rule2218, rule2219, rule2220, rule2221, rule2222, rule2223, rule2224, rule2225, rule2226, rule2227, rule2228, rule2229, rule2230, rule2231, rule2232, rule2233, rule2234, rule2235, rule2236, rule2237, rule2238, rule2239, rule2240, rule2241, rule2242, rule2243, rule2244, rule2245, rule2246, rule2247, rule2248, rule2249, rule2250, rule2251, rule2252, rule2253, rule2254, rule2255, rule2256, rule2257, rule2258, rule2259, rule2260, rule2261, rule2262, rule2263, rule2264, rule2265, rule2266, rule2267, rule2268, rule2269, rule2270, rule2271, rule2272, rule2273, rule2274, rule2275, rule2276, rule2277, rule2278, rule2279, rule2280, rule2281, rule2282, rule2283, rule2284, rule2285, rule2286, rule2287, rule2288, rule2289, rule2290, rule2291, rule2292, rule2293, rule2294, rule2295, rule2296, rule2297, rule2298, rule2299, rule2300, rule2301, rule2302, rule2303, rule2304, rule2305, rule2306, rule2307, rule2308, rule2309, rule2310, rule2311, rule2312, rule2313, rule2314, rule2315, rule2316, rule2317, rule2318, rule2319, rule2320, rule2321, rule2322, rule2323, rule2324, rule2325, rule2326, rule2327, rule2328, rule2329, rule2330, rule2331, rule2332, rule2333, rule2334, rule2335, rule2336, rule2337, rule2338, rule2339, rule2340, rule2341, rule2342, rule2343, rule2344, rule2345, rule2346, rule2347, rule2348, rule2349, rule2350, rule2351, rule2352, rule2353, rule2354, rule2355, rule2356, rule2357, rule2358, rule2359, rule2360, rule2361, rule2362, rule2363, rule2364, rule2365, rule2366, rule2367, rule2368, rule2369, rule2370, rule2371, rule2372, rule2373, rule2374, rule2375, rule2376, rule2377, rule2378, rule2379, rule2380, rule2381, rule2382, rule2383, rule2384, rule2385, rule2386, rule2387, rule2388, rule2389, rule2390, rule2391, rule2392, rule2393, rule2394, rule2395, rule2396, rule2397, rule2398, rule2399, rule2400, rule2401, rule2402, rule2403, rule2404, rule2405, rule2406, rule2407, rule2408, rule2409, rule2410, rule2411, rule2412, rule2413, rule2414, rule2415, rule2416, rule2417, rule2418, rule2419, rule2420, rule2421, rule2422, rule2423, rule2424, rule2425, rule2426, rule2427, rule2428, rule2429, rule2430, rule2431, rule2432, rule2433, rule2434, rule2435, rule2436, rule2437, rule2438, rule2439, rule2440, rule2441, rule2442, rule2443, rule2444, rule2445, rule2446, rule2447, rule2448, rule2449, rule2450, rule2451, rule2452, rule2453, rule2454, rule2455, rule2456, rule2457, rule2458, rule2459, rule2460, rule2461, rule2462, rule2463, rule2464, rule2465, rule2466, rule2467, rule2468, rule2469, rule2470, rule2471, rule2472, rule2473, rule2474, rule2475, rule2476, rule2477, rule2478, rule2479, rule2480, rule2481, rule2482, rule2483, rule2484, rule2485, rule2486, rule2487, rule2488, rule2489, rule2490, rule2491, rule2492, rule2493, rule2494, rule2495, rule2496, rule2497, rule2498, rule2499, rule2500, rule2501, rule2502, rule2503, rule2504, rule2505, rule2506, rule2507, rule2508, rule2509, rule2510, rule2511, rule2512, rule2513, rule2514, rule2515, rule2516, rule2517, rule2518, rule2519, rule2520, rule2521, rule2522, rule2523, rule2524, rule2525, rule2526, rule2527, rule2528, rule2529, rule2530, rule2531, rule2532, rule2533, rule2534, rule2535, rule2536, rule2537, rule2538, rule2539, rule2540, rule2541, rule2542, rule2543, rule2544, rule2545, rule2546, rule2547, rule2548, rule2549, rule2550, rule2551, rule2552, rule2553, rule2554, rule2555, rule2556, rule2557, rule2558, rule2559, rule2560, rule2561, rule2562, rule2563, rule2564, rule2565, rule2566, rule2567, rule2568, rule2569, rule2570, rule2571, rule2572, rule2573, rule2574, rule2575, rule2576, rule2577, rule2578, rule2579, rule2580, rule2581, rule2582, rule2583, rule2584, rule2585, rule2586, rule2587, rule2588, rule2589, rule2590, rule2591, rule2592, rule2593, rule2594, rule2595, rule2596, rule2597, rule2598, rule2599, rule2600, rule2601, rule2602, rule2603, rule2604, rule2605, rule2606, rule2607, rule2608, rule2609, rule2610, rule2611, rule2612, rule2613, rule2614, rule2615, rule2616, rule2617, rule2618, rule2619, rule2620, rule2621, rule2622, rule2623, rule2624, rule2625, rule2626, rule2627, rule2628, rule2629, rule2630, rule2631, rule2632, rule2633, rule2634, rule2635, rule2636, rule2637, rule2638, rule2639, rule2640, rule2641, rule2642, rule2643, rule2644, rule2645, rule2646, rule2647, rule2648, rule2649, rule2650, rule2651, rule2652, rule2653, rule2654, rule2655, rule2656, rule2657, rule2658, rule2659, rule2660, rule2661, rule2662, rule2663, rule2664, rule2665, rule2666, rule2667, rule2668, rule2669, rule2670, rule2671, rule2672, rule2673, rule2674, rule2675, rule2676, rule2677, rule2678, rule2679, rule2680, rule2681, rule2682, rule2683, rule2684, rule2685, rule2686, rule2687, rule2688, rule2689, rule2690, rule2691, rule2692, rule2693, rule2694, rule2695, rule2696, rule2697, rule2698, rule2699, rule2700, rule2701, rule2702, rule2703, rule2704, rule2705, rule2706, rule2707, rule2708, rule2709, rule2710, rule2711, rule2712, rule2713, rule2714, rule2715, rule2716, rule2717, rule2718, rule2719, rule2720, rule2721, rule2722, rule2723, rule2724, rule2725, rule2726, rule2727, rule2728, rule2729, rule2730, rule2731, rule2732, rule2733, rule2734, rule2735, rule2736, rule2737, rule2738, rule2739, rule2740, rule2741, rule2742, rule2743, rule2744, rule2745, rule2746, rule2747, rule2748, rule2749, rule2750, rule2751, rule2752, rule2753, rule2754, rule2755, rule2756, rule2757, rule2758, rule2759, rule2760, rule2761, rule2762, rule2763, rule2764, rule2765, rule2766, rule2767, rule2768, rule2769, rule2770, rule2771, rule2772, rule2773, rule2774, rule2775, rule2776, rule2777, rule2778, rule2779, rule2780, rule2781, rule2782, rule2783, rule2784, rule2785, rule2786, rule2787, rule2788, rule2789, rule2790, rule2791, rule2792, rule2793, rule2794, rule2795, rule2796, rule2797, rule2798, rule2799, rule2800, rule2801, rule2802, rule2803, rule2804, rule2805, rule2806, rule2807, rule2808, rule2809, rule2810, rule2811, rule2812, rule2813, rule2814, rule2815, rule2816, rule2817, rule2818, rule2819, rule2820, rule2821, rule2822, rule2823, rule2824, rule2825, rule2826, rule2827, rule2828, rule2829, rule2830, rule2831, rule2832, rule2833, rule2834, rule2835, rule2836, rule2837, rule2838, rule2839, rule2840, rule2841, rule2842, rule2843, rule2844, rule2845, rule2846, rule2847, rule2848, rule2849, rule2850, rule2851, rule2852, rule2853, rule2854, rule2855, rule2856, rule2857, rule2858, rule2859, rule2860, rule2861, rule2862, rule2863, rule2864, rule2865, rule2866, rule2867, rule2868, rule2869, rule2870, rule2871, rule2872, rule2873, rule2874, rule2875, rule2876, rule2877, rule2878, rule2879, rule2880, rule2881, rule2882, rule2883, rule2884, rule2885, rule2886, rule2887, rule2888, rule2889, rule2890, rule2891, rule2892, rule2893, rule2894, rule2895, rule2896, rule2897, rule2898, rule2899, rule2900, rule2901, rule2902, rule2903, rule2904, rule2905, rule2906, rule2907, rule2908, rule2909, rule2910, rule2911, rule2912, rule2913, rule2914, rule2915, rule2916, rule2917, rule2918, rule2919, rule2920, rule2921, rule2922, rule2923, rule2924, rule2925, rule2926, rule2927, rule2928, rule2929, rule2930, rule2931, rule2932, rule2933, rule2934, rule2935, rule2936, rule2937, rule2938, rule2939, rule2940, rule2941, rule2942, rule2943, rule2944, rule2945, rule2946, rule2947, rule2948, rule2949, rule2950, rule2951, rule2952, rule2953, rule2954, rule2955, rule2956, rule2957, rule2958, rule2959, rule2960, rule2961, rule2962, rule2963, rule2964, rule2965, rule2966, rule2967, rule2968, rule2969, rule2970, rule2971, rule2972, rule2973, rule2974, rule2975, rule2976, rule2977, rule2978, rule2979, rule2980, rule2981, rule2982, rule2983, rule2984, rule2985, rule2986, rule2987, rule2988, rule2989, rule2990, rule2991, rule2992, rule2993, rule2994, rule2995, rule2996, rule2997, rule2998, rule2999, rule3000, rule3001, rule3002, rule3003, rule3004, rule3005, rule3006, rule3007, rule3008, rule3009, rule3010, rule3011, rule3012, rule3013, rule3014, rule3015, rule3016, rule3017, rule3018, rule3019, rule3020, rule3021, rule3022, rule3023, rule3024, rule3025, rule3026, rule3027, rule3028, rule3029, rule3030, rule3031, rule3032, rule3033, rule3034, rule3035, rule3036, rule3037, rule3038, rule3039, rule3040, rule3041, rule3042, rule3043, rule3044, rule3045, rule3046, rule3047, rule3048, rule3049, rule3050, rule3051, rule3052, rule3053, rule3054, rule3055, rule3056, rule3057, rule3058, rule3059, rule3060, rule3061, rule3062, rule3063, rule3064, rule3065, rule3066, rule3067, rule3068, rule3069, rule3070, rule3071, rule3072, rule3073, rule3074, rule3075, rule3076, rule3077, rule3078, rule3079, rule3080, rule3081, rule3082, rule3083, rule3084, rule3085, rule3086, rule3087, rule3088, rule3089, rule3090, rule3091, rule3092, rule3093, rule3094, rule3095, rule3096, rule3097, rule3098, rule3099, rule3100, rule3101, rule3102, rule3103, rule3104, rule3105, rule3106, rule3107, rule3108, rule3109, rule3110, rule3111, rule3112, rule3113, rule3114, rule3115, rule3116, rule3117, rule3118, rule3119, rule3120, rule3121, rule3122, rule3123, rule3124, rule3125, rule3126, rule3127, rule3128, rule3129, rule3130, rule3131, rule3132, rule3133, rule3134, rule3135, rule3136, rule3137, rule3138, rule3139, rule3140, rule3141, rule3142, rule3143, rule3144, rule3145, rule3146, rule3147, rule3148, rule3149, rule3150, rule3151, rule3152, rule3153, rule3154, rule3155, rule3156, rule3157, rule3158, rule3159, rule3160, rule3161, rule3162, rule3163, rule3164, rule3165, rule3166, rule3167, rule3168, rule3169, rule3170, rule3171, rule3172, rule3173, rule3174, rule3175, rule3176, rule3177, rule3178, rule3179, rule3180, rule3181, rule3182, rule3183, rule3184, rule3185, rule3186, rule3187, rule3188, rule3189, rule3190, rule3191, rule3192, rule3193, rule3194, rule3195, rule3196, rule3197, rule3198, rule3199, rule3200, rule3201, rule3202, rule3203, rule3204, rule3205, rule3206, rule3207, rule3208, rule3209, rule3210, rule3211, rule3212, rule3213, rule3214, rule3215, rule3216, rule3217, rule3218, rule3219, rule3220, rule3221, rule3222, rule3223, rule3224, rule3225, rule3226, rule3227, rule3228, rule3229, rule3230, rule3231, rule3232, rule3233, rule3234, rule3235, rule3236, rule3237, rule3238, rule3239, rule3240, rule3241, rule3242, rule3243, rule3244, rule3245, rule3246, rule3247, rule3248, rule3249, rule3250, rule3251, rule3252, rule3253, rule3254, rule3255, rule3256, rule3257, rule3258, rule3259, rule3260, rule3261, rule3262, rule3263, rule3264, rule3265, rule3266, rule3267, rule3268, rule3269, rule3270, rule3271, rule3272, rule3273, rule3274, rule3275, rule3276, rule3277, rule3278, rule3279, rule3280, rule3281, rule3282, rule3283, rule3284, rule3285, rule3286, rule3287, rule3288, rule3289, rule3290, rule3291, rule3292, rule3293, rule3294, rule3295, rule3296, rule3297, rule3298, rule3299, rule3300, rule3301, rule3302, rule3303, rule3304, rule3305, rule3306, rule3307, rule3308, rule3309, rule3310, rule3311, rule3312, rule3313, rule3314, rule3315, rule3316, rule3317, rule3318, rule3319, rule3320, rule3321, rule3322, rule3323, rule3324, rule3325, rule3326, rule3327, rule3328, rule3329, rule3330, rule3331, rule3332, rule3333, rule3334, rule3335, rule3336, rule3337, rule3338, rule3339, rule3340, rule3341, rule3342, rule3343, rule3344, rule3345, rule3346, rule3347, rule3348, rule3349, rule3350, rule3351, rule3352, rule3353, rule3354, rule3355, rule3356, rule3357, rule3358, rule3359, rule3360, rule3361, rule3362, rule3363, rule3364, rule3365, rule3366, rule3367, rule3368, rule3369, rule3370, rule3371, rule3372, rule3373, rule3374, rule3375, rule3376, rule3377, rule3378, rule3379, rule3380, rule3381, rule3382, rule3383, rule3384, rule3385, rule3386, rule3387, rule3388, rule3389, rule3390, rule3391, rule3392, rule3393, rule3394, rule3395, rule3396, rule3397, rule3398, rule3399, rule3400, ] def replacement2167(u, x): return Int(DeactivateTrig(u, x), x) def replacement2168(a, b, e, f, m, n, x): return Simp((a*sin(e + f*x))**(m + S(1))*(b*cos(e + f*x))**(n + S(1))/(a*b*f*(m + S(1))), x) def replacement2169(a, e, f, m, n, x): return Dist(S(1)/(a*f), Subst(Int(x**m*(S(1) - x**S(2)/a**S(2))**(n/S(2) + S(-1)/2), x), x, a*sin(e + f*x)), x) def replacement2170(a, e, f, m, n, x): return -Dist(S(1)/(a*f), Subst(Int(x**m*(S(1) - x**S(2)/a**S(2))**(n/S(2) + S(-1)/2), x), x, a*cos(e + f*x)), x) def replacement2171(a, b, e, f, m, n, x): return Dist(a**S(2)*(m + S(-1))/(b**S(2)*(n + S(1))), Int((a*sin(e + f*x))**(m + S(-2))*(b*cos(e + f*x))**(n + S(2)), x), x) - Simp(a*(a*sin(e + f*x))**(m + S(-1))*(b*cos(e + f*x))**(n + S(1))/(b*f*(n + S(1))), x) def replacement2172(a, b, e, f, m, n, x): return Dist(a**S(2)*(m + S(-1))/(b**S(2)*(n + S(1))), Int((a*cos(e + f*x))**(m + S(-2))*(b*sin(e + f*x))**(n + S(2)), x), x) + Simp(a*(a*cos(e + f*x))**(m + S(-1))*(b*sin(e + f*x))**(n + S(1))/(b*f*(n + S(1))), x) def replacement2173(a, b, e, f, m, n, x): return Dist(a**S(2)*(m + S(-1))/(m + n), Int((a*sin(e + f*x))**(m + S(-2))*(b*cos(e + f*x))**n, x), x) - Simp(a*(a*sin(e + f*x))**(m + S(-1))*(b*cos(e + f*x))**(n + S(1))/(b*f*(m + n)), x) def replacement2174(a, b, e, f, m, n, x): return Dist(a**S(2)*(m + S(-1))/(m + n), Int((a*cos(e + f*x))**(m + S(-2))*(b*sin(e + f*x))**n, x), x) + Simp(a*(a*cos(e + f*x))**(m + S(-1))*(b*sin(e + f*x))**(n + S(1))/(b*f*(m + n)), x) def replacement2175(a, b, e, f, m, n, x): return Dist((m + n + S(2))/(a**S(2)*(m + S(1))), Int((a*sin(e + f*x))**(m + S(2))*(b*cos(e + f*x))**n, x), x) + Simp((a*sin(e + f*x))**(m + S(1))*(b*cos(e + f*x))**(n + S(1))/(a*b*f*(m + S(1))), x) def replacement2176(a, b, e, f, m, n, x): return Dist((m + n + S(2))/(a**S(2)*(m + S(1))), Int((a*cos(e + f*x))**(m + S(2))*(b*sin(e + f*x))**n, x), x) - Simp((a*cos(e + f*x))**(m + S(1))*(b*sin(e + f*x))**(n + S(1))/(a*b*f*(m + S(1))), x) def replacement2177(a, b, e, f, x): return Dist(sqrt(a*sin(e + f*x))*sqrt(b*cos(e + f*x))/sqrt(sin(S(2)*e + S(2)*f*x)), Int(sqrt(sin(S(2)*e + S(2)*f*x)), x), x) def replacement2178(a, b, e, f, x): return Dist(sqrt(sin(S(2)*e + S(2)*f*x))/(sqrt(a*sin(e + f*x))*sqrt(b*cos(e + f*x))), Int(S(1)/sqrt(sin(S(2)*e + S(2)*f*x)), x), x) def With2179(a, b, e, f, m, n, x): k = Denominator(m) return Dist(a*b*k/f, Subst(Int(x**(k*(m + S(1)) + S(-1))/(a**S(2) + b**S(2)*x**(S(2)*k)), x), x, (a*sin(e + f*x))**(S(1)/k)*(b*cos(e + f*x))**(-S(1)/k)), x) def With2180(a, b, e, f, m, n, x): k = Denominator(m) return -Dist(a*b*k/f, Subst(Int(x**(k*(m + S(1)) + S(-1))/(a**S(2) + b**S(2)*x**(S(2)*k)), x), x, (a*cos(e + f*x))**(S(1)/k)*(b*sin(e + f*x))**(-S(1)/k)), x) def replacement2181(a, b, e, f, m, n, x): return Simp(b**(S(2)*IntPart(n/S(2) + S(-1)/2) + S(1))*(a*sin(e + f*x))**(m + S(1))*(b*cos(e + f*x))**(S(2)*FracPart(n/S(2) + S(-1)/2))*(cos(e + f*x)**S(2))**(-FracPart(n/S(2) + S(-1)/2))*Hypergeometric2F1(m/S(2) + S(1)/2, S(1)/2 - n/S(2), m/S(2) + S(3)/2, sin(e + f*x)**S(2))/(a*f*(m + S(1))), x) def replacement2182(a, b, e, f, m, n, x): return -Simp(b**(S(2)*IntPart(n/S(2) + S(-1)/2) + S(1))*(a*cos(e + f*x))**(m + S(1))*(b*sin(e + f*x))**(S(2)*FracPart(n/S(2) + S(-1)/2))*(sin(e + f*x)**S(2))**(-FracPart(n/S(2) + S(-1)/2))*Hypergeometric2F1(m/S(2) + S(1)/2, S(1)/2 - n/S(2), m/S(2) + S(3)/2, cos(e + f*x)**S(2))/(a*f*(m + S(1))), x) def replacement2183(a, b, e, f, m, n, x): return Simp(b*(a*sin(e + f*x))**(m + S(1))*(b/cos(e + f*x))**(n + S(-1))/(a*f*(m + S(1))), x) def replacement2184(a, b, e, f, m, n, x): return -Simp(b*(a*cos(e + f*x))**(m + S(1))*(b/sin(e + f*x))**(n + S(-1))/(a*f*(m + S(1))), x) def replacement2185(a, b, e, f, m, n, x): return -Dist(a**S(2)*b**S(2)*(m + S(-1))/(n + S(-1)), Int((a*sin(e + f*x))**(m + S(-2))*(b/cos(e + f*x))**(n + S(-2)), x), x) + Simp(a*b*(a*sin(e + f*x))**(m + S(-1))*(b/cos(e + f*x))**(n + S(-1))/(f*(n + S(-1))), x) def replacement2186(a, b, e, f, m, n, x): return -Dist(a**S(2)*b**S(2)*(m + S(-1))/(n + S(-1)), Int((a*cos(e + f*x))**(m + S(-2))*(b/sin(e + f*x))**(n + S(-2)), x), x) - Simp(a*b*(a*cos(e + f*x))**(m + S(-1))*(b/sin(e + f*x))**(n + S(-1))/(f*(n + S(-1))), x) def replacement2187(a, b, e, f, m, n, x): return Dist(a**S(2)*(m + S(-1))/(m - n), Int((a*sin(e + f*x))**(m + S(-2))*(b/cos(e + f*x))**n, x), x) - Simp(a*b*(a*sin(e + f*x))**(m + S(-1))*(b/cos(e + f*x))**(n + S(-1))/(f*(m - n)), x) def replacement2188(a, b, e, f, m, n, x): return Dist(a**S(2)*(m + S(-1))/(m - n), Int((a*cos(e + f*x))**(m + S(-2))*(b/sin(e + f*x))**n, x), x) + Simp(a*b*(a*cos(e + f*x))**(m + S(-1))*(b/sin(e + f*x))**(n + S(-1))/(f*(m - n)), x) def replacement2189(a, b, e, f, m, n, x): return Dist((m - n + S(2))/(a**S(2)*(m + S(1))), Int((a*sin(e + f*x))**(m + S(2))*(b/cos(e + f*x))**n, x), x) + Simp(b*(a*sin(e + f*x))**(m + S(1))*(b/cos(e + f*x))**(n + S(-1))/(a*f*(m + S(1))), x) def replacement2190(a, b, e, f, m, n, x): return Dist((m - n + S(2))/(a**S(2)*(m + S(1))), Int((a*cos(e + f*x))**(m + S(2))*(b/sin(e + f*x))**n, x), x) - Simp(b*(a*cos(e + f*x))**(m + S(1))*(b/sin(e + f*x))**(n + S(-1))/(a*f*(m + S(1))), x) def replacement2191(a, b, e, f, m, n, x): return Dist((cos(e + f*x)/b)**(FracPart(n) + S(1))*(b/cos(e + f*x))**(FracPart(n) + S(1)), Int((a*sin(e + f*x))**m*(cos(e + f*x)/b)**(-n), x), x) def replacement2192(a, b, e, f, m, n, x): return Dist((sin(e + f*x)/b)**(FracPart(n) + S(1))*(b/sin(e + f*x))**(FracPart(n) + S(1)), Int((a*cos(e + f*x))**m*(sin(e + f*x)/b)**(-n), x), x) def replacement2193(a, b, e, f, m, n, x): return Dist((a*b)**IntPart(n)*(a*sin(e + f*x))**FracPart(n)*(b/sin(e + f*x))**FracPart(n), Int((a*sin(e + f*x))**(m - n), x), x) def replacement2194(a, b, e, f, m, n, x): return Dist((a*b)**IntPart(n)*(a*cos(e + f*x))**FracPart(n)*(b/cos(e + f*x))**FracPart(n), Int((a*cos(e + f*x))**(m - n), x), x) def replacement2195(c, d, n, x): return -Dist(S(1)/d, Subst(Int((S(1) - x**S(2))**(n/S(2))/sqrt(S(1) - x**S(2)), x), x, cos(c + d*x)), x) def replacement2196(c, d, n, x): return Dist(S(1)/d, Subst(Int((S(1) - x**S(2))**(n/S(2))/sqrt(S(1) - x**S(2)), x), x, sin(c + d*x)), x) def replacement2197(b, c, d, n, x): return Dist(b**S(2)*(n + S(-1))/n, Int((b*sin(c + d*x))**(n + S(-2)), x), x) - Simp(b*(b*sin(c + d*x))**(n + S(-1))*cos(c + d*x)/(d*n), x) def replacement2198(b, c, d, n, x): return Dist(b**S(2)*(n + S(-1))/n, Int((b*cos(c + d*x))**(n + S(-2)), x), x) + Simp(b*(b*cos(c + d*x))**(n + S(-1))*sin(c + d*x)/(d*n), x) def replacement2199(b, c, d, n, x): return Dist((n + S(2))/(b**S(2)*(n + S(1))), Int((b*sin(c + d*x))**(n + S(2)), x), x) + Simp((b*sin(c + d*x))**(n + S(1))*cos(c + d*x)/(b*d*(n + S(1))), x) def replacement2200(b, c, d, n, x): return Dist((n + S(2))/(b**S(2)*(n + S(1))), Int((b*cos(c + d*x))**(n + S(2)), x), x) - Simp((b*cos(c + d*x))**(n + S(1))*sin(c + d*x)/(b*d*(n + S(1))), x) def replacement2201(c, d, x): return -Simp(cos(c + d*x)/d, x) def replacement2202(c, d, x): return Simp(sin(c + d*x)/d, x) def replacement2203(c, d, x): return Simp(S(2)*EllipticE(-Pi/S(4) + c/S(2) + d*x/S(2), S(2))/d, x) def replacement2204(c, d, x): return Simp(S(2)*EllipticE(c/S(2) + d*x/S(2), S(2))/d, x) def replacement2205(b, c, d, x): return Dist(sqrt(b*sin(c + d*x))/sqrt(sin(c + d*x)), Int(sqrt(sin(c + d*x)), x), x) def replacement2206(b, c, d, x): return Dist(sqrt(b*cos(c + d*x))/sqrt(cos(c + d*x)), Int(sqrt(cos(c + d*x)), x), x) def replacement2207(c, d, x): return Simp(S(2)*EllipticF(-Pi/S(4) + c/S(2) + d*x/S(2), S(2))/d, x) def replacement2208(c, d, x): return Simp(S(2)*EllipticF(c/S(2) + d*x/S(2), S(2))/d, x) def replacement2209(b, c, d, x): return Dist(sqrt(sin(c + d*x))/sqrt(b*sin(c + d*x)), Int(S(1)/sqrt(sin(c + d*x)), x), x) def replacement2210(b, c, d, x): return Dist(sqrt(cos(c + d*x))/sqrt(b*cos(c + d*x)), Int(S(1)/sqrt(cos(c + d*x)), x), x) def replacement2211(b, c, d, n, x): return Simp((b*sin(c + d*x))**(n + S(1))*Hypergeometric2F1(S(1)/2, n/S(2) + S(1)/2, n/S(2) + S(3)/2, sin(c + d*x)**S(2))*cos(c + d*x)/(b*d*(n + S(1))*sqrt(cos(c + d*x)**S(2))), x) def replacement2212(b, c, d, n, x): return -Simp((b*cos(c + d*x))**(n + S(1))*Hypergeometric2F1(S(1)/2, n/S(2) + S(1)/2, n/S(2) + S(3)/2, cos(c + d*x)**S(2))*sin(c + d*x)/(b*d*(n + S(1))*sqrt(sin(c + d*x)**S(2))), x) def replacement2213(a, b, c, d, x): return Dist(S(2)*a*b, Int(sin(c + d*x), x), x) + Simp(x*(S(2)*a**S(2) + b**S(2))/S(2), x) - Simp(b**S(2)*sin(c + d*x)*cos(c + d*x)/(S(2)*d), x) def replacement2214(a, b, c, d, x): return Dist(S(2)*a*b, Int(cos(c + d*x), x), x) + Simp(x*(S(2)*a**S(2) + b**S(2))/S(2), x) + Simp(b**S(2)*sin(c + d*x)*cos(c + d*x)/(S(2)*d), x) def replacement2215(a, b, c, d, n, x): return Int(ExpandTrig((a + b*sin(c + d*x))**n, x), x) def replacement2216(a, b, c, d, n, x): return Int(ExpandTrig((a + b*cos(c + d*x))**n, x), x) def replacement2217(a, b, c, d, x): return Simp(-S(2)*b*cos(c + d*x)/(d*sqrt(a + b*sin(c + d*x))), x) def replacement2218(a, b, c, d, x): return Simp(S(2)*b*sin(c + d*x)/(d*sqrt(a + b*cos(c + d*x))), x) def replacement2219(a, b, c, d, n, x): return Dist(a*(S(2)*n + S(-1))/n, Int((a + b*sin(c + d*x))**(n + S(-1)), x), x) - Simp(b*(a + b*sin(c + d*x))**(n + S(-1))*cos(c + d*x)/(d*n), x) def replacement2220(a, b, c, d, n, x): return Dist(a*(S(2)*n + S(-1))/n, Int((a + b*cos(c + d*x))**(n + S(-1)), x), x) + Simp(b*(a + b*cos(c + d*x))**(n + S(-1))*sin(c + d*x)/(d*n), x) def replacement2221(a, b, c, d, x): return -Simp(cos(c + d*x)/(d*(a*sin(c + d*x) + b)), x) def replacement2222(a, b, c, d, x): return Simp(sin(c + d*x)/(d*(a*cos(c + d*x) + b)), x) def replacement2223(a, b, c, d, x): return Dist(-S(2)/d, Subst(Int(S(1)/(S(2)*a - x**S(2)), x), x, b*cos(c + d*x)/sqrt(a + b*sin(c + d*x))), x) def replacement2224(a, b, c, d, x): return Dist(S(2)/d, Subst(Int(S(1)/(S(2)*a - x**S(2)), x), x, b*sin(c + d*x)/sqrt(a + b*cos(c + d*x))), x) def replacement2225(a, b, c, d, n, x): return Dist((n + S(1))/(a*(S(2)*n + S(1))), Int((a + b*sin(c + d*x))**(n + S(1)), x), x) + Simp(b*(a + b*sin(c + d*x))**n*cos(c + d*x)/(a*d*(S(2)*n + S(1))), x) def replacement2226(a, b, c, d, n, x): return Dist((n + S(1))/(a*(S(2)*n + S(1))), Int((a + b*cos(c + d*x))**(n + S(1)), x), x) - Simp(b*(a + b*cos(c + d*x))**n*sin(c + d*x)/(a*d*(S(2)*n + S(1))), x) def replacement2227(a, b, c, d, n, x): return -Simp(S(2)**(n + S(1)/2)*a**(n + S(-1)/2)*b*Hypergeometric2F1(S(1)/2, S(1)/2 - n, S(3)/2, S(1)/2 - b*sin(c + d*x)/(S(2)*a))*cos(c + d*x)/(d*sqrt(a + b*sin(c + d*x))), x) def replacement2228(a, b, c, d, n, x): return Simp(S(2)**(n + S(1)/2)*a**(n + S(-1)/2)*b*Hypergeometric2F1(S(1)/2, S(1)/2 - n, S(3)/2, S(1)/2 - b*cos(c + d*x)/(S(2)*a))*sin(c + d*x)/(d*sqrt(a + b*cos(c + d*x))), x) def replacement2229(a, b, c, d, n, x): return Dist(a**IntPart(n)*(S(1) + b*sin(c + d*x)/a)**(-FracPart(n))*(a + b*sin(c + d*x))**FracPart(n), Int((S(1) + b*sin(c + d*x)/a)**n, x), x) def replacement2230(a, b, c, d, n, x): return Dist(a**IntPart(n)*(S(1) + b*cos(c + d*x)/a)**(-FracPart(n))*(a + b*cos(c + d*x))**FracPart(n), Int((S(1) + b*cos(c + d*x)/a)**n, x), x) def replacement2231(a, b, c, d, x): return Simp(S(2)*sqrt(a + b)*EllipticE(-Pi/S(4) + c/S(2) + d*x/S(2), S(2)*b/(a + b))/d, x) def replacement2232(a, b, c, d, x): return Simp(S(2)*sqrt(a + b)*EllipticE(c/S(2) + d*x/S(2), S(2)*b/(a + b))/d, x) def replacement2233(a, b, c, d, x): return Simp(S(2)*sqrt(a - b)*EllipticE(Pi/S(4) + c/S(2) + d*x/S(2), -S(2)*b/(a - b))/d, x) def replacement2234(a, b, c, d, x): return Simp(S(2)*sqrt(a - b)*EllipticE(Pi/S(2) + c/S(2) + d*x/S(2), -S(2)*b/(a - b))/d, x) def replacement2235(a, b, c, d, x): return Dist(sqrt(a + b*sin(c + d*x))/sqrt((a + b*sin(c + d*x))/(a + b)), Int(sqrt(a/(a + b) + b*sin(c + d*x)/(a + b)), x), x) def replacement2236(a, b, c, d, x): return Dist(sqrt(a + b*cos(c + d*x))/sqrt((a + b*cos(c + d*x))/(a + b)), Int(sqrt(a/(a + b) + b*cos(c + d*x)/(a + b)), x), x) def replacement2237(a, b, c, d, n, x): return Dist(S(1)/n, Int((a + b*sin(c + d*x))**(n + S(-2))*Simp(a**S(2)*n + a*b*(S(2)*n + S(-1))*sin(c + d*x) + b**S(2)*(n + S(-1)), x), x), x) - Simp(b*(a + b*sin(c + d*x))**(n + S(-1))*cos(c + d*x)/(d*n), x) def replacement2238(a, b, c, d, n, x): return Dist(S(1)/n, Int((a + b*cos(c + d*x))**(n + S(-2))*Simp(a**S(2)*n + a*b*(S(2)*n + S(-1))*cos(c + d*x) + b**S(2)*(n + S(-1)), x), x), x) + Simp(b*(a + b*cos(c + d*x))**(n + S(-1))*sin(c + d*x)/(d*n), x) def With2239(a, b, c, d, x): q = Rt(a**S(2) - b**S(2), S(2)) return Simp(x/q, x) + Simp(S(2)*ArcTan(b*cos(c + d*x)/(a + b*sin(c + d*x) + q))/(d*q), x) def With2240(a, b, c, d, x): q = Rt(a**S(2) - b**S(2), S(2)) return Simp(x/q, x) - Simp(S(2)*ArcTan(b*sin(c + d*x)/(a + b*cos(c + d*x) + q))/(d*q), x) def With2241(a, b, c, d, x): q = Rt(a**S(2) - b**S(2), S(2)) return -Simp(x/q, x) - Simp(S(2)*ArcTan(b*cos(c + d*x)/(a + b*sin(c + d*x) - q))/(d*q), x) def With2242(a, b, c, d, x): q = Rt(a**S(2) - b**S(2), S(2)) return -Simp(x/q, x) + Simp(S(2)*ArcTan(b*sin(c + d*x)/(a + b*cos(c + d*x) - q))/(d*q), x) def With2243(a, b, c, d, x): e = FreeFactors(tan(-Pi/S(4) + c/S(2) + d*x/S(2)), x) return Dist(S(2)*e/d, Subst(Int(S(1)/(a + b + e**S(2)*x**S(2)*(a - b)), x), x, tan(-Pi/S(4) + c/S(2) + d*x/S(2))/e), x) def With2244(a, b, c, d, x): e = FreeFactors(tan(c/S(2) + d*x/S(2)), x) return Dist(S(2)*e/d, Subst(Int(S(1)/(a*e**S(2)*x**S(2) + a + S(2)*b*e*x), x), x, tan(c/S(2) + d*x/S(2))/e), x) def With2245(a, b, c, d, x): e = FreeFactors(tan(c/S(2) + d*x/S(2)), x) return Dist(S(2)*e/d, Subst(Int(S(1)/(a + b + e**S(2)*x**S(2)*(a - b)), x), x, tan(c/S(2) + d*x/S(2))/e), x) def replacement2246(a, b, c, d, x): return Simp(S(2)*EllipticF(-Pi/S(4) + c/S(2) + d*x/S(2), S(2)*b/(a + b))/(d*sqrt(a + b)), x) def replacement2247(a, b, c, d, x): return Simp(S(2)*EllipticF(c/S(2) + d*x/S(2), S(2)*b/(a + b))/(d*sqrt(a + b)), x) def replacement2248(a, b, c, d, x): return Simp(S(2)*EllipticF(Pi/S(4) + c/S(2) + d*x/S(2), -S(2)*b/(a - b))/(d*sqrt(a - b)), x) def replacement2249(a, b, c, d, x): return Simp(S(2)*EllipticF(Pi/S(2) + c/S(2) + d*x/S(2), -S(2)*b/(a - b))/(d*sqrt(a - b)), x) def replacement2250(a, b, c, d, x): return Dist(sqrt((a + b*sin(c + d*x))/(a + b))/sqrt(a + b*sin(c + d*x)), Int(S(1)/sqrt(a/(a + b) + b*sin(c + d*x)/(a + b)), x), x) def replacement2251(a, b, c, d, x): return Dist(sqrt((a + b*cos(c + d*x))/(a + b))/sqrt(a + b*cos(c + d*x)), Int(S(1)/sqrt(a/(a + b) + b*cos(c + d*x)/(a + b)), x), x) def replacement2252(a, b, c, d, n, x): return Dist(S(1)/((a**S(2) - b**S(2))*(n + S(1))), Int((a + b*sin(c + d*x))**(n + S(1))*Simp(a*(n + S(1)) - b*(n + S(2))*sin(c + d*x), x), x), x) - Simp(b*(a + b*sin(c + d*x))**(n + S(1))*cos(c + d*x)/(d*(a**S(2) - b**S(2))*(n + S(1))), x) def replacement2253(a, b, c, d, n, x): return Dist(S(1)/((a**S(2) - b**S(2))*(n + S(1))), Int((a + b*cos(c + d*x))**(n + S(1))*Simp(a*(n + S(1)) - b*(n + S(2))*cos(c + d*x), x), x), x) + Simp(b*(a + b*cos(c + d*x))**(n + S(1))*sin(c + d*x)/(d*(a**S(2) - b**S(2))*(n + S(1))), x) def replacement2254(a, b, c, d, n, x): return Dist(cos(c + d*x)/(d*sqrt(S(1) - sin(c + d*x))*sqrt(sin(c + d*x) + S(1))), Subst(Int((a + b*x)**n/(sqrt(S(1) - x)*sqrt(x + S(1))), x), x, sin(c + d*x)), x) def replacement2255(a, b, c, d, n, x): return -Dist(sin(c + d*x)/(d*sqrt(S(1) - cos(c + d*x))*sqrt(cos(c + d*x) + S(1))), Subst(Int((a + b*x)**n/(sqrt(S(1) - x)*sqrt(x + S(1))), x), x, cos(c + d*x)), x) def replacement2256(a, b, c, d, n, x): return Int((a + b*sin(S(2)*c + S(2)*d*x)/S(2))**n, x) def replacement2257(a, b, e, f, m, p, x): return Dist(b**(-p)/f, Subst(Int((a - x)**(p/S(2) + S(-1)/2)*(a + x)**(m + p/S(2) + S(-1)/2), x), x, b*sin(e + f*x)), x) def replacement2258(a, b, e, f, m, p, x): return -Dist(b**(-p)/f, Subst(Int((a - x)**(p/S(2) + S(-1)/2)*(a + x)**(m + p/S(2) + S(-1)/2), x), x, b*cos(e + f*x)), x) def replacement2259(a, b, e, f, m, p, x): return Dist(b**(-p)/f, Subst(Int((a + x)**m*(b**S(2) - x**S(2))**(p/S(2) + S(-1)/2), x), x, b*sin(e + f*x)), x) def replacement2260(a, b, e, f, m, p, x): return -Dist(b**(-p)/f, Subst(Int((a + x)**m*(b**S(2) - x**S(2))**(p/S(2) + S(-1)/2), x), x, b*cos(e + f*x)), x) def replacement2261(a, b, e, f, g, p, x): return Dist(a, Int((g*cos(e + f*x))**p, x), x) - Simp(b*(g*cos(e + f*x))**(p + S(1))/(f*g*(p + S(1))), x) def replacement2262(a, b, e, f, g, p, x): return Dist(a, Int((g*sin(e + f*x))**p, x), x) + Simp(b*(g*sin(e + f*x))**(p + S(1))/(f*g*(p + S(1))), x) def replacement2263(a, b, e, f, g, m, p, x): return Dist((a/g)**(S(2)*m), Int((g*cos(e + f*x))**(S(2)*m + p)*(a - b*sin(e + f*x))**(-m), x), x) def replacement2264(a, b, e, f, g, m, p, x): return Dist((a/g)**(S(2)*m), Int((g*sin(e + f*x))**(S(2)*m + p)*(a - b*cos(e + f*x))**(-m), x), x) def replacement2265(a, b, e, f, g, m, p, x): return Simp(b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**m/(a*f*g*m), x) def replacement2266(a, b, e, f, g, m, p, x): return -Simp(b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**m/(a*f*g*m), x) def replacement2267(a, b, e, f, g, m, p, x): return Dist((m + p + S(1))/(a*(S(2)*m + p + S(1))), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**(m + S(1)), x), x) + Simp(b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**m/(a*f*g*(S(2)*m + p + S(1))), x) def replacement2268(a, b, e, f, g, m, p, x): return Dist((m + p + S(1))/(a*(S(2)*m + p + S(1))), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**(m + S(1)), x), x) - Simp(b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**m/(a*f*g*(S(2)*m + p + S(1))), x) def replacement2269(a, b, e, f, g, m, p, x): return Simp(b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(-1))/(f*g*(m + S(-1))), x) def replacement2270(a, b, e, f, g, m, p, x): return -Simp(b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(-1))/(f*g*(m + S(-1))), x) def replacement2271(a, b, e, f, g, m, p, x): return Dist(a*(S(2)*m + p + S(-1))/(m + p), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**(m + S(-1)), x), x) - Simp(b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(-1))/(f*g*(m + p)), x) def replacement2272(a, b, e, f, g, m, p, x): return Dist(a*(S(2)*m + p + S(-1))/(m + p), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**(m + S(-1)), x), x) + Simp(b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(-1))/(f*g*(m + p)), x) def replacement2273(a, b, e, f, g, m, p, x): return Dist(a*(m + p + S(1))/(g**S(2)*(p + S(1))), Int((g*cos(e + f*x))**(p + S(2))*(a + b*sin(e + f*x))**(m + S(-1)), x), x) - Simp(b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**m/(a*f*g*(p + S(1))), x) def replacement2274(a, b, e, f, g, m, p, x): return Dist(a*(m + p + S(1))/(g**S(2)*(p + S(1))), Int((g*sin(e + f*x))**(p + S(2))*(a + b*cos(e + f*x))**(m + S(-1)), x), x) + Simp(b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**m/(a*f*g*(p + S(1))), x) def replacement2275(a, b, e, f, g, m, p, x): return Dist(b**S(2)*(S(2)*m + p + S(-1))/(g**S(2)*(p + S(1))), Int((g*cos(e + f*x))**(p + S(2))*(a + b*sin(e + f*x))**(m + S(-2)), x), x) + Simp(-S(2)*b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(-1))/(f*g*(p + S(1))), x) def replacement2276(a, b, e, f, g, m, p, x): return Dist(b**S(2)*(S(2)*m + p + S(-1))/(g**S(2)*(p + S(1))), Int((g*sin(e + f*x))**(p + S(2))*(a + b*cos(e + f*x))**(m + S(-2)), x), x) + Simp(S(2)*b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(-1))/(f*g*(p + S(1))), x) def replacement2277(a, b, e, f, g, x): return Dist(a*sqrt(a + b*sin(e + f*x))*sqrt(cos(e + f*x) + S(1))/(a*cos(e + f*x) + a + b*sin(e + f*x)), Int(sqrt(cos(e + f*x) + S(1))/sqrt(g*cos(e + f*x)), x), x) + Dist(b*sqrt(a + b*sin(e + f*x))*sqrt(cos(e + f*x) + S(1))/(a*cos(e + f*x) + a + b*sin(e + f*x)), Int(sin(e + f*x)/(sqrt(g*cos(e + f*x))*sqrt(cos(e + f*x) + S(1))), x), x) def replacement2278(a, b, e, f, g, x): return Dist(a*sqrt(a + b*cos(e + f*x))*sqrt(sin(e + f*x) + S(1))/(a*sin(e + f*x) + a + b*cos(e + f*x)), Int(sqrt(sin(e + f*x) + S(1))/sqrt(g*sin(e + f*x)), x), x) + Dist(b*sqrt(a + b*cos(e + f*x))*sqrt(sin(e + f*x) + S(1))/(a*sin(e + f*x) + a + b*cos(e + f*x)), Int(cos(e + f*x)/(sqrt(g*sin(e + f*x))*sqrt(sin(e + f*x) + S(1))), x), x) def replacement2279(a, b, e, f, g, m, p, x): return Dist(a*(S(2)*m + p + S(-1))/(m + p), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**(m + S(-1)), x), x) - Simp(b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(-1))/(f*g*(m + p)), x) def replacement2280(a, b, e, f, g, m, p, x): return Dist(a*(S(2)*m + p + S(-1))/(m + p), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**(m + S(-1)), x), x) + Simp(b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(-1))/(f*g*(m + p)), x) def replacement2281(a, b, e, f, g, m, p, x): return Dist(g**S(2)*(p + S(-1))/(a*(m + p)), Int((g*cos(e + f*x))**(p + S(-2))*(a + b*sin(e + f*x))**(m + S(1)), x), x) + Simp(g*(g*cos(e + f*x))**(p + S(-1))*(a + b*sin(e + f*x))**(m + S(1))/(b*f*(m + p)), x) def replacement2282(a, b, e, f, g, m, p, x): return Dist(g**S(2)*(p + S(-1))/(a*(m + p)), Int((g*sin(e + f*x))**(p + S(-2))*(a + b*cos(e + f*x))**(m + S(1)), x), x) - Simp(g*(g*sin(e + f*x))**(p + S(-1))*(a + b*cos(e + f*x))**(m + S(1))/(b*f*(m + p)), x) def replacement2283(a, b, e, f, g, m, p, x): return Dist(g**S(2)*(p + S(-1))/(b**S(2)*(S(2)*m + p + S(1))), Int((g*cos(e + f*x))**(p + S(-2))*(a + b*sin(e + f*x))**(m + S(2)), x), x) + Simp(S(2)*g*(g*cos(e + f*x))**(p + S(-1))*(a + b*sin(e + f*x))**(m + S(1))/(b*f*(S(2)*m + p + S(1))), x) def replacement2284(a, b, e, f, g, m, p, x): return Dist(g**S(2)*(p + S(-1))/(b**S(2)*(S(2)*m + p + S(1))), Int((g*sin(e + f*x))**(p + S(-2))*(a + b*cos(e + f*x))**(m + S(2)), x), x) + Simp(-S(2)*g*(g*sin(e + f*x))**(p + S(-1))*(a + b*cos(e + f*x))**(m + S(1))/(b*f*(S(2)*m + p + S(1))), x) def replacement2285(a, b, e, f, g, m, p, x): return Dist((m + p + S(1))/(a*(S(2)*m + p + S(1))), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**(m + S(1)), x), x) + Simp(b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**m/(a*f*g*(S(2)*m + p + S(1))), x) def replacement2286(a, b, e, f, g, m, p, x): return Dist((m + p + S(1))/(a*(S(2)*m + p + S(1))), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**(m + S(1)), x), x) - Simp(b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**m/(a*f*g*(S(2)*m + p + S(1))), x) def replacement2287(a, b, e, f, g, p, x): return Dist(g**S(2)/a, Int((g*cos(e + f*x))**(p + S(-2)), x), x) + Simp(g*(g*cos(e + f*x))**(p + S(-1))/(b*f*(p + S(-1))), x) def replacement2288(a, b, e, f, g, p, x): return Dist(g**S(2)/a, Int((g*sin(e + f*x))**(p + S(-2)), x), x) - Simp(g*(g*sin(e + f*x))**(p + S(-1))/(b*f*(p + S(-1))), x) def replacement2289(a, b, e, f, g, p, x): return Dist(p/(a*(p + S(-1))), Int((g*cos(e + f*x))**p, x), x) + Simp(b*(g*cos(e + f*x))**(p + S(1))/(a*f*g*(a + b*sin(e + f*x))*(p + S(-1))), x) def replacement2290(a, b, e, f, g, p, x): return Dist(p/(a*(p + S(-1))), Int((g*sin(e + f*x))**p, x), x) - Simp(b*(g*sin(e + f*x))**(p + S(1))/(a*f*g*(a + b*cos(e + f*x))*(p + S(-1))), x) def replacement2291(a, b, e, f, g, x): return -Dist(g*sqrt(a + b*sin(e + f*x))*sqrt(cos(e + f*x) + S(1))/(a*sin(e + f*x) + b*cos(e + f*x) + b), Int(sin(e + f*x)/(sqrt(g*cos(e + f*x))*sqrt(cos(e + f*x) + S(1))), x), x) + Dist(g*sqrt(a + b*sin(e + f*x))*sqrt(cos(e + f*x) + S(1))/(a*cos(e + f*x) + a + b*sin(e + f*x)), Int(sqrt(cos(e + f*x) + S(1))/sqrt(g*cos(e + f*x)), x), x) def replacement2292(a, b, e, f, g, x): return Dist(g*sqrt(a + b*cos(e + f*x))*sqrt(sin(e + f*x) + S(1))/(a*sin(e + f*x) + a + b*cos(e + f*x)), Int(sqrt(sin(e + f*x) + S(1))/sqrt(g*sin(e + f*x)), x), x) - Dist(g*sqrt(a + b*cos(e + f*x))*sqrt(sin(e + f*x) + S(1))/(a*cos(e + f*x) + b*sin(e + f*x) + b), Int(cos(e + f*x)/(sqrt(g*sin(e + f*x))*sqrt(sin(e + f*x) + S(1))), x), x) def replacement2293(a, b, e, f, g, x): return Dist(g**S(2)/(S(2)*a), Int(sqrt(a + b*sin(e + f*x))/sqrt(g*cos(e + f*x)), x), x) + Simp(g*sqrt(g*cos(e + f*x))*sqrt(a + b*sin(e + f*x))/(b*f), x) def replacement2294(a, b, e, f, g, x): return Dist(g**S(2)/(S(2)*a), Int(sqrt(a + b*cos(e + f*x))/sqrt(g*sin(e + f*x)), x), x) - Simp(g*sqrt(g*sin(e + f*x))*sqrt(a + b*cos(e + f*x))/(b*f), x) def replacement2295(a, b, e, f, g, p, x): return Dist(S(2)*a*(p + S(-2))/(S(2)*p + S(-1)), Int((g*cos(e + f*x))**p/(a + b*sin(e + f*x))**(S(3)/2), x), x) + Simp(-S(2)*b*(g*cos(e + f*x))**(p + S(1))/(f*g*(a + b*sin(e + f*x))**(S(3)/2)*(S(2)*p + S(-1))), x) def replacement2296(a, b, e, f, g, p, x): return Dist(S(2)*a*(p + S(-2))/(S(2)*p + S(-1)), Int((g*sin(e + f*x))**p/(a + b*cos(e + f*x))**(S(3)/2), x), x) + Simp(S(2)*b*(g*sin(e + f*x))**(p + S(1))/(f*g*(a + b*cos(e + f*x))**(S(3)/2)*(S(2)*p + S(-1))), x) def replacement2297(a, b, e, f, g, p, x): return Dist(a*(S(2)*p + S(1))/(S(2)*g**S(2)*(p + S(1))), Int((g*cos(e + f*x))**(p + S(2))/(a + b*sin(e + f*x))**(S(3)/2), x), x) - Simp(b*(g*cos(e + f*x))**(p + S(1))/(a*f*g*sqrt(a + b*sin(e + f*x))*(p + S(1))), x) def replacement2298(a, b, e, f, g, p, x): return Dist(a*(S(2)*p + S(1))/(S(2)*g**S(2)*(p + S(1))), Int((g*sin(e + f*x))**(p + S(2))/(a + b*cos(e + f*x))**(S(3)/2), x), x) + Simp(b*(g*sin(e + f*x))**(p + S(1))/(a*f*g*sqrt(a + b*cos(e + f*x))*(p + S(1))), x) def replacement2299(a, b, e, f, g, m, p, x): return Dist(a**m*(g*cos(e + f*x))**(p + S(1))*(S(1) - sin(e + f*x))**(-p/S(2) + S(-1)/2)*(sin(e + f*x) + S(1))**(-p/S(2) + S(-1)/2)/(f*g), Subst(Int((S(1) - b*x/a)**(p/S(2) + S(-1)/2)*(S(1) + b*x/a)**(m + p/S(2) + S(-1)/2), x), x, sin(e + f*x)), x) def replacement2300(a, b, e, f, g, m, p, x): return -Dist(a**m*(g*sin(e + f*x))**(p + S(1))*(S(1) - cos(e + f*x))**(-p/S(2) + S(-1)/2)*(cos(e + f*x) + S(1))**(-p/S(2) + S(-1)/2)/(f*g), Subst(Int((S(1) - b*x/a)**(p/S(2) + S(-1)/2)*(S(1) + b*x/a)**(m + p/S(2) + S(-1)/2), x), x, cos(e + f*x)), x) def replacement2301(a, b, e, f, g, m, p, x): return Dist(a**S(2)*(g*cos(e + f*x))**(p + S(1))*(a - b*sin(e + f*x))**(-p/S(2) + S(-1)/2)*(a + b*sin(e + f*x))**(-p/S(2) + S(-1)/2)/(f*g), Subst(Int((a - b*x)**(p/S(2) + S(-1)/2)*(a + b*x)**(m + p/S(2) + S(-1)/2), x), x, sin(e + f*x)), x) def replacement2302(a, b, e, f, g, m, p, x): return -Dist(a**S(2)*(g*sin(e + f*x))**(p + S(1))*(a - b*cos(e + f*x))**(-p/S(2) + S(-1)/2)*(a + b*cos(e + f*x))**(-p/S(2) + S(-1)/2)/(f*g), Subst(Int((a - b*x)**(p/S(2) + S(-1)/2)*(a + b*x)**(m + p/S(2) + S(-1)/2), x), x, cos(e + f*x)), x) def replacement2303(a, b, e, f, g, m, p, x): return Dist(S(1)/(g**S(2)*(p + S(1))), Int((g*cos(e + f*x))**(p + S(2))*(a + b*sin(e + f*x))**(m + S(-1))*(a*(p + S(2)) + b*(m + p + S(2))*sin(e + f*x)), x), x) - Simp((g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**m*sin(e + f*x)/(f*g*(p + S(1))), x) def replacement2304(a, b, e, f, g, m, p, x): return Dist(S(1)/(g**S(2)*(p + S(1))), Int((g*sin(e + f*x))**(p + S(2))*(a + b*cos(e + f*x))**(m + S(-1))*(a*(p + S(2)) + b*(m + p + S(2))*cos(e + f*x)), x), x) + Simp((g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**m*cos(e + f*x)/(f*g*(p + S(1))), x) def replacement2305(a, b, e, f, g, m, p, x): return Dist(S(1)/(g**S(2)*(p + S(1))), Int((g*cos(e + f*x))**(p + S(2))*(a + b*sin(e + f*x))**(m + S(-2))*(a**S(2)*(p + S(2)) + a*b*(m + p + S(1))*sin(e + f*x) + b**S(2)*(m + S(-1))), x), x) - Simp((g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(-1))*(a*sin(e + f*x) + b)/(f*g*(p + S(1))), x) def replacement2306(a, b, e, f, g, m, p, x): return Dist(S(1)/(g**S(2)*(p + S(1))), Int((g*sin(e + f*x))**(p + S(2))*(a + b*cos(e + f*x))**(m + S(-2))*(a**S(2)*(p + S(2)) + a*b*(m + p + S(1))*cos(e + f*x) + b**S(2)*(m + S(-1))), x), x) + Simp((g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(-1))*(a*cos(e + f*x) + b)/(f*g*(p + S(1))), x) def replacement2307(a, b, e, f, g, m, p, x): return Dist(S(1)/(m + p), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**(m + S(-2))*(a**S(2)*(m + p) + a*b*(S(2)*m + p + S(-1))*sin(e + f*x) + b**S(2)*(m + S(-1))), x), x) - Simp(b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(-1))/(f*g*(m + p)), x) def replacement2308(a, b, e, f, g, m, p, x): return Dist(S(1)/(m + p), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**(m + S(-2))*(a**S(2)*(m + p) + a*b*(S(2)*m + p + S(-1))*cos(e + f*x) + b**S(2)*(m + S(-1))), x), x) + Simp(b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(-1))/(f*g*(m + p)), x) def replacement2309(a, b, e, f, g, m, p, x): return Dist(g**S(2)*(p + S(-1))/(b*(m + S(1))), Int((g*cos(e + f*x))**(p + S(-2))*(a + b*sin(e + f*x))**(m + S(1))*sin(e + f*x), x), x) + Simp(g*(g*cos(e + f*x))**(p + S(-1))*(a + b*sin(e + f*x))**(m + S(1))/(b*f*(m + S(1))), x) def replacement2310(a, b, e, f, g, m, p, x): return Dist(g**S(2)*(p + S(-1))/(b*(m + S(1))), Int((g*sin(e + f*x))**(p + S(-2))*(a + b*cos(e + f*x))**(m + S(1))*cos(e + f*x), x), x) - Simp(g*(g*sin(e + f*x))**(p + S(-1))*(a + b*cos(e + f*x))**(m + S(1))/(b*f*(m + S(1))), x) def replacement2311(a, b, e, f, g, m, p, x): return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**(m + S(1))*(a*(m + S(1)) - b*(m + p + S(2))*sin(e + f*x)), x), x) - Simp(b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(1))/(f*g*(a**S(2) - b**S(2))*(m + S(1))), x) def replacement2312(a, b, e, f, g, m, p, x): return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**(m + S(1))*(a*(m + S(1)) - b*(m + p + S(2))*cos(e + f*x)), x), x) + Simp(b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(1))/(f*g*(a**S(2) - b**S(2))*(m + S(1))), x) def replacement2313(a, b, e, f, g, m, p, x): return Dist(g**S(2)*(p + S(-1))/(b*(m + p)), Int((g*cos(e + f*x))**(p + S(-2))*(a + b*sin(e + f*x))**m*(a*sin(e + f*x) + b), x), x) + Simp(g*(g*cos(e + f*x))**(p + S(-1))*(a + b*sin(e + f*x))**(m + S(1))/(b*f*(m + p)), x) def replacement2314(a, b, e, f, g, m, p, x): return Dist(g**S(2)*(p + S(-1))/(b*(m + p)), Int((g*sin(e + f*x))**(p + S(-2))*(a + b*cos(e + f*x))**m*(a*cos(e + f*x) + b), x), x) - Simp(g*(g*sin(e + f*x))**(p + S(-1))*(a + b*cos(e + f*x))**(m + S(1))/(b*f*(m + p)), x) def replacement2315(a, b, e, f, g, m, p, x): return Dist(S(1)/(g**S(2)*(a**S(2) - b**S(2))*(p + S(1))), Int((g*cos(e + f*x))**(p + S(2))*(a + b*sin(e + f*x))**m*(a**S(2)*(p + S(2)) + a*b*(m + p + S(3))*sin(e + f*x) - b**S(2)*(m + p + S(2))), x), x) + Simp((g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(1))*(-a*sin(e + f*x) + b)/(f*g*(a**S(2) - b**S(2))*(p + S(1))), x) def replacement2316(a, b, e, f, g, m, p, x): return Dist(S(1)/(g**S(2)*(a**S(2) - b**S(2))*(p + S(1))), Int((g*sin(e + f*x))**(p + S(2))*(a + b*cos(e + f*x))**m*(a**S(2)*(p + S(2)) + a*b*(m + p + S(3))*cos(e + f*x) - b**S(2)*(m + p + S(2))), x), x) - Simp((g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(1))*(-a*cos(e + f*x) + b)/(f*g*(a**S(2) - b**S(2))*(p + S(1))), x) def replacement2317(a, b, e, f, g, x): return Dist(S(2)*sqrt(S(2))*sqrt(g*cos(e + f*x))*sqrt((a + b*sin(e + f*x))/((S(1) - sin(e + f*x))*(a - b)))/(f*g*sqrt((sin(e + f*x) + cos(e + f*x) + S(1))/(-sin(e + f*x) + cos(e + f*x) + S(1)))*sqrt(a + b*sin(e + f*x))), Subst(Int(S(1)/sqrt(x**S(4)*(a + b)/(a - b) + S(1)), x), x, sqrt((sin(e + f*x) + cos(e + f*x) + S(1))/(-sin(e + f*x) + cos(e + f*x) + S(1)))), x) def replacement2318(a, b, e, f, g, x): return Dist(-S(2)*sqrt(S(2))*sqrt(g*sin(e + f*x))*sqrt((a + b*cos(e + f*x))/((S(1) - cos(e + f*x))*(a - b)))/(f*g*sqrt((sin(e + f*x) + cos(e + f*x) + S(1))/(sin(e + f*x) - cos(e + f*x) + S(1)))*sqrt(a + b*cos(e + f*x))), Subst(Int(S(1)/sqrt(x**S(4)*(a + b)/(a - b) + S(1)), x), x, sqrt((sin(e + f*x) + cos(e + f*x) + S(1))/(sin(e + f*x) - cos(e + f*x) + S(1)))), x) def replacement2319(a, b, e, f, g, m, p, x): return Simp(g*(g*cos(e + f*x))**(p + S(-1))*(-(S(1) - sin(e + f*x))*(a - b)/((a + b)*(sin(e + f*x) + S(1))))**(m/S(2))*(S(1) - sin(e + f*x))*(a + b*sin(e + f*x))**(m + S(1))*Hypergeometric2F1(m + S(1), m/S(2) + S(1), m + S(2), S(2)*(a + b*sin(e + f*x))/((a + b)*(sin(e + f*x) + S(1))))/(f*(a + b)*(m + S(1))), x) def replacement2320(a, b, e, f, g, m, p, x): return -Simp(g*(g*sin(e + f*x))**(p + S(-1))*(-(S(1) - cos(e + f*x))*(a - b)/((a + b)*(cos(e + f*x) + S(1))))**(m/S(2))*(S(1) - cos(e + f*x))*(a + b*cos(e + f*x))**(m + S(1))*Hypergeometric2F1(m + S(1), m/S(2) + S(1), m + S(2), S(2)*(a + b*cos(e + f*x))/((a + b)*(cos(e + f*x) + S(1))))/(f*(a + b)*(m + S(1))), x) def replacement2321(a, b, e, f, g, m, p, x): return Dist(a/(g**S(2)*(a - b)), Int((g*cos(e + f*x))**(p + S(2))*(a + b*sin(e + f*x))**m/(S(1) - sin(e + f*x)), x), x) + Simp((g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(1))/(f*g*(a - b)*(p + S(1))), x) def replacement2322(a, b, e, f, g, m, p, x): return Dist(a/(g**S(2)*(a - b)), Int((g*sin(e + f*x))**(p + S(2))*(a + b*cos(e + f*x))**m/(S(1) - cos(e + f*x)), x), x) - Simp((g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(1))/(f*g*(a - b)*(p + S(1))), x) def replacement2323(a, b, e, f, g, m, p, x): return Dist(a/(g**S(2)*(a - b)), Int((g*cos(e + f*x))**(p + S(2))*(a + b*sin(e + f*x))**m/(S(1) - sin(e + f*x)), x), x) - Dist(b*(m + p + S(2))/(g**S(2)*(a - b)*(p + S(1))), Int((g*cos(e + f*x))**(p + S(2))*(a + b*sin(e + f*x))**m, x), x) + Simp((g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(1))/(f*g*(a - b)*(p + S(1))), x) def replacement2324(a, b, e, f, g, m, p, x): return Dist(a/(g**S(2)*(a - b)), Int((g*sin(e + f*x))**(p + S(2))*(a + b*cos(e + f*x))**m/(S(1) - cos(e + f*x)), x), x) - Dist(b*(m + p + S(2))/(g**S(2)*(a - b)*(p + S(1))), Int((g*sin(e + f*x))**(p + S(2))*(a + b*cos(e + f*x))**m, x), x) - Simp((g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(1))/(f*g*(a - b)*(p + S(1))), x) def With2325(a, b, e, f, g, x): q = Rt(-a**S(2) + b**S(2), S(2)) return -Dist(a*g/(S(2)*b), Int(S(1)/(sqrt(g*cos(e + f*x))*(-b*cos(e + f*x) + q)), x), x) + Dist(a*g/(S(2)*b), Int(S(1)/(sqrt(g*cos(e + f*x))*(b*cos(e + f*x) + q)), x), x) + Dist(b*g/f, Subst(Int(sqrt(x)/(b**S(2)*x**S(2) + g**S(2)*(a**S(2) - b**S(2))), x), x, g*cos(e + f*x)), x) def With2326(a, b, e, f, g, x): q = Rt(-a**S(2) + b**S(2), S(2)) return -Dist(a*g/(S(2)*b), Int(S(1)/(sqrt(g*sin(e + f*x))*(-b*sin(e + f*x) + q)), x), x) + Dist(a*g/(S(2)*b), Int(S(1)/(sqrt(g*sin(e + f*x))*(b*sin(e + f*x) + q)), x), x) - Dist(b*g/f, Subst(Int(sqrt(x)/(b**S(2)*x**S(2) + g**S(2)*(a**S(2) - b**S(2))), x), x, g*sin(e + f*x)), x) def With2327(a, b, e, f, g, x): q = Rt(-a**S(2) + b**S(2), S(2)) return -Dist(a/(S(2)*q), Int(S(1)/(sqrt(g*cos(e + f*x))*(-b*cos(e + f*x) + q)), x), x) - Dist(a/(S(2)*q), Int(S(1)/(sqrt(g*cos(e + f*x))*(b*cos(e + f*x) + q)), x), x) + Dist(b*g/f, Subst(Int(S(1)/(sqrt(x)*(b**S(2)*x**S(2) + g**S(2)*(a**S(2) - b**S(2)))), x), x, g*cos(e + f*x)), x) def With2328(a, b, e, f, g, x): q = Rt(-a**S(2) + b**S(2), S(2)) return -Dist(a/(S(2)*q), Int(S(1)/(sqrt(g*sin(e + f*x))*(-b*sin(e + f*x) + q)), x), x) - Dist(a/(S(2)*q), Int(S(1)/(sqrt(g*sin(e + f*x))*(b*sin(e + f*x) + q)), x), x) - Dist(b*g/f, Subst(Int(S(1)/(sqrt(x)*(b**S(2)*x**S(2) + g**S(2)*(a**S(2) - b**S(2)))), x), x, g*sin(e + f*x)), x) def replacement2329(a, b, e, f, g, m, p, x): return Simp(g*(g*cos(e + f*x))**(p + S(-1))*(b*(sin(e + f*x) + S(1))/(a + b*sin(e + f*x)))**(S(1)/2 - p/S(2))*(-b*(S(1) - sin(e + f*x))/(a + b*sin(e + f*x)))**(S(1)/2 - p/S(2))*(a + b*sin(e + f*x))**(m + S(1))*AppellF1(-m - p, S(1)/2 - p/S(2), S(1)/2 - p/S(2), -m - p + S(1), (a + b)/(a + b*sin(e + f*x)), (a - b)/(a + b*sin(e + f*x)))/(b*f*(m + p)), x) def replacement2330(a, b, e, f, g, m, p, x): return -Simp(g*(g*sin(e + f*x))**(p + S(-1))*(b*(cos(e + f*x) + S(1))/(a + b*cos(e + f*x)))**(S(1)/2 - p/S(2))*(-b*(S(1) - cos(e + f*x))/(a + b*cos(e + f*x)))**(S(1)/2 - p/S(2))*(a + b*cos(e + f*x))**(m + S(1))*AppellF1(-m - p, S(1)/2 - p/S(2), S(1)/2 - p/S(2), -m - p + S(1), (a + b)/(a + b*cos(e + f*x)), (a - b)/(a + b*cos(e + f*x)))/(b*f*(m + p)), x) def replacement2331(a, b, e, f, g, m, p, x): return Dist(g*(g*cos(e + f*x))**(p + S(-1))*(S(1) - (a + b*sin(e + f*x))/(a - b))**(S(1)/2 - p/S(2))*(S(1) - (a + b*sin(e + f*x))/(a + b))**(S(1)/2 - p/S(2))/f, Subst(Int((a + b*x)**m*(-b*x/(a - b) - b/(a - b))**(p/S(2) + S(-1)/2)*(-b*x/(a + b) + b/(a + b))**(p/S(2) + S(-1)/2), x), x, sin(e + f*x)), x) def replacement2332(a, b, e, f, g, m, p, x): return -Dist(g*(g*sin(e + f*x))**(p + S(-1))*(S(1) - (a + b*cos(e + f*x))/(a - b))**(S(1)/2 - p/S(2))*(S(1) - (a + b*cos(e + f*x))/(a + b))**(S(1)/2 - p/S(2))/f, Subst(Int((a + b*x)**m*(-b*x/(a - b) - b/(a - b))**(p/S(2) + S(-1)/2)*(-b*x/(a + b) + b/(a + b))**(p/S(2) + S(-1)/2), x), x, cos(e + f*x)), x) def replacement2333(a, b, e, f, g, m, p, x): return Dist(g**(S(2)*IntPart(p))*(g/cos(e + f*x))**FracPart(p)*(g*cos(e + f*x))**FracPart(p), Int((g*cos(e + f*x))**(-p)*(a + b*sin(e + f*x))**m, x), x) def replacement2334(a, b, e, f, g, m, p, x): return Dist(g**(S(2)*IntPart(p))*(g/sin(e + f*x))**FracPart(p)*(g*sin(e + f*x))**FracPart(p), Int((g*sin(e + f*x))**(-p)*(a + b*cos(e + f*x))**m, x), x) def replacement2335(a, b, e, f, g, p, x): return Dist(S(1)/a, Int((g*tan(e + f*x))**p/cos(e + f*x)**S(2), x), x) - Dist(S(1)/(b*g), Int((g*tan(e + f*x))**(p + S(1))/cos(e + f*x), x), x) def replacement2336(a, b, e, f, g, p, x): return Dist(S(1)/a, Int((g/tan(e + f*x))**p/sin(e + f*x)**S(2), x), x) - Dist(S(1)/(b*g), Int((g/tan(e + f*x))**(p + S(1))/sin(e + f*x), x), x) def replacement2337(a, b, e, f, m, p, x): return Dist(S(1)/f, Subst(Int(x**p*(a - x)**(-p/S(2) + S(-1)/2)*(a + x)**(m - p/S(2) + S(-1)/2), x), x, b*sin(e + f*x)), x) def replacement2338(a, b, e, f, m, p, x): return -Dist(S(1)/f, Subst(Int(x**p*(a - x)**(-p/S(2) + S(-1)/2)*(a + x)**(m - p/S(2) + S(-1)/2), x), x, b*cos(e + f*x)), x) def replacement2339(a, b, e, f, m, p, x): return Dist(a**p, Int((a - b*sin(e + f*x))**(-m)*sin(e + f*x)**p, x), x) def replacement2340(a, b, e, f, m, p, x): return Dist(a**p, Int((a - b*cos(e + f*x))**(-m)*cos(e + f*x)**p, x), x) def replacement2341(a, b, e, f, m, p, x): return Dist(a**p, Int(ExpandIntegrand((a - b*sin(e + f*x))**(-p/S(2))*(a + b*sin(e + f*x))**(m - p/S(2))*sin(e + f*x)**p, x), x), x) def replacement2342(a, b, e, f, m, p, x): return Dist(a**p, Int(ExpandIntegrand((a - b*cos(e + f*x))**(-p/S(2))*(a + b*cos(e + f*x))**(m - p/S(2))*cos(e + f*x)**p, x), x), x) def replacement2343(a, b, e, f, g, m, p, x): return Int(ExpandIntegrand((g*tan(e + f*x))**p, (a + b*sin(e + f*x))**m, x), x) def replacement2344(a, b, e, f, g, m, p, x): return Int(ExpandIntegrand((g/tan(e + f*x))**p, (a + b*cos(e + f*x))**m, x), x) def replacement2345(a, b, e, f, g, m, p, x): return Dist(a**(S(2)*m), Int(ExpandIntegrand((g*tan(e + f*x))**p*(S(1)/cos(e + f*x))**(-m), (a/cos(e + f*x) - b*tan(e + f*x))**(-m), x), x), x) def replacement2346(a, b, e, f, g, m, p, x): return Dist(a**(S(2)*m), Int(ExpandIntegrand((g/tan(e + f*x))**p*(S(1)/sin(e + f*x))**(-m), (a/sin(e + f*x) - b/tan(e + f*x))**(-m), x), x), x) def replacement2347(a, b, e, f, m, x): return -Dist(S(1)/(a**S(2)*(S(2)*m + S(-1))), Int((a + b*sin(e + f*x))**(m + S(1))*(a*m - b*(S(2)*m + S(-1))*sin(e + f*x))/cos(e + f*x)**S(2), x), x) + Simp(b*(a + b*sin(e + f*x))**m/(a*f*(S(2)*m + S(-1))*cos(e + f*x)), x) def replacement2348(a, b, e, f, m, x): return -Dist(S(1)/(a**S(2)*(S(2)*m + S(-1))), Int((a + b*cos(e + f*x))**(m + S(1))*(a*m - b*(S(2)*m + S(-1))*cos(e + f*x))/sin(e + f*x)**S(2), x), x) - Simp(b*(a + b*cos(e + f*x))**m/(a*f*(S(2)*m + S(-1))*sin(e + f*x)), x) def replacement2349(a, b, e, f, m, x): return Dist(S(1)/(b*m), Int((a + b*sin(e + f*x))**m*(a*sin(e + f*x) + b*(m + S(1)))/cos(e + f*x)**S(2), x), x) - Simp((a + b*sin(e + f*x))**(m + S(1))/(b*f*m*cos(e + f*x)), x) def replacement2350(a, b, e, f, m, x): return Dist(S(1)/(b*m), Int((a + b*cos(e + f*x))**m*(a*cos(e + f*x) + b*(m + S(1)))/sin(e + f*x)**S(2), x), x) + Simp((a + b*cos(e + f*x))**(m + S(1))/(b*f*m*sin(e + f*x)), x) def replacement2351(a, b, e, f, m, x): return -Int((S(1) - S(2)*sin(e + f*x)**S(2))*(a + b*sin(e + f*x))**m/cos(e + f*x)**S(4), x) + Int((a + b*sin(e + f*x))**m, x) def replacement2352(a, b, e, f, m, x): return -Int((S(1) - S(2)*cos(e + f*x)**S(2))*(a + b*cos(e + f*x))**m/sin(e + f*x)**S(4), x) + Int((a + b*cos(e + f*x))**m, x) def replacement2353(a, b, e, f, m, x): return Dist(b**(S(-2)), Int((a + b*sin(e + f*x))**(m + S(1))*(-a*(m + S(1))*sin(e + f*x) + b*m)/sin(e + f*x), x), x) - Simp((a + b*sin(e + f*x))**(m + S(1))/(a*f*tan(e + f*x)), x) def replacement2354(a, b, e, f, m, x): return Dist(b**(S(-2)), Int((a + b*cos(e + f*x))**(m + S(1))*(-a*(m + S(1))*cos(e + f*x) + b*m)/cos(e + f*x), x), x) + Simp((a + b*cos(e + f*x))**(m + S(1))*tan(e + f*x)/(a*f), x) def replacement2355(a, b, e, f, m, x): return Dist(S(1)/a, Int((a + b*sin(e + f*x))**m*(-a*(m + S(1))*sin(e + f*x) + b*m)/sin(e + f*x), x), x) - Simp((a + b*sin(e + f*x))**m/(f*tan(e + f*x)), x) def replacement2356(a, b, e, f, m, x): return Dist(S(1)/a, Int((a + b*cos(e + f*x))**m*(-a*(m + S(1))*cos(e + f*x) + b*m)/cos(e + f*x), x), x) + Simp((a + b*cos(e + f*x))**m*tan(e + f*x)/f, x) def replacement2357(a, b, e, f, m, x): return Dist(a**(S(-2)), Int((a + b*sin(e + f*x))**(m + S(2))*(sin(e + f*x)**S(2) + S(1))/sin(e + f*x)**S(4), x), x) + Dist(-S(2)/(a*b), Int((a + b*sin(e + f*x))**(m + S(2))/sin(e + f*x)**S(3), x), x) def replacement2358(a, b, e, f, m, x): return Dist(a**(S(-2)), Int((a + b*cos(e + f*x))**(m + S(2))*(cos(e + f*x)**S(2) + S(1))/cos(e + f*x)**S(4), x), x) + Dist(-S(2)/(a*b), Int((a + b*cos(e + f*x))**(m + S(2))/cos(e + f*x)**S(3), x), x) def replacement2359(a, b, e, f, m, x): return Int((S(1) - S(2)*sin(e + f*x)**S(2))*(a + b*sin(e + f*x))**m/sin(e + f*x)**S(4), x) + Int((a + b*sin(e + f*x))**m, x) def replacement2360(a, b, e, f, m, x): return Int((S(1) - S(2)*cos(e + f*x)**S(2))*(a + b*cos(e + f*x))**m/cos(e + f*x)**S(4), x) + Int((a + b*cos(e + f*x))**m, x) def replacement2361(a, b, e, f, m, p, x): return Dist(sqrt(a - b*sin(e + f*x))*sqrt(a + b*sin(e + f*x))/(b*f*cos(e + f*x)), Subst(Int(x**p*(a - x)**(-p/S(2) + S(-1)/2)*(a + x)**(m - p/S(2) + S(-1)/2), x), x, b*sin(e + f*x)), x) def replacement2362(a, b, e, f, m, p, x): return -Dist(sqrt(a - b*cos(e + f*x))*sqrt(a + b*cos(e + f*x))/(b*f*sin(e + f*x)), Subst(Int(x**p*(a - x)**(-p/S(2) + S(-1)/2)*(a + x)**(m - p/S(2) + S(-1)/2), x), x, b*cos(e + f*x)), x) def replacement2363(a, b, e, f, g, m, p, x): return Dist((b*sin(e + f*x))**(-p + S(-1))*(g*tan(e + f*x))**(p + S(1))*(a - b*sin(e + f*x))**(p/S(2) + S(1)/2)*(a + b*sin(e + f*x))**(p/S(2) + S(1)/2)/(f*g), Subst(Int(x**p*(a - x)**(-p/S(2) + S(-1)/2)*(a + x)**(m - p/S(2) + S(-1)/2), x), x, b*sin(e + f*x)), x) def replacement2364(a, b, e, f, g, m, p, x): return -Dist((b*cos(e + f*x))**(-p + S(-1))*(g/tan(e + f*x))**(p + S(1))*(a - b*cos(e + f*x))**(p/S(2) + S(1)/2)*(a + b*cos(e + f*x))**(p/S(2) + S(1)/2)/(f*g), Subst(Int(x**p*(a - x)**(-p/S(2) + S(-1)/2)*(a + x)**(m - p/S(2) + S(-1)/2), x), x, b*cos(e + f*x)), x) def replacement2365(a, b, e, f, m, p, x): return Dist(S(1)/f, Subst(Int(x**p*(a + x)**m*(b**S(2) - x**S(2))**(-p/S(2) + S(-1)/2), x), x, b*sin(e + f*x)), x) def replacement2366(a, b, e, f, m, p, x): return -Dist(S(1)/f, Subst(Int(x**p*(a + x)**m*(b**S(2) - x**S(2))**(-p/S(2) + S(-1)/2), x), x, b*cos(e + f*x)), x) def replacement2367(a, b, e, f, g, m, p, x): return Int(ExpandIntegrand((g*tan(e + f*x))**p, (a + b*sin(e + f*x))**m, x), x) def replacement2368(a, b, e, f, g, m, p, x): return Int(ExpandIntegrand((g/tan(e + f*x))**p, (a + b*cos(e + f*x))**m, x), x) def replacement2369(a, b, e, f, m, x): return Int((S(1) - sin(e + f*x)**S(2))*(a + b*sin(e + f*x))**m/sin(e + f*x)**S(2), x) def replacement2370(a, b, e, f, m, x): return Int((S(1) - cos(e + f*x)**S(2))*(a + b*cos(e + f*x))**m/cos(e + f*x)**S(2), x) def replacement2371(a, b, e, f, m, x): return -Dist(S(1)/(S(3)*a**S(2)*b*(m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*Simp(S(6)*a**S(2) + a*b*(m + S(1))*sin(e + f*x) - b**S(2)*(m + S(-2))*(m + S(-1)) - (S(3)*a**S(2) - b**S(2)*m*(m + S(-2)))*sin(e + f*x)**S(2), x)/sin(e + f*x)**S(3), x), x) - Simp((a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(S(3)*a*f*sin(e + f*x)**S(3)), x) - Simp((a + b*sin(e + f*x))**(m + S(1))*(S(3)*a**S(2) + b**S(2)*(m + S(-2)))*cos(e + f*x)/(S(3)*a**S(2)*b*f*(m + S(1))*sin(e + f*x)**S(2)), x) def replacement2372(a, b, e, f, m, x): return -Dist(S(1)/(S(3)*a**S(2)*b*(m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*Simp(S(6)*a**S(2) + a*b*(m + S(1))*cos(e + f*x) - b**S(2)*(m + S(-2))*(m + S(-1)) - (S(3)*a**S(2) - b**S(2)*m*(m + S(-2)))*cos(e + f*x)**S(2), x)/cos(e + f*x)**S(3), x), x) + Simp((a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(S(3)*a*f*cos(e + f*x)**S(3)), x) + Simp((a + b*cos(e + f*x))**(m + S(1))*(S(3)*a**S(2) + b**S(2)*(m + S(-2)))*sin(e + f*x)/(S(3)*a**S(2)*b*f*(m + S(1))*cos(e + f*x)**S(2)), x) def replacement2373(a, b, e, f, m, x): return -Dist(S(1)/(S(6)*a**S(2)), Int((a + b*sin(e + f*x))**m*Simp(S(8)*a**S(2) + a*b*m*sin(e + f*x) - b**S(2)*(m + S(-2))*(m + S(-1)) - (S(6)*a**S(2) - b**S(2)*m*(m + S(-2)))*sin(e + f*x)**S(2), x)/sin(e + f*x)**S(2), x), x) - Simp((a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(S(3)*a*f*sin(e + f*x)**S(3)), x) - Simp(b*(a + b*sin(e + f*x))**(m + S(1))*(m + S(-2))*cos(e + f*x)/(S(6)*a**S(2)*f*sin(e + f*x)**S(2)), x) def replacement2374(a, b, e, f, m, x): return -Dist(S(1)/(S(6)*a**S(2)), Int((a + b*cos(e + f*x))**m*Simp(S(8)*a**S(2) + a*b*m*cos(e + f*x) - b**S(2)*(m + S(-2))*(m + S(-1)) - (S(6)*a**S(2) - b**S(2)*m*(m + S(-2)))*cos(e + f*x)**S(2), x)/cos(e + f*x)**S(2), x), x) + Simp((a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(S(3)*a*f*cos(e + f*x)**S(3)), x) + Simp(b*(a + b*cos(e + f*x))**(m + S(1))*(m + S(-2))*sin(e + f*x)/(S(6)*a**S(2)*f*cos(e + f*x)**S(2)), x) def replacement2375(a, b, e, f, m, x): return Dist(S(1)/(S(20)*a**S(2)*b**S(2)*m*(m + S(-1))), Int((a + b*sin(e + f*x))**m*Simp(S(60)*a**S(4) - S(44)*a**S(2)*b**S(2)*m*(m + S(-1)) + a*b*m*(S(20)*a**S(2) - b**S(2)*m*(m + S(-1)))*sin(e + f*x) + b**S(4)*m*(m + S(-4))*(m + S(-3))*(m + S(-1)) - (S(40)*a**S(4) - S(20)*a**S(2)*b**S(2)*(m + S(-1))*(S(2)*m + S(1)) + b**S(4)*m*(m + S(-4))*(m + S(-2))*(m + S(-1)))*sin(e + f*x)**S(2), x)/sin(e + f*x)**S(4), x), x) - Simp((a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(S(5)*a*f*sin(e + f*x)**S(5)), x) + Simp((a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(b*f*m*sin(e + f*x)**S(2)), x) - Simp(b*(a + b*sin(e + f*x))**(m + S(1))*(m + S(-4))*cos(e + f*x)/(S(20)*a**S(2)*f*sin(e + f*x)**S(4)), x) + Simp(a*(a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(b**S(2)*f*m*(m + S(-1))*sin(e + f*x)**S(3)), x) def replacement2376(a, b, e, f, m, x): return Dist(S(1)/(S(20)*a**S(2)*b**S(2)*m*(m + S(-1))), Int((a + b*cos(e + f*x))**m*Simp(S(60)*a**S(4) - S(44)*a**S(2)*b**S(2)*m*(m + S(-1)) + a*b*m*(S(20)*a**S(2) - b**S(2)*m*(m + S(-1)))*cos(e + f*x) + b**S(4)*m*(m + S(-4))*(m + S(-3))*(m + S(-1)) - (S(40)*a**S(4) - S(20)*a**S(2)*b**S(2)*(m + S(-1))*(S(2)*m + S(1)) + b**S(4)*m*(m + S(-4))*(m + S(-2))*(m + S(-1)))*cos(e + f*x)**S(2), x)/cos(e + f*x)**S(4), x), x) + Simp((a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(S(5)*a*f*cos(e + f*x)**S(5)), x) - Simp((a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(b*f*m*cos(e + f*x)**S(2)), x) + Simp(b*(a + b*cos(e + f*x))**(m + S(1))*(m + S(-4))*sin(e + f*x)/(S(20)*a**S(2)*f*cos(e + f*x)**S(4)), x) - Simp(a*(a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(b**S(2)*f*m*(m + S(-1))*cos(e + f*x)**S(3)), x) def replacement2377(a, b, e, f, g, p, x): return Dist(a/(a**S(2) - b**S(2)), Int((g*tan(e + f*x))**p/sin(e + f*x)**S(2), x), x) - Dist(a**S(2)*g**S(2)/(a**S(2) - b**S(2)), Int((g*tan(e + f*x))**(p + S(-2))/(a + b*sin(e + f*x)), x), x) - Dist(b*g/(a**S(2) - b**S(2)), Int((g*tan(e + f*x))**(p + S(-1))/cos(e + f*x), x), x) def replacement2378(a, b, e, f, g, p, x): return Dist(a/(a**S(2) - b**S(2)), Int((g/tan(e + f*x))**p/cos(e + f*x)**S(2), x), x) - Dist(a**S(2)*g**S(2)/(a**S(2) - b**S(2)), Int((g/tan(e + f*x))**(p + S(-2))/(a + b*cos(e + f*x)), x), x) - Dist(b*g/(a**S(2) - b**S(2)), Int((g/tan(e + f*x))**(p + S(-1))/sin(e + f*x), x), x) def replacement2379(a, b, e, f, g, p, x): return Dist(S(1)/a, Int((g*tan(e + f*x))**p/cos(e + f*x)**S(2), x), x) - Dist(b/(a**S(2)*g), Int((g*tan(e + f*x))**(p + S(1))/cos(e + f*x), x), x) - Dist((a**S(2) - b**S(2))/(a**S(2)*g**S(2)), Int((g*tan(e + f*x))**(p + S(2))/(a + b*sin(e + f*x)), x), x) def replacement2380(a, b, e, f, g, p, x): return Dist(S(1)/a, Int((g/tan(e + f*x))**p/sin(e + f*x)**S(2), x), x) - Dist(b/(a**S(2)*g), Int((g/tan(e + f*x))**(p + S(1))/sin(e + f*x), x), x) - Dist((a**S(2) - b**S(2))/(a**S(2)*g**S(2)), Int((g/tan(e + f*x))**(p + S(2))/(a + b*cos(e + f*x)), x), x) def replacement2381(a, b, e, f, g, x): return Dist(sqrt(g*tan(e + f*x))*sqrt(cos(e + f*x))/sqrt(sin(e + f*x)), Int(sqrt(sin(e + f*x))/((a + b*sin(e + f*x))*sqrt(cos(e + f*x))), x), x) def replacement2382(a, b, e, f, g, x): return Dist(sqrt(g/tan(e + f*x))*sqrt(sin(e + f*x))/sqrt(cos(e + f*x)), Int(sqrt(cos(e + f*x))/((a + b*cos(e + f*x))*sqrt(sin(e + f*x))), x), x) def replacement2383(a, b, e, f, g, x): return Dist(sqrt(sin(e + f*x))/(sqrt(g*tan(e + f*x))*sqrt(cos(e + f*x))), Int(sqrt(cos(e + f*x))/((a + b*sin(e + f*x))*sqrt(sin(e + f*x))), x), x) def replacement2384(a, b, e, f, g, x): return Dist(sqrt(cos(e + f*x))/(sqrt(g/tan(e + f*x))*sqrt(sin(e + f*x))), Int(sqrt(sin(e + f*x))/((a + b*cos(e + f*x))*sqrt(cos(e + f*x))), x), x) def replacement2385(a, b, e, f, m, p, x): return Int(ExpandIntegrand((S(1) - sin(e + f*x)**S(2))**(-p/S(2))*(a + b*sin(e + f*x))**m*sin(e + f*x)**p, x), x) def replacement2386(a, b, e, f, m, p, x): return Int(ExpandIntegrand((S(1) - cos(e + f*x)**S(2))**(-p/S(2))*(a + b*cos(e + f*x))**m*cos(e + f*x)**p, x), x) def replacement2387(a, b, e, f, g, m, p, x): return Int((g*tan(e + f*x))**p*(a + b*sin(e + f*x))**m, x) def replacement2388(a, b, e, f, g, m, p, x): return Int((g/tan(e + f*x))**p*(a + b*cos(e + f*x))**m, x) def replacement2389(a, b, e, f, g, m, p, x): return Dist(g**(S(2)*IntPart(p))*(g/tan(e + f*x))**FracPart(p)*(g*tan(e + f*x))**FracPart(p), Int((g*tan(e + f*x))**(-p)*(a + b*sin(e + f*x))**m, x), x) def replacement2390(a, b, e, f, g, m, p, x): return Dist(g**(S(2)*IntPart(p))*(g/tan(e + f*x))**FracPart(p)*(g*tan(e + f*x))**FracPart(p), Int((g/tan(e + f*x))**(-p)*(a + b*cos(e + f*x))**m, x), x) def replacement2391(a, b, c, d, e, f, x): return Simp(x*(S(2)*a*c + b*d)/S(2), x) - Simp((a*d + b*c)*cos(e + f*x)/f, x) - Simp(b*d*sin(e + f*x)*cos(e + f*x)/(S(2)*f), x) def replacement2392(a, b, c, d, e, f, x): return Simp(x*(S(2)*a*c + b*d)/S(2), x) + Simp((a*d + b*c)*sin(e + f*x)/f, x) + Simp(b*d*sin(e + f*x)*cos(e + f*x)/(S(2)*f), x) def replacement2393(a, b, c, d, e, f, x): return -Dist((-a*d + b*c)/d, Int(S(1)/(c + d*sin(e + f*x)), x), x) + Simp(b*x/d, x) def replacement2394(a, b, c, d, e, f, x): return -Dist((-a*d + b*c)/d, Int(S(1)/(c + d*cos(e + f*x)), x), x) + Simp(b*x/d, x) def replacement2395(a, b, c, d, e, f, m, n, x): return Dist(a**m*c**m, Int((c + d*sin(e + f*x))**(-m + n)*cos(e + f*x)**(S(2)*m), x), x) def replacement2396(a, b, c, d, e, f, m, n, x): return Dist(a**m*c**m, Int((c + d*cos(e + f*x))**(-m + n)*sin(e + f*x)**(S(2)*m), x), x) def replacement2397(a, b, c, d, e, f, x): return Dist(a*c*cos(e + f*x)/(sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))), Int(cos(e + f*x)/(c + d*sin(e + f*x)), x), x) def replacement2398(a, b, c, d, e, f, x): return Dist(a*c*sin(e + f*x)/(sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x))), Int(sin(e + f*x)/(c + d*cos(e + f*x)), x), x) def replacement2399(a, b, c, d, e, f, n, x): return Simp(-S(2)*b*(c + d*sin(e + f*x))**n*cos(e + f*x)/(f*sqrt(a + b*sin(e + f*x))*(S(2)*n + S(1))), x) def replacement2400(a, b, c, d, e, f, n, x): return Simp(S(2)*b*(c + d*cos(e + f*x))**n*sin(e + f*x)/(f*sqrt(a + b*cos(e + f*x))*(S(2)*n + S(1))), x) def replacement2401(a, b, c, d, e, f, m, n, x): return -Dist(b*(S(2)*m + S(-1))/(d*(S(2)*n + S(1))), Int((a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**(n + S(1)), x), x) + Simp(-S(2)*b*(a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**n*cos(e + f*x)/(f*(S(2)*n + S(1))), x) def replacement2402(a, b, c, d, e, f, m, n, x): return -Dist(b*(S(2)*m + S(-1))/(d*(S(2)*n + S(1))), Int((a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**(n + S(1)), x), x) + Simp(S(2)*b*(a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**n*sin(e + f*x)/(f*(S(2)*n + S(1))), x) def replacement2403(a, b, c, d, e, f, m, n, x): return Dist(a*(S(2)*m + S(-1))/(m + n), Int((a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**n, x), x) - Simp(b*(a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**n*cos(e + f*x)/(f*(m + n)), x) def replacement2404(a, b, c, d, e, f, m, n, x): return Dist(a*(S(2)*m + S(-1))/(m + n), Int((a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**n, x), x) + Simp(b*(a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**n*sin(e + f*x)/(f*(m + n)), x) def replacement2405(a, b, c, d, e, f, x): return Dist(cos(e + f*x)/(sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))), Int(S(1)/cos(e + f*x), x), x) def replacement2406(a, b, c, d, e, f, x): return Dist(sin(e + f*x)/(sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x))), Int(S(1)/sin(e + f*x), x), x) def replacement2407(a, b, c, d, e, f, m, n, x): return Simp(b*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n*cos(e + f*x)/(a*f*(S(2)*m + S(1))), x) def replacement2408(a, b, c, d, e, f, m, n, x): return -Simp(b*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n*sin(e + f*x)/(a*f*(S(2)*m + S(1))), x) def replacement2409(a, b, c, d, e, f, m, n, x): return Dist((m + n + S(1))/(a*(S(2)*m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n, x), x) + Simp(b*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n*cos(e + f*x)/(a*f*(S(2)*m + S(1))), x) def replacement2410(a, b, c, d, e, f, m, n, x): return Dist((m + n + S(1))/(a*(S(2)*m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n, x), x) - Simp(b*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n*sin(e + f*x)/(a*f*(S(2)*m + S(1))), x) def replacement2411(a, b, c, d, e, f, m, n, x): return Dist((m + n + S(1))/(a*(S(2)*m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n, x), x) + Simp(b*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n*cos(e + f*x)/(a*f*(S(2)*m + S(1))), x) def replacement2412(a, b, c, d, e, f, m, n, x): return Dist((m + n + S(1))/(a*(S(2)*m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n, x), x) - Simp(b*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n*sin(e + f*x)/(a*f*(S(2)*m + S(1))), x) def replacement2413(a, b, c, d, e, f, m, n, x): return Dist(a**IntPart(m)*c**IntPart(m)*(a + b*sin(e + f*x))**FracPart(m)*(c + d*sin(e + f*x))**FracPart(m)*cos(e + f*x)**(-S(2)*FracPart(m)), Int((c + d*sin(e + f*x))**(-m + n)*cos(e + f*x)**(S(2)*m), x), x) def replacement2414(a, b, c, d, e, f, m, n, x): return Dist(a**IntPart(m)*c**IntPart(m)*(a + b*cos(e + f*x))**FracPart(m)*(c + d*cos(e + f*x))**FracPart(m)*sin(e + f*x)**(-S(2)*FracPart(m)), Int((c + d*cos(e + f*x))**(-m + n)*sin(e + f*x)**(S(2)*m), x), x) def replacement2415(a, b, c, d, e, f, x): return Dist(S(1)/d, Int(Simp(a**S(2)*d - b*(-S(2)*a*d + b*c)*sin(e + f*x), x)/(c + d*sin(e + f*x)), x), x) - Simp(b**S(2)*cos(e + f*x)/(d*f), x) def replacement2416(a, b, c, d, e, f, x): return Dist(S(1)/d, Int(Simp(a**S(2)*d - b*(-S(2)*a*d + b*c)*cos(e + f*x), x)/(c + d*cos(e + f*x)), x), x) + Simp(b**S(2)*sin(e + f*x)/(d*f), x) def replacement2417(a, b, c, d, e, f, x): return Dist(b/(-a*d + b*c), Int(S(1)/(a + b*sin(e + f*x)), x), x) - Dist(d/(-a*d + b*c), Int(S(1)/(c + d*sin(e + f*x)), x), x) def replacement2418(a, b, c, d, e, f, x): return Dist(b/(-a*d + b*c), Int(S(1)/(a + b*cos(e + f*x)), x), x) - Dist(d/(-a*d + b*c), Int(S(1)/(c + d*cos(e + f*x)), x), x) def replacement2419(b, c, d, e, f, m, x): return Dist(c, Int((b*sin(e + f*x))**m, x), x) + Dist(d/b, Int((b*sin(e + f*x))**(m + S(1)), x), x) def replacement2420(b, c, d, e, f, m, x): return Dist(c, Int((b*cos(e + f*x))**m, x), x) + Dist(d/b, Int((b*cos(e + f*x))**(m + S(1)), x), x) def replacement2421(a, b, c, d, e, f, m, x): return -Simp(d*(a + b*sin(e + f*x))**m*cos(e + f*x)/(f*(m + S(1))), x) def replacement2422(a, b, c, d, e, f, m, x): return Simp(d*(a + b*cos(e + f*x))**m*sin(e + f*x)/(f*(m + S(1))), x) def replacement2423(a, b, c, d, e, f, m, x): return Dist((a*d*m + b*c*(m + S(1)))/(a*b*(S(2)*m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1)), x), x) + Simp((a + b*sin(e + f*x))**m*(-a*d + b*c)*cos(e + f*x)/(a*f*(S(2)*m + S(1))), x) def replacement2424(a, b, c, d, e, f, m, x): return Dist((a*d*m + b*c*(m + S(1)))/(a*b*(S(2)*m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1)), x), x) - Simp((a + b*cos(e + f*x))**m*(-a*d + b*c)*sin(e + f*x)/(a*f*(S(2)*m + S(1))), x) def replacement2425(a, b, c, d, e, f, m, x): return Dist((a*d*m + b*c*(m + S(1)))/(b*(m + S(1))), Int((a + b*sin(e + f*x))**m, x), x) - Simp(d*(a + b*sin(e + f*x))**m*cos(e + f*x)/(f*(m + S(1))), x) def replacement2426(a, b, c, d, e, f, m, x): return Dist((a*d*m + b*c*(m + S(1)))/(b*(m + S(1))), Int((a + b*cos(e + f*x))**m, x), x) + Simp(d*(a + b*cos(e + f*x))**m*sin(e + f*x)/(f*(m + S(1))), x) def replacement2427(a, b, c, d, e, f, x): return Dist(d/b, Int(sqrt(a + b*sin(e + f*x)), x), x) + Dist((-a*d + b*c)/b, Int(S(1)/sqrt(a + b*sin(e + f*x)), x), x) def replacement2428(a, b, c, d, e, f, x): return Dist(d/b, Int(sqrt(a + b*cos(e + f*x)), x), x) + Dist((-a*d + b*c)/b, Int(S(1)/sqrt(a + b*cos(e + f*x)), x), x) def replacement2429(a, b, c, d, e, f, m, x): return Dist(S(1)/(m + S(1)), Int((a + b*sin(e + f*x))**(m + S(-1))*Simp(a*c*(m + S(1)) + b*d*m + (a*d*m + b*c*(m + S(1)))*sin(e + f*x), x), x), x) - Simp(d*(a + b*sin(e + f*x))**m*cos(e + f*x)/(f*(m + S(1))), x) def replacement2430(a, b, c, d, e, f, m, x): return Dist(S(1)/(m + S(1)), Int((a + b*cos(e + f*x))**(m + S(-1))*Simp(a*c*(m + S(1)) + b*d*m + (a*d*m + b*c*(m + S(1)))*cos(e + f*x), x), x), x) + Simp(d*(a + b*cos(e + f*x))**m*sin(e + f*x)/(f*(m + S(1))), x) def replacement2431(a, b, c, d, e, f, m, x): return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*Simp((m + S(1))*(a*c - b*d) - (m + S(2))*(-a*d + b*c)*sin(e + f*x), x), x), x) - Simp((a + b*sin(e + f*x))**(m + S(1))*(-a*d + b*c)*cos(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))), x) def replacement2432(a, b, c, d, e, f, m, x): return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*Simp((m + S(1))*(a*c - b*d) - (m + S(2))*(-a*d + b*c)*cos(e + f*x), x), x), x) + Simp((a + b*cos(e + f*x))**(m + S(1))*(-a*d + b*c)*sin(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))), x) def replacement2433(a, b, c, d, e, f, m, x): return Dist(c*cos(e + f*x)/(f*sqrt(S(1) - sin(e + f*x))*sqrt(sin(e + f*x) + S(1))), Subst(Int(sqrt(S(1) + d*x/c)*(a + b*x)**m/sqrt(S(1) - d*x/c), x), x, sin(e + f*x)), x) def replacement2434(a, b, c, d, e, f, m, x): return -Dist(c*sin(e + f*x)/(f*sqrt(S(1) - cos(e + f*x))*sqrt(cos(e + f*x) + S(1))), Subst(Int(sqrt(S(1) + d*x/c)*(a + b*x)**m/sqrt(S(1) - d*x/c), x), x, cos(e + f*x)), x) def replacement2435(a, b, c, d, e, f, m, x): return Dist(d/b, Int((a + b*sin(e + f*x))**(m + S(1)), x), x) + Dist((-a*d + b*c)/b, Int((a + b*sin(e + f*x))**m, x), x) def replacement2436(a, b, c, d, e, f, m, x): return Dist(d/b, Int((a + b*cos(e + f*x))**(m + S(1)), x), x) + Dist((-a*d + b*c)/b, Int((a + b*cos(e + f*x))**m, x), x) def replacement2437(a, b, d, e, f, m, n, x): return Int(ExpandTrig((d*sin(e + f*x))**n*(a + b*sin(e + f*x))**m, x), x) def replacement2438(a, b, d, e, f, m, n, x): return Int(ExpandTrig((d*cos(e + f*x))**n*(a + b*cos(e + f*x))**m, x), x) def replacement2439(a, b, e, f, m, x): return -Dist(S(1)/(a**S(2)*(S(2)*m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*(a*m - b*(S(2)*m + S(1))*sin(e + f*x)), x), x) + Simp(b*(a + b*sin(e + f*x))**m*cos(e + f*x)/(a*f*(S(2)*m + S(1))), x) def replacement2440(a, b, e, f, m, x): return -Dist(S(1)/(a**S(2)*(S(2)*m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*(a*m - b*(S(2)*m + S(1))*cos(e + f*x)), x), x) - Simp(b*(a + b*cos(e + f*x))**m*sin(e + f*x)/(a*f*(S(2)*m + S(1))), x) def replacement2441(a, b, e, f, m, x): return Dist(S(1)/(b*(m + S(2))), Int((a + b*sin(e + f*x))**m*(-a*sin(e + f*x) + b*(m + S(1))), x), x) - Simp((a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(b*f*(m + S(2))), x) def replacement2442(a, b, e, f, m, x): return Dist(S(1)/(b*(m + S(2))), Int((a + b*cos(e + f*x))**m*(-a*cos(e + f*x) + b*(m + S(1))), x), x) + Simp((a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(b*f*(m + S(2))), x) def replacement2443(a, b, c, d, e, f, m, x): return Dist(S(1)/(a*b*(S(2)*m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*Simp(a*c*d*(m + S(-1)) + b*(c**S(2)*(m + S(1)) + d**S(2)) + d*(a*d*(m + S(-1)) + b*c*(m + S(2)))*sin(e + f*x), x), x), x) + Simp((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))*(-a*d + b*c)*cos(e + f*x)/(a*f*(S(2)*m + S(1))), x) def replacement2444(a, b, c, d, e, f, m, x): return Dist(S(1)/(a*b*(S(2)*m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*Simp(a*c*d*(m + S(-1)) + b*(c**S(2)*(m + S(1)) + d**S(2)) + d*(a*d*(m + S(-1)) + b*c*(m + S(2)))*cos(e + f*x), x), x), x) - Simp((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))*(-a*d + b*c)*sin(e + f*x)/(a*f*(S(2)*m + S(1))), x) def replacement2445(a, b, c, d, e, f, m, x): return Dist(S(1)/(b*(m + S(2))), Int((a + b*sin(e + f*x))**m*Simp(b*(c**S(2)*(m + S(2)) + d**S(2)*(m + S(1))) - d*(a*d - S(2)*b*c*(m + S(2)))*sin(e + f*x), x), x), x) - Simp(d**S(2)*(a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(b*f*(m + S(2))), x) def replacement2446(a, b, c, d, e, f, m, x): return Dist(S(1)/(b*(m + S(2))), Int((a + b*cos(e + f*x))**m*Simp(b*(c**S(2)*(m + S(2)) + d**S(2)*(m + S(1))) - d*(a*d - S(2)*b*c*(m + S(2)))*cos(e + f*x), x), x), x) + Simp(d**S(2)*(a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(b*f*(m + S(2))), x) def replacement2447(a, b, c, d, e, f, m, n, x): return Dist(b**S(2)/(d*(n + S(1))*(a*d + b*c)), Int((a + b*sin(e + f*x))**(m + S(-2))*(c + d*sin(e + f*x))**(n + S(1))*Simp(a*c*(m + S(-2)) - b*d*(m - S(2)*n + S(-4)) - (-a*d*(m + S(2)*n + S(1)) + b*c*(m + S(-1)))*sin(e + f*x), x), x), x) - Simp(b**S(2)*(a + b*sin(e + f*x))**(m + S(-2))*(c + d*sin(e + f*x))**(n + S(1))*(-a*d + b*c)*cos(e + f*x)/(d*f*(n + S(1))*(a*d + b*c)), x) def replacement2448(a, b, c, d, e, f, m, n, x): return Dist(b**S(2)/(d*(n + S(1))*(a*d + b*c)), Int((a + b*cos(e + f*x))**(m + S(-2))*(c + d*cos(e + f*x))**(n + S(1))*Simp(a*c*(m + S(-2)) - b*d*(m - S(2)*n + S(-4)) - (-a*d*(m + S(2)*n + S(1)) + b*c*(m + S(-1)))*cos(e + f*x), x), x), x) + Simp(b**S(2)*(a + b*cos(e + f*x))**(m + S(-2))*(c + d*cos(e + f*x))**(n + S(1))*(-a*d + b*c)*sin(e + f*x)/(d*f*(n + S(1))*(a*d + b*c)), x) def replacement2449(a, b, c, d, e, f, m, n, x): return Dist(S(1)/(d*(m + n)), Int((a + b*sin(e + f*x))**(m + S(-2))*(c + d*sin(e + f*x))**n*Simp(a**S(2)*d*(m + n) + a*b*c*(m + S(-2)) + b**S(2)*d*(n + S(1)) - b*(-a*d*(S(3)*m + S(2)*n + S(-2)) + b*c*(m + S(-1)))*sin(e + f*x), x), x), x) - Simp(b**S(2)*(a + b*sin(e + f*x))**(m + S(-2))*(c + d*sin(e + f*x))**(n + S(1))*cos(e + f*x)/(d*f*(m + n)), x) def replacement2450(a, b, c, d, e, f, m, n, x): return Dist(S(1)/(d*(m + n)), Int((a + b*cos(e + f*x))**(m + S(-2))*(c + d*cos(e + f*x))**n*Simp(a**S(2)*d*(m + n) + a*b*c*(m + S(-2)) + b**S(2)*d*(n + S(1)) - b*(-a*d*(S(3)*m + S(2)*n + S(-2)) + b*c*(m + S(-1)))*cos(e + f*x), x), x), x) + Simp(b**S(2)*(a + b*cos(e + f*x))**(m + S(-2))*(c + d*cos(e + f*x))**(n + S(1))*sin(e + f*x)/(d*f*(m + n)), x) def replacement2451(a, b, c, d, e, f, m, n, x): return -Dist(S(1)/(a*b*(S(2)*m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**(n + S(-1))*Simp(a*d*n - b*c*(m + S(1)) - b*d*(m + n + S(1))*sin(e + f*x), x), x), x) + Simp(b*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n*cos(e + f*x)/(a*f*(S(2)*m + S(1))), x) def replacement2452(a, b, c, d, e, f, m, n, x): return -Dist(S(1)/(a*b*(S(2)*m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**(n + S(-1))*Simp(a*d*n - b*c*(m + S(1)) - b*d*(m + n + S(1))*cos(e + f*x), x), x), x) - Simp(b*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n*sin(e + f*x)/(a*f*(S(2)*m + S(1))), x) def replacement2453(a, b, c, d, e, f, m, n, x): return Dist(S(1)/(a*b*(S(2)*m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**(n + S(-2))*Simp(a*c*d*(m - n + S(1)) + b*(c**S(2)*(m + S(1)) + d**S(2)*(n + S(-1))) + d*(a*d*(m - n + S(1)) + b*c*(m + n))*sin(e + f*x), x), x), x) + Simp((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(-1))*(-a*d + b*c)*cos(e + f*x)/(a*f*(S(2)*m + S(1))), x) def replacement2454(a, b, c, d, e, f, m, n, x): return Dist(S(1)/(a*b*(S(2)*m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**(n + S(-2))*Simp(a*c*d*(m - n + S(1)) + b*(c**S(2)*(m + S(1)) + d**S(2)*(n + S(-1))) + d*(a*d*(m - n + S(1)) + b*c*(m + n))*cos(e + f*x), x), x), x) - Simp((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(-1))*(-a*d + b*c)*sin(e + f*x)/(a*f*(S(2)*m + S(1))), x) def replacement2455(a, b, c, d, e, f, m, n, x): return Dist(S(1)/(a*(S(2)*m + S(1))*(-a*d + b*c)), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n*Simp(-a*d*(S(2)*m + n + S(2)) + b*c*(m + S(1)) + b*d*(m + n + S(2))*sin(e + f*x), x), x), x) + Simp(b**S(2)*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*cos(e + f*x)/(a*f*(S(2)*m + S(1))*(-a*d + b*c)), x) def replacement2456(a, b, c, d, e, f, m, n, x): return Dist(S(1)/(a*(S(2)*m + S(1))*(-a*d + b*c)), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n*Simp(-a*d*(S(2)*m + n + S(2)) + b*c*(m + S(1)) + b*d*(m + n + S(2))*cos(e + f*x), x), x), x) - Simp(b**S(2)*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*sin(e + f*x)/(a*f*(S(2)*m + S(1))*(-a*d + b*c)), x) def replacement2457(a, b, c, d, e, f, n, x): return -Dist(d/(a*b), Int((c + d*sin(e + f*x))**(n + S(-2))*Simp(-a*c*n + b*d*(n + S(-1)) + (-a*d*n + b*c*(n + S(-1)))*sin(e + f*x), x), x), x) - Simp((c + d*sin(e + f*x))**(n + S(-1))*(-a*d + b*c)*cos(e + f*x)/(a*f*(a + b*sin(e + f*x))), x) def replacement2458(a, b, c, d, e, f, n, x): return -Dist(d/(a*b), Int((c + d*cos(e + f*x))**(n + S(-2))*Simp(-a*c*n + b*d*(n + S(-1)) + (-a*d*n + b*c*(n + S(-1)))*cos(e + f*x), x), x), x) + Simp((c + d*cos(e + f*x))**(n + S(-1))*(-a*d + b*c)*sin(e + f*x)/(a*f*(a + b*cos(e + f*x))), x) def replacement2459(a, b, c, d, e, f, n, x): return Dist(d/(a*(-a*d + b*c)), Int((c + d*sin(e + f*x))**n*(a*n - b*(n + S(1))*sin(e + f*x)), x), x) - Simp(b**S(2)*(c + d*sin(e + f*x))**(n + S(1))*cos(e + f*x)/(a*f*(a + b*sin(e + f*x))*(-a*d + b*c)), x) def replacement2460(a, b, c, d, e, f, n, x): return Dist(d/(a*(-a*d + b*c)), Int((c + d*cos(e + f*x))**n*(a*n - b*(n + S(1))*cos(e + f*x)), x), x) + Simp(b**S(2)*(c + d*cos(e + f*x))**(n + S(1))*sin(e + f*x)/(a*f*(a + b*cos(e + f*x))*(-a*d + b*c)), x) def replacement2461(a, b, c, d, e, f, n, x): return Dist(d*n/(a*b), Int((a - b*sin(e + f*x))*(c + d*sin(e + f*x))**(n + S(-1)), x), x) - Simp(b*(c + d*sin(e + f*x))**n*cos(e + f*x)/(a*f*(a + b*sin(e + f*x))), x) def replacement2462(a, b, c, d, e, f, n, x): return Dist(d*n/(a*b), Int((a - b*cos(e + f*x))*(c + d*cos(e + f*x))**(n + S(-1)), x), x) + Simp(b*(c + d*cos(e + f*x))**n*sin(e + f*x)/(a*f*(a + b*cos(e + f*x))), x) def replacement2463(a, b, c, d, e, f, n, x): return Dist(S(2)*n*(a*d + b*c)/(b*(S(2)*n + S(1))), Int(sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))**(n + S(-1)), x), x) + Simp(-S(2)*b*(c + d*sin(e + f*x))**n*cos(e + f*x)/(f*sqrt(a + b*sin(e + f*x))*(S(2)*n + S(1))), x) def replacement2464(a, b, c, d, e, f, n, x): return Dist(S(2)*n*(a*d + b*c)/(b*(S(2)*n + S(1))), Int(sqrt(a + b*cos(e + f*x))*(c + d*cos(e + f*x))**(n + S(-1)), x), x) + Simp(S(2)*b*(c + d*cos(e + f*x))**n*sin(e + f*x)/(f*sqrt(a + b*cos(e + f*x))*(S(2)*n + S(1))), x) def replacement2465(a, b, c, d, e, f, x): return Simp(-S(2)*b**S(2)*cos(e + f*x)/(f*sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))*(a*d + b*c)), x) def replacement2466(a, b, c, d, e, f, x): return Simp(S(2)*b**S(2)*sin(e + f*x)/(f*sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x))*(a*d + b*c)), x) def replacement2467(a, b, c, d, e, f, n, x): return Dist((S(2)*n + S(3))*(-a*d + b*c)/(S(2)*b*(c**S(2) - d**S(2))*(n + S(1))), Int(sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))**(n + S(1)), x), x) + Simp((c + d*sin(e + f*x))**(n + S(1))*(-a*d + b*c)*cos(e + f*x)/(f*sqrt(a + b*sin(e + f*x))*(c**S(2) - d**S(2))*(n + S(1))), x) def replacement2468(a, b, c, d, e, f, n, x): return Dist((S(2)*n + S(3))*(-a*d + b*c)/(S(2)*b*(c**S(2) - d**S(2))*(n + S(1))), Int(sqrt(a + b*cos(e + f*x))*(c + d*cos(e + f*x))**(n + S(1)), x), x) - Simp((c + d*cos(e + f*x))**(n + S(1))*(-a*d + b*c)*sin(e + f*x)/(f*sqrt(a + b*cos(e + f*x))*(c**S(2) - d**S(2))*(n + S(1))), x) def replacement2469(a, b, c, d, e, f, x): return Dist(-S(2)*b/f, Subst(Int(S(1)/(a*d + b*c - d*x**S(2)), x), x, b*cos(e + f*x)/sqrt(a + b*sin(e + f*x))), x) def replacement2470(a, b, c, d, e, f, x): return Dist(S(2)*b/f, Subst(Int(S(1)/(a*d + b*c - d*x**S(2)), x), x, b*sin(e + f*x)/sqrt(a + b*cos(e + f*x))), x) def replacement2471(a, b, d, e, f, x): return Dist(-S(2)/f, Subst(Int(S(1)/sqrt(S(1) - x**S(2)/a), x), x, b*cos(e + f*x)/sqrt(a + b*sin(e + f*x))), x) def replacement2472(a, b, d, e, f, x): return Dist(S(2)/f, Subst(Int(S(1)/sqrt(S(1) - x**S(2)/a), x), x, b*sin(e + f*x)/sqrt(a + b*cos(e + f*x))), x) def replacement2473(a, b, c, d, e, f, x): return Dist(-S(2)*b/f, Subst(Int(S(1)/(b + d*x**S(2)), x), x, b*cos(e + f*x)/(sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))), x) def replacement2474(a, b, c, d, e, f, x): return Dist(S(2)*b/f, Subst(Int(S(1)/(b + d*x**S(2)), x), x, b*sin(e + f*x)/(sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x)))), x) def replacement2475(a, b, c, d, e, f, n, x): return Dist(a**S(2)*cos(e + f*x)/(f*sqrt(a - b*sin(e + f*x))*sqrt(a + b*sin(e + f*x))), Subst(Int((c + d*x)**n/sqrt(a - b*x), x), x, sin(e + f*x)), x) def replacement2476(a, b, c, d, e, f, n, x): return -Dist(a**S(2)*sin(e + f*x)/(f*sqrt(a - b*cos(e + f*x))*sqrt(a + b*cos(e + f*x))), Subst(Int((c + d*x)**n/sqrt(a - b*x), x), x, cos(e + f*x)), x) def replacement2477(a, b, c, d, e, f, x): return Dist(d/b, Int(sqrt(a + b*sin(e + f*x))/sqrt(c + d*sin(e + f*x)), x), x) + Dist((-a*d + b*c)/b, Int(S(1)/(sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))), x), x) def replacement2478(a, b, c, d, e, f, x): return Dist(d/b, Int(sqrt(a + b*cos(e + f*x))/sqrt(c + d*cos(e + f*x)), x), x) + Dist((-a*d + b*c)/b, Int(S(1)/(sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x))), x), x) def replacement2479(a, b, c, d, e, f, n, x): return -Dist(S(1)/(b*(S(2)*n + S(-1))), Int((c + d*sin(e + f*x))**(n + S(-2))*Simp(a*c*d - b*(c**S(2)*(S(2)*n + S(-1)) + S(2)*d**S(2)*(n + S(-1))) + d*(a*d - b*c*(S(4)*n + S(-3)))*sin(e + f*x), x)/sqrt(a + b*sin(e + f*x)), x), x) + Simp(-S(2)*d*(c + d*sin(e + f*x))**(n + S(-1))*cos(e + f*x)/(f*sqrt(a + b*sin(e + f*x))*(S(2)*n + S(-1))), x) def replacement2480(a, b, c, d, e, f, n, x): return -Dist(S(1)/(b*(S(2)*n + S(-1))), Int((c + d*cos(e + f*x))**(n + S(-2))*Simp(a*c*d - b*(c**S(2)*(S(2)*n + S(-1)) + S(2)*d**S(2)*(n + S(-1))) + d*(a*d - b*c*(S(4)*n + S(-3)))*cos(e + f*x), x)/sqrt(a + b*cos(e + f*x)), x), x) + Simp(S(2)*d*(c + d*cos(e + f*x))**(n + S(-1))*sin(e + f*x)/(f*sqrt(a + b*cos(e + f*x))*(S(2)*n + S(-1))), x) def replacement2481(a, b, c, d, e, f, n, x): return -Dist(S(1)/(S(2)*b*(c**S(2) - d**S(2))*(n + S(1))), Int((c + d*sin(e + f*x))**(n + S(1))*Simp(a*d - S(2)*b*c*(n + S(1)) + b*d*(S(2)*n + S(3))*sin(e + f*x), x)/sqrt(a + b*sin(e + f*x)), x), x) - Simp(d*(c + d*sin(e + f*x))**(n + S(1))*cos(e + f*x)/(f*sqrt(a + b*sin(e + f*x))*(c**S(2) - d**S(2))*(n + S(1))), x) def replacement2482(a, b, c, d, e, f, n, x): return -Dist(S(1)/(S(2)*b*(c**S(2) - d**S(2))*(n + S(1))), Int((c + d*cos(e + f*x))**(n + S(1))*Simp(a*d - S(2)*b*c*(n + S(1)) + b*d*(S(2)*n + S(3))*cos(e + f*x), x)/sqrt(a + b*cos(e + f*x)), x), x) + Simp(d*(c + d*cos(e + f*x))**(n + S(1))*sin(e + f*x)/(f*sqrt(a + b*cos(e + f*x))*(c**S(2) - d**S(2))*(n + S(1))), x) def replacement2483(a, b, c, d, e, f, x): return Dist(b/(-a*d + b*c), Int(S(1)/sqrt(a + b*sin(e + f*x)), x), x) - Dist(d/(-a*d + b*c), Int(sqrt(a + b*sin(e + f*x))/(c + d*sin(e + f*x)), x), x) def replacement2484(a, b, c, d, e, f, x): return Dist(b/(-a*d + b*c), Int(S(1)/sqrt(a + b*cos(e + f*x)), x), x) - Dist(d/(-a*d + b*c), Int(sqrt(a + b*cos(e + f*x))/(c + d*cos(e + f*x)), x), x) def replacement2485(a, b, d, e, f, x): return -Dist(sqrt(S(2))/(sqrt(a)*f), Subst(Int(S(1)/sqrt(S(1) - x**S(2)), x), x, b*cos(e + f*x)/(a + b*sin(e + f*x))), x) def replacement2486(a, b, d, e, f, x): return Dist(sqrt(S(2))/(sqrt(a)*f), Subst(Int(S(1)/sqrt(S(1) - x**S(2)), x), x, b*sin(e + f*x)/(a + b*cos(e + f*x))), x) def replacement2487(a, b, c, d, e, f, x): return Dist(-S(2)*a/f, Subst(Int(S(1)/(S(2)*b**S(2) - x**S(2)*(a*c - b*d)), x), x, b*cos(e + f*x)/(sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))), x) def replacement2488(a, b, c, d, e, f, x): return Dist(S(2)*a/f, Subst(Int(S(1)/(S(2)*b**S(2) - x**S(2)*(a*c - b*d)), x), x, b*sin(e + f*x)/(sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x)))), x) def replacement2489(a, b, c, d, e, f, m, n, x): return Dist(S(1)/(b*(m + n)), Int((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(-2))*Simp(b*c**S(2)*(m + n) + d*(a*c*m + b*d*(n + S(-1))) + d*(a*d*m + b*c*(m + S(2)*n + S(-1)))*sin(e + f*x), x), x), x) - Simp(d*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(-1))*cos(e + f*x)/(f*(m + n)), x) def replacement2490(a, b, c, d, e, f, m, n, x): return Dist(S(1)/(b*(m + n)), Int((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(-2))*Simp(b*c**S(2)*(m + n) + d*(a*c*m + b*d*(n + S(-1))) + d*(a*d*m + b*c*(m + S(2)*n + S(-1)))*cos(e + f*x), x), x), x) + Simp(d*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(-1))*sin(e + f*x)/(f*(m + n)), x) def replacement2491(a, b, c, d, e, f, m, n, x): return Dist(a**m*cos(e + f*x)/(f*sqrt(S(1) - sin(e + f*x))*sqrt(sin(e + f*x) + S(1))), Subst(Int((S(1) + b*x/a)**(m + S(-1)/2)*(c + d*x)**n/sqrt(S(1) - b*x/a), x), x, sin(e + f*x)), x) def replacement2492(a, b, c, d, e, f, m, n, x): return -Dist(a**m*sin(e + f*x)/(f*sqrt(S(1) - cos(e + f*x))*sqrt(cos(e + f*x) + S(1))), Subst(Int((S(1) + b*x/a)**(m + S(-1)/2)*(c + d*x)**n/sqrt(S(1) - b*x/a), x), x, cos(e + f*x)), x) def replacement2493(a, b, d, e, f, m, n, x): return -Dist(b*(d/b)**n*cos(e + f*x)/(f*sqrt(a - b*sin(e + f*x))*sqrt(a + b*sin(e + f*x))), Subst(Int((a - x)**n*(S(2)*a - x)**(m + S(-1)/2)/sqrt(x), x), x, a - b*sin(e + f*x)), x) def replacement2494(a, b, d, e, f, m, n, x): return Dist(b*(d/b)**n*sin(e + f*x)/(f*sqrt(a - b*cos(e + f*x))*sqrt(a + b*cos(e + f*x))), Subst(Int((a - x)**n*(S(2)*a - x)**(m + S(-1)/2)/sqrt(x), x), x, a - b*cos(e + f*x)), x) def replacement2495(a, b, d, e, f, m, n, x): return Dist((d/b)**IntPart(n)*(b*sin(e + f*x))**(-FracPart(n))*(d*sin(e + f*x))**FracPart(n), Int((b*sin(e + f*x))**n*(a + b*sin(e + f*x))**m, x), x) def replacement2496(a, b, d, e, f, m, n, x): return Dist((d/b)**IntPart(n)*(b*cos(e + f*x))**(-FracPart(n))*(d*cos(e + f*x))**FracPart(n), Int((b*cos(e + f*x))**n*(a + b*cos(e + f*x))**m, x), x) def replacement2497(a, b, d, e, f, m, n, x): return Dist(a**IntPart(m)*(S(1) + b*sin(e + f*x)/a)**(-FracPart(m))*(a + b*sin(e + f*x))**FracPart(m), Int((d*sin(e + f*x))**n*(S(1) + b*sin(e + f*x)/a)**m, x), x) def replacement2498(a, b, d, e, f, m, n, x): return Dist(a**IntPart(m)*(S(1) + b*cos(e + f*x)/a)**(-FracPart(m))*(a + b*cos(e + f*x))**FracPart(m), Int((d*cos(e + f*x))**n*(S(1) + b*cos(e + f*x)/a)**m, x), x) def replacement2499(a, b, c, d, e, f, m, n, x): return Dist(a**S(2)*cos(e + f*x)/(f*sqrt(a - b*sin(e + f*x))*sqrt(a + b*sin(e + f*x))), Subst(Int((a + b*x)**(m + S(-1)/2)*(c + d*x)**n/sqrt(a - b*x), x), x, sin(e + f*x)), x) def replacement2500(a, b, c, d, e, f, m, n, x): return -Dist(a**S(2)*sin(e + f*x)/(f*sqrt(a - b*cos(e + f*x))*sqrt(a + b*cos(e + f*x))), Subst(Int((a + b*x)**(m + S(-1)/2)*(c + d*x)**n/sqrt(a - b*x), x), x, cos(e + f*x)), x) def replacement2501(b, c, d, e, f, m, x): return Dist(S(2)*c*d/b, Int((b*sin(e + f*x))**(m + S(1)), x), x) + Int((b*sin(e + f*x))**m*(c**S(2) + d**S(2)*sin(e + f*x)**S(2)), x) def replacement2502(b, c, d, e, f, m, x): return Dist(S(2)*c*d/b, Int((b*cos(e + f*x))**(m + S(1)), x), x) + Int((b*cos(e + f*x))**m*(c**S(2) + d**S(2)*cos(e + f*x)**S(2)), x) def replacement2503(a, b, c, d, e, f, m, x): return -Dist(S(1)/(b*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*Simp(b*(m + S(1))*(-a*(c**S(2) + d**S(2)) + S(2)*b*c*d) + (a**S(2)*d**S(2) - S(2)*a*b*c*d*(m + S(2)) + b**S(2)*(c**S(2)*(m + S(2)) + d**S(2)*(m + S(1))))*sin(e + f*x), x), x), x) - Simp((a + b*sin(e + f*x))**(m + S(1))*(a**S(2)*d**S(2) - S(2)*a*b*c*d + b**S(2)*c**S(2))*cos(e + f*x)/(b*f*(a**S(2) - b**S(2))*(m + S(1))), x) def replacement2504(a, b, c, d, e, f, m, x): return -Dist(S(1)/(b*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*Simp(b*(m + S(1))*(-a*(c**S(2) + d**S(2)) + S(2)*b*c*d) + (a**S(2)*d**S(2) - S(2)*a*b*c*d*(m + S(2)) + b**S(2)*(c**S(2)*(m + S(2)) + d**S(2)*(m + S(1))))*cos(e + f*x), x), x), x) + Simp((a + b*cos(e + f*x))**(m + S(1))*(a**S(2)*d**S(2) - S(2)*a*b*c*d + b**S(2)*c**S(2))*sin(e + f*x)/(b*f*(a**S(2) - b**S(2))*(m + S(1))), x) def replacement2505(a, b, c, d, e, f, m, x): return Dist(S(1)/(b*(m + S(2))), Int((a + b*sin(e + f*x))**m*Simp(b*(c**S(2)*(m + S(2)) + d**S(2)*(m + S(1))) - d*(a*d - S(2)*b*c*(m + S(2)))*sin(e + f*x), x), x), x) - Simp(d**S(2)*(a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(b*f*(m + S(2))), x) def replacement2506(a, b, c, d, e, f, m, x): return Dist(S(1)/(b*(m + S(2))), Int((a + b*cos(e + f*x))**m*Simp(b*(c**S(2)*(m + S(2)) + d**S(2)*(m + S(1))) - d*(a*d - S(2)*b*c*(m + S(2)))*cos(e + f*x), x), x), x) + Simp(d**S(2)*(a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(b*f*(m + S(2))), x) def replacement2507(a, b, c, d, e, f, m, n, x): return Dist(S(1)/(d*(c**S(2) - d**S(2))*(n + S(1))), Int((a + b*sin(e + f*x))**(m + S(-3))*(c + d*sin(e + f*x))**(n + S(1))*Simp(a*d*(n + S(1))*(-S(2)*a*b*d + c*(a**S(2) + b**S(2))) + b*(m + S(-2))*(-a*d + b*c)**S(2) + b*(b**S(2)*(c**S(2) - d**S(2)) + d*n*(S(2)*a*b*c - d*(a**S(2) + b**S(2))) - m*(-a*d + b*c)**S(2))*sin(e + f*x)**S(2) + (-a*(n + S(2))*(-a*d + b*c)**S(2) + b*(n + S(1))*(a*b*c**S(2) - S(3)*a*b*d**S(2) + c*d*(a**S(2) + b**S(2))))*sin(e + f*x), x), x), x) - Simp((a + b*sin(e + f*x))**(m + S(-2))*(c + d*sin(e + f*x))**(n + S(1))*(a**S(2)*d**S(2) - S(2)*a*b*c*d + b**S(2)*c**S(2))*cos(e + f*x)/(d*f*(c**S(2) - d**S(2))*(n + S(1))), x) def replacement2508(a, b, c, d, e, f, m, n, x): return Dist(S(1)/(d*(c**S(2) - d**S(2))*(n + S(1))), Int((a + b*cos(e + f*x))**(m + S(-3))*(c + d*cos(e + f*x))**(n + S(1))*Simp(a*d*(n + S(1))*(-S(2)*a*b*d + c*(a**S(2) + b**S(2))) + b*(m + S(-2))*(-a*d + b*c)**S(2) + b*(b**S(2)*(c**S(2) - d**S(2)) + d*n*(S(2)*a*b*c - d*(a**S(2) + b**S(2))) - m*(-a*d + b*c)**S(2))*cos(e + f*x)**S(2) + (-a*(n + S(2))*(-a*d + b*c)**S(2) + b*(n + S(1))*(a*b*c**S(2) - S(3)*a*b*d**S(2) + c*d*(a**S(2) + b**S(2))))*cos(e + f*x), x), x), x) + Simp((a + b*cos(e + f*x))**(m + S(-2))*(c + d*cos(e + f*x))**(n + S(1))*(a**S(2)*d**S(2) - S(2)*a*b*c*d + b**S(2)*c**S(2))*sin(e + f*x)/(d*f*(c**S(2) - d**S(2))*(n + S(1))), x) def replacement2509(a, b, c, d, e, f, m, n, x): return Dist(S(1)/(d*(m + n)), Int((a + b*sin(e + f*x))**(m + S(-3))*(c + d*sin(e + f*x))**n*Simp(a**S(3)*d*(m + n) + b**S(2)*(a*d*(n + S(1)) + b*c*(m + S(-2))) - b**S(2)*(-a*d*(S(3)*m + S(2)*n + S(-2)) + b*c*(m + S(-1)))*sin(e + f*x)**S(2) - b*(-S(3)*a**S(2)*d*(m + n) + a*b*c - b**S(2)*d*(m + n + S(-1)))*sin(e + f*x), x), x), x) - Simp(b**S(2)*(a + b*sin(e + f*x))**(m + S(-2))*(c + d*sin(e + f*x))**(n + S(1))*cos(e + f*x)/(d*f*(m + n)), x) def replacement2510(a, b, c, d, e, f, m, n, x): return Dist(S(1)/(d*(m + n)), Int((a + b*cos(e + f*x))**(m + S(-3))*(c + d*cos(e + f*x))**n*Simp(a**S(3)*d*(m + n) + b**S(2)*(a*d*(n + S(1)) + b*c*(m + S(-2))) - b**S(2)*(-a*d*(S(3)*m + S(2)*n + S(-2)) + b*c*(m + S(-1)))*cos(e + f*x)**S(2) - b*(-S(3)*a**S(2)*d*(m + n) + a*b*c - b**S(2)*d*(m + n + S(-1)))*cos(e + f*x), x), x), x) + Simp(b**S(2)*(a + b*cos(e + f*x))**(m + S(-2))*(c + d*cos(e + f*x))**(n + S(1))*sin(e + f*x)/(d*f*(m + n)), x) def replacement2511(a, b, d, e, f, x): return -Dist(d**S(2)/(a**S(2) - b**S(2)), Int(sqrt(a + b*sin(e + f*x))/(d*sin(e + f*x))**(S(3)/2), x), x) + Simp(-S(2)*a*d*cos(e + f*x)/(f*sqrt(d*sin(e + f*x))*sqrt(a + b*sin(e + f*x))*(a**S(2) - b**S(2))), x) def replacement2512(a, b, d, e, f, x): return -Dist(d**S(2)/(a**S(2) - b**S(2)), Int(sqrt(a + b*cos(e + f*x))/(d*cos(e + f*x))**(S(3)/2), x), x) + Simp(S(2)*a*d*sin(e + f*x)/(f*sqrt(d*cos(e + f*x))*sqrt(a + b*cos(e + f*x))*(a**S(2) - b**S(2))), x) def replacement2513(a, b, c, d, e, f, x): return Dist((c - d)/(a - b), Int(S(1)/(sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))), x), x) - Dist((-a*d + b*c)/(a - b), Int((sin(e + f*x) + S(1))/((a + b*sin(e + f*x))**(S(3)/2)*sqrt(c + d*sin(e + f*x))), x), x) def replacement2514(a, b, c, d, e, f, x): return Dist((c - d)/(a - b), Int(S(1)/(sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x))), x), x) - Dist((-a*d + b*c)/(a - b), Int((cos(e + f*x) + S(1))/((a + b*cos(e + f*x))**(S(3)/2)*sqrt(c + d*cos(e + f*x))), x), x) def replacement2515(a, b, c, d, e, f, m, n, x): return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**(n + S(-1))*Simp(a*c*(m + S(1)) + b*d*n - b*d*(m + n + S(2))*sin(e + f*x)**S(2) + (a*d*(m + S(1)) - b*c*(m + S(2)))*sin(e + f*x), x), x), x) - Simp(b*(a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n*cos(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))), x) def replacement2516(a, b, c, d, e, f, m, n, x): return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**(n + S(-1))*Simp(a*c*(m + S(1)) + b*d*n - b*d*(m + n + S(2))*cos(e + f*x)**S(2) + (a*d*(m + S(1)) - b*c*(m + S(2)))*cos(e + f*x), x), x), x) + Simp(b*(a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n*sin(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))), x) def replacement2517(a, b, d, e, f, x): return Dist(d/b, Int(sqrt(d*sin(e + f*x))/sqrt(a + b*sin(e + f*x)), x), x) - Dist(a*d/b, Int(sqrt(d*sin(e + f*x))/(a + b*sin(e + f*x))**(S(3)/2), x), x) def replacement2518(a, b, d, e, f, x): return Dist(d/b, Int(sqrt(d*cos(e + f*x))/sqrt(a + b*cos(e + f*x)), x), x) - Dist(a*d/b, Int(sqrt(d*cos(e + f*x))/(a + b*cos(e + f*x))**(S(3)/2), x), x) def replacement2519(a, b, c, d, e, f, x): return Dist(d**S(2)/b**S(2), Int(sqrt(a + b*sin(e + f*x))/sqrt(c + d*sin(e + f*x)), x), x) + Dist((-a*d + b*c)/b**S(2), Int(Simp(a*d + b*c + S(2)*b*d*sin(e + f*x), x)/((a + b*sin(e + f*x))**(S(3)/2)*sqrt(c + d*sin(e + f*x))), x), x) def replacement2520(a, b, c, d, e, f, x): return Dist(d**S(2)/b**S(2), Int(sqrt(a + b*cos(e + f*x))/sqrt(c + d*cos(e + f*x)), x), x) + Dist((-a*d + b*c)/b**S(2), Int(Simp(a*d + b*c + S(2)*b*d*cos(e + f*x), x)/((a + b*cos(e + f*x))**(S(3)/2)*sqrt(c + d*cos(e + f*x))), x), x) def replacement2521(a, b, c, d, e, f, m, n, x): return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**(n + S(-2))*Simp(c*(m + S(1))*(a*c - b*d) + d*(n + S(-1))*(-a*d + b*c) - d*(-a*d + b*c)*(m + n + S(1))*sin(e + f*x)**S(2) + (-c*(m + S(2))*(-a*d + b*c) + d*(m + S(1))*(a*c - b*d))*sin(e + f*x), x), x), x) - Simp((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**(n + S(-1))*(-a*d + b*c)*cos(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))), x) def replacement2522(a, b, c, d, e, f, m, n, x): return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**(n + S(-2))*Simp(c*(m + S(1))*(a*c - b*d) + d*(n + S(-1))*(-a*d + b*c) - d*(-a*d + b*c)*(m + n + S(1))*cos(e + f*x)**S(2) + (-c*(m + S(2))*(-a*d + b*c) + d*(m + S(1))*(a*c - b*d))*cos(e + f*x), x), x), x) + Simp((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**(n + S(-1))*(-a*d + b*c)*sin(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))), x) def replacement2523(a, b, d, e, f, x): return Dist(d/(a**S(2) - b**S(2)), Int((a*sin(e + f*x) + b)/((d*sin(e + f*x))**(S(3)/2)*sqrt(a + b*sin(e + f*x))), x), x) + Simp(S(2)*b*cos(e + f*x)/(f*sqrt(d*sin(e + f*x))*sqrt(a + b*sin(e + f*x))*(a**S(2) - b**S(2))), x) def replacement2524(a, b, d, e, f, x): return Dist(d/(a**S(2) - b**S(2)), Int((a*cos(e + f*x) + b)/((d*cos(e + f*x))**(S(3)/2)*sqrt(a + b*cos(e + f*x))), x), x) + Simp(-S(2)*b*sin(e + f*x)/(f*sqrt(d*cos(e + f*x))*sqrt(a + b*cos(e + f*x))*(a**S(2) - b**S(2))), x) def replacement2525(a, b, c, d, e, f, x): return -Dist(b/(a - b), Int((sin(e + f*x) + S(1))/((a + b*sin(e + f*x))**(S(3)/2)*sqrt(c + d*sin(e + f*x))), x), x) + Dist(S(1)/(a - b), Int(S(1)/(sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))), x), x) def replacement2526(a, b, c, d, e, f, x): return -Dist(b/(a - b), Int((cos(e + f*x) + S(1))/((a + b*cos(e + f*x))**(S(3)/2)*sqrt(c + d*cos(e + f*x))), x), x) + Dist(S(1)/(a - b), Int(S(1)/(sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x))), x), x) def replacement2527(a, b, c, d, e, f, m, n, x): return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))*(-a*d + b*c)), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n*Simp(a*(m + S(1))*(-a*d + b*c) + b**S(2)*d*(m + n + S(2)) - b**S(2)*d*(m + n + S(3))*sin(e + f*x)**S(2) - (b**S(2)*c + b*(m + S(1))*(-a*d + b*c))*sin(e + f*x), x), x), x) - Simp(b**S(2)*(a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**(n + S(1))*cos(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))*(-a*d + b*c)), x) def replacement2528(a, b, c, d, e, f, m, n, x): return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))*(-a*d + b*c)), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n*Simp(a*(m + S(1))*(-a*d + b*c) + b**S(2)*d*(m + n + S(2)) - b**S(2)*d*(m + n + S(3))*cos(e + f*x)**S(2) - (b**S(2)*c + b*(m + S(1))*(-a*d + b*c))*cos(e + f*x), x), x), x) + Simp(b**S(2)*(a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**(n + S(1))*sin(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))*(-a*d + b*c)), x) def replacement2529(a, b, c, d, e, f, x): return Dist(d/b, Int(S(1)/sqrt(c + d*sin(e + f*x)), x), x) + Dist((-a*d + b*c)/b, Int(S(1)/((a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))), x), x) def replacement2530(a, b, c, d, e, f, x): return Dist(d/b, Int(S(1)/sqrt(c + d*cos(e + f*x)), x), x) + Dist((-a*d + b*c)/b, Int(S(1)/((a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x))), x), x) def replacement2531(a, b, c, d, e, f, x): return Dist(b/d, Int(sqrt(a + b*sin(e + f*x)), x), x) - Dist((-a*d + b*c)/d, Int(sqrt(a + b*sin(e + f*x))/(c + d*sin(e + f*x)), x), x) def replacement2532(a, b, c, d, e, f, x): return Dist(b/d, Int(sqrt(a + b*cos(e + f*x)), x), x) - Dist((-a*d + b*c)/d, Int(sqrt(a + b*cos(e + f*x))/(c + d*cos(e + f*x)), x), x) def replacement2533(a, b, c, d, e, f, x): return Simp(S(2)*EllipticPi(S(2)*b/(a + b), -Pi/S(4) + e/S(2) + f*x/S(2), S(2)*d/(c + d))/(f*(a + b)*sqrt(c + d)), x) def replacement2534(a, b, c, d, e, f, x): return Simp(S(2)*EllipticPi(S(2)*b/(a + b), e/S(2) + f*x/S(2), S(2)*d/(c + d))/(f*(a + b)*sqrt(c + d)), x) def replacement2535(a, b, c, d, e, f, x): return Simp(S(2)*EllipticPi(-S(2)*b/(a - b), Pi/S(4) + e/S(2) + f*x/S(2), -S(2)*d/(c - d))/(f*(a - b)*sqrt(c - d)), x) def replacement2536(a, b, c, d, e, f, x): return Simp(S(2)*EllipticPi(-S(2)*b/(a - b), Pi/S(2) + e/S(2) + f*x/S(2), -S(2)*d/(c - d))/(f*(a - b)*sqrt(c - d)), x) def replacement2537(a, b, c, d, e, f, x): return Dist(sqrt((c + d*sin(e + f*x))/(c + d))/sqrt(c + d*sin(e + f*x)), Int(S(1)/((a + b*sin(e + f*x))*sqrt(c/(c + d) + d*sin(e + f*x)/(c + d))), x), x) def replacement2538(a, b, c, d, e, f, x): return Dist(sqrt((c + d*cos(e + f*x))/(c + d))/sqrt(c + d*cos(e + f*x)), Int(S(1)/((a + b*cos(e + f*x))*sqrt(c/(c + d) + d*cos(e + f*x)/(c + d))), x), x) def replacement2539(b, c, d, e, f, x): return Simp(S(2)*c*sqrt(S(1) - S(1)/sin(e + f*x))*sqrt(S(1) + S(1)/sin(e + f*x))*EllipticPi((c + d)/d, asin(sqrt(c + d*sin(e + f*x))/(sqrt(b*sin(e + f*x))*Rt((c + d)/b, S(2)))), -(c + d)/(c - d))*Rt(b*(c + d), S(2))*tan(e + f*x)/(d*f*sqrt(c**S(2) - d**S(2))), x) def replacement2540(b, c, d, e, f, x): return Simp(-S(2)*c*sqrt(S(1) - S(1)/cos(e + f*x))*sqrt(S(1) + S(1)/cos(e + f*x))*EllipticPi((c + d)/d, asin(sqrt(c + d*cos(e + f*x))/(sqrt(b*cos(e + f*x))*Rt((c + d)/b, S(2)))), -(c + d)/(c - d))*Rt(b*(c + d), S(2))/(d*f*sqrt(c**S(2) - d**S(2))*tan(e + f*x)), x) def replacement2541(b, c, d, e, f, x): return Simp(S(2)*b*sqrt(c*(S(1) - S(1)/sin(e + f*x))/(c + d))*sqrt(c*(S(1) + S(1)/sin(e + f*x))/(c - d))*EllipticPi((c + d)/d, asin(sqrt(c + d*sin(e + f*x))/(sqrt(b*sin(e + f*x))*Rt((c + d)/b, S(2)))), -(c + d)/(c - d))*Rt((c + d)/b, S(2))*tan(e + f*x)/(d*f), x) def replacement2542(b, c, d, e, f, x): return Simp(-S(2)*b*sqrt(c*(S(1) - S(1)/cos(e + f*x))/(c + d))*sqrt(c*(S(1) + S(1)/cos(e + f*x))/(c - d))*EllipticPi((c + d)/d, asin(sqrt(c + d*cos(e + f*x))/(sqrt(b*cos(e + f*x))*Rt((c + d)/b, S(2)))), -(c + d)/(c - d))*Rt((c + d)/b, S(2))/(d*f*tan(e + f*x)), x) def replacement2543(b, c, d, e, f, x): return Dist(sqrt(b*sin(e + f*x))/sqrt(-b*sin(e + f*x)), Int(sqrt(-b*sin(e + f*x))/sqrt(c + d*sin(e + f*x)), x), x) def replacement2544(b, c, d, e, f, x): return Dist(sqrt(b*cos(e + f*x))/sqrt(-b*cos(e + f*x)), Int(sqrt(-b*cos(e + f*x))/sqrt(c + d*cos(e + f*x)), x), x) def replacement2545(a, b, c, d, e, f, x): return Simp(S(2)*sqrt((-a*d + b*c)*(sin(e + f*x) + S(1))/((a + b*sin(e + f*x))*(c - d)))*sqrt(-(S(1) - sin(e + f*x))*(-a*d + b*c)/((a + b*sin(e + f*x))*(c + d)))*(a + b*sin(e + f*x))*EllipticPi(b*(c + d)/(d*(a + b)), asin(sqrt(c + d*sin(e + f*x))*Rt((a + b)/(c + d), S(2))/sqrt(a + b*sin(e + f*x))), (a - b)*(c + d)/((a + b)*(c - d)))/(d*f*Rt((a + b)/(c + d), S(2))*cos(e + f*x)), x) def replacement2546(a, b, c, d, e, f, x): return Simp(-S(2)*sqrt((-a*d + b*c)*(cos(e + f*x) + S(1))/((a + b*cos(e + f*x))*(c - d)))*sqrt(-(S(1) - cos(e + f*x))*(-a*d + b*c)/((a + b*cos(e + f*x))*(c + d)))*(a + b*cos(e + f*x))*EllipticPi(b*(c + d)/(d*(a + b)), asin(sqrt(c + d*cos(e + f*x))*Rt((a + b)/(c + d), S(2))/sqrt(a + b*cos(e + f*x))), (a - b)*(c + d)/((a + b)*(c - d)))/(d*f*Rt((a + b)/(c + d), S(2))*sin(e + f*x)), x) def replacement2547(a, b, c, d, e, f, x): return Dist(sqrt(-c - d*sin(e + f*x))/sqrt(c + d*sin(e + f*x)), Int(sqrt(a + b*sin(e + f*x))/sqrt(-c - d*sin(e + f*x)), x), x) def replacement2548(a, b, c, d, e, f, x): return Dist(sqrt(-c - d*cos(e + f*x))/sqrt(c + d*cos(e + f*x)), Int(sqrt(a + b*cos(e + f*x))/sqrt(-c - d*cos(e + f*x)), x), x) def replacement2549(a, b, d, e, f, x): return Simp(-S(2)*d*EllipticF(asin(cos(e + f*x)/(d*sin(e + f*x) + S(1))), -(a - b*d)/(a + b*d))/(f*sqrt(a + b*d)), x) def replacement2550(a, b, d, e, f, x): return Simp(S(2)*d*EllipticF(asin(sin(e + f*x)/(d*cos(e + f*x) + S(1))), -(a - b*d)/(a + b*d))/(f*sqrt(a + b*d)), x) def replacement2551(a, b, d, e, f, x): return Dist(sqrt(sin(e + f*x)*sign(b))/sqrt(d*sin(e + f*x)), Int(S(1)/(sqrt(sin(e + f*x)*sign(b))*sqrt(a + b*sin(e + f*x))), x), x) def replacement2552(a, b, d, e, f, x): return Dist(sqrt(cos(e + f*x)*sign(b))/sqrt(d*cos(e + f*x)), Int(S(1)/(sqrt(cos(e + f*x)*sign(b))*sqrt(a + b*cos(e + f*x))), x), x) def replacement2553(a, b, d, e, f, x): return Simp(-S(2)*sqrt(-S(1)/tan(e + f*x)**S(2))*sqrt(a**S(2))*EllipticF(asin(sqrt(a + b*sin(e + f*x))/(sqrt(d*sin(e + f*x))*Rt((a + b)/d, S(2)))), -(a + b)/(a - b))*Rt((a + b)/d, S(2))*tan(e + f*x)/(a*f*sqrt(a**S(2) - b**S(2))), x) def replacement2554(a, b, d, e, f, x): return Simp(S(2)*sqrt(-tan(e + f*x)**S(2))*sqrt(a**S(2))*EllipticF(asin(sqrt(a + b*cos(e + f*x))/(sqrt(d*cos(e + f*x))*Rt((a + b)/d, S(2)))), -(a + b)/(a - b))*Rt((a + b)/d, S(2))/(a*f*sqrt(a**S(2) - b**S(2))*tan(e + f*x)), x) def replacement2555(a, b, d, e, f, x): return Simp(-S(2)*sqrt(a*(S(1) - S(1)/sin(e + f*x))/(a + b))*sqrt(a*(S(1) + S(1)/sin(e + f*x))/(a - b))*EllipticF(asin(sqrt(a + b*sin(e + f*x))/(sqrt(d*sin(e + f*x))*Rt((a + b)/d, S(2)))), -(a + b)/(a - b))*Rt((a + b)/d, S(2))*tan(e + f*x)/(a*f), x) def replacement2556(a, b, d, e, f, x): return Simp(S(2)*sqrt(a*(S(1) - S(1)/cos(e + f*x))/(a + b))*sqrt(a*(S(1) + S(1)/cos(e + f*x))/(a - b))*EllipticF(asin(sqrt(a + b*cos(e + f*x))/(sqrt(d*cos(e + f*x))*Rt((a + b)/d, S(2)))), -(a + b)/(a - b))*Rt((a + b)/d, S(2))/(a*f*tan(e + f*x)), x) def replacement2557(a, b, d, e, f, x): return Dist(sqrt(-d*sin(e + f*x))/sqrt(d*sin(e + f*x)), Int(S(1)/(sqrt(-d*sin(e + f*x))*sqrt(a + b*sin(e + f*x))), x), x) def replacement2558(a, b, d, e, f, x): return Dist(sqrt(-d*cos(e + f*x))/sqrt(d*cos(e + f*x)), Int(S(1)/(sqrt(-d*cos(e + f*x))*sqrt(a + b*cos(e + f*x))), x), x) def replacement2559(a, b, c, d, e, f, x): return Simp(S(2)*sqrt((S(1) - sin(e + f*x))*(-a*d + b*c)/((a + b)*(c + d*sin(e + f*x))))*sqrt(-(-a*d + b*c)*(sin(e + f*x) + S(1))/((a - b)*(c + d*sin(e + f*x))))*(c + d*sin(e + f*x))*EllipticF(asin(sqrt(a + b*sin(e + f*x))*Rt((c + d)/(a + b), S(2))/sqrt(c + d*sin(e + f*x))), (a + b)*(c - d)/((a - b)*(c + d)))/(f*(-a*d + b*c)*Rt((c + d)/(a + b), S(2))*cos(e + f*x)), x) def replacement2560(a, b, c, d, e, f, x): return Simp(-S(2)*sqrt((S(1) - cos(e + f*x))*(-a*d + b*c)/((a + b)*(c + d*cos(e + f*x))))*sqrt(-(-a*d + b*c)*(cos(e + f*x) + S(1))/((a - b)*(c + d*cos(e + f*x))))*(c + d*cos(e + f*x))*EllipticF(asin(sqrt(a + b*cos(e + f*x))*Rt((c + d)/(a + b), S(2))/sqrt(c + d*cos(e + f*x))), (a + b)*(c - d)/((a - b)*(c + d)))/(f*(-a*d + b*c)*Rt((c + d)/(a + b), S(2))*sin(e + f*x)), x) def replacement2561(a, b, c, d, e, f, x): return Dist(sqrt(-a - b*sin(e + f*x))/sqrt(a + b*sin(e + f*x)), Int(S(1)/(sqrt(-a - b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))), x), x) def replacement2562(a, b, c, d, e, f, x): return Dist(sqrt(-a - b*cos(e + f*x))/sqrt(a + b*cos(e + f*x)), Int(S(1)/(sqrt(-a - b*cos(e + f*x))*sqrt(c + d*cos(e + f*x))), x), x) def replacement2563(a, b, d, e, f, x): return Dist(d/(S(2)*b), Int(sqrt(d*sin(e + f*x))*(a + S(2)*b*sin(e + f*x))/sqrt(a + b*sin(e + f*x)), x), x) - Dist(a*d/(S(2)*b), Int(sqrt(d*sin(e + f*x))/sqrt(a + b*sin(e + f*x)), x), x) def replacement2564(a, b, d, e, f, x): return Dist(d/(S(2)*b), Int(sqrt(d*cos(e + f*x))*(a + S(2)*b*cos(e + f*x))/sqrt(a + b*cos(e + f*x)), x), x) - Dist(a*d/(S(2)*b), Int(sqrt(d*cos(e + f*x))/sqrt(a + b*cos(e + f*x)), x), x) def replacement2565(a, b, c, d, e, f, m, n, x): return Dist(S(1)/(d*(m + n)), Int((a + b*sin(e + f*x))**(m + S(-2))*(c + d*sin(e + f*x))**(n + S(-1))*Simp(a**S(2)*c*d*(m + n) + b*d*(a*d*n + b*c*(m + S(-1))) + b*d*(a*d*(S(2)*m + n + S(-1)) + b*c*n)*sin(e + f*x)**S(2) + (a*d*(m + n)*(a*d + S(2)*b*c) - b*d*(a*c - b*d*(m + n + S(-1))))*sin(e + f*x), x), x), x) - Simp(b*(a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**n*cos(e + f*x)/(f*(m + n)), x) def replacement2566(a, b, c, d, e, f, m, n, x): return Dist(S(1)/(d*(m + n)), Int((a + b*cos(e + f*x))**(m + S(-2))*(c + d*cos(e + f*x))**(n + S(-1))*Simp(a**S(2)*c*d*(m + n) + b*d*(a*d*n + b*c*(m + S(-1))) + b*d*(a*d*(S(2)*m + n + S(-1)) + b*c*n)*cos(e + f*x)**S(2) + (a*d*(m + n)*(a*d + S(2)*b*c) - b*d*(a*c - b*d*(m + n + S(-1))))*cos(e + f*x), x), x), x) + Simp(b*(a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**n*sin(e + f*x)/(f*(m + n)), x) def replacement2567(a, b, c, d, e, f, m, n, x): return Dist(b/d, Int((a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**(n + S(1)), x), x) - Dist((-a*d + b*c)/d, Int((a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**n, x), x) def replacement2568(a, b, c, d, e, f, m, n, x): return Dist(b/d, Int((a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**(n + S(1)), x), x) - Dist((-a*d + b*c)/d, Int((a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**n, x), x) def replacement2569(a, b, c, d, e, f, m, n, x): return Int((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n, x) def replacement2570(a, b, c, d, e, f, m, n, x): return Int((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n, x) def replacement2571(a, b, c, d, e, f, m, n, p, x): return Dist(c**IntPart(n)*(c*(d*sin(e + f*x))**p)**FracPart(n)*(d*sin(e + f*x))**(-p*FracPart(n)), Int((d*sin(e + f*x))**(n*p)*(a + b*sin(e + f*x))**m, x), x) def replacement2572(a, b, c, d, e, f, m, n, p, x): return Dist(c**IntPart(n)*(c*(d*cos(e + f*x))**p)**FracPart(n)*(d*cos(e + f*x))**(-p*FracPart(n)), Int((d*cos(e + f*x))**(n*p)*(a + b*cos(e + f*x))**m, x), x) def replacement2573(a, b, c, d, e, f, m, n, x): return Int((a + b*sin(e + f*x))**m*(c*sin(e + f*x) + d)**n*sin(e + f*x)**(-n), x) def replacement2574(a, b, c, d, e, f, m, n, x): return Int((a + b*cos(e + f*x))**m*(c*cos(e + f*x) + d)**n*cos(e + f*x)**(-n), x) def replacement2575(a, b, c, d, e, f, m, n, x): return Int((c + d/sin(e + f*x))**n*(a/sin(e + f*x) + b)**m*(S(1)/sin(e + f*x))**(-m), x) def replacement2576(a, b, c, d, e, f, m, n, x): return Int((c + d/cos(e + f*x))**n*(a/cos(e + f*x) + b)**m*(S(1)/cos(e + f*x))**(-m), x) def replacement2577(a, b, c, d, e, f, m, n, x): return Dist((c + d/sin(e + f*x))**n*(c*sin(e + f*x) + d)**(-n)*sin(e + f*x)**n, Int((a + b*sin(e + f*x))**m*(c*sin(e + f*x) + d)**n*sin(e + f*x)**(-n), x), x) def replacement2578(a, b, c, d, e, f, m, n, x): return Dist((c + d/cos(e + f*x))**n*(c*cos(e + f*x) + d)**(-n)*cos(e + f*x)**n, Int((a + b*cos(e + f*x))**m*(c*cos(e + f*x) + d)**n*cos(e + f*x)**(-n), x), x) def replacement2579(a, b, c, d, e, f, m, n, x): return Dist(S(1)/(b*f), Subst(Int((a + x)**m*(c + d*x/b)**n, x), x, b*sin(e + f*x)), x) def replacement2580(a, b, c, d, e, f, m, n, x): return -Dist(S(1)/(b*f), Subst(Int((a + x)**m*(c + d*x/b)**n, x), x, b*cos(e + f*x)), x) def replacement2581(a, b, d, e, f, n, p, x): return Dist(a, Int((d*sin(e + f*x))**n*cos(e + f*x)**p, x), x) + Dist(b/d, Int((d*sin(e + f*x))**(n + S(1))*cos(e + f*x)**p, x), x) def replacement2582(a, b, d, e, f, n, p, x): return Dist(a, Int((d*cos(e + f*x))**n*sin(e + f*x)**p, x), x) + Dist(b/d, Int((d*cos(e + f*x))**(n + S(1))*sin(e + f*x)**p, x), x) def replacement2583(a, b, d, e, f, n, p, x): return Dist(S(1)/a, Int((d*sin(e + f*x))**n*cos(e + f*x)**(p + S(-2)), x), x) - Dist(S(1)/(b*d), Int((d*sin(e + f*x))**(n + S(1))*cos(e + f*x)**(p + S(-2)), x), x) def replacement2584(a, b, d, e, f, n, p, x): return Dist(S(1)/a, Int((d*cos(e + f*x))**n*sin(e + f*x)**(p + S(-2)), x), x) - Dist(S(1)/(b*d), Int((d*cos(e + f*x))**(n + S(1))*sin(e + f*x)**(p + S(-2)), x), x) def replacement2585(a, b, c, d, e, f, m, n, p, x): return Dist(b**(-p)/f, Subst(Int((a - x)**(p/S(2) + S(-1)/2)*(a + x)**(m + p/S(2) + S(-1)/2)*(c + d*x/b)**n, x), x, b*sin(e + f*x)), x) def replacement2586(a, b, c, d, e, f, m, n, p, x): return -Dist(b**(-p)/f, Subst(Int((a - x)**(p/S(2) + S(-1)/2)*(a + x)**(m + p/S(2) + S(-1)/2)*(c + d*x/b)**n, x), x, b*cos(e + f*x)), x) def replacement2587(a, b, c, d, e, f, m, n, p, x): return Dist(b**(-p)/f, Subst(Int((a + x)**m*(b**S(2) - x**S(2))**(p/S(2) + S(-1)/2)*(c + d*x/b)**n, x), x, b*sin(e + f*x)), x) def replacement2588(a, b, c, d, e, f, m, n, p, x): return -Dist(b**(-p)/f, Subst(Int((a + x)**m*(b**S(2) - x**S(2))**(p/S(2) + S(-1)/2)*(c + d*x/b)**n, x), x, b*cos(e + f*x)), x) def replacement2589(a, b, d, e, f, g, n, p, x): return Dist(a, Int((d*sin(e + f*x))**n*(g*cos(e + f*x))**p, x), x) + Dist(b/d, Int((d*sin(e + f*x))**(n + S(1))*(g*cos(e + f*x))**p, x), x) def replacement2590(a, b, d, e, f, g, n, p, x): return Dist(a, Int((d*cos(e + f*x))**n*(g*sin(e + f*x))**p, x), x) + Dist(b/d, Int((d*cos(e + f*x))**(n + S(1))*(g*sin(e + f*x))**p, x), x) def replacement2591(a, b, d, e, f, g, n, p, x): return Dist(g**S(2)/a, Int((d*sin(e + f*x))**n*(g*cos(e + f*x))**(p + S(-2)), x), x) - Dist(g**S(2)/(b*d), Int((d*sin(e + f*x))**(n + S(1))*(g*cos(e + f*x))**(p + S(-2)), x), x) def replacement2592(a, b, d, e, f, g, n, p, x): return Dist(g**S(2)/a, Int((d*cos(e + f*x))**n*(g*sin(e + f*x))**(p + S(-2)), x), x) - Dist(g**S(2)/(b*d), Int((d*cos(e + f*x))**(n + S(1))*(g*sin(e + f*x))**(p + S(-2)), x), x) def replacement2593(a, b, c, d, e, f, g, m, n, p, x): return Dist(a**m*c**m*g**(-S(2)*m), Int((g*cos(e + f*x))**(S(2)*m + p)*(c + d*sin(e + f*x))**(-m + n), x), x) def replacement2594(a, b, c, d, e, f, g, m, n, p, x): return Dist(a**m*c**m*g**(-S(2)*m), Int((g*sin(e + f*x))**(S(2)*m + p)*(c + d*cos(e + f*x))**(-m + n), x), x) def replacement2595(a, b, c, d, e, f, m, n, p, x): return Dist(a**(-p/S(2))*c**(-p/S(2)), Int((a + b*sin(e + f*x))**(m + p/S(2))*(c + d*sin(e + f*x))**(n + p/S(2)), x), x) def replacement2596(a, b, c, d, e, f, m, n, p, x): return Dist(a**(-p/S(2))*c**(-p/S(2)), Int((a + b*cos(e + f*x))**(m + p/S(2))*(c + d*cos(e + f*x))**(n + p/S(2)), x), x) def replacement2597(a, b, c, d, e, f, g, p, x): return Dist(g*cos(e + f*x)/(sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))), Int((g*cos(e + f*x))**(p + S(-1)), x), x) def replacement2598(a, b, c, d, e, f, g, p, x): return Dist(g*sin(e + f*x)/(sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x))), Int((g*sin(e + f*x))**(p + S(-1)), x), x) def replacement2599(a, b, c, d, e, f, g, m, n, p, x): return Dist(a**IntPart(m)*c**IntPart(m)*g**(-S(2)*IntPart(m))*(g*cos(e + f*x))**(-S(2)*FracPart(m))*(a + b*sin(e + f*x))**FracPart(m)*(c + d*sin(e + f*x))**FracPart(m), Int((g*cos(e + f*x))**(S(2)*m + p)/(c + d*sin(e + f*x)), x), x) def replacement2600(a, b, c, d, e, f, g, m, n, p, x): return Dist(a**IntPart(m)*c**IntPart(m)*g**(-S(2)*IntPart(m))*(g*sin(e + f*x))**(-S(2)*FracPart(m))*(a + b*cos(e + f*x))**FracPart(m)*(c + d*cos(e + f*x))**FracPart(m), Int((g*sin(e + f*x))**(S(2)*m + p)/(c + d*cos(e + f*x)), x), x) def replacement2601(a, b, c, d, e, f, g, m, n, p, x): return Simp(b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**n/(f*g*(m - n + S(-1))), x) def replacement2602(a, b, c, d, e, f, g, m, n, p, x): return -Simp(b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**n/(f*g*(m - n + S(-1))), x) def replacement2603(a, b, c, d, e, f, g, m, n, p, x): return -Dist(b*(S(2)*m + p + S(-1))/(d*(S(2)*n + p + S(1))), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**(n + S(1)), x), x) + Simp(-S(2)*b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**n/(f*g*(S(2)*n + p + S(1))), x) def replacement2604(a, b, c, d, e, f, g, m, n, p, x): return -Dist(b*(S(2)*m + p + S(-1))/(d*(S(2)*n + p + S(1))), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**(n + S(1)), x), x) + Simp(S(2)*b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**n/(f*g*(S(2)*n + p + S(1))), x) def replacement2605(a, b, c, d, e, f, g, m, n, p, x): return Dist(a*(S(2)*m + p + S(-1))/(m + n + p), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**n, x), x) - Simp(b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**n/(f*g*(m + n + p)), x) def replacement2606(a, b, c, d, e, f, g, m, n, p, x): return Dist(a*(S(2)*m + p + S(-1))/(m + n + p), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**n, x), x) + Simp(b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**n/(f*g*(m + n + p)), x) def replacement2607(a, b, c, d, e, f, g, m, p, x): return Dist(a**IntPart(m)*c**IntPart(m)*g**(-S(2)*IntPart(m))*(g*cos(e + f*x))**(-S(2)*FracPart(m))*(a + b*sin(e + f*x))**FracPart(m)*(c + d*sin(e + f*x))**FracPart(m), Int((g*cos(e + f*x))**(S(2)*m + p), x), x) def replacement2608(a, b, c, d, e, f, g, m, p, x): return Dist(a**IntPart(m)*c**IntPart(m)*g**(-S(2)*IntPart(m))*(g*sin(e + f*x))**(-S(2)*FracPart(m))*(a + b*cos(e + f*x))**FracPart(m)*(c + d*cos(e + f*x))**FracPart(m), Int((g*sin(e + f*x))**(S(2)*m + p), x), x) def replacement2609(a, b, c, d, e, f, g, m, n, p, x): return Simp(b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n/(a*f*g*(m - n)), x) def replacement2610(a, b, c, d, e, f, g, m, n, p, x): return -Simp(b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n/(a*f*g*(m - n)), x) def replacement2611(a, b, c, d, e, f, g, m, n, p, x): return Dist((m + n + p + S(1))/(a*(S(2)*m + p + S(1))), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n, x), x) + Simp(b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n/(a*f*g*(S(2)*m + p + S(1))), x) def replacement2612(a, b, c, d, e, f, g, m, n, p, x): return Dist((m + n + p + S(1))/(a*(S(2)*m + p + S(1))), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n, x), x) - Simp(b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n/(a*f*g*(S(2)*m + p + S(1))), x) def replacement2613(a, b, c, d, e, f, g, m, n, p, x): return -Dist(b*(S(2)*m + p + S(-1))/(d*(S(2)*n + p + S(1))), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**(n + S(1)), x), x) + Simp(-S(2)*b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**n/(f*g*(S(2)*n + p + S(1))), x) def replacement2614(a, b, c, d, e, f, g, m, n, p, x): return -Dist(b*(S(2)*m + p + S(-1))/(d*(S(2)*n + p + S(1))), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**(n + S(1)), x), x) + Simp(S(2)*b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**n/(f*g*(S(2)*n + p + S(1))), x) def replacement2615(a, b, c, d, e, f, g, m, n, p, x): return Dist(a*(S(2)*m + p + S(-1))/(m + n + p), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**n, x), x) - Simp(b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**n/(f*g*(m + n + p)), x) def replacement2616(a, b, c, d, e, f, g, m, n, p, x): return Dist(a*(S(2)*m + p + S(-1))/(m + n + p), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**n, x), x) + Simp(b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**n/(f*g*(m + n + p)), x) def replacement2617(a, b, c, d, e, f, g, m, n, p, x): return Dist((m + n + p + S(1))/(a*(S(2)*m + p + S(1))), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n, x), x) + Simp(b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n/(a*f*g*(S(2)*m + p + S(1))), x) def replacement2618(a, b, c, d, e, f, g, m, n, p, x): return Dist((m + n + p + S(1))/(a*(S(2)*m + p + S(1))), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n, x), x) - Simp(b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n/(a*f*g*(S(2)*m + p + S(1))), x) def replacement2619(a, b, c, d, e, f, g, m, n, p, x): return Dist(a**IntPart(m)*c**IntPart(m)*g**(-S(2)*IntPart(m))*(g*cos(e + f*x))**(-S(2)*FracPart(m))*(a + b*sin(e + f*x))**FracPart(m)*(c + d*sin(e + f*x))**FracPart(m), Int((g*cos(e + f*x))**(S(2)*m + p)*(c + d*sin(e + f*x))**(-m + n), x), x) def replacement2620(a, b, c, d, e, f, g, m, n, p, x): return Dist(a**IntPart(m)*c**IntPart(m)*g**(-S(2)*IntPart(m))*(g*sin(e + f*x))**(-S(2)*FracPart(m))*(a + b*cos(e + f*x))**FracPart(m)*(c + d*cos(e + f*x))**FracPart(m), Int((g*sin(e + f*x))**(S(2)*m + p)*(c + d*cos(e + f*x))**(-m + n), x), x) def replacement2621(a, b, c, d, e, f, g, m, p, x): return -Simp(d*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**m/(f*g*(m + p + S(1))), x) def replacement2622(a, b, c, d, e, f, g, m, p, x): return Simp(d*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**m/(f*g*(m + p + S(1))), x) def replacement2623(a, b, c, d, e, f, g, m, p, x): return Dist(b*(a*d*m + b*c*(m + p + S(1)))/(a*g**S(2)*(p + S(1))), Int((g*cos(e + f*x))**(p + S(2))*(a + b*sin(e + f*x))**(m + S(-1)), x), x) - Simp((g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**m*(a*d + b*c)/(a*f*g*(p + S(1))), x) def replacement2624(a, b, c, d, e, f, g, m, p, x): return Dist(b*(a*d*m + b*c*(m + p + S(1)))/(a*g**S(2)*(p + S(1))), Int((g*sin(e + f*x))**(p + S(2))*(a + b*cos(e + f*x))**(m + S(-1)), x), x) + Simp((g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**m*(a*d + b*c)/(a*f*g*(p + S(1))), x) def replacement2625(a, b, c, d, e, f, g, m, p, x): return Dist((a*d*m + b*c*(m + p + S(1)))/(b*(m + p + S(1))), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**m, x), x) - Simp(d*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**m/(f*g*(m + p + S(1))), x) def replacement2626(a, b, c, d, e, f, g, m, p, x): return Dist((a*d*m + b*c*(m + p + S(1)))/(b*(m + p + S(1))), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**m, x), x) + Simp(d*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**m/(f*g*(m + p + S(1))), x) def replacement2627(a, b, c, d, e, f, m, x): return Dist(S(1)/(b**S(3)*(S(2)*m + S(3))), Int((a + b*sin(e + f*x))**(m + S(2))*(S(2)*a*d*(m + S(1)) + b*c - b*d*(S(2)*m + S(3))*sin(e + f*x)), x), x) + Simp(S(2)*(a + b*sin(e + f*x))**(m + S(1))*(-a*d + b*c)*cos(e + f*x)/(b**S(2)*f*(S(2)*m + S(3))), x) def replacement2628(a, b, c, d, e, f, m, x): return Dist(S(1)/(b**S(3)*(S(2)*m + S(3))), Int((a + b*cos(e + f*x))**(m + S(2))*(S(2)*a*d*(m + S(1)) + b*c - b*d*(S(2)*m + S(3))*cos(e + f*x)), x), x) + Simp(-S(2)*(a + b*cos(e + f*x))**(m + S(1))*(-a*d + b*c)*sin(e + f*x)/(b**S(2)*f*(S(2)*m + S(3))), x) def replacement2629(a, b, c, d, e, f, m, x): return -Dist(S(1)/(b**S(2)*(m + S(3))), Int((a + b*sin(e + f*x))**(m + S(1))*(-a*c*(m + S(3)) + b*d*(m + S(2)) + (-a*d*(m + S(4)) + b*c*(m + S(3)))*sin(e + f*x)), x), x) + Simp(d*(a + b*sin(e + f*x))**(m + S(2))*cos(e + f*x)/(b**S(2)*f*(m + S(3))), x) def replacement2630(a, b, c, d, e, f, m, x): return -Dist(S(1)/(b**S(2)*(m + S(3))), Int((a + b*cos(e + f*x))**(m + S(1))*(-a*c*(m + S(3)) + b*d*(m + S(2)) + (-a*d*(m + S(4)) + b*c*(m + S(3)))*cos(e + f*x)), x), x) - Simp(d*(a + b*cos(e + f*x))**(m + S(2))*sin(e + f*x)/(b**S(2)*f*(m + S(3))), x) def replacement2631(a, b, c, d, e, f, g, m, p, x): return Dist((a*d*m + b*c*(m + p + S(1)))/(a*b*(S(2)*m + p + S(1))), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**(m + S(1)), x), x) + Simp((g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**m*(-a*d + b*c)/(a*f*g*(S(2)*m + p + S(1))), x) def replacement2632(a, b, c, d, e, f, g, m, p, x): return Dist((a*d*m + b*c*(m + p + S(1)))/(a*b*(S(2)*m + p + S(1))), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**(m + S(1)), x), x) - Simp((g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**m*(-a*d + b*c)/(a*f*g*(S(2)*m + p + S(1))), x) def replacement2633(a, b, c, d, e, f, g, m, p, x): return Dist((a*d*m + b*c*(m + p + S(1)))/(b*(m + p + S(1))), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**m, x), x) - Simp(d*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**m/(f*g*(m + p + S(1))), x) def replacement2634(a, b, c, d, e, f, g, m, p, x): return Dist((a*d*m + b*c*(m + p + S(1)))/(b*(m + p + S(1))), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**m, x), x) + Simp(d*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**m/(f*g*(m + p + S(1))), x) def replacement2635(a, b, c, d, e, f, g, m, p, x): return Dist(S(1)/(g**S(2)*(p + S(1))), Int((g*cos(e + f*x))**(p + S(2))*(a + b*sin(e + f*x))**(m + S(-1))*Simp(a*c*(p + S(2)) + b*c*(m + p + S(2))*sin(e + f*x) + b*d*m, x), x), x) - Simp((g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**m*(c*sin(e + f*x) + d)/(f*g*(p + S(1))), x) def replacement2636(a, b, c, d, e, f, g, m, p, x): return Dist(S(1)/(g**S(2)*(p + S(1))), Int((g*sin(e + f*x))**(p + S(2))*(a + b*cos(e + f*x))**(m + S(-1))*Simp(a*c*(p + S(2)) + b*c*(m + p + S(2))*cos(e + f*x) + b*d*m, x), x), x) + Simp((g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**m*(c*cos(e + f*x) + d)/(f*g*(p + S(1))), x) def replacement2637(a, b, c, d, e, f, g, m, p, x): return Dist(S(1)/(m + p + S(1)), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**(m + S(-1))*Simp(a*c*(m + p + S(1)) + b*d*m + (a*d*m + b*c*(m + p + S(1)))*sin(e + f*x), x), x), x) - Simp(d*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**m/(f*g*(m + p + S(1))), x) def replacement2638(a, b, c, d, e, f, g, m, p, x): return Dist(S(1)/(m + p + S(1)), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**(m + S(-1))*Simp(a*c*(m + p + S(1)) + b*d*m + (a*d*m + b*c*(m + p + S(1)))*cos(e + f*x), x), x), x) + Simp(d*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**m/(f*g*(m + p + S(1))), x) def replacement2639(a, b, c, d, e, f, g, m, p, x): return Dist(g**S(2)*(p + S(-1))/(b**S(2)*(m + S(1))*(m + p + S(1))), Int((g*cos(e + f*x))**(p + S(-2))*(a + b*sin(e + f*x))**(m + S(1))*Simp(b*d*(m + S(1)) + (-a*d*p + b*c*(m + p + S(1)))*sin(e + f*x), x), x), x) + Simp(g*(g*cos(e + f*x))**(p + S(-1))*(a + b*sin(e + f*x))**(m + S(1))*(-a*d*p + b*c*(m + p + S(1)) + b*d*(m + S(1))*sin(e + f*x))/(b**S(2)*f*(m + S(1))*(m + p + S(1))), x) def replacement2640(a, b, c, d, e, f, g, m, p, x): return Dist(g**S(2)*(p + S(-1))/(b**S(2)*(m + S(1))*(m + p + S(1))), Int((g*sin(e + f*x))**(p + S(-2))*(a + b*cos(e + f*x))**(m + S(1))*Simp(b*d*(m + S(1)) + (-a*d*p + b*c*(m + p + S(1)))*cos(e + f*x), x), x), x) - Simp(g*(g*sin(e + f*x))**(p + S(-1))*(a + b*cos(e + f*x))**(m + S(1))*(-a*d*p + b*c*(m + p + S(1)) + b*d*(m + S(1))*cos(e + f*x))/(b**S(2)*f*(m + S(1))*(m + p + S(1))), x) def replacement2641(a, b, c, d, e, f, g, m, p, x): return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**(m + S(1))*Simp((m + S(1))*(a*c - b*d) - (-a*d + b*c)*(m + p + S(2))*sin(e + f*x), x), x), x) - Simp((g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(1))*(-a*d + b*c)/(f*g*(a**S(2) - b**S(2))*(m + S(1))), x) def replacement2642(a, b, c, d, e, f, g, m, p, x): return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**(m + S(1))*Simp((m + S(1))*(a*c - b*d) - (-a*d + b*c)*(m + p + S(2))*cos(e + f*x), x), x), x) + Simp((g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(1))*(-a*d + b*c)/(f*g*(a**S(2) - b**S(2))*(m + S(1))), x) def replacement2643(a, b, c, d, e, f, g, m, p, x): return Dist(g**S(2)*(p + S(-1))/(b**S(2)*(m + p)*(m + p + S(1))), Int((g*cos(e + f*x))**(p + S(-2))*(a + b*sin(e + f*x))**m*Simp(b*(a*d*m + b*c*(m + p + S(1))) + (a*b*c*(m + p + S(1)) - d*(a**S(2)*p - b**S(2)*(m + p)))*sin(e + f*x), x), x), x) + Simp(g*(g*cos(e + f*x))**(p + S(-1))*(a + b*sin(e + f*x))**(m + S(1))*(-a*d*p + b*c*(m + p + S(1)) + b*d*(m + p)*sin(e + f*x))/(b**S(2)*f*(m + p)*(m + p + S(1))), x) def replacement2644(a, b, c, d, e, f, g, m, p, x): return Dist(g**S(2)*(p + S(-1))/(b**S(2)*(m + p)*(m + p + S(1))), Int((g*sin(e + f*x))**(p + S(-2))*(a + b*cos(e + f*x))**m*Simp(b*(a*d*m + b*c*(m + p + S(1))) + (a*b*c*(m + p + S(1)) - d*(a**S(2)*p - b**S(2)*(m + p)))*cos(e + f*x), x), x), x) - Simp(g*(g*sin(e + f*x))**(p + S(-1))*(a + b*cos(e + f*x))**(m + S(1))*(-a*d*p + b*c*(m + p + S(1)) + b*d*(m + p)*cos(e + f*x))/(b**S(2)*f*(m + p)*(m + p + S(1))), x) def replacement2645(a, b, c, d, e, f, g, m, p, x): return Dist(S(1)/(g**S(2)*(a**S(2) - b**S(2))*(p + S(1))), Int((g*cos(e + f*x))**(p + S(2))*(a + b*sin(e + f*x))**m*Simp(a*b*d*m + b*(a*c - b*d)*(m + p + S(3))*sin(e + f*x) + c*(a**S(2)*(p + S(2)) - b**S(2)*(m + p + S(2))), x), x), x) + Simp((g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(1))*(-a*d + b*c - (a*c - b*d)*sin(e + f*x))/(f*g*(a**S(2) - b**S(2))*(p + S(1))), x) def replacement2646(a, b, c, d, e, f, g, m, p, x): return Dist(S(1)/(g**S(2)*(a**S(2) - b**S(2))*(p + S(1))), Int((g*sin(e + f*x))**(p + S(2))*(a + b*cos(e + f*x))**m*Simp(a*b*d*m + b*(a*c - b*d)*(m + p + S(3))*cos(e + f*x) + c*(a**S(2)*(p + S(2)) - b**S(2)*(m + p + S(2))), x), x), x) - Simp((g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(1))*(-a*d + b*c - (a*c - b*d)*cos(e + f*x))/(f*g*(a**S(2) - b**S(2))*(p + S(1))), x) def replacement2647(a, b, c, d, e, f, g, p, x): return Dist(d/b, Int((g*cos(e + f*x))**p, x), x) + Dist((-a*d + b*c)/b, Int((g*cos(e + f*x))**p/(a + b*sin(e + f*x)), x), x) def replacement2648(a, b, c, d, e, f, g, p, x): return Dist(d/b, Int((g*sin(e + f*x))**p, x), x) + Dist((-a*d + b*c)/b, Int((g*sin(e + f*x))**p/(a + b*cos(e + f*x)), x), x) def replacement2649(a, b, c, d, e, f, g, m, p, x): return Dist(c*g*(g*cos(e + f*x))**(p + S(-1))*(S(1) - sin(e + f*x))**(S(1)/2 - p/S(2))*(sin(e + f*x) + S(1))**(S(1)/2 - p/S(2))/f, Subst(Int((S(1) - d*x/c)**(p/S(2) + S(-1)/2)*(S(1) + d*x/c)**(p/S(2) + S(1)/2)*(a + b*x)**m, x), x, sin(e + f*x)), x) def replacement2650(a, b, c, d, e, f, g, m, p, x): return -Dist(c*g*(g*sin(e + f*x))**(p + S(-1))*(S(1) - cos(e + f*x))**(S(1)/2 - p/S(2))*(cos(e + f*x) + S(1))**(S(1)/2 - p/S(2))/f, Subst(Int((S(1) - d*x/c)**(p/S(2) + S(-1)/2)*(S(1) + d*x/c)**(p/S(2) + S(1)/2)*(a + b*x)**m, x), x, cos(e + f*x)), x) def replacement2651(a, b, d, e, f, m, n, p, x): return Dist(a**(S(2)*m), Int((d*sin(e + f*x))**n*(a - b*sin(e + f*x))**(-m), x), x) def replacement2652(a, b, d, e, f, m, n, p, x): return Dist(a**(S(2)*m), Int((d*cos(e + f*x))**n*(a - b*cos(e + f*x))**(-m), x), x) def replacement2653(a, b, e, f, g, m, p, x): return Dist(a/(S(2)*g**S(2)), Int((g*cos(e + f*x))**(p + S(2))*(a + b*sin(e + f*x))**(m + S(-1)), x), x) - Simp((g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(1))/(S(2)*b*f*g*(m + S(1))), x) def replacement2654(a, b, e, f, g, m, p, x): return Dist(a/(S(2)*g**S(2)), Int((g*sin(e + f*x))**(p + S(2))*(a + b*cos(e + f*x))**(m + S(-1)), x), x) + Simp((g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(1))/(S(2)*b*f*g*(m + S(1))), x) def replacement2655(a, b, e, f, g, m, p, x): return -Dist(g**(S(-2)), Int((g*cos(e + f*x))**(p + S(2))*(a + b*sin(e + f*x))**m, x), x) + Simp(b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**m/(a*f*g*m), x) def replacement2656(a, b, e, f, g, m, p, x): return -Dist(g**(S(-2)), Int((g*sin(e + f*x))**(p + S(2))*(a + b*cos(e + f*x))**m, x), x) - Simp(b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**m/(a*f*g*m), x) def replacement2657(a, b, d, e, f, m, n, p, x): return Dist(a**(-p), Int(ExpandTrig((d*sin(e + f*x))**n*(a - b*sin(e + f*x))**(p/S(2))*(a + b*sin(e + f*x))**(m + p/S(2)), x), x), x) def replacement2658(a, b, d, e, f, m, n, p, x): return Dist(a**(-p), Int(ExpandTrig((d*cos(e + f*x))**n*(a - b*cos(e + f*x))**(p/S(2))*(a + b*cos(e + f*x))**(m + p/S(2)), x), x), x) def replacement2659(a, b, d, e, f, g, m, n, p, x): return Int(ExpandTrig((g*cos(e + f*x))**p, (d*sin(e + f*x))**n*(a + b*sin(e + f*x))**m, x), x) def replacement2660(a, b, d, e, f, g, m, n, p, x): return Int(ExpandTrig((g*sin(e + f*x))**p, (d*cos(e + f*x))**n*(a + b*cos(e + f*x))**m, x), x) def replacement2661(a, b, d, e, f, m, n, x): return Dist(b**(S(-2)), Int((d*sin(e + f*x))**n*(a - b*sin(e + f*x))*(a + b*sin(e + f*x))**(m + S(1)), x), x) def replacement2662(a, b, d, e, f, m, n, x): return Dist(b**(S(-2)), Int((d*cos(e + f*x))**n*(a - b*cos(e + f*x))*(a + b*cos(e + f*x))**(m + S(1)), x), x) def replacement2663(a, b, d, e, f, g, m, n, p, x): return Dist((a/g)**(S(2)*m), Int((d*sin(e + f*x))**n*(g*cos(e + f*x))**(S(2)*m + p)*(a - b*sin(e + f*x))**(-m), x), x) def replacement2664(a, b, d, e, f, g, m, n, p, x): return Dist((a/g)**(S(2)*m), Int((d*cos(e + f*x))**n*(g*sin(e + f*x))**(S(2)*m + p)*(a - b*cos(e + f*x))**(-m), x), x) def replacement2665(a, b, d, e, f, g, m, n, p, x): return Dist((a/g)**(S(2)*m), Int((d*sin(e + f*x))**n*(g*cos(e + f*x))**(S(2)*m + p)*(a - b*sin(e + f*x))**(-m), x), x) def replacement2666(a, b, d, e, f, g, m, n, p, x): return Dist((a/g)**(S(2)*m), Int((d*cos(e + f*x))**n*(g*sin(e + f*x))**(S(2)*m + p)*(a - b*cos(e + f*x))**(-m), x), x) def replacement2667(a, b, e, f, g, m, p, x): return -Dist(S(1)/(a**S(2)*(S(2)*m + p + S(1))), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**(m + S(1))*(a*m - b*(S(2)*m + p + S(1))*sin(e + f*x)), x), x) + Simp(b*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**m/(a*f*g*(S(2)*m + p + S(1))), x) def replacement2668(a, b, e, f, g, m, p, x): return -Dist(S(1)/(a**S(2)*(S(2)*m + p + S(1))), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**(m + S(1))*(a*m - b*(S(2)*m + p + S(1))*cos(e + f*x)), x), x) - Simp(b*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**m/(a*f*g*(S(2)*m + p + S(1))), x) def replacement2669(a, b, e, f, g, m, p, x): return Dist(S(1)/(b*(m + p + S(2))), Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**m*(-a*(p + S(1))*sin(e + f*x) + b*(m + S(1))), x), x) - Simp((g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**(m + S(1))/(b*f*g*(m + p + S(2))), x) def replacement2670(a, b, e, f, g, m, p, x): return Dist(S(1)/(b*(m + p + S(2))), Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**m*(-a*(p + S(1))*cos(e + f*x) + b*(m + S(1))), x), x) + Simp((g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**(m + S(1))/(b*f*g*(m + p + S(2))), x) def replacement2671(a, b, d, e, f, m, n, x): return Dist(b**(S(-2)), Int((d*sin(e + f*x))**n*(a - b*sin(e + f*x))*(a + b*sin(e + f*x))**(m + S(1)), x), x) def replacement2672(a, b, d, e, f, m, n, x): return Dist(b**(S(-2)), Int((d*cos(e + f*x))**n*(a - b*cos(e + f*x))*(a + b*cos(e + f*x))**(m + S(1)), x), x) def replacement2673(a, b, d, e, f, m, n, x): return Dist(a**(S(-2)), Int((d*sin(e + f*x))**n*(a + b*sin(e + f*x))**(m + S(2))*(sin(e + f*x)**S(2) + S(1)), x), x) + Dist(-S(2)/(a*b*d), Int((d*sin(e + f*x))**(n + S(1))*(a + b*sin(e + f*x))**(m + S(2)), x), x) def replacement2674(a, b, d, e, f, m, n, x): return Dist(a**(S(-2)), Int((d*cos(e + f*x))**n*(a + b*cos(e + f*x))**(m + S(2))*(cos(e + f*x)**S(2) + S(1)), x), x) + Dist(-S(2)/(a*b*d), Int((d*cos(e + f*x))**(n + S(1))*(a + b*cos(e + f*x))**(m + S(2)), x), x) def replacement2675(a, b, d, e, f, m, n, x): return Dist(d**(S(-4)), Int((d*sin(e + f*x))**(n + S(4))*(a + b*sin(e + f*x))**m, x), x) + Int((d*sin(e + f*x))**n*(S(1) - S(2)*sin(e + f*x)**S(2))*(a + b*sin(e + f*x))**m, x) def replacement2676(a, b, d, e, f, m, n, x): return Dist(d**(S(-4)), Int((d*cos(e + f*x))**(n + S(4))*(a + b*cos(e + f*x))**m, x), x) + Int((d*cos(e + f*x))**n*(S(1) - S(2)*cos(e + f*x)**S(2))*(a + b*cos(e + f*x))**m, x) def replacement2677(a, b, d, e, f, m, n, p, x): return Dist(a**m*cos(e + f*x)/(f*sqrt(S(1) - sin(e + f*x))*sqrt(sin(e + f*x) + S(1))), Subst(Int((d*x)**n*(S(1) - b*x/a)**(p/S(2) + S(-1)/2)*(S(1) + b*x/a)**(m + p/S(2) + S(-1)/2), x), x, sin(e + f*x)), x) def replacement2678(a, b, d, e, f, m, n, p, x): return -Dist(a**m*sin(e + f*x)/(f*sqrt(S(1) - cos(e + f*x))*sqrt(cos(e + f*x) + S(1))), Subst(Int((d*x)**n*(S(1) - b*x/a)**(p/S(2) + S(-1)/2)*(S(1) + b*x/a)**(m + p/S(2) + S(-1)/2), x), x, cos(e + f*x)), x) def replacement2679(a, b, d, e, f, m, n, p, x): return Dist(a**(S(2) - p)*cos(e + f*x)/(f*sqrt(a - b*sin(e + f*x))*sqrt(a + b*sin(e + f*x))), Subst(Int((d*x)**n*(a - b*x)**(p/S(2) + S(-1)/2)*(a + b*x)**(m + p/S(2) + S(-1)/2), x), x, sin(e + f*x)), x) def replacement2680(a, b, d, e, f, m, n, p, x): return -Dist(a**(S(2) - p)*sin(e + f*x)/(f*sqrt(a - b*cos(e + f*x))*sqrt(a + b*cos(e + f*x))), Subst(Int((d*x)**n*(a - b*x)**(p/S(2) + S(-1)/2)*(a + b*x)**(m + p/S(2) + S(-1)/2), x), x, cos(e + f*x)), x) def replacement2681(a, b, d, e, f, g, m, n, p, x): return Int(ExpandTrig((g*cos(e + f*x))**p, (d*sin(e + f*x))**n*(a + b*sin(e + f*x))**m, x), x) def replacement2682(a, b, d, e, f, g, m, n, p, x): return Int(ExpandTrig((g*sin(e + f*x))**p, (d*cos(e + f*x))**n*(a + b*cos(e + f*x))**m, x), x) def replacement2683(a, b, d, e, f, g, m, n, p, x): return Dist(a**m*g*(g*cos(e + f*x))**(p + S(-1))*(S(1) - sin(e + f*x))**(S(1)/2 - p/S(2))*(sin(e + f*x) + S(1))**(S(1)/2 - p/S(2))/f, Subst(Int((d*x)**n*(S(1) - b*x/a)**(p/S(2) + S(-1)/2)*(S(1) + b*x/a)**(m + p/S(2) + S(-1)/2), x), x, sin(e + f*x)), x) def replacement2684(a, b, d, e, f, g, m, n, p, x): return -Dist(a**m*g*(g*sin(e + f*x))**(p + S(-1))*(S(1) - cos(e + f*x))**(S(1)/2 - p/S(2))*(cos(e + f*x) + S(1))**(S(1)/2 - p/S(2))/f, Subst(Int((d*x)**n*(S(1) - b*x/a)**(p/S(2) + S(-1)/2)*(S(1) + b*x/a)**(m + p/S(2) + S(-1)/2), x), x, cos(e + f*x)), x) def replacement2685(a, b, d, e, f, g, m, n, p, x): return Dist(g*(g*cos(e + f*x))**(p + S(-1))*(a - b*sin(e + f*x))**(S(1)/2 - p/S(2))*(a + b*sin(e + f*x))**(S(1)/2 - p/S(2))/f, Subst(Int((d*x)**n*(a - b*x)**(p/S(2) + S(-1)/2)*(a + b*x)**(m + p/S(2) + S(-1)/2), x), x, sin(e + f*x)), x) def replacement2686(a, b, d, e, f, g, m, n, p, x): return -Dist(g*(g*sin(e + f*x))**(p + S(-1))*(a - b*cos(e + f*x))**(S(1)/2 - p/S(2))*(a + b*cos(e + f*x))**(S(1)/2 - p/S(2))/f, Subst(Int((d*x)**n*(a - b*x)**(p/S(2) + S(-1)/2)*(a + b*x)**(m + p/S(2) + S(-1)/2), x), x, cos(e + f*x)), x) def replacement2687(a, b, d, e, f, g, m, p, x): return Dist(g**S(2)*(S(2)*m + S(3))/(S(2)*a*(m + S(1))), Int((g*cos(e + f*x))**(p + S(-2))*(a + b*sin(e + f*x))**(m + S(1))/sqrt(d*sin(e + f*x)), x), x) - Simp(g*sqrt(d*sin(e + f*x))*(g*cos(e + f*x))**(p + S(-1))*(a + b*sin(e + f*x))**(m + S(1))/(a*d*f*(m + S(1))), x) def replacement2688(a, b, d, e, f, g, m, p, x): return Dist(g**S(2)*(S(2)*m + S(3))/(S(2)*a*(m + S(1))), Int((g*sin(e + f*x))**(p + S(-2))*(a + b*cos(e + f*x))**(m + S(1))/sqrt(d*cos(e + f*x)), x), x) + Simp(g*sqrt(d*cos(e + f*x))*(g*sin(e + f*x))**(p + S(-1))*(a + b*cos(e + f*x))**(m + S(1))/(a*d*f*(m + S(1))), x) def replacement2689(a, b, d, e, f, g, m, p, x): return Dist(S(2)*a*m/(g**S(2)*(S(2)*m + S(1))), Int((g*cos(e + f*x))**(p + S(2))*(a + b*sin(e + f*x))**(m + S(-1))/sqrt(d*sin(e + f*x)), x), x) + Simp(S(2)*sqrt(d*sin(e + f*x))*(g*cos(e + f*x))**(p + S(1))*(a + b*sin(e + f*x))**m/(d*f*g*(S(2)*m + S(1))), x) def replacement2690(a, b, d, e, f, g, m, p, x): return Dist(S(2)*a*m/(g**S(2)*(S(2)*m + S(1))), Int((g*sin(e + f*x))**(p + S(2))*(a + b*cos(e + f*x))**(m + S(-1))/sqrt(d*cos(e + f*x)), x), x) + Simp(-S(2)*sqrt(d*cos(e + f*x))*(g*sin(e + f*x))**(p + S(1))*(a + b*cos(e + f*x))**m/(d*f*g*(S(2)*m + S(1))), x) def replacement2691(a, b, d, e, f, m, n, x): return Int((d*sin(e + f*x))**n*(S(1) - sin(e + f*x)**S(2))*(a + b*sin(e + f*x))**m, x) def replacement2692(a, b, d, e, f, m, n, x): return Int((d*cos(e + f*x))**n*(S(1) - cos(e + f*x)**S(2))*(a + b*cos(e + f*x))**m, x) def replacement2693(a, b, d, e, f, m, n, x): return Dist(S(1)/(a**S(2)*b*d*(m + S(1))*(n + S(1))), Int((d*sin(e + f*x))**(n + S(1))*(a + b*sin(e + f*x))**(m + S(1))*Simp(a**S(2)*(n + S(1))*(n + S(2)) + a*b*(m + S(1))*sin(e + f*x) - b**S(2)*(m + n + S(2))*(m + n + S(3)) - (a**S(2)*(n + S(1))*(n + S(3)) - b**S(2)*(m + n + S(2))*(m + n + S(4)))*sin(e + f*x)**S(2), x), x), x) + Simp((d*sin(e + f*x))**(n + S(1))*(a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(a*d*f*(n + S(1))), x) - Simp((d*sin(e + f*x))**(n + S(2))*(a + b*sin(e + f*x))**(m + S(1))*(a**S(2)*(n + S(1)) - b**S(2)*(m + n + S(2)))*cos(e + f*x)/(a**S(2)*b*d**S(2)*f*(m + S(1))*(n + S(1))), x) def replacement2694(a, b, d, e, f, m, n, x): return Dist(S(1)/(a**S(2)*b*d*(m + S(1))*(n + S(1))), Int((d*cos(e + f*x))**(n + S(1))*(a + b*cos(e + f*x))**(m + S(1))*Simp(a**S(2)*(n + S(1))*(n + S(2)) + a*b*(m + S(1))*cos(e + f*x) - b**S(2)*(m + n + S(2))*(m + n + S(3)) - (a**S(2)*(n + S(1))*(n + S(3)) - b**S(2)*(m + n + S(2))*(m + n + S(4)))*cos(e + f*x)**S(2), x), x), x) - Simp((d*cos(e + f*x))**(n + S(1))*(a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(a*d*f*(n + S(1))), x) + Simp((d*cos(e + f*x))**(n + S(2))*(a + b*cos(e + f*x))**(m + S(1))*(a**S(2)*(n + S(1)) - b**S(2)*(m + n + S(2)))*sin(e + f*x)/(a**S(2)*b*d**S(2)*f*(m + S(1))*(n + S(1))), x) def replacement2695(a, b, d, e, f, m, n, x): return -Dist(S(1)/(a**S(2)*b**S(2)*(m + S(1))*(m + S(2))), Int((d*sin(e + f*x))**n*(a + b*sin(e + f*x))**(m + S(2))*Simp(a**S(2)*(n + S(1))*(n + S(3)) + a*b*(m + S(2))*sin(e + f*x) - b**S(2)*(m + n + S(2))*(m + n + S(3)) - (a**S(2)*(n + S(2))*(n + S(3)) - b**S(2)*(m + n + S(2))*(m + n + S(4)))*sin(e + f*x)**S(2), x), x), x) + Simp((d*sin(e + f*x))**(n + S(1))*(a + b*sin(e + f*x))**(m + S(1))*(a**S(2) - b**S(2))*cos(e + f*x)/(a*b**S(2)*d*f*(m + S(1))), x) + Simp((d*sin(e + f*x))**(n + S(1))*(a + b*sin(e + f*x))**(m + S(2))*(a**S(2)*(-m + n + S(1)) - b**S(2)*(m + n + S(2)))*cos(e + f*x)/(a**S(2)*b**S(2)*d*f*(m + S(1))*(m + S(2))), x) def replacement2696(a, b, d, e, f, m, n, x): return -Dist(S(1)/(a**S(2)*b**S(2)*(m + S(1))*(m + S(2))), Int((d*cos(e + f*x))**n*(a + b*cos(e + f*x))**(m + S(2))*Simp(a**S(2)*(n + S(1))*(n + S(3)) + a*b*(m + S(2))*cos(e + f*x) - b**S(2)*(m + n + S(2))*(m + n + S(3)) - (a**S(2)*(n + S(2))*(n + S(3)) - b**S(2)*(m + n + S(2))*(m + n + S(4)))*cos(e + f*x)**S(2), x), x), x) - Simp((d*cos(e + f*x))**(n + S(1))*(a + b*cos(e + f*x))**(m + S(1))*(a**S(2) - b**S(2))*sin(e + f*x)/(a*b**S(2)*d*f*(m + S(1))), x) - Simp((d*cos(e + f*x))**(n + S(1))*(a + b*cos(e + f*x))**(m + S(2))*(a**S(2)*(-m + n + S(1)) - b**S(2)*(m + n + S(2)))*sin(e + f*x)/(a**S(2)*b**S(2)*d*f*(m + S(1))*(m + S(2))), x) def replacement2697(a, b, d, e, f, m, n, x): return -Dist(S(1)/(a*b**S(2)*(m + S(1))*(m + n + S(4))), Int((d*sin(e + f*x))**n*(a + b*sin(e + f*x))**(m + S(1))*Simp(a**S(2)*(n + S(1))*(n + S(3)) + a*b*(m + S(1))*sin(e + f*x) - b**S(2)*(m + n + S(2))*(m + n + S(4)) - (a**S(2)*(n + S(2))*(n + S(3)) - b**S(2)*(m + n + S(3))*(m + n + S(4)))*sin(e + f*x)**S(2), x), x), x) - Simp((d*sin(e + f*x))**(n + S(1))*(a + b*sin(e + f*x))**(m + S(2))*cos(e + f*x)/(b**S(2)*d*f*(m + n + S(4))), x) + Simp((d*sin(e + f*x))**(n + S(1))*(a + b*sin(e + f*x))**(m + S(1))*(a**S(2) - b**S(2))*cos(e + f*x)/(a*b**S(2)*d*f*(m + S(1))), x) def replacement2698(a, b, d, e, f, m, n, x): return -Dist(S(1)/(a*b**S(2)*(m + S(1))*(m + n + S(4))), Int((d*cos(e + f*x))**n*(a + b*cos(e + f*x))**(m + S(1))*Simp(a**S(2)*(n + S(1))*(n + S(3)) + a*b*(m + S(1))*cos(e + f*x) - b**S(2)*(m + n + S(2))*(m + n + S(4)) - (a**S(2)*(n + S(2))*(n + S(3)) - b**S(2)*(m + n + S(3))*(m + n + S(4)))*cos(e + f*x)**S(2), x), x), x) + Simp((d*cos(e + f*x))**(n + S(1))*(a + b*cos(e + f*x))**(m + S(2))*sin(e + f*x)/(b**S(2)*d*f*(m + n + S(4))), x) - Simp((d*cos(e + f*x))**(n + S(1))*(a + b*cos(e + f*x))**(m + S(1))*(a**S(2) - b**S(2))*sin(e + f*x)/(a*b**S(2)*d*f*(m + S(1))), x) def replacement2699(a, b, d, e, f, m, n, x): return -Dist(S(1)/(a**S(2)*d**S(2)*(n + S(1))*(n + S(2))), Int((d*sin(e + f*x))**(n + S(2))*(a + b*sin(e + f*x))**m*Simp(a**S(2)*n*(n + S(2)) + a*b*m*sin(e + f*x) - b**S(2)*(m + n + S(2))*(m + n + S(3)) - (a**S(2)*(n + S(1))*(n + S(2)) - b**S(2)*(m + n + S(2))*(m + n + S(4)))*sin(e + f*x)**S(2), x), x), x) + Simp((d*sin(e + f*x))**(n + S(1))*(a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(a*d*f*(n + S(1))), x) - Simp(b*(d*sin(e + f*x))**(n + S(2))*(a + b*sin(e + f*x))**(m + S(1))*(m + n + S(2))*cos(e + f*x)/(a**S(2)*d**S(2)*f*(n + S(1))*(n + S(2))), x) def replacement2700(a, b, d, e, f, m, n, x): return -Dist(S(1)/(a**S(2)*d**S(2)*(n + S(1))*(n + S(2))), Int((d*cos(e + f*x))**(n + S(2))*(a + b*cos(e + f*x))**m*Simp(a**S(2)*n*(n + S(2)) + a*b*m*cos(e + f*x) - b**S(2)*(m + n + S(2))*(m + n + S(3)) - (a**S(2)*(n + S(1))*(n + S(2)) - b**S(2)*(m + n + S(2))*(m + n + S(4)))*cos(e + f*x)**S(2), x), x), x) - Simp((d*cos(e + f*x))**(n + S(1))*(a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(a*d*f*(n + S(1))), x) + Simp(b*(d*cos(e + f*x))**(n + S(2))*(a + b*cos(e + f*x))**(m + S(1))*(m + n + S(2))*sin(e + f*x)/(a**S(2)*d**S(2)*f*(n + S(1))*(n + S(2))), x) def replacement2701(a, b, d, e, f, m, n, x): return Dist(S(1)/(a*b*d*(n + S(1))*(m + n + S(4))), Int((d*sin(e + f*x))**(n + S(1))*(a + b*sin(e + f*x))**m*Simp(a**S(2)*(n + S(1))*(n + S(2)) + a*b*(m + S(3))*sin(e + f*x) - b**S(2)*(m + n + S(2))*(m + n + S(4)) - (a**S(2)*(n + S(1))*(n + S(3)) - b**S(2)*(m + n + S(3))*(m + n + S(4)))*sin(e + f*x)**S(2), x), x), x) + Simp((d*sin(e + f*x))**(n + S(1))*(a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(a*d*f*(n + S(1))), x) - Simp((d*sin(e + f*x))**(n + S(2))*(a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(b*d**S(2)*f*(m + n + S(4))), x) def replacement2702(a, b, d, e, f, m, n, x): return Dist(S(1)/(a*b*d*(n + S(1))*(m + n + S(4))), Int((d*cos(e + f*x))**(n + S(1))*(a + b*cos(e + f*x))**m*Simp(a**S(2)*(n + S(1))*(n + S(2)) + a*b*(m + S(3))*cos(e + f*x) - b**S(2)*(m + n + S(2))*(m + n + S(4)) - (a**S(2)*(n + S(1))*(n + S(3)) - b**S(2)*(m + n + S(3))*(m + n + S(4)))*cos(e + f*x)**S(2), x), x), x) - Simp((d*cos(e + f*x))**(n + S(1))*(a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(a*d*f*(n + S(1))), x) + Simp((d*cos(e + f*x))**(n + S(2))*(a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(b*d**S(2)*f*(m + n + S(4))), x) def replacement2703(a, b, d, e, f, m, n, x): return -Dist(S(1)/(b**S(2)*(m + n + S(3))*(m + n + S(4))), Int((d*sin(e + f*x))**n*(a + b*sin(e + f*x))**m*Simp(a**S(2)*(n + S(1))*(n + S(3)) + a*b*m*sin(e + f*x) - b**S(2)*(m + n + S(3))*(m + n + S(4)) - (a**S(2)*(n + S(2))*(n + S(3)) - b**S(2)*(m + n + S(3))*(m + n + S(5)))*sin(e + f*x)**S(2), x), x), x) - Simp((d*sin(e + f*x))**(n + S(2))*(a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(b*d**S(2)*f*(m + n + S(4))), x) + Simp(a*(d*sin(e + f*x))**(n + S(1))*(a + b*sin(e + f*x))**(m + S(1))*(n + S(3))*cos(e + f*x)/(b**S(2)*d*f*(m + n + S(3))*(m + n + S(4))), x) def replacement2704(a, b, d, e, f, m, n, x): return -Dist(S(1)/(b**S(2)*(m + n + S(3))*(m + n + S(4))), Int((d*cos(e + f*x))**n*(a + b*cos(e + f*x))**m*Simp(a**S(2)*(n + S(1))*(n + S(3)) + a*b*m*cos(e + f*x) - b**S(2)*(m + n + S(3))*(m + n + S(4)) - (a**S(2)*(n + S(2))*(n + S(3)) - b**S(2)*(m + n + S(3))*(m + n + S(5)))*cos(e + f*x)**S(2), x), x), x) + Simp((d*cos(e + f*x))**(n + S(2))*(a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(b*d**S(2)*f*(m + n + S(4))), x) - Simp(a*(d*cos(e + f*x))**(n + S(1))*(a + b*cos(e + f*x))**(m + S(1))*(n + S(3))*sin(e + f*x)/(b**S(2)*d*f*(m + n + S(3))*(m + n + S(4))), x) def replacement2705(a, b, d, e, f, m, n, x): return Dist(S(1)/(a**S(2)*b**S(2)*d**S(2)*(n + S(1))*(n + S(2))*(m + n + S(5))*(m + n + S(6))), Int((d*sin(e + f*x))**(n + S(2))*(a + b*sin(e + f*x))**m*Simp(a**S(4)*(n + S(1))*(n + S(2))*(n + S(3))*(n + S(5)) - a**S(2)*b**S(2)*(n + S(2))*(S(2)*n + S(1))*(m + n + S(5))*(m + n + S(6)) + a*b*m*(a**S(2)*(n + S(1))*(n + S(2)) - b**S(2)*(m + n + S(5))*(m + n + S(6)))*sin(e + f*x) + b**S(4)*(m + n + S(2))*(m + n + S(3))*(m + n + S(5))*(m + n + S(6)) - (a**S(4)*(n + S(1))*(n + S(2))*(n + S(4))*(n + S(5)) - a**S(2)*b**S(2)*(n + S(1))*(n + S(2))*(m + n + S(5))*(S(2)*m + S(2)*n + S(13)) + b**S(4)*(m + n + S(2))*(m + n + S(4))*(m + n + S(5))*(m + n + S(6)))*sin(e + f*x)**S(2), x), x), x) + Simp((d*sin(e + f*x))**(n + S(1))*(a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(a*d*f*(n + S(1))), x) + Simp((d*sin(e + f*x))**(n + S(4))*(a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(b*d**S(4)*f*(m + n + S(6))), x) - Simp(b*(d*sin(e + f*x))**(n + S(2))*(a + b*sin(e + f*x))**(m + S(1))*(m + n + S(2))*cos(e + f*x)/(a**S(2)*d**S(2)*f*(n + S(1))*(n + S(2))), x) - Simp(a*(d*sin(e + f*x))**(n + S(3))*(a + b*sin(e + f*x))**(m + S(1))*(n + S(5))*cos(e + f*x)/(b**S(2)*d**S(3)*f*(m + n + S(5))*(m + n + S(6))), x) def replacement2706(a, b, d, e, f, m, n, x): return Dist(S(1)/(a**S(2)*b**S(2)*d**S(2)*(n + S(1))*(n + S(2))*(m + n + S(5))*(m + n + S(6))), Int((d*cos(e + f*x))**(n + S(2))*(a + b*cos(e + f*x))**m*Simp(a**S(4)*(n + S(1))*(n + S(2))*(n + S(3))*(n + S(5)) - a**S(2)*b**S(2)*(n + S(2))*(S(2)*n + S(1))*(m + n + S(5))*(m + n + S(6)) + a*b*m*(a**S(2)*(n + S(1))*(n + S(2)) - b**S(2)*(m + n + S(5))*(m + n + S(6)))*cos(e + f*x) + b**S(4)*(m + n + S(2))*(m + n + S(3))*(m + n + S(5))*(m + n + S(6)) - (a**S(4)*(n + S(1))*(n + S(2))*(n + S(4))*(n + S(5)) - a**S(2)*b**S(2)*(n + S(1))*(n + S(2))*(m + n + S(5))*(S(2)*m + S(2)*n + S(13)) + b**S(4)*(m + n + S(2))*(m + n + S(4))*(m + n + S(5))*(m + n + S(6)))*cos(e + f*x)**S(2), x), x), x) - Simp((d*cos(e + f*x))**(n + S(1))*(a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(a*d*f*(n + S(1))), x) - Simp((d*cos(e + f*x))**(n + S(4))*(a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(b*d**S(4)*f*(m + n + S(6))), x) + Simp(b*(d*cos(e + f*x))**(n + S(2))*(a + b*cos(e + f*x))**(m + S(1))*(m + n + S(2))*sin(e + f*x)/(a**S(2)*d**S(2)*f*(n + S(1))*(n + S(2))), x) + Simp(a*(d*cos(e + f*x))**(n + S(3))*(a + b*cos(e + f*x))**(m + S(1))*(n + S(5))*sin(e + f*x)/(b**S(2)*d**S(3)*f*(m + n + S(5))*(m + n + S(6))), x) def replacement2707(a, b, d, e, f, m, n, p, x): return Int(ExpandTrig((d*sin(e + f*x))**n*(S(1) - sin(e + f*x)**S(2))**(p/S(2))*(a + b*sin(e + f*x))**m, x), x) def replacement2708(a, b, d, e, f, m, n, p, x): return Int(ExpandTrig((d*cos(e + f*x))**n*(S(1) - cos(e + f*x)**S(2))**(p/S(2))*(a + b*cos(e + f*x))**m, x), x) def replacement2709(a, b, e, f, g, n, p, x): return Int(ExpandTrig((g*cos(e + f*x))**p, sin(e + f*x)**n/(a + b*sin(e + f*x)), x), x) def replacement2710(a, b, e, f, g, n, p, x): return Int(ExpandTrig((g*sin(e + f*x))**p, cos(e + f*x)**n/(a + b*cos(e + f*x)), x), x) def replacement2711(a, b, d, e, f, g, n, p, x): return Dist(g**S(2)/a, Int((d*sin(e + f*x))**n*(g*cos(e + f*x))**(p + S(-2)), x), x) - Dist(b*g**S(2)/(a**S(2)*d), Int((d*sin(e + f*x))**(n + S(1))*(g*cos(e + f*x))**(p + S(-2)), x), x) - Dist(g**S(2)*(a**S(2) - b**S(2))/(a**S(2)*d**S(2)), Int((d*sin(e + f*x))**(n + S(2))*(g*cos(e + f*x))**(p + S(-2))/(a + b*sin(e + f*x)), x), x) def replacement2712(a, b, d, e, f, g, n, p, x): return Dist(g**S(2)/a, Int((d*cos(e + f*x))**n*(g*sin(e + f*x))**(p + S(-2)), x), x) - Dist(b*g**S(2)/(a**S(2)*d), Int((d*cos(e + f*x))**(n + S(1))*(g*sin(e + f*x))**(p + S(-2)), x), x) - Dist(g**S(2)*(a**S(2) - b**S(2))/(a**S(2)*d**S(2)), Int((d*cos(e + f*x))**(n + S(2))*(g*sin(e + f*x))**(p + S(-2))/(a + b*cos(e + f*x)), x), x) def replacement2713(a, b, d, e, f, g, n, p, x): return Dist(g**S(2)/(a*b), Int((d*sin(e + f*x))**n*(g*cos(e + f*x))**(p + S(-2))*(-a*sin(e + f*x) + b), x), x) + Dist(g**S(2)*(a**S(2) - b**S(2))/(a*b*d), Int((d*sin(e + f*x))**(n + S(1))*(g*cos(e + f*x))**(p + S(-2))/(a + b*sin(e + f*x)), x), x) def replacement2714(a, b, d, e, f, g, n, p, x): return Dist(g**S(2)/(a*b), Int((d*cos(e + f*x))**n*(g*sin(e + f*x))**(p + S(-2))*(-a*cos(e + f*x) + b), x), x) + Dist(g**S(2)*(a**S(2) - b**S(2))/(a*b*d), Int((d*cos(e + f*x))**(n + S(1))*(g*sin(e + f*x))**(p + S(-2))/(a + b*cos(e + f*x)), x), x) def replacement2715(a, b, d, e, f, g, n, p, x): return Dist(g**S(2)/b**S(2), Int((d*sin(e + f*x))**n*(g*cos(e + f*x))**(p + S(-2))*(a - b*sin(e + f*x)), x), x) - Dist(g**S(2)*(a**S(2) - b**S(2))/b**S(2), Int((d*sin(e + f*x))**n*(g*cos(e + f*x))**(p + S(-2))/(a + b*sin(e + f*x)), x), x) def replacement2716(a, b, d, e, f, g, n, p, x): return Dist(g**S(2)/b**S(2), Int((d*cos(e + f*x))**n*(g*sin(e + f*x))**(p + S(-2))*(a - b*cos(e + f*x)), x), x) - Dist(g**S(2)*(a**S(2) - b**S(2))/b**S(2), Int((d*cos(e + f*x))**n*(g*sin(e + f*x))**(p + S(-2))/(a + b*cos(e + f*x)), x), x) def replacement2717(a, b, d, e, f, g, n, p, x): return Dist(a*d**S(2)/(a**S(2) - b**S(2)), Int((d*sin(e + f*x))**(n + S(-2))*(g*cos(e + f*x))**p, x), x) - Dist(b*d/(a**S(2) - b**S(2)), Int((d*sin(e + f*x))**(n + S(-1))*(g*cos(e + f*x))**p, x), x) - Dist(a**S(2)*d**S(2)/(g**S(2)*(a**S(2) - b**S(2))), Int((d*sin(e + f*x))**(n + S(-2))*(g*cos(e + f*x))**(p + S(2))/(a + b*sin(e + f*x)), x), x) def replacement2718(a, b, d, e, f, g, n, p, x): return Dist(a*d**S(2)/(a**S(2) - b**S(2)), Int((d*cos(e + f*x))**(n + S(-2))*(g*sin(e + f*x))**p, x), x) - Dist(b*d/(a**S(2) - b**S(2)), Int((d*cos(e + f*x))**(n + S(-1))*(g*sin(e + f*x))**p, x), x) - Dist(a**S(2)*d**S(2)/(g**S(2)*(a**S(2) - b**S(2))), Int((d*cos(e + f*x))**(n + S(-2))*(g*sin(e + f*x))**(p + S(2))/(a + b*cos(e + f*x)), x), x) def replacement2719(a, b, d, e, f, g, n, p, x): return -Dist(d/(a**S(2) - b**S(2)), Int((d*sin(e + f*x))**(n + S(-1))*(g*cos(e + f*x))**p*(-a*sin(e + f*x) + b), x), x) + Dist(a*b*d/(g**S(2)*(a**S(2) - b**S(2))), Int((d*sin(e + f*x))**(n + S(-1))*(g*cos(e + f*x))**(p + S(2))/(a + b*sin(e + f*x)), x), x) def replacement2720(a, b, d, e, f, g, n, p, x): return -Dist(d/(a**S(2) - b**S(2)), Int((d*cos(e + f*x))**(n + S(-1))*(g*sin(e + f*x))**p*(-a*cos(e + f*x) + b), x), x) + Dist(a*b*d/(g**S(2)*(a**S(2) - b**S(2))), Int((d*cos(e + f*x))**(n + S(-1))*(g*sin(e + f*x))**(p + S(2))/(a + b*cos(e + f*x)), x), x) def replacement2721(a, b, d, e, f, g, n, p, x): return -Dist(b**S(2)/(g**S(2)*(a**S(2) - b**S(2))), Int((d*sin(e + f*x))**n*(g*cos(e + f*x))**(p + S(2))/(a + b*sin(e + f*x)), x), x) + Dist(S(1)/(a**S(2) - b**S(2)), Int((d*sin(e + f*x))**n*(g*cos(e + f*x))**p*(a - b*sin(e + f*x)), x), x) def replacement2722(a, b, d, e, f, g, n, p, x): return -Dist(b**S(2)/(g**S(2)*(a**S(2) - b**S(2))), Int((d*cos(e + f*x))**n*(g*sin(e + f*x))**(p + S(2))/(a + b*cos(e + f*x)), x), x) + Dist(S(1)/(a**S(2) - b**S(2)), Int((d*cos(e + f*x))**n*(g*sin(e + f*x))**p*(a - b*cos(e + f*x)), x), x) def replacement2723(a, b, e, f, g, x): return Dist(-S(4)*sqrt(S(2))*g/f, Subst(Int(x**S(2)/(sqrt(S(1) - x**S(4)/g**S(2))*(g**S(2)*(a + b) + x**S(4)*(a - b))), x), x, sqrt(g*cos(e + f*x))/sqrt(sin(e + f*x) + S(1))), x) def replacement2724(a, b, e, f, g, x): return Dist(S(4)*sqrt(S(2))*g/f, Subst(Int(x**S(2)/(sqrt(S(1) - x**S(4)/g**S(2))*(g**S(2)*(a + b) + x**S(4)*(a - b))), x), x, sqrt(g*sin(e + f*x))/sqrt(cos(e + f*x) + S(1))), x) def replacement2725(a, b, d, e, f, g, x): return Dist(sqrt(sin(e + f*x))/sqrt(d*sin(e + f*x)), Int(sqrt(g*cos(e + f*x))/((a + b*sin(e + f*x))*sqrt(sin(e + f*x))), x), x) def replacement2726(a, b, d, e, f, g, x): return Dist(sqrt(cos(e + f*x))/sqrt(d*cos(e + f*x)), Int(sqrt(g*sin(e + f*x))/((a + b*cos(e + f*x))*sqrt(cos(e + f*x))), x), x) def With2727(a, b, d, e, f, x): q = Rt(-a**S(2) + b**S(2), S(2)) return -Dist(S(2)*sqrt(S(2))*d*(b - q)/(f*q), Subst(Int(S(1)/(sqrt(S(1) - x**S(4)/d**S(2))*(a*x**S(2) + d*(b - q))), x), x, sqrt(d*sin(e + f*x))/sqrt(cos(e + f*x) + S(1))), x) + Dist(S(2)*sqrt(S(2))*d*(b + q)/(f*q), Subst(Int(S(1)/(sqrt(S(1) - x**S(4)/d**S(2))*(a*x**S(2) + d*(b + q))), x), x, sqrt(d*sin(e + f*x))/sqrt(cos(e + f*x) + S(1))), x) def With2728(a, b, d, e, f, x): q = Rt(-a**S(2) + b**S(2), S(2)) return Dist(S(2)*sqrt(S(2))*d*(b - q)/(f*q), Subst(Int(S(1)/(sqrt(S(1) - x**S(4)/d**S(2))*(a*x**S(2) + d*(b - q))), x), x, sqrt(d*cos(e + f*x))/sqrt(sin(e + f*x) + S(1))), x) + Dist(-S(2)*sqrt(S(2))*d*(b + q)/(f*q), Subst(Int(S(1)/(sqrt(S(1) - x**S(4)/d**S(2))*(a*x**S(2) + d*(b + q))), x), x, sqrt(d*cos(e + f*x))/sqrt(sin(e + f*x) + S(1))), x) def replacement2729(a, b, d, e, f, g, x): return Dist(sqrt(cos(e + f*x))/sqrt(g*cos(e + f*x)), Int(sqrt(d*sin(e + f*x))/((a + b*sin(e + f*x))*sqrt(cos(e + f*x))), x), x) def replacement2730(a, b, d, e, f, g, x): return Dist(sqrt(sin(e + f*x))/sqrt(g*sin(e + f*x)), Int(sqrt(d*cos(e + f*x))/((a + b*cos(e + f*x))*sqrt(sin(e + f*x))), x), x) def replacement2731(a, b, d, e, f, g, n, p, x): return Dist(d/b, Int((d*sin(e + f*x))**(n + S(-1))*(g*cos(e + f*x))**p, x), x) - Dist(a*d/b, Int((d*sin(e + f*x))**(n + S(-1))*(g*cos(e + f*x))**p/(a + b*sin(e + f*x)), x), x) def replacement2732(a, b, d, e, f, g, n, p, x): return Dist(d/b, Int((d*cos(e + f*x))**(n + S(-1))*(g*sin(e + f*x))**p, x), x) - Dist(a*d/b, Int((d*cos(e + f*x))**(n + S(-1))*(g*sin(e + f*x))**p/(a + b*cos(e + f*x)), x), x) def replacement2733(a, b, d, e, f, g, n, p, x): return Dist(S(1)/a, Int((d*sin(e + f*x))**n*(g*cos(e + f*x))**p, x), x) - Dist(b/(a*d), Int((d*sin(e + f*x))**(n + S(1))*(g*cos(e + f*x))**p/(a + b*sin(e + f*x)), x), x) def replacement2734(a, b, d, e, f, g, n, p, x): return Dist(S(1)/a, Int((d*cos(e + f*x))**n*(g*sin(e + f*x))**p, x), x) - Dist(b/(a*d), Int((d*cos(e + f*x))**(n + S(1))*(g*sin(e + f*x))**p/(a + b*cos(e + f*x)), x), x) def replacement2735(a, b, d, e, f, g, m, n, p, x): return Int(ExpandTrig((g*cos(e + f*x))**p, (d*sin(e + f*x))**n*(a + b*sin(e + f*x))**m, x), x) def replacement2736(a, b, d, e, f, g, m, n, p, x): return Int(ExpandTrig((g*sin(e + f*x))**p, (d*cos(e + f*x))**n*(a + b*cos(e + f*x))**m, x), x) def replacement2737(a, b, d, e, f, g, m, n, p, x): return Dist(g**S(2)/a, Int((d*sin(e + f*x))**n*(g*cos(e + f*x))**(p + S(-2))*(a + b*sin(e + f*x))**(m + S(1)), x), x) - Dist(b*g**S(2)/(a**S(2)*d), Int((d*sin(e + f*x))**(n + S(1))*(g*cos(e + f*x))**(p + S(-2))*(a + b*sin(e + f*x))**(m + S(1)), x), x) - Dist(g**S(2)*(a**S(2) - b**S(2))/(a**S(2)*d**S(2)), Int((d*sin(e + f*x))**(n + S(2))*(g*cos(e + f*x))**(p + S(-2))*(a + b*sin(e + f*x))**m, x), x) def replacement2738(a, b, d, e, f, g, m, n, p, x): return Dist(g**S(2)/a, Int((d*cos(e + f*x))**n*(g*sin(e + f*x))**(p + S(-2))*(a + b*cos(e + f*x))**(m + S(1)), x), x) - Dist(b*g**S(2)/(a**S(2)*d), Int((d*cos(e + f*x))**(n + S(1))*(g*sin(e + f*x))**(p + S(-2))*(a + b*cos(e + f*x))**(m + S(1)), x), x) - Dist(g**S(2)*(a**S(2) - b**S(2))/(a**S(2)*d**S(2)), Int((d*cos(e + f*x))**(n + S(2))*(g*sin(e + f*x))**(p + S(-2))*(a + b*cos(e + f*x))**m, x), x) def replacement2739(a, b, c, d, e, f, m, n, p, x): return Dist(a**(S(2)*m), Int((a - b*sin(e + f*x))**(-m)*(c + d*sin(e + f*x))**n, x), x) def replacement2740(a, b, c, d, e, f, m, n, p, x): return Dist(a**(S(2)*m), Int((a - b*cos(e + f*x))**(-m)*(c + d*cos(e + f*x))**n, x), x) def replacement2741(a, b, c, d, e, f, g, m, n, p, x): return Dist((a/g)**(S(2)*m), Int((g*cos(e + f*x))**(S(2)*m + p)*(a - b*sin(e + f*x))**(-m)*(c + d*sin(e + f*x))**n, x), x) def replacement2742(a, b, c, d, e, f, g, m, n, p, x): return Dist((a/g)**(S(2)*m), Int((g*sin(e + f*x))**(S(2)*m + p)*(a - b*cos(e + f*x))**(-m)*(c + d*cos(e + f*x))**n, x), x) def replacement2743(a, b, c, d, e, f, m, n, x): return Dist(b**(S(-2)), Int((a - b*sin(e + f*x))*(a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n, x), x) def replacement2744(a, b, c, d, e, f, m, n, x): return Dist(b**(S(-2)), Int((a - b*cos(e + f*x))*(a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n, x), x) def replacement2745(a, b, c, d, e, f, m, n, p, x): return Dist(a**m*cos(e + f*x)/(f*sqrt(S(1) - sin(e + f*x))*sqrt(sin(e + f*x) + S(1))), Subst(Int((S(1) - b*x/a)**(p/S(2) + S(-1)/2)*(S(1) + b*x/a)**(m + p/S(2) + S(-1)/2)*(c + d*x)**n, x), x, sin(e + f*x)), x) def replacement2746(a, b, c, d, e, f, m, n, p, x): return -Dist(a**m*sin(e + f*x)/(f*sqrt(S(1) - cos(e + f*x))*sqrt(cos(e + f*x) + S(1))), Subst(Int((S(1) - b*x/a)**(p/S(2) + S(-1)/2)*(S(1) + b*x/a)**(m + p/S(2) + S(-1)/2)*(c + d*x)**n, x), x, cos(e + f*x)), x) def replacement2747(a, b, c, d, e, f, m, n, p, x): return Dist(a**(S(2) - p)*cos(e + f*x)/(f*sqrt(a - b*sin(e + f*x))*sqrt(a + b*sin(e + f*x))), Subst(Int((a - b*x)**(p/S(2) + S(-1)/2)*(a + b*x)**(m + p/S(2) + S(-1)/2)*(c + d*x)**n, x), x, sin(e + f*x)), x) def replacement2748(a, b, c, d, e, f, m, n, p, x): return -Dist(a**(S(2) - p)*sin(e + f*x)/(f*sqrt(a - b*cos(e + f*x))*sqrt(a + b*cos(e + f*x))), Subst(Int((a - b*x)**(p/S(2) + S(-1)/2)*(a + b*x)**(m + p/S(2) + S(-1)/2)*(c + d*x)**n, x), x, cos(e + f*x)), x) def replacement2749(a, b, c, d, e, f, g, m, n, p, x): return Int(ExpandTrig((g*cos(e + f*x))**p, (a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n, x), x) def replacement2750(a, b, c, d, e, f, g, m, n, p, x): return Int(ExpandTrig((g*sin(e + f*x))**p, (a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n, x), x) def replacement2751(a, b, c, d, e, f, g, m, n, p, x): return Dist(a**m*g*(g*cos(e + f*x))**(p + S(-1))*(S(1) - sin(e + f*x))**(S(1)/2 - p/S(2))*(sin(e + f*x) + S(1))**(S(1)/2 - p/S(2))/f, Subst(Int((S(1) - b*x/a)**(p/S(2) + S(-1)/2)*(S(1) + b*x/a)**(m + p/S(2) + S(-1)/2)*(c + d*x)**n, x), x, sin(e + f*x)), x) def replacement2752(a, b, c, d, e, f, g, m, n, p, x): return -Dist(a**m*g*(g*sin(e + f*x))**(p + S(-1))*(S(1) - cos(e + f*x))**(S(1)/2 - p/S(2))*(cos(e + f*x) + S(1))**(S(1)/2 - p/S(2))/f, Subst(Int((S(1) - b*x/a)**(p/S(2) + S(-1)/2)*(S(1) + b*x/a)**(m + p/S(2) + S(-1)/2)*(c + d*x)**n, x), x, cos(e + f*x)), x) def replacement2753(a, b, c, d, e, f, g, m, n, p, x): return Dist(g*(g*cos(e + f*x))**(p + S(-1))*(a - b*sin(e + f*x))**(S(1)/2 - p/S(2))*(a + b*sin(e + f*x))**(S(1)/2 - p/S(2))/f, Subst(Int((a - b*x)**(p/S(2) + S(-1)/2)*(a + b*x)**(m + p/S(2) + S(-1)/2)*(c + d*x)**n, x), x, sin(e + f*x)), x) def replacement2754(a, b, c, d, e, f, g, m, n, p, x): return -Dist(g*(g*sin(e + f*x))**(p + S(-1))*(a - b*cos(e + f*x))**(S(1)/2 - p/S(2))*(a + b*cos(e + f*x))**(S(1)/2 - p/S(2))/f, Subst(Int((a - b*x)**(p/S(2) + S(-1)/2)*(a + b*x)**(m + p/S(2) + S(-1)/2)*(c + d*x)**n, x), x, cos(e + f*x)), x) def replacement2755(a, b, c, d, e, f, m, n, x): return Int((S(1) - sin(e + f*x)**S(2))*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n, x) def replacement2756(a, b, c, d, e, f, m, n, x): return Int((S(1) - cos(e + f*x)**S(2))*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n, x) def replacement2757(a, b, c, d, e, f, m, n, p, x): return Int(ExpandTrig((S(1) - sin(e + f*x)**S(2))**(p/S(2))*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n, x), x) def replacement2758(a, b, c, d, e, f, m, n, p, x): return Int(ExpandTrig((S(1) - cos(e + f*x)**S(2))**(p/S(2))*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n, x), x) def replacement2759(a, b, c, d, e, f, g, m, n, p, x): return Int(ExpandTrig((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n, x), x) def replacement2760(a, b, c, d, e, f, g, m, n, p, x): return Int(ExpandTrig((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n, x), x) def replacement2761(a, b, c, d, e, f, g, m, n, p, x): return Int((g*cos(e + f*x))**p*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n, x) def replacement2762(a, b, c, d, e, f, g, m, n, p, x): return Int((g*sin(e + f*x))**p*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n, x) def replacement2763(a, b, c, d, e, f, g, m, n, p, x): return Dist(g**(S(2)*IntPart(p))*(g/cos(e + f*x))**FracPart(p)*(g*cos(e + f*x))**FracPart(p), Int((g*cos(e + f*x))**(-p)*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n, x), x) def replacement2764(a, b, c, d, e, f, g, m, n, p, x): return Dist(g**(S(2)*IntPart(p))*(g/sin(e + f*x))**FracPart(p)*(g*sin(e + f*x))**FracPart(p), Int((g*sin(e + f*x))**(-p)*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n, x), x) def replacement2765(a, b, c, d, e, f, g, x): return Dist(g/d, Int(sqrt(a + b*sin(e + f*x))/sqrt(g*sin(e + f*x)), x), x) - Dist(c*g/d, Int(sqrt(a + b*sin(e + f*x))/(sqrt(g*sin(e + f*x))*(c + d*sin(e + f*x))), x), x) def replacement2766(a, b, c, d, e, f, g, x): return Dist(g/d, Int(sqrt(a + b*cos(e + f*x))/sqrt(g*cos(e + f*x)), x), x) - Dist(c*g/d, Int(sqrt(a + b*cos(e + f*x))/(sqrt(g*cos(e + f*x))*(c + d*cos(e + f*x))), x), x) def replacement2767(a, b, c, d, e, f, g, x): return Dist(b/d, Int(sqrt(g*sin(e + f*x))/sqrt(a + b*sin(e + f*x)), x), x) - Dist((-a*d + b*c)/d, Int(sqrt(g*sin(e + f*x))/(sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))), x), x) def replacement2768(a, b, c, d, e, f, g, x): return Dist(b/d, Int(sqrt(g*cos(e + f*x))/sqrt(a + b*cos(e + f*x)), x), x) - Dist((-a*d + b*c)/d, Int(sqrt(g*cos(e + f*x))/(sqrt(a + b*cos(e + f*x))*(c + d*cos(e + f*x))), x), x) def replacement2769(a, b, c, d, e, f, g, x): return Dist(-S(2)*b/f, Subst(Int(S(1)/(a*d + b*c + c*g*x**S(2)), x), x, b*cos(e + f*x)/(sqrt(g*sin(e + f*x))*sqrt(a + b*sin(e + f*x)))), x) def replacement2770(a, b, c, d, e, f, g, x): return Dist(S(2)*b/f, Subst(Int(S(1)/(a*d + b*c + c*g*x**S(2)), x), x, b*sin(e + f*x)/(sqrt(g*cos(e + f*x))*sqrt(a + b*cos(e + f*x)))), x) def replacement2771(a, b, c, d, e, f, x): return -Simp(sqrt(a + b)*EllipticE(asin(cos(e + f*x)/(sin(e + f*x) + S(1))), -(a - b)/(a + b))/(c*f), x) def replacement2772(a, b, c, d, e, f, x): return Simp(sqrt(a + b)*EllipticE(asin(sin(e + f*x)/(cos(e + f*x) + S(1))), -(a - b)/(a + b))/(c*f), x) def replacement2773(a, b, c, d, e, f, g, x): return -Simp(sqrt(d*sin(e + f*x)/(c + d*sin(e + f*x)))*sqrt(a + b*sin(e + f*x))*EllipticE(asin(c*cos(e + f*x)/(c + d*sin(e + f*x))), (-a*d + b*c)/(a*d + b*c))/(d*f*sqrt(g*sin(e + f*x))*sqrt(c**S(2)*(a + b*sin(e + f*x))/((c + d*sin(e + f*x))*(a*c + b*d)))), x) def replacement2774(a, b, c, d, e, f, g, x): return Simp(sqrt(d*cos(e + f*x)/(c + d*cos(e + f*x)))*sqrt(a + b*cos(e + f*x))*EllipticE(asin(c*sin(e + f*x)/(c + d*cos(e + f*x))), (-a*d + b*c)/(a*d + b*c))/(d*f*sqrt(g*cos(e + f*x))*sqrt(c**S(2)*(a + b*cos(e + f*x))/((c + d*cos(e + f*x))*(a*c + b*d)))), x) def replacement2775(a, b, c, d, e, f, g, x): return Dist(a/c, Int(S(1)/(sqrt(g*sin(e + f*x))*sqrt(a + b*sin(e + f*x))), x), x) + Dist((-a*d + b*c)/(c*g), Int(sqrt(g*sin(e + f*x))/(sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))), x), x) def replacement2776(a, b, c, d, e, f, g, x): return Dist(a/c, Int(S(1)/(sqrt(g*cos(e + f*x))*sqrt(a + b*cos(e + f*x))), x), x) + Dist((-a*d + b*c)/(c*g), Int(sqrt(g*cos(e + f*x))/(sqrt(a + b*cos(e + f*x))*(c + d*cos(e + f*x))), x), x) def replacement2777(a, b, c, d, e, f, x): return Dist(S(1)/c, Int(sqrt(a + b*sin(e + f*x))/sin(e + f*x), x), x) - Dist(d/c, Int(sqrt(a + b*sin(e + f*x))/(c + d*sin(e + f*x)), x), x) def replacement2778(a, b, c, d, e, f, x): return Dist(S(1)/c, Int(sqrt(a + b*cos(e + f*x))/cos(e + f*x), x), x) - Dist(d/c, Int(sqrt(a + b*cos(e + f*x))/(c + d*cos(e + f*x)), x), x) def replacement2779(a, b, c, d, e, f, x): return Dist(a/c, Int(S(1)/(sqrt(a + b*sin(e + f*x))*sin(e + f*x)), x), x) + Dist((-a*d + b*c)/c, Int(S(1)/(sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))), x), x) def replacement2780(a, b, c, d, e, f, x): return Dist(a/c, Int(S(1)/(sqrt(a + b*cos(e + f*x))*cos(e + f*x)), x), x) + Dist((-a*d + b*c)/c, Int(S(1)/(sqrt(a + b*cos(e + f*x))*(c + d*cos(e + f*x))), x), x) def replacement2781(a, b, c, d, e, f, g, x): return -Dist(a*g/(-a*d + b*c), Int(S(1)/(sqrt(g*sin(e + f*x))*sqrt(a + b*sin(e + f*x))), x), x) + Dist(c*g/(-a*d + b*c), Int(sqrt(a + b*sin(e + f*x))/(sqrt(g*sin(e + f*x))*(c + d*sin(e + f*x))), x), x) def replacement2782(a, b, c, d, e, f, g, x): return -Dist(a*g/(-a*d + b*c), Int(S(1)/(sqrt(g*cos(e + f*x))*sqrt(a + b*cos(e + f*x))), x), x) + Dist(c*g/(-a*d + b*c), Int(sqrt(a + b*cos(e + f*x))/(sqrt(g*cos(e + f*x))*(c + d*cos(e + f*x))), x), x) def replacement2783(a, b, c, d, e, f, g, x): return Simp(S(2)*sqrt(-S(1)/tan(e + f*x)**S(2))*sqrt(g*sin(e + f*x))*sqrt((a/sin(e + f*x) + b)/(a + b))*EllipticPi(S(2)*c/(c + d), asin(sqrt(S(2))*sqrt(S(1) - S(1)/sin(e + f*x))/S(2)), S(2)*a/(a + b))*tan(e + f*x)/(f*sqrt(a + b*sin(e + f*x))*(c + d)), x) def replacement2784(a, b, c, d, e, f, g, x): return Simp(-S(2)*sqrt(-tan(e + f*x)**S(2))*sqrt(g*cos(e + f*x))*sqrt((a/cos(e + f*x) + b)/(a + b))*EllipticPi(S(2)*c/(c + d), asin(sqrt(S(2))*sqrt(S(1) - S(1)/cos(e + f*x))/S(2)), S(2)*a/(a + b))/(f*sqrt(a + b*cos(e + f*x))*(c + d)*tan(e + f*x)), x) def replacement2785(a, b, c, d, e, f, g, x): return Dist(b/(-a*d + b*c), Int(S(1)/(sqrt(g*sin(e + f*x))*sqrt(a + b*sin(e + f*x))), x), x) - Dist(d/(-a*d + b*c), Int(sqrt(a + b*sin(e + f*x))/(sqrt(g*sin(e + f*x))*(c + d*sin(e + f*x))), x), x) def replacement2786(a, b, c, d, e, f, g, x): return Dist(b/(-a*d + b*c), Int(S(1)/(sqrt(g*cos(e + f*x))*sqrt(a + b*cos(e + f*x))), x), x) - Dist(d/(-a*d + b*c), Int(sqrt(a + b*cos(e + f*x))/(sqrt(g*cos(e + f*x))*(c + d*cos(e + f*x))), x), x) def replacement2787(a, b, c, d, e, f, g, x): return Dist(S(1)/c, Int(S(1)/(sqrt(g*sin(e + f*x))*sqrt(a + b*sin(e + f*x))), x), x) - Dist(d/(c*g), Int(sqrt(g*sin(e + f*x))/(sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))), x), x) def replacement2788(a, b, c, d, e, f, g, x): return Dist(S(1)/c, Int(S(1)/(sqrt(g*cos(e + f*x))*sqrt(a + b*cos(e + f*x))), x), x) - Dist(d/(c*g), Int(sqrt(g*cos(e + f*x))/(sqrt(a + b*cos(e + f*x))*(c + d*cos(e + f*x))), x), x) def replacement2789(a, b, c, d, e, f, x): return Dist(S(1)/(c*(-a*d + b*c)), Int((-a*d + b*c - b*d*sin(e + f*x))/(sqrt(a + b*sin(e + f*x))*sin(e + f*x)), x), x) + Dist(d**S(2)/(c*(-a*d + b*c)), Int(sqrt(a + b*sin(e + f*x))/(c + d*sin(e + f*x)), x), x) def replacement2790(a, b, c, d, e, f, x): return Dist(S(1)/(c*(-a*d + b*c)), Int((-a*d + b*c - b*d*cos(e + f*x))/(sqrt(a + b*cos(e + f*x))*cos(e + f*x)), x), x) + Dist(d**S(2)/(c*(-a*d + b*c)), Int(sqrt(a + b*cos(e + f*x))/(c + d*cos(e + f*x)), x), x) def replacement2791(a, b, c, d, e, f, x): return Dist(S(1)/c, Int(S(1)/(sqrt(a + b*sin(e + f*x))*sin(e + f*x)), x), x) - Dist(d/c, Int(S(1)/(sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))), x), x) def replacement2792(a, b, c, d, e, f, x): return Dist(S(1)/c, Int(S(1)/(sqrt(a + b*cos(e + f*x))*cos(e + f*x)), x), x) - Dist(d/c, Int(S(1)/(sqrt(a + b*cos(e + f*x))*(c + d*cos(e + f*x))), x), x) def replacement2793(a, b, c, d, e, f, x): return Dist(S(1)/c, Int(sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))/sin(e + f*x), x), x) - Dist(d/c, Int(sqrt(a + b*sin(e + f*x))/sqrt(c + d*sin(e + f*x)), x), x) def replacement2794(a, b, c, d, e, f, x): return Dist(S(1)/c, Int(sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x))/cos(e + f*x), x), x) - Dist(d/c, Int(sqrt(a + b*cos(e + f*x))/sqrt(c + d*cos(e + f*x)), x), x) def replacement2795(a, b, c, d, e, f, x): return Dist(-S(2)*a/f, Subst(Int(S(1)/(-a*c*x**S(2) + S(1)), x), x, cos(e + f*x)/(sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x)))), x) def replacement2796(a, b, c, d, e, f, x): return Dist(S(2)*a/f, Subst(Int(S(1)/(-a*c*x**S(2) + S(1)), x), x, sin(e + f*x)/(sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x)))), x) def replacement2797(a, b, c, d, e, f, x): return Dist(a/c, Int(sqrt(c + d*sin(e + f*x))/(sqrt(a + b*sin(e + f*x))*sin(e + f*x)), x), x) + Dist((-a*d + b*c)/c, Int(S(1)/(sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))), x), x) def replacement2798(a, b, c, d, e, f, x): return Dist(a/c, Int(sqrt(c + d*cos(e + f*x))/(sqrt(a + b*cos(e + f*x))*cos(e + f*x)), x), x) + Dist((-a*d + b*c)/c, Int(S(1)/(sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x))), x), x) def replacement2799(a, b, c, d, e, f, x): return Simp(-S(2)*sqrt((-a*d + b*c)*(sin(e + f*x) + S(1))/((a + b*sin(e + f*x))*(c - d)))*sqrt(-(S(1) - sin(e + f*x))*(-a*d + b*c)/((a + b*sin(e + f*x))*(c + d)))*(a + b*sin(e + f*x))*EllipticPi(a*(c + d)/(c*(a + b)), asin(sqrt(c + d*sin(e + f*x))*Rt((a + b)/(c + d), S(2))/sqrt(a + b*sin(e + f*x))), (a - b)*(c + d)/((a + b)*(c - d)))/(c*f*Rt((a + b)/(c + d), S(2))*cos(e + f*x)), x) def replacement2800(a, b, c, d, e, f, x): return Simp(S(2)*sqrt((-a*d + b*c)*(cos(e + f*x) + S(1))/((a + b*cos(e + f*x))*(c - d)))*sqrt(-(S(1) - cos(e + f*x))*(-a*d + b*c)/((a + b*cos(e + f*x))*(c + d)))*(a + b*cos(e + f*x))*EllipticPi(a*(c + d)/(c*(a + b)), asin(sqrt(c + d*cos(e + f*x))*Rt((a + b)/(c + d), S(2))/sqrt(a + b*cos(e + f*x))), (a - b)*(c + d)/((a + b)*(c - d)))/(c*f*Rt((a + b)/(c + d), S(2))*sin(e + f*x)), x) def replacement2801(a, b, c, d, e, f, x): return Dist(cos(e + f*x)/(sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))), Int(S(1)/(sin(e + f*x)*cos(e + f*x)), x), x) def replacement2802(a, b, c, d, e, f, x): return Dist(sin(e + f*x)/(sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x))), Int(S(1)/(sin(e + f*x)*cos(e + f*x)), x), x) def replacement2803(a, b, c, d, e, f, x): return Dist(S(1)/a, Int(sqrt(a + b*sin(e + f*x))/(sqrt(c + d*sin(e + f*x))*sin(e + f*x)), x), x) - Dist(b/a, Int(S(1)/(sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))), x), x) def replacement2804(a, b, c, d, e, f, x): return Dist(S(1)/a, Int(sqrt(a + b*cos(e + f*x))/(sqrt(c + d*cos(e + f*x))*cos(e + f*x)), x), x) - Dist(b/a, Int(S(1)/(sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x))), x), x) def replacement2805(a, b, c, d, e, f, x): return Dist(sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))/cos(e + f*x), Int(S(1)/tan(e + f*x), x), x) def replacement2806(a, b, c, d, e, f, x): return Dist(sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x))/sin(e + f*x), Int(tan(e + f*x), x), x) def replacement2807(a, b, c, d, e, f, x): return Dist(c, Int(sqrt(a + b*sin(e + f*x))/(sqrt(c + d*sin(e + f*x))*sin(e + f*x)), x), x) + Dist(d, Int(sqrt(a + b*sin(e + f*x))/sqrt(c + d*sin(e + f*x)), x), x) def replacement2808(a, b, c, d, e, f, x): return Dist(c, Int(sqrt(a + b*cos(e + f*x))/(sqrt(c + d*cos(e + f*x))*cos(e + f*x)), x), x) + Dist(d, Int(sqrt(a + b*cos(e + f*x))/sqrt(c + d*cos(e + f*x)), x), x) def replacement2809(a, b, c, d, e, f, m, n, p, x): return Dist(a**n*c**n, Int((a + b*sin(e + f*x))**(m - n)*tan(e + f*x)**p, x), x) def replacement2810(a, b, c, d, e, f, m, n, p, x): return Dist(a**n*c**n, Int((a + b*cos(e + f*x))**(m - n)*(S(1)/tan(e + f*x))**p, x), x) def replacement2811(a, b, c, d, e, f, g, m, n, p, x): return Dist(sqrt(a - b*sin(e + f*x))*sqrt(a + b*sin(e + f*x))/(f*cos(e + f*x)), Subst(Int((g*x)**p*(a + b*x)**(m + S(-1)/2)*(c + d*x)**n/sqrt(a - b*x), x), x, sin(e + f*x)), x) def replacement2812(a, b, c, d, e, f, g, m, n, p, x): return -Dist(sqrt(a - b*cos(e + f*x))*sqrt(a + b*cos(e + f*x))/(f*sin(e + f*x)), Subst(Int((g*x)**p*(a + b*x)**(m + S(-1)/2)*(c + d*x)**n/sqrt(a - b*x), x), x, cos(e + f*x)), x) def replacement2813(a, b, c, d, e, f, g, m, n, p, x): return Int(ExpandTrig((g*sin(e + f*x))**p*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n, x), x) def replacement2814(a, b, c, d, e, f, g, m, n, p, x): return Int(ExpandTrig((g*cos(e + f*x))**p*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n, x), x) def replacement2815(a, b, c, d, e, f, g, m, n, p, x): return Int((g*sin(e + f*x))**p*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n, x) def replacement2816(a, b, c, d, e, f, g, m, n, p, x): return Int((g*cos(e + f*x))**p*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n, x) def replacement2817(a, b, c, d, e, f, g, m, n, p, x): return Dist(g**(m + n), Int((g*sin(e + f*x))**(-m - n + p)*(a*sin(e + f*x) + b)**m*(c*sin(e + f*x) + d)**n, x), x) def replacement2818(a, b, c, d, e, f, g, m, n, p, x): return Dist(g**(m + n), Int((g*cos(e + f*x))**(-m - n + p)*(a*cos(e + f*x) + b)**m*(c*cos(e + f*x) + d)**n, x), x) def replacement2819(a, b, c, d, e, f, g, m, n, p, x): return Dist((g/sin(e + f*x))**p*(g*sin(e + f*x))**p, Int((g/sin(e + f*x))**(-p)*(a + b/sin(e + f*x))**m*(c + d/sin(e + f*x))**n, x), x) def replacement2820(a, b, c, d, e, f, g, m, n, p, x): return Dist((g/cos(e + f*x))**p*(g*cos(e + f*x))**p, Int((g/cos(e + f*x))**(-p)*(a + b/cos(e + f*x))**m*(c + d/cos(e + f*x))**n, x), x) def replacement2821(a, b, c, d, e, f, g, m, n, p, x): return Dist(g**n, Int((g*sin(e + f*x))**(-n + p)*(a + b*sin(e + f*x))**m*(c*sin(e + f*x) + d)**n, x), x) def replacement2822(a, b, c, d, e, f, g, m, n, p, x): return Dist(g**n, Int((g*cos(e + f*x))**(-n + p)*(a + b*cos(e + f*x))**m*(c*cos(e + f*x) + d)**n, x), x) def replacement2823(a, b, c, d, e, f, m, n, p, x): return Int((c + d/sin(e + f*x))**n*(a/sin(e + f*x) + b)**m*(S(1)/sin(e + f*x))**(-m - p), x) def replacement2824(a, b, c, d, e, f, m, n, p, x): return Int((c + d/cos(e + f*x))**n*(a/cos(e + f*x) + b)**m*(S(1)/cos(e + f*x))**(-m - p), x) def replacement2825(a, b, c, d, e, f, g, m, n, p, x): return Dist((g*sin(e + f*x))**p*(S(1)/sin(e + f*x))**p, Int((c + d/sin(e + f*x))**n*(a/sin(e + f*x) + b)**m*(S(1)/sin(e + f*x))**(-m - p), x), x) def replacement2826(a, b, c, d, e, f, g, m, n, p, x): return Dist((g*cos(e + f*x))**p*(S(1)/cos(e + f*x))**p, Int((c + d/cos(e + f*x))**n*(a/cos(e + f*x) + b)**m*(S(1)/cos(e + f*x))**(-m - p), x), x) def replacement2827(a, b, c, d, e, f, g, m, n, p, x): return Dist((g*sin(e + f*x))**n*(c + d/sin(e + f*x))**n*(c*sin(e + f*x) + d)**(-n), Int((g*sin(e + f*x))**(-n + p)*(a + b*sin(e + f*x))**m*(c*sin(e + f*x) + d)**n, x), x) def replacement2828(a, b, c, d, e, f, g, m, n, p, x): return Dist((g*cos(e + f*x))**n*(c + d/cos(e + f*x))**n*(c*cos(e + f*x) + d)**(-n), Int((g*cos(e + f*x))**(-n + p)*(a + b*cos(e + f*x))**m*(c*cos(e + f*x) + d)**n, x), x) def replacement2829(a, b, c, d, e, f, g, m, n, p, x): return Dist(g**(m + n), Int((g/sin(e + f*x))**(-m - n + p)*(a/sin(e + f*x) + b)**m*(c/sin(e + f*x) + d)**n, x), x) def replacement2830(a, b, c, d, e, f, g, m, n, p, x): return Dist(g**(m + n), Int((g/cos(e + f*x))**(-m - n + p)*(a/cos(e + f*x) + b)**m*(c/cos(e + f*x) + d)**n, x), x) def replacement2831(a, b, c, d, e, f, g, m, n, p, x): return Dist((g/sin(e + f*x))**p*(g*sin(e + f*x))**p, Int((g*sin(e + f*x))**(-p)*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n, x), x) def replacement2832(a, b, c, d, e, f, g, m, n, p, x): return Dist((g/cos(e + f*x))**p*(g*cos(e + f*x))**p, Int((g*cos(e + f*x))**(-p)*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n, x), x) def replacement2833(a, b, c, d, e, f, g, m, n, p, x): return Dist(g**m, Int((g/sin(e + f*x))**(-m + p)*(c + d/sin(e + f*x))**n*(a/sin(e + f*x) + b)**m, x), x) def replacement2834(a, b, c, d, e, f, g, m, n, p, x): return Dist(g**m, Int((g/cos(e + f*x))**(-m + p)*(c + d/cos(e + f*x))**n*(a/cos(e + f*x) + b)**m, x), x) def replacement2835(a, b, c, d, e, f, m, n, p, x): return Int((a + b*sin(e + f*x))**m*(c*sin(e + f*x) + d)**n*sin(e + f*x)**(-n - p), x) def replacement2836(a, b, c, d, e, f, m, n, p, x): return Int((a + b*cos(e + f*x))**m*(c*cos(e + f*x) + d)**n*cos(e + f*x)**(-n - p), x) def replacement2837(a, b, c, d, e, f, g, m, n, p, x): return Dist((g/sin(e + f*x))**p*sin(e + f*x)**p, Int((a + b*sin(e + f*x))**m*(c*sin(e + f*x) + d)**n*sin(e + f*x)**(-n - p), x), x) def replacement2838(a, b, c, d, e, f, g, m, n, p, x): return Dist((g/cos(e + f*x))**p*cos(e + f*x)**p, Int((a + b*cos(e + f*x))**m*(c*cos(e + f*x) + d)**n*cos(e + f*x)**(-n - p), x), x) def replacement2839(a, b, c, d, e, f, g, m, n, p, x): return Dist((g/sin(e + f*x))**m*(a + b*sin(e + f*x))**m*(a/sin(e + f*x) + b)**(-m), Int((g/sin(e + f*x))**(-m + p)*(c + d/sin(e + f*x))**n*(a/sin(e + f*x) + b)**m, x), x) def replacement2840(a, b, c, d, e, f, g, m, n, p, x): return Dist((g/cos(e + f*x))**m*(a + b*cos(e + f*x))**m*(a/cos(e + f*x) + b)**(-m), Int((g/cos(e + f*x))**(-m + p)*(c + d/cos(e + f*x))**n*(a/cos(e + f*x) + b)**m, x), x) def replacement2841(A, B, a, b, e, f, m, n, x): return Int(ExpandTrig((A + B*sin(e + f*x))*(a + b*sin(e + f*x))**m*sin(e + f*x)**n, x), x) def replacement2842(A, B, a, b, e, f, m, n, x): return Int(ExpandTrig((A + B*cos(e + f*x))*(a + b*cos(e + f*x))**m*cos(e + f*x)**n, x), x) def replacement2843(A, B, a, b, c, d, e, f, m, n, x): return Dist(a**m*c**m, Int((A + B*sin(e + f*x))*(c + d*sin(e + f*x))**(-m + n)*cos(e + f*x)**(S(2)*m), x), x) def replacement2844(A, B, a, b, c, d, e, f, m, n, x): return Dist(a**m*c**m, Int((A + B*cos(e + f*x))*(c + d*cos(e + f*x))**(-m + n)*sin(e + f*x)**(S(2)*m), x), x) def replacement2845(A, B, a, b, c, d, e, f, m, x): return Int((a + b*sin(e + f*x))**m*(A*c + B*d*sin(e + f*x)**S(2) + (A*d + B*c)*sin(e + f*x)), x) def replacement2846(A, B, a, b, c, d, e, f, m, x): return Int((a + b*cos(e + f*x))**m*(A*c + B*d*cos(e + f*x)**S(2) + (A*d + B*c)*cos(e + f*x)), x) def replacement2847(A, B, a, b, c, d, e, f, x): return Dist((A*b + B*a)/(S(2)*a*b), Int(sqrt(a + b*sin(e + f*x))/sqrt(c + d*sin(e + f*x)), x), x) + Dist((A*d + B*c)/(S(2)*c*d), Int(sqrt(c + d*sin(e + f*x))/sqrt(a + b*sin(e + f*x)), x), x) def replacement2848(A, B, a, b, c, d, e, f, x): return Dist((A*b + B*a)/(S(2)*a*b), Int(sqrt(a + b*cos(e + f*x))/sqrt(c + d*cos(e + f*x)), x), x) + Dist((A*d + B*c)/(S(2)*c*d), Int(sqrt(c + d*cos(e + f*x))/sqrt(a + b*cos(e + f*x)), x), x) def replacement2849(A, B, a, b, c, d, e, f, m, n, x): return -Simp(B*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n*cos(e + f*x)/(f*(m + n + S(1))), x) def replacement2850(A, B, a, b, c, d, e, f, m, n, x): return Simp(B*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n*sin(e + f*x)/(f*(m + n + S(1))), x) def replacement2851(A, B, a, b, c, d, e, f, n, x): return Dist(B/d, Int(sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))**(n + S(1)), x), x) - Dist((-A*d + B*c)/d, Int(sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))**n, x), x) def replacement2852(A, B, a, b, c, d, e, f, n, x): return Dist(B/d, Int(sqrt(a + b*cos(e + f*x))*(c + d*cos(e + f*x))**(n + S(1)), x), x) - Dist((-A*d + B*c)/d, Int(sqrt(a + b*cos(e + f*x))*(c + d*cos(e + f*x))**n, x), x) def replacement2853(A, B, a, b, c, d, e, f, m, n, x): return Dist((A*b*(m + n + S(1)) + B*a*(m - n))/(a*b*(S(2)*m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n, x), x) + Simp((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n*(A*b - B*a)*cos(e + f*x)/(a*f*(S(2)*m + S(1))), x) def replacement2854(A, B, a, b, c, d, e, f, m, n, x): return Dist((A*b*(m + n + S(1)) + B*a*(m - n))/(a*b*(S(2)*m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n, x), x) - Simp((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n*(A*b - B*a)*sin(e + f*x)/(a*f*(S(2)*m + S(1))), x) def replacement2855(A, B, a, b, c, d, e, f, m, n, x): return -Dist((-A*d*(m + n + S(1)) + B*c*(m - n))/(d*(m + n + S(1))), Int((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n, x), x) - Simp(B*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n*cos(e + f*x)/(f*(m + n + S(1))), x) def replacement2856(A, B, a, b, c, d, e, f, m, n, x): return -Dist((-A*d*(m + n + S(1)) + B*c*(m - n))/(d*(m + n + S(1))), Int((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n, x), x) + Simp(B*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n*sin(e + f*x)/(f*(m + n + S(1))), x) def replacement2857(A, B, a, b, c, d, e, f, m, n, x): return Simp((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*(-A*d + B*c)*cos(e + f*x)/(f*(c**S(2) - d**S(2))*(n + S(1))), x) def replacement2858(A, B, a, b, c, d, e, f, m, n, x): return -Simp((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*(-A*d + B*c)*sin(e + f*x)/(f*(c**S(2) - d**S(2))*(n + S(1))), x) def replacement2859(A, B, a, b, c, d, e, f, m, n, x): return -Dist(b/(d*(n + S(1))*(a*d + b*c)), Int((a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**(n + S(1))*Simp(A*a*d*(m - n + S(-2)) - B*(a*c*(m + S(-1)) + b*d*(n + S(1))) - (A*b*d*(m + n + S(1)) - B*(-a*d*(n + S(1)) + b*c*m))*sin(e + f*x), x), x), x) - Simp(b**S(2)*(a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**(n + S(1))*(-A*d + B*c)*cos(e + f*x)/(d*f*(n + S(1))*(a*d + b*c)), x) def replacement2860(A, B, a, b, c, d, e, f, m, n, x): return -Dist(b/(d*(n + S(1))*(a*d + b*c)), Int((a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**(n + S(1))*Simp(A*a*d*(m - n + S(-2)) - B*(a*c*(m + S(-1)) + b*d*(n + S(1))) - (A*b*d*(m + n + S(1)) - B*(-a*d*(n + S(1)) + b*c*m))*cos(e + f*x), x), x), x) + Simp(b**S(2)*(a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**(n + S(1))*(-A*d + B*c)*sin(e + f*x)/(d*f*(n + S(1))*(a*d + b*c)), x) def replacement2861(A, B, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(d*(m + n + S(1))), Int((a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**n*Simp(A*a*d*(m + n + S(1)) + B*(a*c*(m + S(-1)) + b*d*(n + S(1))) + (A*b*d*(m + n + S(1)) - B*(-a*d*(S(2)*m + n) + b*c*m))*sin(e + f*x), x), x), x) - Simp(B*b*(a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**(n + S(1))*cos(e + f*x)/(d*f*(m + n + S(1))), x) def replacement2862(A, B, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(d*(m + n + S(1))), Int((a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**n*Simp(A*a*d*(m + n + S(1)) + B*(a*c*(m + S(-1)) + b*d*(n + S(1))) + (A*b*d*(m + n + S(1)) - B*(-a*d*(S(2)*m + n) + b*c*m))*cos(e + f*x), x), x), x) + Simp(B*b*(a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**(n + S(1))*sin(e + f*x)/(d*f*(m + n + S(1))), x) def replacement2863(A, B, a, b, c, d, e, f, m, n, x): return -Dist(S(1)/(a*b*(S(2)*m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**(n + S(-1))*Simp(A*(a*d*n - b*c*(m + S(1))) - B*(a*c*m + b*d*n) - d*(A*b*(m + n + S(1)) + B*a*(m - n))*sin(e + f*x), x), x), x) + Simp((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n*(A*b - B*a)*cos(e + f*x)/(a*f*(S(2)*m + S(1))), x) def replacement2864(A, B, a, b, c, d, e, f, m, n, x): return -Dist(S(1)/(a*b*(S(2)*m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**(n + S(-1))*Simp(A*(a*d*n - b*c*(m + S(1))) - B*(a*c*m + b*d*n) - d*(A*b*(m + n + S(1)) + B*a*(m - n))*cos(e + f*x), x), x), x) - Simp((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n*(A*b - B*a)*sin(e + f*x)/(a*f*(S(2)*m + S(1))), x) def replacement2865(A, B, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(a*(S(2)*m + S(1))*(-a*d + b*c)), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n*Simp(A*(-a*d*(S(2)*m + n + S(2)) + b*c*(m + S(1))) + B*(a*c*m + b*d*(n + S(1))) + d*(A*b - B*a)*(m + n + S(2))*sin(e + f*x), x), x), x) + Simp(b*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*(A*b - B*a)*cos(e + f*x)/(a*f*(S(2)*m + S(1))*(-a*d + b*c)), x) def replacement2866(A, B, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(a*(S(2)*m + S(1))*(-a*d + b*c)), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n*Simp(A*(-a*d*(S(2)*m + n + S(2)) + b*c*(m + S(1))) + B*(a*c*m + b*d*(n + S(1))) + d*(A*b - B*a)*(m + n + S(2))*cos(e + f*x), x), x), x) - Simp(b*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*(A*b - B*a)*sin(e + f*x)/(a*f*(S(2)*m + S(1))*(-a*d + b*c)), x) def replacement2867(A, B, a, b, c, d, e, f, n, x): return Simp(-S(2)*B*b*(c + d*sin(e + f*x))**(n + S(1))*cos(e + f*x)/(d*f*sqrt(a + b*sin(e + f*x))*(S(2)*n + S(3))), x) def replacement2868(A, B, a, b, c, d, e, f, n, x): return Simp(S(2)*B*b*(c + d*cos(e + f*x))**(n + S(1))*sin(e + f*x)/(d*f*sqrt(a + b*cos(e + f*x))*(S(2)*n + S(3))), x) def replacement2869(A, B, a, b, c, d, e, f, n, x): return Dist((A*b*d*(S(2)*n + S(3)) - B*(-S(2)*a*d*(n + S(1)) + b*c))/(S(2)*d*(n + S(1))*(a*d + b*c)), Int(sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))**(n + S(1)), x), x) - Simp(b**S(2)*(c + d*sin(e + f*x))**(n + S(1))*(-A*d + B*c)*cos(e + f*x)/(d*f*sqrt(a + b*sin(e + f*x))*(n + S(1))*(a*d + b*c)), x) def replacement2870(A, B, a, b, c, d, e, f, n, x): return Dist((A*b*d*(S(2)*n + S(3)) - B*(-S(2)*a*d*(n + S(1)) + b*c))/(S(2)*d*(n + S(1))*(a*d + b*c)), Int(sqrt(a + b*cos(e + f*x))*(c + d*cos(e + f*x))**(n + S(1)), x), x) + Simp(b**S(2)*(c + d*cos(e + f*x))**(n + S(1))*(-A*d + B*c)*sin(e + f*x)/(d*f*sqrt(a + b*cos(e + f*x))*(n + S(1))*(a*d + b*c)), x) def replacement2871(A, B, a, b, c, d, e, f, n, x): return Dist((A*b*d*(S(2)*n + S(3)) - B*(-S(2)*a*d*(n + S(1)) + b*c))/(b*d*(S(2)*n + S(3))), Int(sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))**n, x), x) + Simp(-S(2)*B*b*(c + d*sin(e + f*x))**(n + S(1))*cos(e + f*x)/(d*f*sqrt(a + b*sin(e + f*x))*(S(2)*n + S(3))), x) def replacement2872(A, B, a, b, c, d, e, f, n, x): return Dist((A*b*d*(S(2)*n + S(3)) - B*(-S(2)*a*d*(n + S(1)) + b*c))/(b*d*(S(2)*n + S(3))), Int(sqrt(a + b*cos(e + f*x))*(c + d*cos(e + f*x))**n, x), x) + Simp(S(2)*B*b*(c + d*cos(e + f*x))**(n + S(1))*sin(e + f*x)/(d*f*sqrt(a + b*cos(e + f*x))*(S(2)*n + S(3))), x) def replacement2873(A, B, a, b, c, d, e, f, x): return Dist(B/b, Int(sqrt(a + b*sin(e + f*x))/sqrt(c + d*sin(e + f*x)), x), x) + Dist((A*b - B*a)/b, Int(S(1)/(sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))), x), x) def replacement2874(A, B, a, b, c, d, e, f, x): return Dist(B/b, Int(sqrt(a + b*cos(e + f*x))/sqrt(c + d*cos(e + f*x)), x), x) + Dist((A*b - B*a)/b, Int(S(1)/(sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x))), x), x) def replacement2875(A, B, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(b*(m + n + S(1))), Int((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(-1))*Simp(A*b*c*(m + n + S(1)) + B*(a*c*m + b*d*n) + (A*b*d*(m + n + S(1)) + B*(a*d*m + b*c*n))*sin(e + f*x), x), x), x) - Simp(B*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n*cos(e + f*x)/(f*(m + n + S(1))), x) def replacement2876(A, B, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(b*(m + n + S(1))), Int((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(-1))*Simp(A*b*c*(m + n + S(1)) + B*(a*c*m + b*d*n) + (A*b*d*(m + n + S(1)) + B*(a*d*m + b*c*n))*cos(e + f*x), x), x), x) + Simp(B*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n*sin(e + f*x)/(f*(m + n + S(1))), x) def replacement2877(A, B, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(b*(c**S(2) - d**S(2))*(n + S(1))), Int((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*Simp(A*(a*d*m + b*c*(n + S(1))) - B*(a*c*m + b*d*(n + S(1))) + b*(-A*d + B*c)*(m + n + S(2))*sin(e + f*x), x), x), x) + Simp((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*(-A*d + B*c)*cos(e + f*x)/(f*(c**S(2) - d**S(2))*(n + S(1))), x) def replacement2878(A, B, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(b*(c**S(2) - d**S(2))*(n + S(1))), Int((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*Simp(A*(a*d*m + b*c*(n + S(1))) - B*(a*c*m + b*d*(n + S(1))) + b*(-A*d + B*c)*(m + n + S(2))*cos(e + f*x), x), x), x) - Simp((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*(-A*d + B*c)*sin(e + f*x)/(f*(c**S(2) - d**S(2))*(n + S(1))), x) def replacement2879(A, B, a, b, c, d, e, f, x): return Dist((A*b - B*a)/(-a*d + b*c), Int(S(1)/sqrt(a + b*sin(e + f*x)), x), x) + Dist((-A*d + B*c)/(-a*d + b*c), Int(sqrt(a + b*sin(e + f*x))/(c + d*sin(e + f*x)), x), x) def replacement2880(A, B, a, b, c, d, e, f, x): return Dist((A*b - B*a)/(-a*d + b*c), Int(S(1)/sqrt(a + b*cos(e + f*x)), x), x) + Dist((-A*d + B*c)/(-a*d + b*c), Int(sqrt(a + b*cos(e + f*x))/(c + d*cos(e + f*x)), x), x) def replacement2881(A, B, a, b, c, d, e, f, m, x): return Dist(B/d, Int((a + b*sin(e + f*x))**m, x), x) - Dist((-A*d + B*c)/d, Int((a + b*sin(e + f*x))**m/(c + d*sin(e + f*x)), x), x) def replacement2882(A, B, a, b, c, d, e, f, m, x): return Dist(B/d, Int((a + b*cos(e + f*x))**m, x), x) - Dist((-A*d + B*c)/d, Int((a + b*cos(e + f*x))**m/(c + d*cos(e + f*x)), x), x) def replacement2883(A, B, a, b, c, d, e, f, m, n, x): return Dist(B/b, Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n, x), x) + Dist((A*b - B*a)/b, Int((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n, x), x) def replacement2884(A, B, a, b, c, d, e, f, m, n, x): return Dist(B/b, Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n, x), x) + Dist((A*b - B*a)/b, Int((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n, x), x) def replacement2885(A, B, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(d*(c**S(2) - d**S(2))*(n + S(1))), Int((a + b*sin(e + f*x))**(m + S(-2))*(c + d*sin(e + f*x))**(n + S(1))*Simp(a*d*(n + S(1))*(A*a*c + B*b*c - d*(A*b + B*a)) + b*(m + S(-1))*(-A*d + B*c)*(-a*d + b*c) + b*(-B*b*(c**S(2)*m + d**S(2)*(n + S(1))) + d*(m + n + S(1))*(-A*a*d + A*b*c + B*a*c))*sin(e + f*x)**S(2) + (-a*(n + S(2))*(-A*d + B*c)*(-a*d + b*c) + b*(n + S(1))*(a*(A*c*d + B*(c**S(2) - S(2)*d**S(2))) + b*d*(-A*d + B*c)))*sin(e + f*x), x), x), x) - Simp((a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**(n + S(1))*(-A*d + B*c)*(-a*d + b*c)*cos(e + f*x)/(d*f*(c**S(2) - d**S(2))*(n + S(1))), x) def replacement2886(A, B, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(d*(c**S(2) - d**S(2))*(n + S(1))), Int((a + b*cos(e + f*x))**(m + S(-2))*(c + d*cos(e + f*x))**(n + S(1))*Simp(a*d*(n + S(1))*(A*a*c + B*b*c - d*(A*b + B*a)) + b*(m + S(-1))*(-A*d + B*c)*(-a*d + b*c) + b*(-B*b*(c**S(2)*m + d**S(2)*(n + S(1))) + d*(m + n + S(1))*(-A*a*d + A*b*c + B*a*c))*cos(e + f*x)**S(2) + (-a*(n + S(2))*(-A*d + B*c)*(-a*d + b*c) + b*(n + S(1))*(a*(A*c*d + B*(c**S(2) - S(2)*d**S(2))) + b*d*(-A*d + B*c)))*cos(e + f*x), x), x), x) + Simp((a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**(n + S(1))*(-A*d + B*c)*(-a*d + b*c)*sin(e + f*x)/(d*f*(c**S(2) - d**S(2))*(n + S(1))), x) def replacement2887(A, B, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(d*(m + n + S(1))), Int((a + b*sin(e + f*x))**(m + S(-2))*(c + d*sin(e + f*x))**n*Simp(A*a**S(2)*d*(m + n + S(1)) + B*b*(a*d*(n + S(1)) + b*c*(m + S(-1))) + b*(A*b*d*(m + n + S(1)) - B*(-a*d*(S(2)*m + n) + b*c*m))*sin(e + f*x)**S(2) + (-B*b*(a*c - b*d*(m + n)) + a*d*(S(2)*A*b + B*a)*(m + n + S(1)))*sin(e + f*x), x), x), x) - Simp(B*b*(a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**(n + S(1))*cos(e + f*x)/(d*f*(m + n + S(1))), x) def replacement2888(A, B, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(d*(m + n + S(1))), Int((a + b*cos(e + f*x))**(m + S(-2))*(c + d*cos(e + f*x))**n*Simp(A*a**S(2)*d*(m + n + S(1)) + B*b*(a*d*(n + S(1)) + b*c*(m + S(-1))) + b*(A*b*d*(m + n + S(1)) - B*(-a*d*(S(2)*m + n) + b*c*m))*cos(e + f*x)**S(2) + (-B*b*(a*c - b*d*(m + n)) + a*d*(S(2)*A*b + B*a)*(m + n + S(1)))*cos(e + f*x), x), x), x) + Simp(B*b*(a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**(n + S(1))*sin(e + f*x)/(d*f*(m + n + S(1))), x) def replacement2889(A, B, b, c, d, e, f, x): return Dist(B*d/b**S(2), Int(sqrt(b*sin(e + f*x))/sqrt(c + d*sin(e + f*x)), x), x) + Int((A*c + (A*d + B*c)*sin(e + f*x))/((b*sin(e + f*x))**(S(3)/2)*sqrt(c + d*sin(e + f*x))), x) def replacement2890(A, B, b, c, d, e, f, x): return Dist(B*d/b**S(2), Int(sqrt(b*cos(e + f*x))/sqrt(c + d*cos(e + f*x)), x), x) + Int((A*c + (A*d + B*c)*cos(e + f*x))/((b*cos(e + f*x))**(S(3)/2)*sqrt(c + d*cos(e + f*x))), x) def replacement2891(A, B, a, b, c, d, e, f, x): return Dist(B/b, Int(sqrt(c + d*sin(e + f*x))/sqrt(a + b*sin(e + f*x)), x), x) + Dist((A*b - B*a)/b, Int(sqrt(c + d*sin(e + f*x))/(a + b*sin(e + f*x))**(S(3)/2), x), x) def replacement2892(A, B, a, b, c, d, e, f, x): return Dist(B/b, Int(sqrt(c + d*cos(e + f*x))/sqrt(a + b*cos(e + f*x)), x), x) + Dist((A*b - B*a)/b, Int(sqrt(c + d*cos(e + f*x))/(a + b*cos(e + f*x))**(S(3)/2), x), x) def replacement2893(A, B, a, b, d, e, f, x): return Dist(d/(a**S(2) - b**S(2)), Int((A*b - B*a + (A*a - B*b)*sin(e + f*x))/((d*sin(e + f*x))**(S(3)/2)*sqrt(a + b*sin(e + f*x))), x), x) + Simp(S(2)*(A*b - B*a)*cos(e + f*x)/(f*sqrt(d*sin(e + f*x))*sqrt(a + b*sin(e + f*x))*(a**S(2) - b**S(2))), x) def replacement2894(A, B, a, b, d, e, f, x): return Dist(d/(a**S(2) - b**S(2)), Int((A*b - B*a + (A*a - B*b)*cos(e + f*x))/((d*cos(e + f*x))**(S(3)/2)*sqrt(a + b*cos(e + f*x))), x), x) + Simp(-S(2)*(A*b - B*a)*sin(e + f*x)/(f*sqrt(d*cos(e + f*x))*sqrt(a + b*cos(e + f*x))*(a**S(2) - b**S(2))), x) def replacement2895(A, B, b, c, d, e, f, x): return Simp(-S(2)*A*sqrt(c*(S(1) - S(1)/sin(e + f*x))/(c + d))*sqrt(c*(S(1) + S(1)/sin(e + f*x))/(c - d))*(c - d)*EllipticE(asin(sqrt(c + d*sin(e + f*x))/(sqrt(b*sin(e + f*x))*Rt((c + d)/b, S(2)))), -(c + d)/(c - d))*Rt((c + d)/b, S(2))*tan(e + f*x)/(b*c**S(2)*f), x) def replacement2896(A, B, b, c, d, e, f, x): return Simp(S(2)*A*sqrt(c*(S(1) - S(1)/cos(e + f*x))/(c + d))*sqrt(c*(S(1) + S(1)/cos(e + f*x))/(c - d))*(c - d)*EllipticE(asin(sqrt(c + d*cos(e + f*x))/(sqrt(b*cos(e + f*x))*Rt((c + d)/b, S(2)))), -(c + d)/(c - d))*Rt((c + d)/b, S(2))/(b*c**S(2)*f*tan(e + f*x)), x) def replacement2897(A, B, b, c, d, e, f, x): return -Dist(sqrt(-b*sin(e + f*x))/sqrt(b*sin(e + f*x)), Int((A + B*sin(e + f*x))/((-b*sin(e + f*x))**(S(3)/2)*sqrt(c + d*sin(e + f*x))), x), x) def replacement2898(A, B, b, c, d, e, f, x): return -Dist(sqrt(-b*cos(e + f*x))/sqrt(b*cos(e + f*x)), Int((A + B*cos(e + f*x))/((-b*cos(e + f*x))**(S(3)/2)*sqrt(c + d*cos(e + f*x))), x), x) def replacement2899(A, B, a, b, c, d, e, f, x): return Simp(-S(2)*A*sqrt((-a*d + b*c)*(sin(e + f*x) + S(1))/((a + b*sin(e + f*x))*(c - d)))*sqrt(-(S(1) - sin(e + f*x))*(-a*d + b*c)/((a + b*sin(e + f*x))*(c + d)))*(a + b*sin(e + f*x))*(c - d)*EllipticE(asin(sqrt(c + d*sin(e + f*x))*Rt((a + b)/(c + d), S(2))/sqrt(a + b*sin(e + f*x))), (a - b)*(c + d)/((a + b)*(c - d)))/(f*(-a*d + b*c)**S(2)*Rt((a + b)/(c + d), S(2))*cos(e + f*x)), x) def replacement2900(A, B, a, b, c, d, e, f, x): return Simp(S(2)*A*sqrt((-a*d + b*c)*(cos(e + f*x) + S(1))/((a + b*cos(e + f*x))*(c - d)))*sqrt(-(S(1) - cos(e + f*x))*(-a*d + b*c)/((a + b*cos(e + f*x))*(c + d)))*(a + b*cos(e + f*x))*(c - d)*EllipticE(asin(sqrt(c + d*cos(e + f*x))*Rt((a + b)/(c + d), S(2))/sqrt(a + b*cos(e + f*x))), (a - b)*(c + d)/((a + b)*(c - d)))/(f*(-a*d + b*c)**S(2)*Rt((a + b)/(c + d), S(2))*sin(e + f*x)), x) def replacement2901(A, B, a, b, c, d, e, f, x): return Dist(sqrt(-c - d*sin(e + f*x))/sqrt(c + d*sin(e + f*x)), Int((A + B*sin(e + f*x))/((a + b*sin(e + f*x))**(S(3)/2)*sqrt(-c - d*sin(e + f*x))), x), x) def replacement2902(A, B, a, b, c, d, e, f, x): return Dist(sqrt(-c - d*cos(e + f*x))/sqrt(c + d*cos(e + f*x)), Int((A + B*cos(e + f*x))/((a + b*cos(e + f*x))**(S(3)/2)*sqrt(-c - d*cos(e + f*x))), x), x) def replacement2903(A, B, a, b, c, d, e, f, x): return Dist((A - B)/(a - b), Int(S(1)/(sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))), x), x) - Dist((A*b - B*a)/(a - b), Int((sin(e + f*x) + S(1))/((a + b*sin(e + f*x))**(S(3)/2)*sqrt(c + d*sin(e + f*x))), x), x) def replacement2904(A, B, a, b, c, d, e, f, x): return Dist((A - B)/(a - b), Int(S(1)/(sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x))), x), x) - Dist((A*b - B*a)/(a - b), Int((cos(e + f*x) + S(1))/((a + b*cos(e + f*x))**(S(3)/2)*sqrt(c + d*cos(e + f*x))), x), x) def replacement2905(A, B, a, b, c, d, e, f, m, n, x): return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**(n + S(-1))*Simp(c*(m + S(1))*(A*a - B*b) + d*n*(A*b - B*a) - d*(A*b - B*a)*(m + n + S(2))*sin(e + f*x)**S(2) + (-c*(m + S(2))*(A*b - B*a) + d*(m + S(1))*(A*a - B*b))*sin(e + f*x), x), x), x) + Simp((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n*(-A*b + B*a)*cos(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))), x) def replacement2906(A, B, a, b, c, d, e, f, m, n, x): return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**(n + S(-1))*Simp(c*(m + S(1))*(A*a - B*b) + d*n*(A*b - B*a) - d*(A*b - B*a)*(m + n + S(2))*cos(e + f*x)**S(2) + (-c*(m + S(2))*(A*b - B*a) + d*(m + S(1))*(A*a - B*b))*cos(e + f*x), x), x), x) - Simp((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n*(-A*b + B*a)*sin(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))), x) def replacement2907(A, B, a, b, c, d, e, f, m, n, x): return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))*(-a*d + b*c)), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n*Simp(b*d*(A*b - B*a)*(m + n + S(2)) - b*d*(A*b - B*a)*(m + n + S(3))*sin(e + f*x)**S(2) + (m + S(1))*(A*a - B*b)*(-a*d + b*c) + (A*b - B*a)*(a*d*(m + S(1)) - b*c*(m + S(2)))*sin(e + f*x), x), x), x) - Simp((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**(n + S(1))*(A*b**S(2) - B*a*b)*cos(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))*(-a*d + b*c)), x) def replacement2908(A, B, a, b, c, d, e, f, m, n, x): return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))*(-a*d + b*c)), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n*Simp(b*d*(A*b - B*a)*(m + n + S(2)) - b*d*(A*b - B*a)*(m + n + S(3))*cos(e + f*x)**S(2) + (m + S(1))*(A*a - B*b)*(-a*d + b*c) + (A*b - B*a)*(a*d*(m + S(1)) - b*c*(m + S(2)))*cos(e + f*x), x), x), x) + Simp((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**(n + S(1))*(A*b**S(2) - B*a*b)*sin(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))*(-a*d + b*c)), x) def replacement2909(A, B, a, b, c, d, e, f, x): return Dist((A*b - B*a)/(-a*d + b*c), Int(S(1)/(a + b*sin(e + f*x)), x), x) + Dist((-A*d + B*c)/(-a*d + b*c), Int(S(1)/(c + d*sin(e + f*x)), x), x) def replacement2910(A, B, a, b, c, d, e, f, x): return Dist((A*b - B*a)/(-a*d + b*c), Int(S(1)/(a + b*cos(e + f*x)), x), x) + Dist((-A*d + B*c)/(-a*d + b*c), Int(S(1)/(c + d*cos(e + f*x)), x), x) def replacement2911(A, B, a, b, c, d, e, f, m, x): return Dist(B/d, Int((a + b*sin(e + f*x))**m, x), x) - Dist((-A*d + B*c)/d, Int((a + b*sin(e + f*x))**m/(c + d*sin(e + f*x)), x), x) def replacement2912(A, B, a, b, c, d, e, f, m, x): return Dist(B/d, Int((a + b*cos(e + f*x))**m, x), x) - Dist((-A*d + B*c)/d, Int((a + b*cos(e + f*x))**m/(c + d*cos(e + f*x)), x), x) def replacement2913(A, B, a, b, c, d, e, f, n, x): return Dist(S(1)/(S(2)*n + S(3)), Int((c + d*sin(e + f*x))**(n + S(-1))*Simp(A*a*c*(S(2)*n + S(3)) + B*(S(2)*a*d*n + b*c) + (A*(S(2)*n + S(3))*(a*d + b*c) + B*(S(2)*n + S(1))*(a*c + b*d))*sin(e + f*x) + (A*b*d*(S(2)*n + S(3)) + B*(a*d + S(2)*b*c*n))*sin(e + f*x)**S(2), x)/sqrt(a + b*sin(e + f*x)), x), x) + Simp(-S(2)*B*sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))**n*cos(e + f*x)/(f*(S(2)*n + S(3))), x) def replacement2914(A, B, a, b, c, d, e, f, n, x): return Dist(S(1)/(S(2)*n + S(3)), Int((c + d*cos(e + f*x))**(n + S(-1))*Simp(A*a*c*(S(2)*n + S(3)) + B*(S(2)*a*d*n + b*c) + (A*(S(2)*n + S(3))*(a*d + b*c) + B*(S(2)*n + S(1))*(a*c + b*d))*cos(e + f*x) + (A*b*d*(S(2)*n + S(3)) + B*(a*d + S(2)*b*c*n))*cos(e + f*x)**S(2), x)/sqrt(a + b*cos(e + f*x)), x), x) + Simp(S(2)*B*sqrt(a + b*cos(e + f*x))*(c + d*cos(e + f*x))**n*sin(e + f*x)/(f*(S(2)*n + S(3))), x) def replacement2915(A, B, a, b, e, f, x): return Simp(S(4)*A*EllipticPi(S(-1), -asin(cos(e + f*x)/(sin(e + f*x) + S(1))), -(a - b)/(a + b))/(f*sqrt(a + b)), x) def replacement2916(A, B, a, b, e, f, x): return Simp(S(4)*A*EllipticPi(S(-1), asin(sin(e + f*x)/(cos(e + f*x) + S(1))), -(a - b)/(a + b))/(f*sqrt(a + b)), x) def replacement2917(A, B, a, b, d, e, f, x): return Dist(sqrt(sin(e + f*x))/sqrt(d*sin(e + f*x)), Int((A + B*sin(e + f*x))/(sqrt(a + b*sin(e + f*x))*sqrt(sin(e + f*x))), x), x) def replacement2918(A, B, a, b, d, e, f, x): return Dist(sqrt(cos(e + f*x))/sqrt(d*cos(e + f*x)), Int((A + B*cos(e + f*x))/(sqrt(a + b*cos(e + f*x))*sqrt(cos(e + f*x))), x), x) def replacement2919(A, B, a, b, c, d, e, f, x): return Dist(B/d, Int(sqrt(c + d*sin(e + f*x))/sqrt(a + b*sin(e + f*x)), x), x) - Dist((-A*d + B*c)/d, Int(S(1)/(sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))), x), x) def replacement2920(A, B, a, b, c, d, e, f, x): return Dist(B/d, Int(sqrt(c + d*cos(e + f*x))/sqrt(a + b*cos(e + f*x)), x), x) - Dist((-A*d + B*c)/d, Int(S(1)/(sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x))), x), x) def replacement2921(A, B, a, b, c, d, e, f, m, n, x): return Int((A + B*sin(e + f*x))*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n, x) def replacement2922(A, B, a, b, c, d, e, f, m, n, x): return Int((A + B*cos(e + f*x))*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n, x) def replacement2923(A, B, a, b, c, d, e, f, m, n, p, x): return Dist(sqrt(a + b*sin(e + f*x))*sqrt(c + d*sin(e + f*x))/(f*cos(e + f*x)), Subst(Int((A + B*x)**p*(a + b*x)**(m + S(-1)/2)*(c + d*x)**(n + S(-1)/2), x), x, sin(e + f*x)), x) def replacement2924(A, B, a, b, c, d, e, f, m, n, p, x): return -Dist(sqrt(a + b*cos(e + f*x))*sqrt(c + d*cos(e + f*x))/(f*sin(e + f*x)), Subst(Int((A + B*x)**p*(a + b*x)**(m + S(-1)/2)*(c + d*x)**(n + S(-1)/2), x), x, cos(e + f*x)), x) def replacement2925(B, C, b, e, f, m, x): return Dist(S(1)/b, Int((b*sin(e + f*x))**(m + S(1))*(B + C*sin(e + f*x)), x), x) def replacement2926(B, C, b, e, f, m, x): return Dist(S(1)/b, Int((b*cos(e + f*x))**(m + S(1))*(B + C*cos(e + f*x)), x), x) def replacement2927(A, C, e, f, m, x): return -Dist(S(1)/f, Subst(Int((S(1) - x**S(2))**(m/S(2) + S(-1)/2)*(A - C*x**S(2) + C), x), x, cos(e + f*x)), x) def replacement2928(A, C, e, f, m, x): return Dist(S(1)/f, Subst(Int((S(1) - x**S(2))**(m/S(2) + S(-1)/2)*(A - C*x**S(2) + C), x), x, sin(e + f*x)), x) def replacement2929(A, C, b, e, f, m, x): return Simp(A*(b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(b*f*(m + S(1))), x) def replacement2930(A, C, b, e, f, m, x): return -Simp(A*(b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(b*f*(m + S(1))), x) def replacement2931(A, C, b, e, f, m, x): return Dist((A*(m + S(2)) + C*(m + S(1)))/(b**S(2)*(m + S(1))), Int((b*sin(e + f*x))**(m + S(2)), x), x) + Simp(A*(b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(b*f*(m + S(1))), x) def replacement2932(A, C, b, e, f, m, x): return Dist((A*(m + S(2)) + C*(m + S(1)))/(b**S(2)*(m + S(1))), Int((b*cos(e + f*x))**(m + S(2)), x), x) - Simp(A*(b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(b*f*(m + S(1))), x) def replacement2933(A, C, b, e, f, m, x): return Dist((A*(m + S(2)) + C*(m + S(1)))/(m + S(2)), Int((b*sin(e + f*x))**m, x), x) - Simp(C*(b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(b*f*(m + S(2))), x) def replacement2934(A, C, b, e, f, m, x): return Dist((A*(m + S(2)) + C*(m + S(1)))/(m + S(2)), Int((b*cos(e + f*x))**m, x), x) + Simp(C*(b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(b*f*(m + S(2))), x) def replacement2935(A, B, C, a, b, e, f, m, x): return Dist(b**(S(-2)), Int((a + b*sin(e + f*x))**(m + S(1))*Simp(B*b - C*a + C*b*sin(e + f*x), x), x), x) def replacement2936(A, B, C, a, b, e, f, m, x): return Dist(b**(S(-2)), Int((a + b*cos(e + f*x))**(m + S(1))*Simp(B*b - C*a + C*b*cos(e + f*x), x), x), x) def replacement2937(A, C, a, b, e, f, m, x): return Dist(C/b**S(2), Int((a + b*sin(e + f*x))**(m + S(1))*Simp(-a + b*sin(e + f*x), x), x), x) def replacement2938(A, C, a, b, e, f, m, x): return Dist(C/b**S(2), Int((a + b*cos(e + f*x))**(m + S(1))*Simp(-a + b*cos(e + f*x), x), x), x) def replacement2939(A, B, C, a, b, e, f, m, x): return Dist(C, Int((a + b*sin(e + f*x))**m*(sin(e + f*x) + S(1))**S(2), x), x) + Dist(A - C, Int((a + b*sin(e + f*x))**m*(sin(e + f*x) + S(1)), x), x) def replacement2940(A, B, C, a, b, e, f, m, x): return Dist(C, Int((a + b*cos(e + f*x))**m*(cos(e + f*x) + S(1))**S(2), x), x) + Dist(A - C, Int((a + b*cos(e + f*x))**m*(cos(e + f*x) + S(1)), x), x) def replacement2941(A, C, a, b, e, f, m, x): return Dist(C, Int((a + b*sin(e + f*x))**m*(sin(e + f*x) + S(1))**S(2), x), x) + Dist(A - C, Int((a + b*sin(e + f*x))**m*(sin(e + f*x) + S(1)), x), x) def replacement2942(A, C, a, b, e, f, m, x): return Dist(C, Int((a + b*cos(e + f*x))**m*(cos(e + f*x) + S(1))**S(2), x), x) + Dist(A - C, Int((a + b*cos(e + f*x))**m*(cos(e + f*x) + S(1)), x), x) def replacement2943(A, B, C, a, b, e, f, m, x): return Dist(S(1)/(a**S(2)*(S(2)*m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*Simp(A*a*(m + S(1)) + C*b*(S(2)*m + S(1))*sin(e + f*x) + m*(B*b - C*a), x), x), x) + Simp((a + b*sin(e + f*x))**m*(A*b - B*a + C*b)*cos(e + f*x)/(a*f*(S(2)*m + S(1))), x) def replacement2944(A, B, C, a, b, e, f, m, x): return Dist(S(1)/(a**S(2)*(S(2)*m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*Simp(A*a*(m + S(1)) + C*b*(S(2)*m + S(1))*cos(e + f*x) + m*(B*b - C*a), x), x), x) - Simp((a + b*cos(e + f*x))**m*(A*b - B*a + C*b)*sin(e + f*x)/(a*f*(S(2)*m + S(1))), x) def replacement2945(A, C, a, b, e, f, m, x): return Dist(S(1)/(a**S(2)*(S(2)*m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*Simp(A*a*(m + S(1)) - C*a*m + C*b*(S(2)*m + S(1))*sin(e + f*x), x), x), x) + Simp(b*(A + C)*(a + b*sin(e + f*x))**m*cos(e + f*x)/(a*f*(S(2)*m + S(1))), x) def replacement2946(A, C, a, b, e, f, m, x): return Dist(S(1)/(a**S(2)*(S(2)*m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*Simp(A*a*(m + S(1)) - C*a*m + C*b*(S(2)*m + S(1))*cos(e + f*x), x), x), x) - Simp(b*(A + C)*(a + b*cos(e + f*x))**m*sin(e + f*x)/(a*f*(S(2)*m + S(1))), x) def replacement2947(A, B, C, a, b, e, f, m, x): return Dist(S(1)/(b*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*Simp(b*(m + S(1))*(A*a - B*b + C*a) - (A*b**S(2) - B*a*b + C*a**S(2) + b*(m + S(1))*(A*b - B*a + C*b))*sin(e + f*x), x), x), x) - Simp((a + b*sin(e + f*x))**(m + S(1))*(A*b**S(2) - B*a*b + C*a**S(2))*cos(e + f*x)/(b*f*(a**S(2) - b**S(2))*(m + S(1))), x) def replacement2948(A, B, C, a, b, e, f, m, x): return Dist(S(1)/(b*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*Simp(b*(m + S(1))*(A*a - B*b + C*a) - (A*b**S(2) - B*a*b + C*a**S(2) + b*(m + S(1))*(A*b - B*a + C*b))*cos(e + f*x), x), x), x) + Simp((a + b*cos(e + f*x))**(m + S(1))*(A*b**S(2) - B*a*b + C*a**S(2))*sin(e + f*x)/(b*f*(a**S(2) - b**S(2))*(m + S(1))), x) def replacement2949(A, C, a, b, e, f, m, x): return Dist(S(1)/(b*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*Simp(a*b*(A + C)*(m + S(1)) - (A*b**S(2) + C*a**S(2) + b**S(2)*(A + C)*(m + S(1)))*sin(e + f*x), x), x), x) - Simp((a + b*sin(e + f*x))**(m + S(1))*(A*b**S(2) + C*a**S(2))*cos(e + f*x)/(b*f*(a**S(2) - b**S(2))*(m + S(1))), x) def replacement2950(A, C, a, b, e, f, m, x): return Dist(S(1)/(b*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*Simp(a*b*(A + C)*(m + S(1)) - (A*b**S(2) + C*a**S(2) + b**S(2)*(A + C)*(m + S(1)))*cos(e + f*x), x), x), x) + Simp((a + b*cos(e + f*x))**(m + S(1))*(A*b**S(2) + C*a**S(2))*sin(e + f*x)/(b*f*(a**S(2) - b**S(2))*(m + S(1))), x) def replacement2951(A, B, C, a, b, e, f, m, x): return Dist(S(1)/(b*(m + S(2))), Int((a + b*sin(e + f*x))**m*Simp(A*b*(m + S(2)) + C*b*(m + S(1)) + (B*b*(m + S(2)) - C*a)*sin(e + f*x), x), x), x) - Simp(C*(a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(b*f*(m + S(2))), x) def replacement2952(A, B, C, a, b, e, f, m, x): return Dist(S(1)/(b*(m + S(2))), Int((a + b*cos(e + f*x))**m*Simp(A*b*(m + S(2)) + C*b*(m + S(1)) + (B*b*(m + S(2)) - C*a)*cos(e + f*x), x), x), x) + Simp(C*(a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(b*f*(m + S(2))), x) def replacement2953(A, C, a, b, e, f, m, x): return Dist(S(1)/(b*(m + S(2))), Int((a + b*sin(e + f*x))**m*Simp(A*b*(m + S(2)) - C*a*sin(e + f*x) + C*b*(m + S(1)), x), x), x) - Simp(C*(a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(b*f*(m + S(2))), x) def replacement2954(A, C, a, b, e, f, m, x): return Dist(S(1)/(b*(m + S(2))), Int((a + b*cos(e + f*x))**m*Simp(A*b*(m + S(2)) - C*a*cos(e + f*x) + C*b*(m + S(1)), x), x), x) + Simp(C*(a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(b*f*(m + S(2))), x) def replacement2955(A, B, C, b, e, f, m, p, x): return Dist((b*sin(e + f*x))**(-m*p)*(b*sin(e + f*x)**p)**m, Int((b*sin(e + f*x))**(m*p)*(A + B*sin(e + f*x) + C*sin(e + f*x)**S(2)), x), x) def replacement2956(A, B, C, b, e, f, m, p, x): return Dist((b*cos(e + f*x))**(-m*p)*(b*cos(e + f*x)**p)**m, Int((b*cos(e + f*x))**(m*p)*(A + B*cos(e + f*x) + C*cos(e + f*x)**S(2)), x), x) def replacement2957(A, C, b, e, f, m, p, x): return Dist((b*sin(e + f*x))**(-m*p)*(b*sin(e + f*x)**p)**m, Int((b*sin(e + f*x))**(m*p)*(A + C*sin(e + f*x)**S(2)), x), x) def replacement2958(A, C, b, e, f, m, p, x): return Dist((b*cos(e + f*x))**(-m*p)*(b*cos(e + f*x)**p)**m, Int((b*cos(e + f*x))**(m*p)*(A + C*cos(e + f*x)**S(2)), x), x) def replacement2959(A, B, C, a, b, c, d, e, f, m, x): return -Dist(S(1)/(b**S(2)*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*Simp(-C*b*d*(a**S(2) - b**S(2))*(m + S(1))*sin(e + f*x)**S(2) + b*(m + S(1))*(-A*b*(a*c - b*d) + (B*b - C*a)*(-a*d + b*c)) + (B*b*(a**S(2)*d - a*b*c*(m + S(2)) + b**S(2)*d*(m + S(1))) + (-a*d + b*c)*(A*b**S(2)*(m + S(2)) + C*(a**S(2) + b**S(2)*(m + S(1)))))*sin(e + f*x), x), x), x) - Simp((a + b*sin(e + f*x))**(m + S(1))*(-a*d + b*c)*(A*b**S(2) - B*a*b + C*a**S(2))*cos(e + f*x)/(b**S(2)*f*(a**S(2) - b**S(2))*(m + S(1))), x) def replacement2960(A, B, C, a, b, c, d, e, f, m, x): return -Dist(S(1)/(b**S(2)*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*Simp(-C*b*d*(a**S(2) - b**S(2))*(m + S(1))*cos(e + f*x)**S(2) + b*(m + S(1))*(-A*b*(a*c - b*d) + (B*b - C*a)*(-a*d + b*c)) + (B*b*(a**S(2)*d - a*b*c*(m + S(2)) + b**S(2)*d*(m + S(1))) + (-a*d + b*c)*(A*b**S(2)*(m + S(2)) + C*(a**S(2) + b**S(2)*(m + S(1)))))*cos(e + f*x), x), x), x) + Simp((a + b*cos(e + f*x))**(m + S(1))*(-a*d + b*c)*(A*b**S(2) - B*a*b + C*a**S(2))*sin(e + f*x)/(b**S(2)*f*(a**S(2) - b**S(2))*(m + S(1))), x) def replacement2961(A, C, a, b, c, d, e, f, m, x): return Dist(S(1)/(b**S(2)*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*Simp(C*b*d*(a**S(2) - b**S(2))*(m + S(1))*sin(e + f*x)**S(2) + b*(m + S(1))*(A*b*(a*c - b*d) + C*a*(-a*d + b*c)) - (-a*d + b*c)*(A*b**S(2)*(m + S(2)) + C*(a**S(2) + b**S(2)*(m + S(1))))*sin(e + f*x), x), x), x) - Simp((a + b*sin(e + f*x))**(m + S(1))*(A*b**S(2) + C*a**S(2))*(-a*d + b*c)*cos(e + f*x)/(b**S(2)*f*(a**S(2) - b**S(2))*(m + S(1))), x) def replacement2962(A, C, a, b, c, d, e, f, m, x): return Dist(S(1)/(b**S(2)*(a**S(2) - b**S(2))*(m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*Simp(C*b*d*(a**S(2) - b**S(2))*(m + S(1))*cos(e + f*x)**S(2) + b*(m + S(1))*(A*b*(a*c - b*d) + C*a*(-a*d + b*c)) - (-a*d + b*c)*(A*b**S(2)*(m + S(2)) + C*(a**S(2) + b**S(2)*(m + S(1))))*cos(e + f*x), x), x), x) + Simp((a + b*cos(e + f*x))**(m + S(1))*(A*b**S(2) + C*a**S(2))*(-a*d + b*c)*sin(e + f*x)/(b**S(2)*f*(a**S(2) - b**S(2))*(m + S(1))), x) def replacement2963(A, B, C, a, b, c, d, e, f, m, x): return Dist(S(1)/(b*(m + S(3))), Int((a + b*sin(e + f*x))**m*Simp(A*b*c*(m + S(3)) + C*a*d + b*(B*c*(m + S(3)) + d*(A*(m + S(3)) + C*(m + S(2))))*sin(e + f*x) - (S(2)*C*a*d - b*(m + S(3))*(B*d + C*c))*sin(e + f*x)**S(2), x), x), x) - Simp(C*d*(a + b*sin(e + f*x))**(m + S(1))*sin(e + f*x)*cos(e + f*x)/(b*f*(m + S(3))), x) def replacement2964(A, B, C, a, b, c, d, e, f, m, x): return Dist(S(1)/(b*(m + S(3))), Int((a + b*cos(e + f*x))**m*Simp(A*b*c*(m + S(3)) + C*a*d + b*(B*c*(m + S(3)) + d*(A*(m + S(3)) + C*(m + S(2))))*cos(e + f*x) - (S(2)*C*a*d - b*(m + S(3))*(B*d + C*c))*cos(e + f*x)**S(2), x), x), x) + Simp(C*d*(a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)*cos(e + f*x)/(b*f*(m + S(3))), x) def replacement2965(A, C, a, b, c, d, e, f, m, x): return Dist(S(1)/(b*(m + S(3))), Int((a + b*sin(e + f*x))**m*Simp(A*b*c*(m + S(3)) + C*a*d + b*d*(A*(m + S(3)) + C*(m + S(2)))*sin(e + f*x) - (S(2)*C*a*d - C*b*c*(m + S(3)))*sin(e + f*x)**S(2), x), x), x) - Simp(C*d*(a + b*sin(e + f*x))**(m + S(1))*sin(e + f*x)*cos(e + f*x)/(b*f*(m + S(3))), x) def replacement2966(A, C, a, b, c, d, e, f, m, x): return Dist(S(1)/(b*(m + S(3))), Int((a + b*cos(e + f*x))**m*Simp(A*b*c*(m + S(3)) + C*a*d + b*d*(A*(m + S(3)) + C*(m + S(2)))*cos(e + f*x) - (S(2)*C*a*d - C*b*c*(m + S(3)))*cos(e + f*x)**S(2), x), x), x) + Simp(C*d*(a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)*cos(e + f*x)/(b*f*(m + S(3))), x) def replacement2967(A, B, C, a, b, c, d, e, f, m, n, x): return -Dist(S(1)/(S(2)*b*c*d*(S(2)*m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n*Simp(A*(c**S(2)*(m + S(1)) + d**S(2)*(S(2)*m + n + S(2))) - B*c*d*(m - n + S(-1)) - C*(c**S(2)*m - d**S(2)*(n + S(1))) + d*(-C*c*(S(3)*m - n) + (A*c + B*d)*(m + n + S(2)))*sin(e + f*x), x), x), x) + Simp((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*(A*a - B*b + C*a)*cos(e + f*x)/(S(2)*b*c*f*(S(2)*m + S(1))), x) def replacement2968(A, B, C, a, b, c, d, e, f, m, n, x): return -Dist(S(1)/(S(2)*b*c*d*(S(2)*m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n*Simp(A*(c**S(2)*(m + S(1)) + d**S(2)*(S(2)*m + n + S(2))) - B*c*d*(m - n + S(-1)) - C*(c**S(2)*m - d**S(2)*(n + S(1))) + d*(-C*c*(S(3)*m - n) + (A*c + B*d)*(m + n + S(2)))*cos(e + f*x), x), x), x) - Simp((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*(A*a - B*b + C*a)*sin(e + f*x)/(S(2)*b*c*f*(S(2)*m + S(1))), x) def replacement2969(A, C, a, b, c, d, e, f, m, n, x): return -Dist(S(1)/(S(2)*b*c*d*(S(2)*m + S(1))), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n*Simp(A*(c**S(2)*(m + S(1)) + d**S(2)*(S(2)*m + n + S(2))) - C*(c**S(2)*m - d**S(2)*(n + S(1))) + d*(A*c*(m + n + S(2)) - C*c*(S(3)*m - n))*sin(e + f*x), x), x), x) + Simp((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*(A*a + C*a)*cos(e + f*x)/(S(2)*b*c*f*(S(2)*m + S(1))), x) def replacement2970(A, C, a, b, c, d, e, f, m, n, x): return -Dist(S(1)/(S(2)*b*c*d*(S(2)*m + S(1))), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n*Simp(A*(c**S(2)*(m + S(1)) + d**S(2)*(S(2)*m + n + S(2))) - C*(c**S(2)*m - d**S(2)*(n + S(1))) + d*(A*c*(m + n + S(2)) - C*c*(S(3)*m - n))*cos(e + f*x), x), x), x) - Simp((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*(A*a + C*a)*sin(e + f*x)/(S(2)*b*c*f*(S(2)*m + S(1))), x) def replacement2971(A, B, C, a, b, c, d, e, f, m, x): return Int((a + b*sin(e + f*x))**m*Simp(A + B*sin(e + f*x) + C, x)/sqrt(c + d*sin(e + f*x)), x) + Simp(-S(2)*C*(a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(b*f*sqrt(c + d*sin(e + f*x))*(S(2)*m + S(3))), x) def replacement2972(A, B, C, a, b, c, d, e, f, m, x): return Int((a + b*cos(e + f*x))**m*Simp(A + B*cos(e + f*x) + C, x)/sqrt(c + d*cos(e + f*x)), x) + Simp(S(2)*C*(a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(b*f*sqrt(c + d*cos(e + f*x))*(S(2)*m + S(3))), x) def replacement2973(A, C, a, b, c, d, e, f, m, x): return Dist(A + C, Int((a + b*sin(e + f*x))**m/sqrt(c + d*sin(e + f*x)), x), x) + Simp(-S(2)*C*(a + b*sin(e + f*x))**(m + S(1))*cos(e + f*x)/(b*f*sqrt(c + d*sin(e + f*x))*(S(2)*m + S(3))), x) def replacement2974(A, C, a, b, c, d, e, f, m, x): return Dist(A + C, Int((a + b*cos(e + f*x))**m/sqrt(c + d*cos(e + f*x)), x), x) + Simp(S(2)*C*(a + b*cos(e + f*x))**(m + S(1))*sin(e + f*x)/(b*f*sqrt(c + d*cos(e + f*x))*(S(2)*m + S(3))), x) def replacement2975(A, B, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(b*d*(m + n + S(2))), Int((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n*Simp(A*b*d*(m + n + S(2)) + C*(a*c*m + b*d*(n + S(1))) + (B*b*d*(m + n + S(2)) - C*b*c*(S(2)*m + S(1)))*sin(e + f*x), x), x), x) - Simp(C*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*cos(e + f*x)/(d*f*(m + n + S(2))), x) def replacement2976(A, B, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(b*d*(m + n + S(2))), Int((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n*Simp(A*b*d*(m + n + S(2)) + C*(a*c*m + b*d*(n + S(1))) + (B*b*d*(m + n + S(2)) - C*b*c*(S(2)*m + S(1)))*cos(e + f*x), x), x), x) + Simp(C*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*sin(e + f*x)/(d*f*(m + n + S(2))), x) def replacement2977(A, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(b*d*(m + n + S(2))), Int((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n*Simp(A*b*d*(m + n + S(2)) - C*b*c*(S(2)*m + S(1))*sin(e + f*x) + C*(a*c*m + b*d*(n + S(1))), x), x), x) - Simp(C*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*cos(e + f*x)/(d*f*(m + n + S(2))), x) def replacement2978(A, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(b*d*(m + n + S(2))), Int((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n*Simp(A*b*d*(m + n + S(2)) - C*b*c*(S(2)*m + S(1))*cos(e + f*x) + C*(a*c*m + b*d*(n + S(1))), x), x), x) + Simp(C*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*sin(e + f*x)/(d*f*(m + n + S(2))), x) def replacement2979(A, B, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(b*(S(2)*m + S(1))*(-a*d + b*c)), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n*Simp(A*(a*c*(m + S(1)) - b*d*(S(2)*m + n + S(2))) + B*(a*d*(n + S(1)) + b*c*m) - C*(a*c*m + b*d*(n + S(1))) + (C*(-a*d*(m - n + S(-1)) + b*c*(S(2)*m + S(1))) + d*(A*a - B*b)*(m + n + S(2)))*sin(e + f*x), x), x), x) + Simp((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*(A*a - B*b + C*a)*cos(e + f*x)/(f*(S(2)*m + S(1))*(-a*d + b*c)), x) def replacement2980(A, B, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(b*(S(2)*m + S(1))*(-a*d + b*c)), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n*Simp(A*(a*c*(m + S(1)) - b*d*(S(2)*m + n + S(2))) + B*(a*d*(n + S(1)) + b*c*m) - C*(a*c*m + b*d*(n + S(1))) + (C*(-a*d*(m - n + S(-1)) + b*c*(S(2)*m + S(1))) + d*(A*a - B*b)*(m + n + S(2)))*cos(e + f*x), x), x), x) - Simp((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*(A*a - B*b + C*a)*sin(e + f*x)/(f*(S(2)*m + S(1))*(-a*d + b*c)), x) def replacement2981(A, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(b*(S(2)*m + S(1))*(-a*d + b*c)), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n*Simp(A*(a*c*(m + S(1)) - b*d*(S(2)*m + n + S(2))) - C*(a*c*m + b*d*(n + S(1))) + (A*a*d*(m + n + S(2)) + C*(-a*d*(m - n + S(-1)) + b*c*(S(2)*m + S(1))))*sin(e + f*x), x), x), x) + Simp(a*(A + C)*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*cos(e + f*x)/(f*(S(2)*m + S(1))*(-a*d + b*c)), x) def replacement2982(A, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(b*(S(2)*m + S(1))*(-a*d + b*c)), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n*Simp(A*(a*c*(m + S(1)) - b*d*(S(2)*m + n + S(2))) - C*(a*c*m + b*d*(n + S(1))) + (A*a*d*(m + n + S(2)) + C*(-a*d*(m - n + S(-1)) + b*c*(S(2)*m + S(1))))*cos(e + f*x), x), x), x) - Simp(a*(A + C)*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*sin(e + f*x)/(f*(S(2)*m + S(1))*(-a*d + b*c)), x) def replacement2983(A, B, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(b*d*(c**S(2) - d**S(2))*(n + S(1))), Int((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*Simp(A*d*(a*d*m + b*c*(n + S(1))) + b*(-C*(c**S(2)*(m + S(1)) + d**S(2)*(n + S(1))) + d*(-A*d + B*c)*(m + n + S(2)))*sin(e + f*x) + (-B*d + C*c)*(a*c*m + b*d*(n + S(1))), x), x), x) - Simp((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*(A*d**S(2) - B*c*d + C*c**S(2))*cos(e + f*x)/(d*f*(c**S(2) - d**S(2))*(n + S(1))), x) def replacement2984(A, B, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(b*d*(c**S(2) - d**S(2))*(n + S(1))), Int((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*Simp(A*d*(a*d*m + b*c*(n + S(1))) + b*(-C*(c**S(2)*(m + S(1)) + d**S(2)*(n + S(1))) + d*(-A*d + B*c)*(m + n + S(2)))*cos(e + f*x) + (-B*d + C*c)*(a*c*m + b*d*(n + S(1))), x), x), x) + Simp((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*(A*d**S(2) - B*c*d + C*c**S(2))*sin(e + f*x)/(d*f*(c**S(2) - d**S(2))*(n + S(1))), x) def replacement2985(A, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(b*d*(c**S(2) - d**S(2))*(n + S(1))), Int((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*Simp(A*d*(a*d*m + b*c*(n + S(1))) + C*c*(a*c*m + b*d*(n + S(1))) - b*(A*d**S(2)*(m + n + S(2)) + C*(c**S(2)*(m + S(1)) + d**S(2)*(n + S(1))))*sin(e + f*x), x), x), x) - Simp((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*(A*d**S(2) + C*c**S(2))*cos(e + f*x)/(d*f*(c**S(2) - d**S(2))*(n + S(1))), x) def replacement2986(A, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(b*d*(c**S(2) - d**S(2))*(n + S(1))), Int((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*Simp(A*d*(a*d*m + b*c*(n + S(1))) + C*c*(a*c*m + b*d*(n + S(1))) - b*(A*d**S(2)*(m + n + S(2)) + C*(c**S(2)*(m + S(1)) + d**S(2)*(n + S(1))))*cos(e + f*x), x), x), x) + Simp((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*(A*d**S(2) + C*c**S(2))*sin(e + f*x)/(d*f*(c**S(2) - d**S(2))*(n + S(1))), x) def replacement2987(A, B, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(b*d*(m + n + S(2))), Int((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n*Simp(A*b*d*(m + n + S(2)) + C*(a*c*m + b*d*(n + S(1))) + (B*b*d*(m + n + S(2)) + C*(a*d*m - b*c*(m + S(1))))*sin(e + f*x), x), x), x) - Simp(C*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*cos(e + f*x)/(d*f*(m + n + S(2))), x) def replacement2988(A, B, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(b*d*(m + n + S(2))), Int((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n*Simp(A*b*d*(m + n + S(2)) + C*(a*c*m + b*d*(n + S(1))) + (B*b*d*(m + n + S(2)) + C*(a*d*m - b*c*(m + S(1))))*cos(e + f*x), x), x), x) + Simp(C*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*sin(e + f*x)/(d*f*(m + n + S(2))), x) def replacement2989(A, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(b*d*(m + n + S(2))), Int((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n*Simp(A*b*d*(m + n + S(2)) + C*(a*c*m + b*d*(n + S(1))) + C*(a*d*m - b*c*(m + S(1)))*sin(e + f*x), x), x), x) - Simp(C*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*cos(e + f*x)/(d*f*(m + n + S(2))), x) def replacement2990(A, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(b*d*(m + n + S(2))), Int((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n*Simp(A*b*d*(m + n + S(2)) + C*(a*c*m + b*d*(n + S(1))) + C*(a*d*m - b*c*(m + S(1)))*cos(e + f*x), x), x), x) + Simp(C*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*sin(e + f*x)/(d*f*(m + n + S(2))), x) def replacement2991(A, B, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(d*(c**S(2) - d**S(2))*(n + S(1))), Int((a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**(n + S(1))*Simp(A*d*(a*c*(n + S(1)) + b*d*m) + b*(-C*(c**S(2)*(m + S(1)) + d**S(2)*(n + S(1))) + d*(-A*d + B*c)*(m + n + S(2)))*sin(e + f*x)**S(2) + (-B*d + C*c)*(a*d*(n + S(1)) + b*c*m) - (-C*(-a*(c**S(2) + d**S(2)*(n + S(1))) + b*c*d*(n + S(1))) + d*(A*(a*d*(n + S(2)) - b*c*(n + S(1))) + B*(-a*c*(n + S(2)) + b*d*(n + S(1)))))*sin(e + f*x), x), x), x) - Simp((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*(A*d**S(2) - B*c*d + C*c**S(2))*cos(e + f*x)/(d*f*(c**S(2) - d**S(2))*(n + S(1))), x) def replacement2992(A, B, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(d*(c**S(2) - d**S(2))*(n + S(1))), Int((a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**(n + S(1))*Simp(A*d*(a*c*(n + S(1)) + b*d*m) + b*(-C*(c**S(2)*(m + S(1)) + d**S(2)*(n + S(1))) + d*(-A*d + B*c)*(m + n + S(2)))*cos(e + f*x)**S(2) + (-B*d + C*c)*(a*d*(n + S(1)) + b*c*m) - (-C*(-a*(c**S(2) + d**S(2)*(n + S(1))) + b*c*d*(n + S(1))) + d*(A*(a*d*(n + S(2)) - b*c*(n + S(1))) + B*(-a*c*(n + S(2)) + b*d*(n + S(1)))))*cos(e + f*x), x), x), x) + Simp((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*(A*d**S(2) - B*c*d + C*c**S(2))*sin(e + f*x)/(d*f*(c**S(2) - d**S(2))*(n + S(1))), x) def replacement2993(A, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(d*(c**S(2) - d**S(2))*(n + S(1))), Int((a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**(n + S(1))*Simp(A*d*(a*c*(n + S(1)) + b*d*m) + C*c*(a*d*(n + S(1)) + b*c*m) - b*(A*d**S(2)*(m + n + S(2)) + C*(c**S(2)*(m + S(1)) + d**S(2)*(n + S(1))))*sin(e + f*x)**S(2) - (A*d*(a*d*(n + S(2)) - b*c*(n + S(1))) - C*(-a*(c**S(2) + d**S(2)*(n + S(1))) + b*c*d*(n + S(1))))*sin(e + f*x), x), x), x) - Simp((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*(A*d**S(2) + C*c**S(2))*cos(e + f*x)/(d*f*(c**S(2) - d**S(2))*(n + S(1))), x) def replacement2994(A, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(d*(c**S(2) - d**S(2))*(n + S(1))), Int((a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**(n + S(1))*Simp(A*d*(a*c*(n + S(1)) + b*d*m) + C*c*(a*d*(n + S(1)) + b*c*m) - b*(A*d**S(2)*(m + n + S(2)) + C*(c**S(2)*(m + S(1)) + d**S(2)*(n + S(1))))*cos(e + f*x)**S(2) - (A*d*(a*d*(n + S(2)) - b*c*(n + S(1))) - C*(-a*(c**S(2) + d**S(2)*(n + S(1))) + b*c*d*(n + S(1))))*cos(e + f*x), x), x), x) + Simp((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*(A*d**S(2) + C*c**S(2))*sin(e + f*x)/(d*f*(c**S(2) - d**S(2))*(n + S(1))), x) def replacement2995(A, B, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(d*(m + n + S(2))), Int((a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**n*Simp(A*a*d*(m + n + S(2)) + C*(a*d*(n + S(1)) + b*c*m) + (-C*(a*c - b*d*(m + n + S(1))) + d*(A*b + B*a)*(m + n + S(2)))*sin(e + f*x) + (B*b*d*(m + n + S(2)) + C*(a*d*m - b*c*(m + S(1))))*sin(e + f*x)**S(2), x), x), x) - Simp(C*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*cos(e + f*x)/(d*f*(m + n + S(2))), x) def replacement2996(A, B, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(d*(m + n + S(2))), Int((a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**n*Simp(A*a*d*(m + n + S(2)) + C*(a*d*(n + S(1)) + b*c*m) + (-C*(a*c - b*d*(m + n + S(1))) + d*(A*b + B*a)*(m + n + S(2)))*cos(e + f*x) + (B*b*d*(m + n + S(2)) + C*(a*d*m - b*c*(m + S(1))))*cos(e + f*x)**S(2), x), x), x) + Simp(C*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*sin(e + f*x)/(d*f*(m + n + S(2))), x) def replacement2997(A, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(d*(m + n + S(2))), Int((a + b*sin(e + f*x))**(m + S(-1))*(c + d*sin(e + f*x))**n*Simp(A*a*d*(m + n + S(2)) + C*(a*d*m - b*c*(m + S(1)))*sin(e + f*x)**S(2) + C*(a*d*(n + S(1)) + b*c*m) + (A*b*d*(m + n + S(2)) - C*(a*c - b*d*(m + n + S(1))))*sin(e + f*x), x), x), x) - Simp(C*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**(n + S(1))*cos(e + f*x)/(d*f*(m + n + S(2))), x) def replacement2998(A, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(d*(m + n + S(2))), Int((a + b*cos(e + f*x))**(m + S(-1))*(c + d*cos(e + f*x))**n*Simp(A*a*d*(m + n + S(2)) + C*(a*d*m - b*c*(m + S(1)))*cos(e + f*x)**S(2) + C*(a*d*(n + S(1)) + b*c*m) + (A*b*d*(m + n + S(2)) - C*(a*c - b*d*(m + n + S(1))))*cos(e + f*x), x), x), x) + Simp(C*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**(n + S(1))*sin(e + f*x)/(d*f*(m + n + S(2))), x) def replacement2999(A, B, C, a, b, d, e, f, x): return Dist(S(1)/b, Int((A*b + (B*b - C*a)*sin(e + f*x))/(sqrt(d*sin(e + f*x))*(a + b*sin(e + f*x))**(S(3)/2)), x), x) + Dist(C/(b*d), Int(sqrt(d*sin(e + f*x))/sqrt(a + b*sin(e + f*x)), x), x) def replacement3000(A, B, C, a, b, d, e, f, x): return Dist(S(1)/b, Int((A*b + (B*b - C*a)*cos(e + f*x))/(sqrt(d*cos(e + f*x))*(a + b*cos(e + f*x))**(S(3)/2)), x), x) + Dist(C/(b*d), Int(sqrt(d*cos(e + f*x))/sqrt(a + b*cos(e + f*x)), x), x) def replacement3001(A, C, a, b, d, e, f, x): return Dist(S(1)/b, Int((A*b - C*a*sin(e + f*x))/(sqrt(d*sin(e + f*x))*(a + b*sin(e + f*x))**(S(3)/2)), x), x) + Dist(C/(b*d), Int(sqrt(d*sin(e + f*x))/sqrt(a + b*sin(e + f*x)), x), x) def replacement3002(A, C, a, b, d, e, f, x): return Dist(S(1)/b, Int((A*b - C*a*cos(e + f*x))/(sqrt(d*cos(e + f*x))*(a + b*cos(e + f*x))**(S(3)/2)), x), x) + Dist(C/(b*d), Int(sqrt(d*cos(e + f*x))/sqrt(a + b*cos(e + f*x)), x), x) def replacement3003(A, B, C, a, b, c, d, e, f, x): return Dist(b**(S(-2)), Int((A*b**S(2) - C*a**S(2) + b*(B*b - S(2)*C*a)*sin(e + f*x))/((a + b*sin(e + f*x))**(S(3)/2)*sqrt(c + d*sin(e + f*x))), x), x) + Dist(C/b**S(2), Int(sqrt(a + b*sin(e + f*x))/sqrt(c + d*sin(e + f*x)), x), x) def replacement3004(A, B, C, a, b, c, d, e, f, x): return Dist(b**(S(-2)), Int((A*b**S(2) - C*a**S(2) + b*(B*b - S(2)*C*a)*cos(e + f*x))/((a + b*cos(e + f*x))**(S(3)/2)*sqrt(c + d*cos(e + f*x))), x), x) + Dist(C/b**S(2), Int(sqrt(a + b*cos(e + f*x))/sqrt(c + d*cos(e + f*x)), x), x) def replacement3005(A, C, a, b, c, d, e, f, x): return Dist(b**(S(-2)), Int((A*b**S(2) - C*a**S(2) - S(2)*C*a*b*sin(e + f*x))/((a + b*sin(e + f*x))**(S(3)/2)*sqrt(c + d*sin(e + f*x))), x), x) + Dist(C/b**S(2), Int(sqrt(a + b*sin(e + f*x))/sqrt(c + d*sin(e + f*x)), x), x) def replacement3006(A, C, a, b, c, d, e, f, x): return Dist(b**(S(-2)), Int((A*b**S(2) - C*a**S(2) - S(2)*C*a*b*cos(e + f*x))/((a + b*cos(e + f*x))**(S(3)/2)*sqrt(c + d*cos(e + f*x))), x), x) + Dist(C/b**S(2), Int(sqrt(a + b*cos(e + f*x))/sqrt(c + d*cos(e + f*x)), x), x) def replacement3007(A, B, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))*(-a*d + b*c)), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n*Simp(d*(m + n + S(2))*(A*b**S(2) - B*a*b + C*a**S(2)) - d*(m + n + S(3))*(A*b**S(2) - B*a*b + C*a**S(2))*sin(e + f*x)**S(2) + (m + S(1))*(-a*d + b*c)*(A*a - B*b + C*a) - (c*(A*b**S(2) - B*a*b + C*a**S(2)) + (m + S(1))*(-a*d + b*c)*(A*b - B*a + C*b))*sin(e + f*x), x), x), x) - Simp((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**(n + S(1))*(A*b**S(2) - B*a*b + C*a**S(2))*cos(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))*(-a*d + b*c)), x) def replacement3008(A, B, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))*(-a*d + b*c)), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n*Simp(d*(m + n + S(2))*(A*b**S(2) - B*a*b + C*a**S(2)) - d*(m + n + S(3))*(A*b**S(2) - B*a*b + C*a**S(2))*cos(e + f*x)**S(2) + (m + S(1))*(-a*d + b*c)*(A*a - B*b + C*a) - (c*(A*b**S(2) - B*a*b + C*a**S(2)) + (m + S(1))*(-a*d + b*c)*(A*b - B*a + C*b))*cos(e + f*x), x), x), x) + Simp((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**(n + S(1))*(A*b**S(2) - B*a*b + C*a**S(2))*sin(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))*(-a*d + b*c)), x) def replacement3009(A, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))*(-a*d + b*c)), Int((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**n*Simp(a*(A + C)*(m + S(1))*(-a*d + b*c) + d*(A*b**S(2) + C*a**S(2))*(m + n + S(2)) - d*(A*b**S(2) + C*a**S(2))*(m + n + S(3))*sin(e + f*x)**S(2) - (b*(A + C)*(m + S(1))*(-a*d + b*c) + c*(A*b**S(2) + C*a**S(2)))*sin(e + f*x), x), x), x) - Simp((a + b*sin(e + f*x))**(m + S(1))*(c + d*sin(e + f*x))**(n + S(1))*(A*b**S(2) + C*a**S(2))*cos(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))*(-a*d + b*c)), x) def replacement3010(A, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/((a**S(2) - b**S(2))*(m + S(1))*(-a*d + b*c)), Int((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**n*Simp(a*(A + C)*(m + S(1))*(-a*d + b*c) + d*(A*b**S(2) + C*a**S(2))*(m + n + S(2)) - d*(A*b**S(2) + C*a**S(2))*(m + n + S(3))*cos(e + f*x)**S(2) - (b*(A + C)*(m + S(1))*(-a*d + b*c) + c*(A*b**S(2) + C*a**S(2)))*cos(e + f*x), x), x), x) + Simp((a + b*cos(e + f*x))**(m + S(1))*(c + d*cos(e + f*x))**(n + S(1))*(A*b**S(2) + C*a**S(2))*sin(e + f*x)/(f*(a**S(2) - b**S(2))*(m + S(1))*(-a*d + b*c)), x) def replacement3011(A, B, C, a, b, c, d, e, f, x): return Dist((A*b**S(2) - B*a*b + C*a**S(2))/(b*(-a*d + b*c)), Int(S(1)/(a + b*sin(e + f*x)), x), x) - Dist((A*d**S(2) - B*c*d + C*c**S(2))/(d*(-a*d + b*c)), Int(S(1)/(c + d*sin(e + f*x)), x), x) + Simp(C*x/(b*d), x) def replacement3012(A, B, C, a, b, c, d, e, f, x): return Dist((A*b**S(2) - B*a*b + C*a**S(2))/(b*(-a*d + b*c)), Int(S(1)/(a + b*cos(e + f*x)), x), x) - Dist((A*d**S(2) - B*c*d + C*c**S(2))/(d*(-a*d + b*c)), Int(S(1)/(c + d*cos(e + f*x)), x), x) + Simp(C*x/(b*d), x) def replacement3013(A, C, a, b, c, d, e, f, x): return Dist((A*b**S(2) + C*a**S(2))/(b*(-a*d + b*c)), Int(S(1)/(a + b*sin(e + f*x)), x), x) - Dist((A*d**S(2) + C*c**S(2))/(d*(-a*d + b*c)), Int(S(1)/(c + d*sin(e + f*x)), x), x) + Simp(C*x/(b*d), x) def replacement3014(A, C, a, b, c, d, e, f, x): return Dist((A*b**S(2) + C*a**S(2))/(b*(-a*d + b*c)), Int(S(1)/(a + b*cos(e + f*x)), x), x) - Dist((A*d**S(2) + C*c**S(2))/(d*(-a*d + b*c)), Int(S(1)/(c + d*cos(e + f*x)), x), x) + Simp(C*x/(b*d), x) def replacement3015(A, B, C, a, b, c, d, e, f, x): return -Dist(S(1)/(b*d), Int(Simp(-A*b*d + C*a*c + (-B*b*d + C*a*d + C*b*c)*sin(e + f*x), x)/(sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))), x), x) + Dist(C/(b*d), Int(sqrt(a + b*sin(e + f*x)), x), x) def replacement3016(A, B, C, a, b, c, d, e, f, x): return -Dist(S(1)/(b*d), Int(Simp(-A*b*d + C*a*c + (-B*b*d + C*a*d + C*b*c)*cos(e + f*x), x)/(sqrt(a + b*cos(e + f*x))*(c + d*cos(e + f*x))), x), x) + Dist(C/(b*d), Int(sqrt(a + b*cos(e + f*x)), x), x) def replacement3017(A, C, a, b, c, d, e, f, x): return -Dist(S(1)/(b*d), Int(Simp(-A*b*d + C*a*c + (C*a*d + C*b*c)*sin(e + f*x), x)/(sqrt(a + b*sin(e + f*x))*(c + d*sin(e + f*x))), x), x) + Dist(C/(b*d), Int(sqrt(a + b*sin(e + f*x)), x), x) def replacement3018(A, C, a, b, c, d, e, f, x): return -Dist(S(1)/(b*d), Int(Simp(-A*b*d + C*a*c + (C*a*d + C*b*c)*cos(e + f*x), x)/(sqrt(a + b*cos(e + f*x))*(c + d*cos(e + f*x))), x), x) + Dist(C/(b*d), Int(sqrt(a + b*cos(e + f*x)), x), x) def replacement3019(A, B, C, a, b, c, d, e, f, x): return Dist(S(1)/(S(2)*d), Int(Simp(S(2)*A*a*d - C*(-a*d + b*c) + (S(2)*B*b*d - C*(a*d + b*c))*sin(e + f*x)**S(2) - S(2)*(C*a*c - d*(A*b + B*a))*sin(e + f*x), x)/((a + b*sin(e + f*x))**(S(3)/2)*sqrt(c + d*sin(e + f*x))), x), x) - Simp(C*sqrt(c + d*sin(e + f*x))*cos(e + f*x)/(d*f*sqrt(a + b*sin(e + f*x))), x) def replacement3020(A, B, C, a, b, c, d, e, f, x): return Dist(S(1)/(S(2)*d), Int(Simp(S(2)*A*a*d - C*(-a*d + b*c) + (S(2)*B*b*d - C*(a*d + b*c))*cos(e + f*x)**S(2) - S(2)*(C*a*c - d*(A*b + B*a))*cos(e + f*x), x)/((a + b*cos(e + f*x))**(S(3)/2)*sqrt(c + d*cos(e + f*x))), x), x) + Simp(C*sqrt(c + d*cos(e + f*x))*sin(e + f*x)/(d*f*sqrt(a + b*cos(e + f*x))), x) def replacement3021(A, C, a, b, c, d, e, f, x): return Dist(S(1)/(S(2)*d), Int(Simp(S(2)*A*a*d - C*(-a*d + b*c) - C*(a*d + b*c)*sin(e + f*x)**S(2) - S(2)*(-A*b*d + C*a*c)*sin(e + f*x), x)/((a + b*sin(e + f*x))**(S(3)/2)*sqrt(c + d*sin(e + f*x))), x), x) - Simp(C*sqrt(c + d*sin(e + f*x))*cos(e + f*x)/(d*f*sqrt(a + b*sin(e + f*x))), x) def replacement3022(A, C, a, b, c, d, e, f, x): return Dist(S(1)/(S(2)*d), Int(Simp(S(2)*A*a*d - C*(-a*d + b*c) - C*(a*d + b*c)*cos(e + f*x)**S(2) - S(2)*(-A*b*d + C*a*c)*cos(e + f*x), x)/((a + b*cos(e + f*x))**(S(3)/2)*sqrt(c + d*cos(e + f*x))), x), x) + Simp(C*sqrt(c + d*cos(e + f*x))*sin(e + f*x)/(d*f*sqrt(a + b*cos(e + f*x))), x) def replacement3023(A, B, C, a, b, c, d, e, f, m, n, x): return Int((a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n*(A + B*sin(e + f*x) + C*sin(e + f*x)**S(2)), x) def replacement3024(A, B, C, a, b, c, d, e, f, m, n, x): return Int((a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n*(A + B*cos(e + f*x) + C*cos(e + f*x)**S(2)), x) def replacement3025(A, C, a, b, c, d, e, f, m, n, x): return Int((A + C*sin(e + f*x)**S(2))*(a + b*sin(e + f*x))**m*(c + d*sin(e + f*x))**n, x) def replacement3026(A, C, a, b, c, d, e, f, m, n, x): return Int((A + C*cos(e + f*x)**S(2))*(a + b*cos(e + f*x))**m*(c + d*cos(e + f*x))**n, x) def replacement3027(A, B, C, b, c, d, e, f, m, n, p, x): return Dist((b*sin(e + f*x))**(-m*p)*(b*sin(e + f*x)**p)**m, Int((b*sin(e + f*x))**(m*p)*(c + d*sin(e + f*x))**n*(A + B*sin(e + f*x) + C*sin(e + f*x)**S(2)), x), x) def replacement3028(A, B, C, b, c, d, e, f, m, n, p, x): return Dist((b*cos(e + f*x))**(-m*p)*(b*cos(e + f*x)**p)**m, Int((b*cos(e + f*x))**(m*p)*(c + d*cos(e + f*x))**n*(A + B*cos(e + f*x) + C*cos(e + f*x)**S(2)), x), x) def replacement3029(A, C, b, c, d, e, f, m, n, p, x): return Dist((b*sin(e + f*x))**(-m*p)*(b*sin(e + f*x)**p)**m, Int((b*sin(e + f*x))**(m*p)*(A + C*sin(e + f*x)**S(2))*(c + d*sin(e + f*x))**n, x), x) def replacement3030(A, C, b, c, d, e, f, m, n, p, x): return Dist((b*cos(e + f*x))**(-m*p)*(b*cos(e + f*x)**p)**m, Int((b*cos(e + f*x))**(m*p)*(A + C*cos(e + f*x)**S(2))*(c + d*cos(e + f*x))**n, x), x) def replacement3031(a, b, c, d, n, x): return Simp(a*(a*cos(c + d*x) + b*sin(c + d*x))**n/(b*d*n), x) def replacement3032(a, b, c, d, n, x): return -Dist(S(1)/d, Subst(Int((a**S(2) + b**S(2) - x**S(2))**(n/S(2) + S(-1)/2), x), x, -a*sin(c + d*x) + b*cos(c + d*x)), x) def replacement3033(a, b, c, d, n, x): return Dist((a**S(2) + b**S(2))*(n + S(-1))/n, Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(-2)), x), x) - Simp((-a*sin(c + d*x) + b*cos(c + d*x))*(a*cos(c + d*x) + b*sin(c + d*x))**(n + S(-1))/(d*n), x) def replacement3034(a, b, c, d, x): return -Dist(S(1)/d, Subst(Int(S(1)/(a**S(2) + b**S(2) - x**S(2)), x), x, -a*sin(c + d*x) + b*cos(c + d*x)), x) def replacement3035(a, b, c, d, x): return Simp(sin(c + d*x)/(a*d*(a*cos(c + d*x) + b*sin(c + d*x))), x) def replacement3036(a, b, c, d, n, x): return Dist((n + S(2))/((a**S(2) + b**S(2))*(n + S(1))), Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(2)), x), x) + Simp((-a*sin(c + d*x) + b*cos(c + d*x))*(a*cos(c + d*x) + b*sin(c + d*x))**(n + S(1))/(d*(a**S(2) + b**S(2))*(n + S(1))), x) def replacement3037(a, b, c, d, n, x): return Dist((a**S(2) + b**S(2))**(n/S(2)), Int(cos(c + d*x - ArcTan(a, b))**n, x), x) def replacement3038(a, b, c, d, n, x): return Dist(((a*cos(c + d*x) + b*sin(c + d*x))/sqrt(a**S(2) + b**S(2)))**(-n)*(a*cos(c + d*x) + b*sin(c + d*x))**n, Int(cos(c + d*x - ArcTan(a, b))**n, x), x) def replacement3039(a, b, c, d, m, n, x): return Dist(S(2)*b, Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(-1))*sin(c + d*x)**(S(1) - n), x), x) - Simp(a*(a*cos(c + d*x) + b*sin(c + d*x))**(n + S(-1))*sin(c + d*x)**(S(1) - n)/(d*(n + S(-1))), x) def replacement3040(a, b, c, d, m, n, x): return Dist(S(2)*a, Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(-1))*cos(c + d*x)**(S(1) - n), x), x) + Simp(b*(a*cos(c + d*x) + b*sin(c + d*x))**(n + S(-1))*cos(c + d*x)**(S(1) - n)/(d*(n + S(-1))), x) def replacement3041(a, b, c, d, m, n, x): return Dist(S(1)/(S(2)*b), Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(1))*sin(c + d*x)**(-n + S(-1)), x), x) + Simp(a*(a*cos(c + d*x) + b*sin(c + d*x))**n*sin(c + d*x)**(-n)/(S(2)*b*d*n), x) def replacement3042(a, b, c, d, m, n, x): return Dist(S(1)/(S(2)*a), Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(1))*cos(c + d*x)**(-n + S(-1)), x), x) - Simp(b*(a*cos(c + d*x) + b*sin(c + d*x))**n*cos(c + d*x)**(-n)/(S(2)*a*d*n), x) def replacement3043(a, b, c, d, m, n, x): return Simp(a*(a*cos(c + d*x) + b*sin(c + d*x))**n*Hypergeometric2F1(S(1), n, n + S(1), (a/tan(c + d*x) + b)/(S(2)*b))*sin(c + d*x)**(-n)/(S(2)*b*d*n), x) def replacement3044(a, b, c, d, m, n, x): return -Simp(b*(a*cos(c + d*x) + b*sin(c + d*x))**n*Hypergeometric2F1(S(1), n, n + S(1), (a + b*tan(c + d*x))/(S(2)*a))*cos(c + d*x)**(-n)/(S(2)*a*d*n), x) def replacement3045(a, b, c, d, m, n, x): return Int((a/tan(c + d*x) + b)**n, x) def replacement3046(a, b, c, d, m, n, x): return Int((a + b*tan(c + d*x))**n, x) def replacement3047(a, b, c, d, m, n, x): return Dist(S(1)/d, Subst(Int(x**m*(a + b*x)**n*(x**S(2) + S(1))**(-m/S(2) - n/S(2) + S(-1)), x), x, tan(c + d*x)), x) def replacement3048(a, b, c, d, m, n, x): return -Dist(S(1)/d, Subst(Int(x**m*(x**S(2) + S(1))**(-m/S(2) - n/S(2) + S(-1))*(a*x + b)**n, x), x, S(1)/tan(c + d*x)), x) def replacement3049(a, b, c, d, m, n, x): return Int(ExpandTrig((a*cos(c + d*x) + b*sin(c + d*x))**n*sin(c + d*x)**m, x), x) def replacement3050(a, b, c, d, m, n, x): return Int(ExpandTrig((a*cos(c + d*x) + b*sin(c + d*x))**n*cos(c + d*x)**m, x), x) def replacement3051(a, b, c, d, m, n, x): return Dist(a**n*b**n, Int((a*sin(c + d*x) + b*cos(c + d*x))**(-n)*sin(c + d*x)**m, x), x) def replacement3052(a, b, c, d, m, n, x): return Dist(a**n*b**n, Int((a*sin(c + d*x) + b*cos(c + d*x))**(-n)*cos(c + d*x)**m, x), x) def replacement3053(a, b, c, d, n, x): return Dist(a**(S(-2)), Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(2))/sin(c + d*x), x), x) - Dist(b/a**S(2), Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(1)), x), x) - Simp((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(1))/(a*d*(n + S(1))), x) def replacement3054(a, b, c, d, n, x): return Dist(b**(S(-2)), Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(2))/cos(c + d*x), x), x) - Dist(a/b**S(2), Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(1)), x), x) + Simp((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(1))/(b*d*(n + S(1))), x) def replacement3055(a, b, c, d, m, n, x): return Dist(a**S(2), Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(-2))*sin(c + d*x)**m, x), x) + Dist(S(2)*b, Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(-1))*sin(c + d*x)**(m + S(1)), x), x) - Dist(a**S(2) + b**S(2), Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(-2))*sin(c + d*x)**(m + S(2)), x), x) def replacement3056(a, b, c, d, m, n, x): return Dist(S(2)*a, Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(-1))*cos(c + d*x)**(m + S(1)), x), x) + Dist(b**S(2), Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(-2))*cos(c + d*x)**m, x), x) - Dist(a**S(2) + b**S(2), Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(-2))*cos(c + d*x)**(m + S(2)), x), x) def replacement3057(a, b, c, d, x): return -Dist(a/(a**S(2) + b**S(2)), Int((-a*sin(c + d*x) + b*cos(c + d*x))/(a*cos(c + d*x) + b*sin(c + d*x)), x), x) + Simp(b*x/(a**S(2) + b**S(2)), x) def replacement3058(a, b, c, d, x): return Dist(b/(a**S(2) + b**S(2)), Int((-a*sin(c + d*x) + b*cos(c + d*x))/(a*cos(c + d*x) + b*sin(c + d*x)), x), x) + Simp(a*x/(a**S(2) + b**S(2)), x) def replacement3059(a, b, c, d, m, x): return Dist(a**S(2)/(a**S(2) + b**S(2)), Int(sin(c + d*x)**(m + S(-2))/(a*cos(c + d*x) + b*sin(c + d*x)), x), x) + Dist(b/(a**S(2) + b**S(2)), Int(sin(c + d*x)**(m + S(-1)), x), x) - Simp(a*sin(c + d*x)**(m + S(-1))/(d*(a**S(2) + b**S(2))*(m + S(-1))), x) def replacement3060(a, b, c, d, m, x): return Dist(a/(a**S(2) + b**S(2)), Int(cos(c + d*x)**(m + S(-1)), x), x) + Dist(b**S(2)/(a**S(2) + b**S(2)), Int(cos(c + d*x)**(m + S(-2))/(a*cos(c + d*x) + b*sin(c + d*x)), x), x) + Simp(b*cos(c + d*x)**(m + S(-1))/(d*(a**S(2) + b**S(2))*(m + S(-1))), x) def replacement3061(a, b, c, d, x): return -Dist(S(1)/a, Int((-a*sin(c + d*x) + b*cos(c + d*x))/(a*cos(c + d*x) + b*sin(c + d*x)), x), x) + Dist(S(1)/a, Int(S(1)/tan(c + d*x), x), x) def replacement3062(a, b, c, d, x): return Dist(S(1)/b, Int((-a*sin(c + d*x) + b*cos(c + d*x))/(a*cos(c + d*x) + b*sin(c + d*x)), x), x) + Dist(S(1)/b, Int(tan(c + d*x), x), x) def replacement3063(a, b, c, d, m, x): return -Dist(b/a**S(2), Int(sin(c + d*x)**(m + S(1)), x), x) + Dist((a**S(2) + b**S(2))/a**S(2), Int(sin(c + d*x)**(m + S(2))/(a*cos(c + d*x) + b*sin(c + d*x)), x), x) + Simp(sin(c + d*x)**(m + S(1))/(a*d*(m + S(1))), x) def replacement3064(a, b, c, d, m, x): return -Dist(a/b**S(2), Int(cos(c + d*x)**(m + S(1)), x), x) + Dist((a**S(2) + b**S(2))/b**S(2), Int(cos(c + d*x)**(m + S(2))/(a*cos(c + d*x) + b*sin(c + d*x)), x), x) - Simp(cos(c + d*x)**(m + S(1))/(b*d*(m + S(1))), x) def replacement3065(a, b, c, d, m, n, x): return Dist(a**(S(-2)), Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(2))*sin(c + d*x)**m, x), x) - Dist(S(2)*b/a**S(2), Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(1))*sin(c + d*x)**(m + S(1)), x), x) + Dist((a**S(2) + b**S(2))/a**S(2), Int((a*cos(c + d*x) + b*sin(c + d*x))**n*sin(c + d*x)**(m + S(2)), x), x) def replacement3066(a, b, c, d, m, n, x): return Dist(b**(S(-2)), Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(2))*cos(c + d*x)**m, x), x) - Dist(S(2)*a/b**S(2), Int((a*cos(c + d*x) + b*sin(c + d*x))**(n + S(1))*cos(c + d*x)**(m + S(1)), x), x) + Dist((a**S(2) + b**S(2))/b**S(2), Int((a*cos(c + d*x) + b*sin(c + d*x))**n*cos(c + d*x)**(m + S(2)), x), x) def replacement3067(a, b, c, d, m, n, p, x): return Int(ExpandTrig((a*cos(c + d*x) + b*sin(c + d*x))**p*sin(c + d*x)**n*cos(c + d*x)**m, x), x) def replacement3068(a, b, c, d, m, n, p, x): return Dist(a**p*b**p, Int((a*sin(c + d*x) + b*cos(c + d*x))**(-p)*sin(c + d*x)**n*cos(c + d*x)**m, x), x) def replacement3069(a, b, c, d, m, n, x): return Dist(a/(a**S(2) + b**S(2)), Int(sin(c + d*x)**n*cos(c + d*x)**(m + S(-1)), x), x) + Dist(b/(a**S(2) + b**S(2)), Int(sin(c + d*x)**(n + S(-1))*cos(c + d*x)**m, x), x) - Dist(a*b/(a**S(2) + b**S(2)), Int(sin(c + d*x)**(n + S(-1))*cos(c + d*x)**(m + S(-1))/(a*cos(c + d*x) + b*sin(c + d*x)), x), x) def replacement3070(a, b, c, d, m, n, x): return Int(ExpandTrig(sin(c + d*x)**n*cos(c + d*x)**m/(a*cos(c + d*x) + b*sin(c + d*x)), x), x) def replacement3071(a, b, c, d, m, n, p, x): return Dist(a/(a**S(2) + b**S(2)), Int((a*cos(c + d*x) + b*sin(c + d*x))**(p + S(1))*sin(c + d*x)**n*cos(c + d*x)**(m + S(-1)), x), x) + Dist(b/(a**S(2) + b**S(2)), Int((a*cos(c + d*x) + b*sin(c + d*x))**(p + S(1))*sin(c + d*x)**(n + S(-1))*cos(c + d*x)**m, x), x) - Dist(a*b/(a**S(2) + b**S(2)), Int((a*cos(c + d*x) + b*sin(c + d*x))**p*sin(c + d*x)**(n + S(-1))*cos(c + d*x)**(m + S(-1)), x), x) def replacement3072(a, b, c, d, e, x): return Simp(-S(2)*(-b*sin(d + e*x) + c*cos(d + e*x))/(e*sqrt(a + b*cos(d + e*x) + c*sin(d + e*x))), x) def replacement3073(a, b, c, d, e, n, x): return Dist(a*(S(2)*n + S(-1))/n, Int((a + b*cos(d + e*x) + c*sin(d + e*x))**(n + S(-1)), x), x) - Simp((-b*sin(d + e*x) + c*cos(d + e*x))*(a + b*cos(d + e*x) + c*sin(d + e*x))**(n + S(-1))/(e*n), x) def replacement3074(a, b, c, d, e, x): return -Simp((-a*sin(d + e*x) + c)/(c*e*(-b*sin(d + e*x) + c*cos(d + e*x))), x) def replacement3075(a, b, c, d, e, x): return Int(S(1)/sqrt(a + sqrt(b**S(2) + c**S(2))*cos(d + e*x - ArcTan(b, c))), x) def replacement3076(a, b, c, d, e, n, x): return Dist((n + S(1))/(a*(S(2)*n + S(1))), Int((a + b*cos(d + e*x) + c*sin(d + e*x))**(n + S(1)), x), x) + Simp((-b*sin(d + e*x) + c*cos(d + e*x))*(a + b*cos(d + e*x) + c*sin(d + e*x))**n/(a*e*(S(2)*n + S(1))), x) def replacement3077(a, b, c, d, e, x): return Dist(b/(c*e), Subst(Int(sqrt(a + x)/x, x), x, b*cos(d + e*x) + c*sin(d + e*x)), x) def replacement3078(a, b, c, d, e, x): return Int(sqrt(a + sqrt(b**S(2) + c**S(2))*cos(d + e*x - ArcTan(b, c))), x) def replacement3079(a, b, c, d, e, x): return Dist(sqrt(a + b*cos(d + e*x) + c*sin(d + e*x))/sqrt((a + b*cos(d + e*x) + c*sin(d + e*x))/(a + sqrt(b**S(2) + c**S(2)))), Int(sqrt(a/(a + sqrt(b**S(2) + c**S(2))) + sqrt(b**S(2) + c**S(2))*cos(d + e*x - ArcTan(b, c))/(a + sqrt(b**S(2) + c**S(2)))), x), x) def replacement3080(a, b, c, d, e, n, x): return Dist(S(1)/n, Int((a + b*cos(d + e*x) + c*sin(d + e*x))**(n + S(-2))*Simp(a**S(2)*n + a*b*(S(2)*n + S(-1))*cos(d + e*x) + a*c*(S(2)*n + S(-1))*sin(d + e*x) + (b**S(2) + c**S(2))*(n + S(-1)), x), x), x) - Simp((-b*sin(d + e*x) + c*cos(d + e*x))*(a + b*cos(d + e*x) + c*sin(d + e*x))**(n + S(-1))/(e*n), x) def With3081(a, b, c, d, e, x): f = FreeFactors(S(1)/tan(d/S(2) + e*x/S(2)), x) return -Dist(f/e, Subst(Int(S(1)/(a + c*f*x), x), x, S(1)/(f*tan(d/S(2) + e*x/S(2)))), x) def With3082(a, b, c, d, e, x): f = FreeFactors(tan(Pi/S(4) + d/S(2) + e*x/S(2)), x) return Dist(f/e, Subst(Int(S(1)/(a + b*f*x), x), x, tan(Pi/S(4) + d/S(2) + e*x/S(2))/f), x) def With3083(a, b, c, d, e, x): f = FreeFactors(S(1)/tan(Pi/S(4) + d/S(2) + e*x/S(2)), x) return -Dist(f/e, Subst(Int(S(1)/(a + b*f*x), x), x, S(1)/(f*tan(Pi/S(4) + d/S(2) + e*x/S(2)))), x) def With3084(a, b, c, d, e, x): f = FreeFactors(tan(d/S(2) + e*x/S(2)), x) return Dist(S(2)*f/e, Subst(Int(S(1)/(a + b + S(2)*c*f*x + f**S(2)*x**S(2)*(a - b)), x), x, tan(d/S(2) + e*x/S(2))/f), x) def replacement3085(a, b, c, d, e, x): return Dist(b/(c*e), Subst(Int(S(1)/(x*sqrt(a + x)), x), x, b*cos(d + e*x) + c*sin(d + e*x)), x) def replacement3086(a, b, c, d, e, x): return Int(S(1)/sqrt(a + sqrt(b**S(2) + c**S(2))*cos(d + e*x - ArcTan(b, c))), x) def replacement3087(a, b, c, d, e, x): return Dist(sqrt((a + b*cos(d + e*x) + c*sin(d + e*x))/(a + sqrt(b**S(2) + c**S(2))))/sqrt(a + b*cos(d + e*x) + c*sin(d + e*x)), Int(S(1)/sqrt(a/(a + sqrt(b**S(2) + c**S(2))) + sqrt(b**S(2) + c**S(2))*cos(d + e*x - ArcTan(b, c))/(a + sqrt(b**S(2) + c**S(2)))), x), x) def replacement3088(a, b, c, d, e, x): return Dist(S(1)/(a**S(2) - b**S(2) - c**S(2)), Int(sqrt(a + b*cos(d + e*x) + c*sin(d + e*x)), x), x) + Simp(S(2)*(-b*sin(d + e*x) + c*cos(d + e*x))/(e*sqrt(a + b*cos(d + e*x) + c*sin(d + e*x))*(a**S(2) - b**S(2) - c**S(2))), x) def replacement3089(a, b, c, d, e, n, x): return Dist(S(1)/((n + S(1))*(a**S(2) - b**S(2) - c**S(2))), Int((a + b*cos(d + e*x) + c*sin(d + e*x))**(n + S(1))*(a*(n + S(1)) - b*(n + S(2))*cos(d + e*x) - c*(n + S(2))*sin(d + e*x)), x), x) + Simp((b*sin(d + e*x) - c*cos(d + e*x))*(a + b*cos(d + e*x) + c*sin(d + e*x))**(n + S(1))/(e*(n + S(1))*(a**S(2) - b**S(2) - c**S(2))), x) def replacement3090(A, B, C, a, b, c, d, e, x): return Simp(x*(S(2)*A*a - B*b - C*c)/(S(2)*a**S(2)), x) + Simp((-S(2)*A*a*b**S(2) + a**S(2)*(B*b - C*c) + b**S(2)*(B*b + C*c))*log(RemoveContent(a + b*cos(d + e*x) + c*sin(d + e*x), x))/(S(2)*a**S(2)*b*c*e), x) - Simp((B*b + C*c)*(b*cos(d + e*x) - c*sin(d + e*x))/(S(2)*a*b*c*e), x) def replacement3091(A, C, a, b, c, d, e, x): return Simp(x*(S(2)*A*a - C*c)/(S(2)*a**S(2)), x) - Simp(C*cos(d + e*x)/(S(2)*a*e), x) + Simp((S(2)*A*a*c - C*a**S(2) + C*b**S(2))*log(RemoveContent(a + b*cos(d + e*x) + c*sin(d + e*x), x))/(S(2)*a**S(2)*b*e), x) + Simp(C*c*sin(d + e*x)/(S(2)*a*b*e), x) def replacement3092(A, B, a, b, c, d, e, x): return Simp(x*(S(2)*A*a - B*b)/(S(2)*a**S(2)), x) + Simp(B*sin(d + e*x)/(S(2)*a*e), x) + Simp((-S(2)*A*a*b + B*a**S(2) + B*b**S(2))*log(RemoveContent(a + b*cos(d + e*x) + c*sin(d + e*x), x))/(S(2)*a**S(2)*c*e), x) - Simp(B*b*cos(d + e*x)/(S(2)*a*c*e), x) def replacement3093(A, B, C, a, b, c, d, e, x): return Simp(x*(B*b + C*c)/(b**S(2) + c**S(2)), x) + Simp((B*c - C*b)*log(a + b*cos(d + e*x) + c*sin(d + e*x))/(e*(b**S(2) + c**S(2))), x) def replacement3094(A, C, a, b, c, d, e, x): return Simp(C*c*x/(b**S(2) + c**S(2)), x) - Simp(C*b*log(a + b*cos(d + e*x) + c*sin(d + e*x))/(e*(b**S(2) + c**S(2))), x) def replacement3095(A, B, a, b, c, d, e, x): return Simp(B*b*x/(b**S(2) + c**S(2)), x) + Simp(B*c*log(a + b*cos(d + e*x) + c*sin(d + e*x))/(e*(b**S(2) + c**S(2))), x) def replacement3096(A, B, C, a, b, c, d, e, x): return Dist((A*(b**S(2) + c**S(2)) - a*(B*b + C*c))/(b**S(2) + c**S(2)), Int(S(1)/(a + b*cos(d + e*x) + c*sin(d + e*x)), x), x) + Simp(x*(B*b + C*c)/(b**S(2) + c**S(2)), x) + Simp((B*c - C*b)*log(a + b*cos(d + e*x) + c*sin(d + e*x))/(e*(b**S(2) + c**S(2))), x) def replacement3097(A, C, a, b, c, d, e, x): return Dist((A*(b**S(2) + c**S(2)) - C*a*c)/(b**S(2) + c**S(2)), Int(S(1)/(a + b*cos(d + e*x) + c*sin(d + e*x)), x), x) - Simp(C*b*log(a + b*cos(d + e*x) + c*sin(d + e*x))/(e*(b**S(2) + c**S(2))), x) + Simp(C*c*(d + e*x)/(e*(b**S(2) + c**S(2))), x) def replacement3098(A, B, a, b, c, d, e, x): return Dist((A*(b**S(2) + c**S(2)) - B*a*b)/(b**S(2) + c**S(2)), Int(S(1)/(a + b*cos(d + e*x) + c*sin(d + e*x)), x), x) + Simp(B*b*(d + e*x)/(e*(b**S(2) + c**S(2))), x) + Simp(B*c*log(a + b*cos(d + e*x) + c*sin(d + e*x))/(e*(b**S(2) + c**S(2))), x) def replacement3099(A, B, C, a, b, c, d, e, n, x): return Simp((a + b*cos(d + e*x) + c*sin(d + e*x))**n*(B*a*sin(d + e*x) + B*c - C*a*cos(d + e*x) - C*b)/(a*e*(n + S(1))), x) def replacement3100(A, C, a, b, c, d, e, n, x): return -Simp((C*a*cos(d + e*x) + C*b)*(a + b*cos(d + e*x) + c*sin(d + e*x))**n/(a*e*(n + S(1))), x) def replacement3101(A, B, a, b, c, d, e, n, x): return Simp((B*a*sin(d + e*x) + B*c)*(a + b*cos(d + e*x) + c*sin(d + e*x))**n/(a*e*(n + S(1))), x) def replacement3102(A, B, C, a, b, c, d, e, n, x): return Dist((A*a*(n + S(1)) + n*(B*b + C*c))/(a*(n + S(1))), Int((a + b*cos(d + e*x) + c*sin(d + e*x))**n, x), x) + Simp((a + b*cos(d + e*x) + c*sin(d + e*x))**n*(B*a*sin(d + e*x) + B*c - C*a*cos(d + e*x) - C*b)/(a*e*(n + S(1))), x) def replacement3103(A, C, a, b, c, d, e, n, x): return Dist((A*a*(n + S(1)) + C*c*n)/(a*(n + S(1))), Int((a + b*cos(d + e*x) + c*sin(d + e*x))**n, x), x) - Simp((C*a*cos(d + e*x) + C*b)*(a + b*cos(d + e*x) + c*sin(d + e*x))**n/(a*e*(n + S(1))), x) def replacement3104(A, B, a, b, c, d, e, n, x): return Dist((A*a*(n + S(1)) + B*b*n)/(a*(n + S(1))), Int((a + b*cos(d + e*x) + c*sin(d + e*x))**n, x), x) + Simp((B*a*sin(d + e*x) + B*c)*(a + b*cos(d + e*x) + c*sin(d + e*x))**n/(a*e*(n + S(1))), x) def replacement3105(B, C, b, c, d, e, n, x): return Simp((B*c - C*b)*(b*cos(d + e*x) + c*sin(d + e*x))**(n + S(1))/(e*(b**S(2) + c**S(2))*(n + S(1))), x) def replacement3106(A, B, C, a, b, c, d, e, n, x): return Dist(S(1)/(a*(n + S(1))), Int((a + b*cos(d + e*x) + c*sin(d + e*x))**(n + S(-1))*Simp(A*a**S(2)*(n + S(1)) + a*n*(B*b + C*c) + (A*a*b*(n + S(1)) + n*(B*a**S(2) - B*c**S(2) + C*b*c))*cos(d + e*x) + (A*a*c*(n + S(1)) + n*(B*b*c + C*a**S(2) - C*b**S(2)))*sin(d + e*x), x), x), x) + Simp((a + b*cos(d + e*x) + c*sin(d + e*x))**n*(B*a*sin(d + e*x) + B*c - C*a*cos(d + e*x) - C*b)/(a*e*(n + S(1))), x) def replacement3107(A, C, a, b, c, d, e, n, x): return Dist(S(1)/(a*(n + S(1))), Int((a + b*cos(d + e*x) + c*sin(d + e*x))**(n + S(-1))*Simp(A*a**S(2)*(n + S(1)) + C*a*c*n + (A*a*b*(n + S(1)) + C*b*c*n)*cos(d + e*x) + (A*a*c*(n + S(1)) + C*a**S(2)*n - C*b**S(2)*n)*sin(d + e*x), x), x), x) - Simp((C*a*cos(d + e*x) + C*b)*(a + b*cos(d + e*x) + c*sin(d + e*x))**n/(a*e*(n + S(1))), x) def replacement3108(A, B, a, b, c, d, e, n, x): return Dist(S(1)/(a*(n + S(1))), Int((a + b*cos(d + e*x) + c*sin(d + e*x))**(n + S(-1))*Simp(A*a**S(2)*(n + S(1)) + B*a*b*n + (A*a*c*(n + S(1)) + B*b*c*n)*sin(d + e*x) + (A*a*b*(n + S(1)) + B*a**S(2)*n - B*c**S(2)*n)*cos(d + e*x), x), x), x) + Simp((B*a*sin(d + e*x) + B*c)*(a + b*cos(d + e*x) + c*sin(d + e*x))**n/(a*e*(n + S(1))), x) def replacement3109(A, B, C, a, b, c, d, e, x): return Dist(B/b, Int(sqrt(a + b*cos(d + e*x) + c*sin(d + e*x)), x), x) + Dist((A*b - B*a)/b, Int(S(1)/sqrt(a + b*cos(d + e*x) + c*sin(d + e*x)), x), x) def replacement3110(A, B, C, a, b, c, d, e, x): return Simp((B*c - C*b + (-A*b + B*a)*sin(d + e*x) - (-A*c + C*a)*cos(d + e*x))/(e*(a + b*cos(d + e*x) + c*sin(d + e*x))*(a**S(2) - b**S(2) - c**S(2))), x) def replacement3111(A, C, a, b, c, d, e, x): return -Simp((A*b*sin(d + e*x) + C*b + (-A*c + C*a)*cos(d + e*x))/(e*(a + b*cos(d + e*x) + c*sin(d + e*x))*(a**S(2) - b**S(2) - c**S(2))), x) def replacement3112(A, B, a, b, c, d, e, x): return Simp((A*c*cos(d + e*x) + B*c + (-A*b + B*a)*sin(d + e*x))/(e*(a + b*cos(d + e*x) + c*sin(d + e*x))*(a**S(2) - b**S(2) - c**S(2))), x) def replacement3113(A, B, C, a, b, c, d, e, x): return Dist((A*a - B*b - C*c)/(a**S(2) - b**S(2) - c**S(2)), Int(S(1)/(a + b*cos(d + e*x) + c*sin(d + e*x)), x), x) + Simp((B*c - C*b + (-A*b + B*a)*sin(d + e*x) - (-A*c + C*a)*cos(d + e*x))/(e*(a + b*cos(d + e*x) + c*sin(d + e*x))*(a**S(2) - b**S(2) - c**S(2))), x) def replacement3114(A, C, a, b, c, d, e, x): return Dist((A*a - C*c)/(a**S(2) - b**S(2) - c**S(2)), Int(S(1)/(a + b*cos(d + e*x) + c*sin(d + e*x)), x), x) - Simp((A*b*sin(d + e*x) + C*b + (-A*c + C*a)*cos(d + e*x))/(e*(a + b*cos(d + e*x) + c*sin(d + e*x))*(a**S(2) - b**S(2) - c**S(2))), x) def replacement3115(A, B, a, b, c, d, e, x): return Dist((A*a - B*b)/(a**S(2) - b**S(2) - c**S(2)), Int(S(1)/(a + b*cos(d + e*x) + c*sin(d + e*x)), x), x) + Simp((A*c*cos(d + e*x) + B*c + (-A*b + B*a)*sin(d + e*x))/(e*(a + b*cos(d + e*x) + c*sin(d + e*x))*(a**S(2) - b**S(2) - c**S(2))), x) def replacement3116(A, B, C, a, b, c, d, e, n, x): return Dist(S(1)/((n + S(1))*(a**S(2) - b**S(2) - c**S(2))), Int((a + b*cos(d + e*x) + c*sin(d + e*x))**(n + S(1))*Simp((n + S(1))*(A*a - B*b - C*c) + (n + S(2))*(-A*b + B*a)*cos(d + e*x) + (n + S(2))*(-A*c + C*a)*sin(d + e*x), x), x), x) - Simp((a + b*cos(d + e*x) + c*sin(d + e*x))**(n + S(1))*(B*c - C*b + (-A*b + B*a)*sin(d + e*x) - (-A*c + C*a)*cos(d + e*x))/(e*(n + S(1))*(a**S(2) - b**S(2) - c**S(2))), x) def replacement3117(A, C, a, b, c, d, e, n, x): return Dist(S(1)/((n + S(1))*(a**S(2) - b**S(2) - c**S(2))), Int((a + b*cos(d + e*x) + c*sin(d + e*x))**(n + S(1))*Simp(-A*b*(n + S(2))*cos(d + e*x) + (n + S(1))*(A*a - C*c) + (n + S(2))*(-A*c + C*a)*sin(d + e*x), x), x), x) + Simp((a + b*cos(d + e*x) + c*sin(d + e*x))**(n + S(1))*(A*b*sin(d + e*x) + C*b + (-A*c + C*a)*cos(d + e*x))/(e*(n + S(1))*(a**S(2) - b**S(2) - c**S(2))), x) def replacement3118(A, B, a, b, c, d, e, n, x): return Dist(S(1)/((n + S(1))*(a**S(2) - b**S(2) - c**S(2))), Int((a + b*cos(d + e*x) + c*sin(d + e*x))**(n + S(1))*Simp(-A*c*(n + S(2))*sin(d + e*x) + (n + S(1))*(A*a - B*b) + (n + S(2))*(-A*b + B*a)*cos(d + e*x), x), x), x) - Simp((a + b*cos(d + e*x) + c*sin(d + e*x))**(n + S(1))*(A*c*cos(d + e*x) + B*c + (-A*b + B*a)*sin(d + e*x))/(e*(n + S(1))*(a**S(2) - b**S(2) - c**S(2))), x) def replacement3119(a, b, c, d, e, x): return Int(cos(d + e*x)/(a*cos(d + e*x) + b + c*sin(d + e*x)), x) def replacement3120(a, b, c, d, e, x): return Int(sin(d + e*x)/(a*sin(d + e*x) + b + c*cos(d + e*x)), x) def replacement3121(a, b, c, d, e, n, x): return Int((a*cos(d + e*x) + b + c*sin(d + e*x))**n, x) def replacement3122(a, b, c, d, e, n, x): return Int((a*sin(d + e*x) + b + c*cos(d + e*x))**n, x) def replacement3123(a, b, c, d, e, n, x): return Dist((a + b/cos(d + e*x) + c*tan(d + e*x))**n*(a*cos(d + e*x) + b + c*sin(d + e*x))**(-n)*cos(d + e*x)**n, Int((a*cos(d + e*x) + b + c*sin(d + e*x))**n, x), x) def replacement3124(a, b, c, d, e, n, x): return Dist((a + b/sin(d + e*x) + c/tan(d + e*x))**n*(a*sin(d + e*x) + b + c*cos(d + e*x))**(-n)*sin(d + e*x)**n, Int((a*sin(d + e*x) + b + c*cos(d + e*x))**n, x), x) def replacement3125(a, b, c, d, e, m, n, x): return Int((a*cos(d + e*x) + b + c*sin(d + e*x))**(-n), x) def replacement3126(a, b, c, d, e, m, n, x): return Int((a*sin(d + e*x) + b + c*cos(d + e*x))**(-n), x) def replacement3127(a, b, c, d, e, m, n, x): return Dist((a + b/cos(d + e*x) + c*tan(d + e*x))**(-n)*(a*cos(d + e*x) + b + c*sin(d + e*x))**n*(S(1)/cos(d + e*x))**n, Int((a*cos(d + e*x) + b + c*sin(d + e*x))**(-n), x), x) def replacement3128(a, b, c, d, e, m, n, x): return Dist((a + b/sin(d + e*x) + c/tan(d + e*x))**(-n)*(a*sin(d + e*x) + b + c*cos(d + e*x))**n*(S(1)/sin(d + e*x))**n, Int((a*sin(d + e*x) + b + c*cos(d + e*x))**(-n), x), x) def replacement3129(b, c, d, p, x): return Dist(b*(S(2)*p + S(-1))/(S(2)*p), Int((b*sin(c + d*x)**S(2))**(p + S(-1)), x), x) - Simp((b*sin(c + d*x)**S(2))**p/(S(2)*d*p*tan(c + d*x)), x) def replacement3130(b, c, d, p, x): return Dist(b*(S(2)*p + S(-1))/(S(2)*p), Int((b*cos(c + d*x)**S(2))**(p + S(-1)), x), x) + Simp((b*cos(c + d*x)**S(2))**p*tan(c + d*x)/(S(2)*d*p), x) def replacement3131(b, c, d, p, x): return Dist(S(2)*(p + S(1))/(b*(S(2)*p + S(1))), Int((b*sin(c + d*x)**S(2))**(p + S(1)), x), x) + Simp((b*sin(c + d*x)**S(2))**(p + S(1))/(b*d*(S(2)*p + S(1))*tan(c + d*x)), x) def replacement3132(b, c, d, p, x): return Dist(S(2)*(p + S(1))/(b*(S(2)*p + S(1))), Int((b*cos(c + d*x)**S(2))**(p + S(1)), x), x) - Simp((b*cos(c + d*x)**S(2))**(p + S(1))*tan(c + d*x)/(b*d*(S(2)*p + S(1))), x) def replacement3133(a, b, c, d, p, x): return Dist(a**p, Int(cos(c + d*x)**(S(2)*p), x), x) def replacement3134(a, b, c, d, p, x): return Dist(a**p, Int(sin(c + d*x)**(S(2)*p), x), x) def replacement3135(a, b, c, d, p, x): return Int((a*cos(c + d*x)**S(2))**p, x) def replacement3136(a, b, c, d, p, x): return Int((a*sin(c + d*x)**S(2))**p, x) def With3137(a, b, c, d, p, x): e = FreeFactors(tan(c + d*x), x) return Dist(e/d, Subst(Int((a + e**S(2)*x**S(2)*(a + b))**p*(e**S(2)*x**S(2) + S(1))**(-p + S(-1)), x), x, tan(c + d*x)/e), x) def With3138(a, b, c, d, p, x): e = FreeFactors(tan(c + d*x), x) return Dist(e/d, Subst(Int((e**S(2)*x**S(2) + S(1))**(-p + S(-1))*(a*e**S(2)*x**S(2) + a + b)**p, x), x, tan(c + d*x)/e), x) def replacement3139(a, b, c, d, p, x): return Dist(S(2)**(-p), Int((S(2)*a - b*cos(S(2)*c + S(2)*d*x) + b)**p, x), x) def replacement3140(a, b, c, d, p, x): return Dist(S(2)**(-p), Int((S(2)*a + b*cos(S(2)*c + S(2)*d*x) + b)**p, x), x) def replacement3141(a, b, c, d, n, p, x): return Int((a + b*sin(c + d*x)**n)**p, x) def replacement3142(a, b, c, d, n, p, x): return Int((a + b*cos(c + d*x)**n)**p, x) def With3143(a, b, c, d, n, p, x): f = FreeFactors(S(1)/tan(c + d*x), x) return -Dist(f/d, Subst(Int((f**S(2)*x**S(2) + S(1))**(-n*p/S(2) + S(-1))*ExpandToSum(a*(f**S(2)*x**S(2) + S(1))**(n/S(2)) + b, x)**p, x), x, S(1)/(f*tan(c + d*x))), x) def With3144(a, b, c, d, n, p, x): f = FreeFactors(tan(c + d*x), x) return Dist(f/d, Subst(Int((f**S(2)*x**S(2) + S(1))**(-n*p/S(2) + S(-1))*ExpandToSum(a*(f**S(2)*x**S(2) + S(1))**(n/S(2)) + b, x)**p, x), x, tan(c + d*x)/f), x) def replacement3145(a, b, c, d, p, u, x): return Dist(a**p, Int(ActivateTrig(u)*cos(c + d*x)**(S(2)*p), x), x) def replacement3146(a, b, c, d, p, u, x): return Dist(a**p, Int(ActivateTrig(u)*sin(c + d*x)**(S(2)*p), x), x) def replacement3147(a, b, c, d, p, u, x): return Dist((a*cos(c + d*x)**S(2))**p*cos(c + d*x)**(-S(2)*p), Int(ActivateTrig(u)*cos(c + d*x)**(S(2)*p), x), x) def replacement3148(a, b, c, d, p, u, x): return Dist((a*sin(c + d*x)**S(2))**p*sin(c + d*x)**(-S(2)*p), Int(ActivateTrig(u)*sin(c + d*x)**(S(2)*p), x), x) def With3149(a, b, c, d, m, n, p, x): f = FreeFactors(S(1)/tan(c + d*x), x) return -Dist(f/d, Subst(Int((f**S(2)*x**S(2) + S(1))**(-m/S(2) - n*p/S(2) + S(-1))*ExpandToSum(a*(f**S(2)*x**S(2) + S(1))**(n/S(2)) + b, x)**p, x), x, S(1)/(f*tan(c + d*x))), x) def With3150(a, b, c, d, m, n, p, x): f = FreeFactors(tan(c + d*x), x) return Dist(f/d, Subst(Int((f**S(2)*x**S(2) + S(1))**(-m/S(2) - n*p/S(2) + S(-1))*ExpandToSum(a*(f**S(2)*x**S(2) + S(1))**(n/S(2)) + b, x)**p, x), x, tan(c + d*x)/f), x) def With3151(a, b, c, d, m, n, p, x): f = FreeFactors(cos(c + d*x), x) return -Dist(f/d, Subst(Int((-f**S(2)*x**S(2) + S(1))**(m/S(2) + S(-1)/2)*ExpandToSum(a + b*(-f**S(2)*x**S(2) + S(1))**(n/S(2)), x)**p, x), x, cos(c + d*x)/f), x) def With3152(a, b, c, d, m, n, p, x): f = FreeFactors(sin(c + d*x), x) return Dist(f/d, Subst(Int((-f**S(2)*x**S(2) + S(1))**(m/S(2) + S(-1)/2)*ExpandToSum(a + b*(-f**S(2)*x**S(2) + S(1))**(n/S(2)), x)**p, x), x, sin(c + d*x)/f), x) def replacement3153(a, b, c, d, m, n, p, x): return Int(ExpandTrig((a + b*sin(c + d*x)**n)**p*sin(c + d*x)**m, x), x) def replacement3154(a, b, c, d, m, n, p, x): return Int(ExpandTrig((a + b*cos(c + d*x)**n)**p*cos(c + d*x)**m, x), x) def With3155(a, b, c, d, m, n, p, x): f = FreeFactors(tan(c + d*x), x) return Dist(f/d, Subst(Int((f**S(2)*x**S(2) + S(1))**(-m/S(2) - n*p/S(2) + S(-1))*ExpandToSum(a*(f**S(2)*x**S(2) + S(1))**(n/S(2)) + b*f**n*x**n, x)**p, x), x, tan(c + d*x)/f), x) def With3156(a, b, c, d, m, n, p, x): f = FreeFactors(S(1)/tan(c + d*x), x) return -Dist(f/d, Subst(Int((f**S(2)*x**S(2) + S(1))**(-m/S(2) - n*p/S(2) + S(-1))*ExpandToSum(a*(f**S(2)*x**S(2) + S(1))**(n/S(2)) + b*f**n*x**n, x)**p, x), x, S(1)/(f*tan(c + d*x))), x) def replacement3157(a, b, c, d, m, n, p, x): return Int((S(1) - sin(c + d*x)**S(2))**(m/S(2))*(a + b*sin(c + d*x)**n)**p, x) def replacement3158(a, b, c, d, m, n, p, x): return Int((S(1) - cos(c + d*x)**S(2))**(m/S(2))*(a + b*cos(c + d*x)**n)**p, x) def replacement3159(a, b, c, d, m, n, p, x): return Int(ExpandTrig((S(1) - sin(c + d*x)**S(2))**(m/S(2))*(a + b*sin(c + d*x)**n)**p, x), x) def replacement3160(a, b, c, d, m, n, p, x): return Int(ExpandTrig((S(1) - cos(c + d*x)**S(2))**(m/S(2))*(a + b*cos(c + d*x)**n)**p, x), x) def With3161(a, b, c, d, e, m, n, p, x): f = FreeFactors(sin(c + d*x), x) return Dist(f/d, Subst(Int((a + b*(e*f*x)**n)**p*(-f**S(2)*x**S(2) + S(1))**(m/S(2) + S(-1)/2), x), x, sin(c + d*x)/f), x) def With3162(a, b, c, d, e, m, n, p, x): f = FreeFactors(cos(c + d*x), x) return -Dist(f/d, Subst(Int((a + b*(e*f*x)**n)**p*(-f**S(2)*x**S(2) + S(1))**(m/S(2) + S(-1)/2), x), x, cos(c + d*x)/f), x) def With3163(a, b, c, d, e, m, n, p, x): f = FreeFactors(sin(c + d*x), x) return Dist(f**(m + S(1))/d, Subst(Int(x**m*(a + b*(e*f*x)**n)**p*(-f**S(2)*x**S(2) + S(1))**(-m/S(2) + S(-1)/2), x), x, sin(c + d*x)/f), x) def With3164(a, b, c, d, e, m, n, p, x): f = FreeFactors(cos(c + d*x), x) return -Dist(f**(m + S(1))/d, Subst(Int(x**m*(a + b*(e*f*x)**n)**p*(-f**S(2)*x**S(2) + S(1))**(-m/S(2) + S(-1)/2), x), x, cos(c + d*x)/f), x) def With3165(a, b, c, d, m, n, p, x): f = FreeFactors(tan(c + d*x), x) return Dist(f**(m + S(1))/d, Subst(Int(x**m*(f**S(2)*x**S(2) + S(1))**(-n*p/S(2) + S(-1))*ExpandToSum(a*(f**S(2)*x**S(2) + S(1))**(n/S(2)) + b*f**n*x**n, x)**p, x), x, tan(c + d*x)/f), x) def With3166(a, b, c, d, m, n, p, x): f = FreeFactors(S(1)/tan(c + d*x), x) return -Dist(f**(m + S(1))/d, Subst(Int(x**m*(f**S(2)*x**S(2) + S(1))**(-n*p/S(2) + S(-1))*ExpandToSum(a*(f**S(2)*x**S(2) + S(1))**(n/S(2)) + b*f**n*x**n, x)**p, x), x, S(1)/(f*tan(c + d*x))), x) def With3167(a, b, c, d, m, n, p, x): f = FreeFactors(tan(c + d*x), x) return Dist(f**(m + S(1))/d, Subst(Int(x**m*((f**S(2)*x**S(2) + S(1))**(-n/S(2))*ExpandToSum(a*(f**S(2)*x**S(2) + S(1))**(n/S(2)) + b*f**n*x**n, x))**p/(f**S(2)*x**S(2) + S(1)), x), x, tan(c + d*x)/f), x) def With3168(a, b, c, d, m, n, p, x): f = FreeFactors(S(1)/tan(c + d*x), x) return -Dist(f**(m + S(1))/d, Subst(Int(x**m*((f**S(2)*x**S(2) + S(1))**(-n/S(2))*ExpandToSum(a*(f**S(2)*x**S(2) + S(1))**(n/S(2)) + b*f**n*x**n, x))**p/(f**S(2)*x**S(2) + S(1)), x), x, S(1)/(f*tan(c + d*x))), x) def With3169(a, b, c, d, e, m, n, p, q, x): f = FreeFactors(S(1)/tan(d + e*x), x) return -Dist(f/e, Subst(Int((f**S(2)*x**S(2) + S(1))**(-m/S(2) - n*q/S(2) + S(-1))*ExpandToSum(a*(f**S(2)*x**S(2) + S(1))**(q/S(2)) + b*(f**S(2)*x**S(2) + S(1))**(-p/S(2) + q/S(2)) + c, x)**n, x), x, S(1)/(f*tan(d + e*x))), x) def With3170(a, b, c, d, e, m, n, p, q, x): f = FreeFactors(tan(d + e*x), x) return Dist(f/e, Subst(Int((f**S(2)*x**S(2) + S(1))**(-m/S(2) - n*q/S(2) + S(-1))*ExpandToSum(a*(f**S(2)*x**S(2) + S(1))**(q/S(2)) + b*(f**S(2)*x**S(2) + S(1))**(-p/S(2) + q/S(2)) + c, x)**n, x), x, tan(d + e*x)/f), x) def With3171(a, b, c, d, e, m, n, p, q, x): f = FreeFactors(S(1)/tan(d + e*x), x) return -Dist(f/e, Subst(Int((f**S(2)*x**S(2) + S(1))**(-m/S(2) - n*p/S(2) + S(-1))*ExpandToSum(a*(f**S(2)*x**S(2) + S(1))**(p/S(2)) + b*f**p*x**p + c*(f**S(2)*x**S(2) + S(1))**(p/S(2) - q/S(2)), x)**n, x), x, S(1)/(f*tan(d + e*x))), x) def With3172(a, b, c, d, e, m, n, p, q, x): f = FreeFactors(tan(d + e*x), x) return Dist(f/e, Subst(Int((f**S(2)*x**S(2) + S(1))**(-m/S(2) - n*p/S(2) + S(-1))*ExpandToSum(a*(f**S(2)*x**S(2) + S(1))**(p/S(2)) + b*f**p*x**p + c*(f**S(2)*x**S(2) + S(1))**(p/S(2) - q/S(2)), x)**n, x), x, tan(d + e*x)/f), x) def replacement3173(a, b, c, d, e, n, n2, p, x): return Dist(S(4)**(-p)*c**(-p), Int((b + S(2)*c*sin(d + e*x)**n)**(S(2)*p), x), x) def replacement3174(a, b, c, d, e, n, n2, p, x): return Dist(S(4)**(-p)*c**(-p), Int((b + S(2)*c*cos(d + e*x)**n)**(S(2)*p), x), x) def replacement3175(a, b, c, d, e, n, n2, p, x): return Dist((b + S(2)*c*sin(d + e*x)**n)**(-S(2)*p)*(a + b*sin(d + e*x)**n + c*sin(d + e*x)**(S(2)*n))**p, Int(u*(b + S(2)*c*sin(d + e*x)**n)**(S(2)*p), x), x) def replacement3176(a, b, c, d, e, n, n2, p, x): return Dist((b + S(2)*c*cos(d + e*x)**n)**(-S(2)*p)*(a + b*cos(d + e*x)**n + c*cos(d + e*x)**(S(2)*n))**p, Int(u*(b + S(2)*c*cos(d + e*x)**n)**(S(2)*p), x), x) def With3177(a, b, c, d, e, n, n2, x): q = Rt(-S(4)*a*c + b**S(2), S(2)) return Dist(S(2)*c/q, Int(S(1)/(b + S(2)*c*sin(d + e*x)**n - q), x), x) - Dist(S(2)*c/q, Int(S(1)/(b + S(2)*c*sin(d + e*x)**n + q), x), x) def With3178(a, b, c, d, e, n, n2, x): q = Rt(-S(4)*a*c + b**S(2), S(2)) return Dist(S(2)*c/q, Int(S(1)/(b + S(2)*c*cos(d + e*x)**n - q), x), x) - Dist(S(2)*c/q, Int(S(1)/(b + S(2)*c*cos(d + e*x)**n + q), x), x) def replacement3179(a, b, c, d, e, m, n, n2, p, x): return Dist(S(4)**(-p)*c**(-p), Int((b + S(2)*c*sin(d + e*x)**n)**(S(2)*p)*sin(d + e*x)**m, x), x) def replacement3180(a, b, c, d, e, m, n, n2, p, x): return Dist(S(4)**(-p)*c**(-p), Int((b + S(2)*c*cos(d + e*x)**n)**(S(2)*p)*cos(d + e*x)**m, x), x) def replacement3181(a, b, c, d, e, m, n, n2, p, x): return Dist((b + S(2)*c*sin(d + e*x)**n)**(-S(2)*p)*(a + b*sin(d + e*x)**n + c*sin(d + e*x)**(S(2)*n))**p, Int((b + S(2)*c*sin(d + e*x)**n)**(S(2)*p)*sin(d + e*x)**m, x), x) def replacement3182(a, b, c, d, e, m, n, n2, p, x): return Dist((b + S(2)*c*cos(d + e*x)**n)**(-S(2)*p)*(a + b*cos(d + e*x)**n + c*cos(d + e*x)**(S(2)*n))**p, Int((b + S(2)*c*cos(d + e*x)**n)**(S(2)*p)*cos(d + e*x)**m, x), x) def With3183(a, b, c, d, e, m, n, n2, p, x): f = FreeFactors(S(1)/tan(d + e*x), x) return -Dist(f/e, Subst(Int((f**S(2)*x**S(2) + S(1))**(-m/S(2) - n*p + S(-1))*ExpandToSum(a*(x**S(2) + S(1))**n + b*(x**S(2) + S(1))**(n/S(2)) + c, x)**p, x), x, S(1)/(f*tan(d + e*x))), x) def With3184(a, b, c, d, e, m, n, n2, p, x): f = FreeFactors(tan(d + e*x), x) return Dist(f/e, Subst(Int((f**S(2)*x**S(2) + S(1))**(-m/S(2) - n*p + S(-1))*ExpandToSum(a*(x**S(2) + S(1))**n + b*(x**S(2) + S(1))**(n/S(2)) + c, x)**p, x), x, tan(d + e*x)/f), x) def replacement3185(a, b, c, d, e, m, n, n2, p, x): return Int(ExpandTrig((a + b*sin(d + e*x)**n + c*sin(d + e*x)**(S(2)*n))**p*sin(d + e*x)**m, x), x) def replacement3186(a, b, c, d, e, m, n, n2, p, x): return Int(ExpandTrig((a + b*cos(d + e*x)**n + c*cos(d + e*x)**(S(2)*n))**p*cos(d + e*x)**m, x), x) def With3187(a, b, c, d, e, f, m, n, n2, p, x): g = FreeFactors(sin(d + e*x), x) return Dist(g/e, Subst(Int((-g**S(2)*x**S(2) + S(1))**(m/S(2) + S(-1)/2)*(a + b*(f*g*x)**n + c*(f*g*x)**(S(2)*n))**p, x), x, sin(d + e*x)/g), x) def With3188(a, b, c, d, e, f, m, n, n2, p, x): g = FreeFactors(cos(d + e*x), x) return -Dist(g/e, Subst(Int((-g**S(2)*x**S(2) + S(1))**(m/S(2) + S(-1)/2)*(a + b*(f*g*x)**n + c*(f*g*x)**(S(2)*n))**p, x), x, cos(d + e*x)/g), x) def replacement3189(a, b, c, d, e, m, n, n2, p, x): return Dist(S(4)**(-p)*c**(-p), Int((b + S(2)*c*sin(d + e*x)**n)**(S(2)*p)*cos(d + e*x)**m, x), x) def replacement3190(a, b, c, d, e, m, n, n2, p, x): return Dist(S(4)**(-p)*c**(-p), Int((b + S(2)*c*cos(d + e*x)**n)**(S(2)*p)*sin(d + e*x)**m, x), x) def replacement3191(a, b, c, d, e, m, n, n2, p, x): return Dist((b + S(2)*c*sin(d + e*x)**n)**(-S(2)*p)*(a + b*sin(d + e*x)**n + c*sin(d + e*x)**(S(2)*n))**p, Int((b + S(2)*c*sin(d + e*x)**n)**(S(2)*p)*cos(d + e*x)**m, x), x) def replacement3192(a, b, c, d, e, m, n, n2, p, x): return Dist((b + S(2)*c*cos(d + e*x)**n)**(-S(2)*p)*(a + b*cos(d + e*x)**n + c*cos(d + e*x)**(S(2)*n))**p, Int((b + S(2)*c*cos(d + e*x)**n)**(S(2)*p)*sin(d + e*x)**m, x), x) def With3193(a, b, c, d, e, m, n, n2, p, x): f = FreeFactors(S(1)/tan(d + e*x), x) return -Dist(f**(m + S(1))/e, Subst(Int(x**m*(f**S(2)*x**S(2) + S(1))**(-m/S(2) - n*p + S(-1))*ExpandToSum(a*(x**S(2) + S(1))**n + b*(x**S(2) + S(1))**(n/S(2)) + c, x)**p, x), x, S(1)/(f*tan(d + e*x))), x) def With3194(a, b, c, d, e, m, n, n2, p, x): f = FreeFactors(tan(d + e*x), x) return Dist(f**(m + S(1))/e, Subst(Int(x**m*(f**S(2)*x**S(2) + S(1))**(-m/S(2) - n*p + S(-1))*ExpandToSum(a*(x**S(2) + S(1))**n + b*(x**S(2) + S(1))**(n/S(2)) + c, x)**p, x), x, tan(d + e*x)/f), x) def replacement3195(a, b, c, d, e, m, n, n2, p, x): return Int(ExpandTrig((S(1) - sin(d + e*x)**S(2))**(m/S(2))*(a + b*sin(d + e*x)**n + c*sin(d + e*x)**(S(2)*n))**p, x), x) def replacement3196(a, b, c, d, e, m, n, n2, p, x): return Int(ExpandTrig((S(1) - cos(d + e*x)**S(2))**(m/S(2))*(a + b*cos(d + e*x)**n + c*cos(d + e*x)**(S(2)*n))**p, x), x) def With3197(a, b, c, d, e, f, m, n, n2, p, x): g = FreeFactors(sin(d + e*x), x) return Dist(g**(m + S(1))/e, Subst(Int(x**m*(-g**S(2)*x**S(2) + S(1))**(-m/S(2) + S(-1)/2)*(a + b*(f*g*x)**n + c*(f*g*x)**(S(2)*n))**p, x), x, sin(d + e*x)/g), x) def With3198(a, b, c, d, e, f, m, n, n2, p, x): g = FreeFactors(cos(d + e*x), x) return -Dist(g**(m + S(1))/e, Subst(Int(x**m*(-g**S(2)*x**S(2) + S(1))**(-m/S(2) + S(-1)/2)*(a + b*(f*g*x)**n + c*(f*g*x)**(S(2)*n))**p, x), x, cos(d + e*x)/g), x) def replacement3199(a, b, c, d, e, m, n, n2, p, x): return Dist(S(4)**(-p)*c**(-p), Int((b + S(2)*c*sin(d + e*x)**n)**(S(2)*p)*tan(d + e*x)**m, x), x) def replacement3200(a, b, c, d, e, m, n, n2, p, x): return Dist(S(4)**(-p)*c**(-p), Int((b + S(2)*c*cos(d + e*x)**n)**(S(2)*p)*(S(1)/tan(d + e*x))**m, x), x) def replacement3201(a, b, c, d, e, m, n, n2, p, x): return Dist((b + S(2)*c*sin(d + e*x)**n)**(-S(2)*p)*(a + b*sin(d + e*x)**n + c*sin(d + e*x)**(S(2)*n))**p, Int((b + S(2)*c*sin(d + e*x)**n)**(S(2)*p)*tan(d + e*x)**m, x), x) def replacement3202(a, b, c, d, e, m, n, n2, p, x): return Dist((b + S(2)*c*cos(d + e*x)**n)**(-S(2)*p)*(a + b*cos(d + e*x)**n + c*cos(d + e*x)**(S(2)*n))**p, Int((b + S(2)*c*cos(d + e*x)**n)**(S(2)*p)*(S(1)/tan(d + e*x))**m, x), x) def With3203(a, b, c, d, e, m, n, n2, p, x): f = FreeFactors(tan(d + e*x), x) return Dist(f**(m + S(1))/e, Subst(Int(x**m*(f**S(2)*x**S(2) + S(1))**(-n*p + S(-1))*ExpandToSum(a*(x**S(2) + S(1))**n + b*x**n*(x**S(2) + S(1))**(n/S(2)) + c*x**(S(2)*n), x)**p, x), x, tan(d + e*x)/f), x) def With3204(a, b, c, d, e, m, n, n2, p, x): f = FreeFactors(S(1)/tan(d + e*x), x) return -Dist(f**(m + S(1))/e, Subst(Int(x**m*(f**S(2)*x**S(2) + S(1))**(-n*p + S(-1))*ExpandToSum(a*(x**S(2) + S(1))**n + b*x**n*(x**S(2) + S(1))**(n/S(2)) + c*x**(S(2)*n), x)**p, x), x, S(1)/(f*tan(d + e*x))), x) def replacement3205(a, b, c, d, e, m, n, n2, p, x): return Int(ExpandTrig((S(1) - sin(d + e*x)**S(2))**(-m/S(2))*(a + b*sin(d + e*x)**n + c*sin(d + e*x)**(S(2)*n))**p*sin(d + e*x)**m, x), x) def replacement3206(a, b, c, d, e, m, n, n2, p, x): return Int(ExpandTrig((S(1) - cos(d + e*x)**S(2))**(-m/S(2))*(a + b*cos(d + e*x)**n + c*cos(d + e*x)**(S(2)*n))**p*cos(d + e*x)**m, x), x) def With3207(a, b, c, d, e, f, m, n, n2, p, x): g = FreeFactors(sin(d + e*x), x) return Dist(g**(m + S(1))/e, Subst(Int(x**(-m)*(-g**S(2)*x**S(2) + S(1))**(m/S(2) + S(-1)/2)*(a + b*(f*g*x)**n + c*(f*g*x)**(S(2)*n))**p, x), x, sin(d + e*x)/g), x) def With3208(a, b, c, d, e, f, m, n, n2, p, x): g = FreeFactors(cos(d + e*x), x) return -Dist(g**(m + S(1))/e, Subst(Int(x**(-m)*(-g**S(2)*x**S(2) + S(1))**(m/S(2) + S(-1)/2)*(a + b*(f*g*x)**n + c*(f*g*x)**(S(2)*n))**p, x), x, cos(d + e*x)/g), x) def replacement3209(a, b, c, d, e, m, n, n2, p, x): return Dist(S(4)**(-p)*c**(-p), Int((b + S(2)*c*sin(d + e*x)**n)**(S(2)*p)*(S(1)/tan(d + e*x))**m, x), x) def replacement3210(a, b, c, d, e, m, n, n2, p, x): return Dist(S(4)**(-p)*c**(-p), Int((b + S(2)*c*cos(d + e*x)**n)**(S(2)*p)*tan(d + e*x)**m, x), x) def replacement3211(a, b, c, d, e, m, n, n2, p, x): return Dist((b + S(2)*c*sin(d + e*x)**n)**(-S(2)*p)*(a + b*sin(d + e*x)**n + c*sin(d + e*x)**(S(2)*n))**p, Int((b + S(2)*c*sin(d + e*x)**n)**(S(2)*p)*(S(1)/tan(d + e*x))**m, x), x) def replacement3212(a, b, c, d, e, m, n, n2, p, x): return Dist((b + S(2)*c*cos(d + e*x)**n)**(-S(2)*p)*(a + b*cos(d + e*x)**n + c*cos(d + e*x)**(S(2)*n))**p, Int((b + S(2)*c*cos(d + e*x)**n)**(S(2)*p)*tan(d + e*x)**m, x), x) def With3213(a, b, c, d, e, m, n, n2, p, x): f = FreeFactors(S(1)/tan(d + e*x), x) return -Dist(f**(m + S(1))/e, Subst(Int(x**m*(f**S(2)*x**S(2) + S(1))**(-n*p + S(-1))*ExpandToSum(a*(f**S(2)*x**S(2) + S(1))**n + b*(f**S(2)*x**S(2) + S(1))**(n/S(2)) + c, x)**p, x), x, S(1)/(f*tan(d + e*x))), x) def With3214(a, b, c, d, e, m, n, n2, p, x): f = FreeFactors(tan(d + e*x), x) return Dist(f**(m + S(1))/e, Subst(Int(x**m*(f**S(2)*x**S(2) + S(1))**(-n*p + S(-1))*ExpandToSum(a*(f**S(2)*x**S(2) + S(1))**n + b*(f**S(2)*x**S(2) + S(1))**(n/S(2)) + c, x)**p, x), x, tan(d + e*x)/f), x) def replacement3215(a, b, c, d, e, m, n, n2, p, x): return Int(ExpandTrig((S(1) - sin(d + e*x)**S(2))**(m/S(2))*(a + b*sin(d + e*x)**n + c*sin(d + e*x)**(S(2)*n))**p*sin(d + e*x)**(-m), x), x) def replacement3216(a, b, c, d, e, m, n, n2, p, x): return Int(ExpandTrig((S(1) - cos(d + e*x)**S(2))**(m/S(2))*(a + b*cos(d + e*x)**n + c*cos(d + e*x)**(S(2)*n))**p*cos(d + e*x)**(-m), x), x) def replacement3217(A, B, a, b, c, d, e, n, x): return Dist(S(4)**(-n)*c**(-n), Int((A + B*sin(d + e*x))*(b + S(2)*c*sin(d + e*x))**(S(2)*n), x), x) def replacement3218(A, B, a, b, c, d, e, n, x): return Dist(S(4)**(-n)*c**(-n), Int((A + B*cos(d + e*x))*(b + S(2)*c*cos(d + e*x))**(S(2)*n), x), x) def replacement3219(A, B, a, b, c, d, e, n, x): return Dist((b + S(2)*c*sin(d + e*x))**(-S(2)*n)*(a + b*sin(d + e*x) + c*sin(d + e*x)**S(2))**n, Int((A + B*sin(d + e*x))*(b + S(2)*c*sin(d + e*x))**(S(2)*n), x), x) def replacement3220(A, B, a, b, c, d, e, n, x): return Dist((b + S(2)*c*cos(d + e*x))**(-S(2)*n)*(a + b*cos(d + e*x) + c*cos(d + e*x)**S(2))**n, Int((A + B*cos(d + e*x))*(b + S(2)*c*cos(d + e*x))**(S(2)*n), x), x) def With3221(A, B, a, b, c, d, e, x): q = Rt(-S(4)*a*c + b**S(2), S(2)) return Dist(B - (-S(2)*A*c + B*b)/q, Int(S(1)/(b + S(2)*c*sin(d + e*x) - q), x), x) + Dist(B + (-S(2)*A*c + B*b)/q, Int(S(1)/(b + S(2)*c*sin(d + e*x) + q), x), x) def With3222(A, B, a, b, c, d, e, x): q = Rt(-S(4)*a*c + b**S(2), S(2)) return Dist(B - (-S(2)*A*c + B*b)/q, Int(S(1)/(b + S(2)*c*cos(d + e*x) - q), x), x) + Dist(B + (-S(2)*A*c + B*b)/q, Int(S(1)/(b + S(2)*c*cos(d + e*x) + q), x), x) def replacement3223(A, B, a, b, c, d, e, n, x): return Int(ExpandTrig((A + B*sin(d + e*x))*(a + b*sin(d + e*x) + c*sin(d + e*x)**S(2))**n, x), x) def replacement3224(A, B, a, b, c, d, e, n, x): return Int(ExpandTrig((A + B*cos(d + e*x))*(a + b*cos(d + e*x) + c*cos(d + e*x)**S(2))**n, x), x) def replacement3225(c, d, e, f, m, x): return Dist(d*m/f, Int((c + d*x)**(m + S(-1))*cos(e + f*x), x), x) - Simp((c + d*x)**m*cos(e + f*x)/f, x) def replacement3226(c, d, e, f, m, x): return -Dist(d*m/f, Int((c + d*x)**(m + S(-1))*sin(e + f*x), x), x) + Simp((c + d*x)**m*sin(e + f*x)/f, x) def replacement3227(c, d, e, f, m, x): return -Dist(f/(d*(m + S(1))), Int((c + d*x)**(m + S(1))*cos(e + f*x), x), x) + Simp((c + d*x)**(m + S(1))*sin(e + f*x)/(d*(m + S(1))), x) def replacement3228(c, d, e, f, m, x): return Dist(f/(d*(m + S(1))), Int((c + d*x)**(m + S(1))*sin(e + f*x), x), x) + Simp((c + d*x)**(m + S(1))*cos(e + f*x)/(d*(m + S(1))), x) def replacement3229(c, d, e, f, x): return Simp(SinIntegral(e + f*x)/d, x) def replacement3230(c, d, e, f, x): return Simp(CosIntegral(e + f*x)/d, x) def replacement3231(c, d, e, f, x): return Dist(sin((-c*f + d*e)/d), Int(cos(c*f/d + f*x)/(c + d*x), x), x) + Dist(cos((-c*f + d*e)/d), Int(sin(c*f/d + f*x)/(c + d*x), x), x) def replacement3232(c, d, e, f, x): return -Dist(sin((-c*f + d*e)/d), Int(sin(c*f/d + f*x)/(c + d*x), x), x) + Dist(cos((-c*f + d*e)/d), Int(cos(c*f/d + f*x)/(c + d*x), x), x) def replacement3233(c, d, e, f, x): return Dist(S(2)/d, Subst(Int(sin(f*x**S(2)/d), x), x, sqrt(c + d*x)), x) def replacement3234(c, d, e, f, x): return Dist(S(2)/d, Subst(Int(cos(f*x**S(2)/d), x), x, sqrt(c + d*x)), x) def replacement3235(c, d, e, f, x): return Dist(sin((-c*f + d*e)/d), Int(cos(c*f/d + f*x)/sqrt(c + d*x), x), x) + Dist(cos((-c*f + d*e)/d), Int(sin(c*f/d + f*x)/sqrt(c + d*x), x), x) def replacement3236(c, d, e, f, x): return -Dist(sin((-c*f + d*e)/d), Int(sin(c*f/d + f*x)/sqrt(c + d*x), x), x) + Dist(cos((-c*f + d*e)/d), Int(cos(c*f/d + f*x)/sqrt(c + d*x), x), x) def replacement3237(c, d, e, f, m, x): return Dist(I/S(2), Int((c + d*x)**m*exp(-I*(e + f*x)), x), x) - Dist(I/S(2), Int((c + d*x)**m*exp(I*(e + f*x)), x), x) def replacement3238(c, d, e, f, m, x): return Dist(S(1)/2, Int((c + d*x)**m*exp(-I*(e + f*x)), x), x) + Dist(S(1)/2, Int((c + d*x)**m*exp(I*(e + f*x)), x), x) def replacement3239(b, c, d, e, f, n, x): return Dist(b**S(2)*(n + S(-1))/n, Int((b*sin(e + f*x))**(n + S(-2))*(c + d*x), x), x) + Simp(d*(b*sin(e + f*x))**n/(f**S(2)*n**S(2)), x) - Simp(b*(b*sin(e + f*x))**(n + S(-1))*(c + d*x)*cos(e + f*x)/(f*n), x) def replacement3240(b, c, d, e, f, n, x): return Dist(b**S(2)*(n + S(-1))/n, Int((b*cos(e + f*x))**(n + S(-2))*(c + d*x), x), x) + Simp(d*(b*cos(e + f*x))**n/(f**S(2)*n**S(2)), x) + Simp(b*(b*cos(e + f*x))**(n + S(-1))*(c + d*x)*sin(e + f*x)/(f*n), x) def replacement3241(b, c, d, e, f, m, n, x): return Dist(b**S(2)*(n + S(-1))/n, Int((b*sin(e + f*x))**(n + S(-2))*(c + d*x)**m, x), x) - Dist(d**S(2)*m*(m + S(-1))/(f**S(2)*n**S(2)), Int((b*sin(e + f*x))**n*(c + d*x)**(m + S(-2)), x), x) - Simp(b*(b*sin(e + f*x))**(n + S(-1))*(c + d*x)**m*cos(e + f*x)/(f*n), x) + Simp(d*m*(b*sin(e + f*x))**n*(c + d*x)**(m + S(-1))/(f**S(2)*n**S(2)), x) def replacement3242(b, c, d, e, f, m, n, x): return Dist(b**S(2)*(n + S(-1))/n, Int((b*cos(e + f*x))**(n + S(-2))*(c + d*x)**m, x), x) - Dist(d**S(2)*m*(m + S(-1))/(f**S(2)*n**S(2)), Int((b*cos(e + f*x))**n*(c + d*x)**(m + S(-2)), x), x) + Simp(b*(b*cos(e + f*x))**(n + S(-1))*(c + d*x)**m*sin(e + f*x)/(f*n), x) + Simp(d*m*(b*cos(e + f*x))**n*(c + d*x)**(m + S(-1))/(f**S(2)*n**S(2)), x) def replacement3243(c, d, e, f, m, n, x): return Int(ExpandTrigReduce((c + d*x)**m, sin(e + f*x)**n, x), x) def replacement3244(c, d, e, f, m, n, x): return Int(ExpandTrigReduce((c + d*x)**m, cos(e + f*x)**n, x), x) def replacement3245(c, d, e, f, m, n, x): return -Dist(f*n/(d*(m + S(1))), Int(ExpandTrigReduce((c + d*x)**(m + S(1)), sin(e + f*x)**(n + S(-1))*cos(e + f*x), x), x), x) + Simp((c + d*x)**(m + S(1))*sin(e + f*x)**n/(d*(m + S(1))), x) def replacement3246(c, d, e, f, m, n, x): return Dist(f*n/(d*(m + S(1))), Int(ExpandTrigReduce((c + d*x)**(m + S(1)), sin(e + f*x)*cos(e + f*x)**(n + S(-1)), x), x), x) + Simp((c + d*x)**(m + S(1))*cos(e + f*x)**n/(d*(m + S(1))), x) def replacement3247(b, c, d, e, f, m, n, x): return -Dist(f**S(2)*n**S(2)/(d**S(2)*(m + S(1))*(m + S(2))), Int((b*sin(e + f*x))**n*(c + d*x)**(m + S(2)), x), x) + Dist(b**S(2)*f**S(2)*n*(n + S(-1))/(d**S(2)*(m + S(1))*(m + S(2))), Int((b*sin(e + f*x))**(n + S(-2))*(c + d*x)**(m + S(2)), x), x) + Simp((b*sin(e + f*x))**n*(c + d*x)**(m + S(1))/(d*(m + S(1))), x) - Simp(b*f*n*(b*sin(e + f*x))**(n + S(-1))*(c + d*x)**(m + S(2))*cos(e + f*x)/(d**S(2)*(m + S(1))*(m + S(2))), x) def replacement3248(b, c, d, e, f, m, n, x): return -Dist(f**S(2)*n**S(2)/(d**S(2)*(m + S(1))*(m + S(2))), Int((b*cos(e + f*x))**n*(c + d*x)**(m + S(2)), x), x) + Dist(b**S(2)*f**S(2)*n*(n + S(-1))/(d**S(2)*(m + S(1))*(m + S(2))), Int((b*cos(e + f*x))**(n + S(-2))*(c + d*x)**(m + S(2)), x), x) + Simp((b*cos(e + f*x))**n*(c + d*x)**(m + S(1))/(d*(m + S(1))), x) + Simp(b*f*n*(b*cos(e + f*x))**(n + S(-1))*(c + d*x)**(m + S(2))*sin(e + f*x)/(d**S(2)*(m + S(1))*(m + S(2))), x) def replacement3249(b, c, d, e, f, n, x): return Dist((n + S(2))/(b**S(2)*(n + S(1))), Int((b*sin(e + f*x))**(n + S(2))*(c + d*x), x), x) - Simp(d*(b*sin(e + f*x))**(n + S(2))/(b**S(2)*f**S(2)*(n + S(1))*(n + S(2))), x) + Simp((b*sin(e + f*x))**(n + S(1))*(c + d*x)*cos(e + f*x)/(b*f*(n + S(1))), x) def replacement3250(b, c, d, e, f, n, x): return Dist((n + S(2))/(b**S(2)*(n + S(1))), Int((b*cos(e + f*x))**(n + S(2))*(c + d*x), x), x) - Simp(d*(b*cos(e + f*x))**(n + S(2))/(b**S(2)*f**S(2)*(n + S(1))*(n + S(2))), x) - Simp((b*cos(e + f*x))**(n + S(1))*(c + d*x)*sin(e + f*x)/(b*f*(n + S(1))), x) def replacement3251(b, c, d, e, f, m, n, x): return Dist((n + S(2))/(b**S(2)*(n + S(1))), Int((b*sin(e + f*x))**(n + S(2))*(c + d*x)**m, x), x) + Dist(d**S(2)*m*(m + S(-1))/(b**S(2)*f**S(2)*(n + S(1))*(n + S(2))), Int((b*sin(e + f*x))**(n + S(2))*(c + d*x)**(m + S(-2)), x), x) + Simp((b*sin(e + f*x))**(n + S(1))*(c + d*x)**m*cos(e + f*x)/(b*f*(n + S(1))), x) - Simp(d*m*(b*sin(e + f*x))**(n + S(2))*(c + d*x)**(m + S(-1))/(b**S(2)*f**S(2)*(n + S(1))*(n + S(2))), x) def replacement3252(b, c, d, e, f, m, n, x): return Dist((n + S(2))/(b**S(2)*(n + S(1))), Int((b*cos(e + f*x))**(n + S(2))*(c + d*x)**m, x), x) + Dist(d**S(2)*m*(m + S(-1))/(b**S(2)*f**S(2)*(n + S(1))*(n + S(2))), Int((b*cos(e + f*x))**(n + S(2))*(c + d*x)**(m + S(-2)), x), x) - Simp((b*cos(e + f*x))**(n + S(1))*(c + d*x)**m*sin(e + f*x)/(b*f*(n + S(1))), x) - Simp(d*m*(b*cos(e + f*x))**(n + S(2))*(c + d*x)**(m + S(-1))/(b**S(2)*f**S(2)*(n + S(1))*(n + S(2))), x) def replacement3253(a, b, c, d, e, f, m, n, x): return Int(ExpandIntegrand((c + d*x)**m, (a + b*sin(e + f*x))**n, x), x) def replacement3254(a, b, c, d, e, f, m, n, x): return Int(ExpandIntegrand((c + d*x)**m, (a + b*cos(e + f*x))**n, x), x) def replacement3255(a, b, c, d, e, f, m, n, x): return Dist((S(2)*a)**n, Int((c + d*x)**m*cos(-Pi*a/(S(4)*b) + e/S(2) + f*x/S(2))**(S(2)*n), x), x) def replacement3256(a, b, c, d, e, f, m, n, x): return Dist((S(2)*a)**IntPart(n)*(a + b*sin(e + f*x))**FracPart(n)*cos(-Pi*a/(S(4)*b) + e/S(2) + f*x/S(2))**(-S(2)*FracPart(n)), Int((c + d*x)**m*cos(-Pi*a/(S(4)*b) + e/S(2) + f*x/S(2))**(S(2)*n), x), x) def replacement3257(a, b, c, d, e, f, m, n, x): return Dist((S(2)*a)**n, Int((c + d*x)**m*cos(e/S(2) + f*x/S(2))**(S(2)*n), x), x) def replacement3258(a, b, c, d, e, f, m, n, x): return Dist((S(2)*a)**n, Int((c + d*x)**m*sin(e/S(2) + f*x/S(2))**(S(2)*n), x), x) def replacement3259(a, b, c, d, e, f, m, n, x): return Dist((S(2)*a)**IntPart(n)*(a + b*cos(e + f*x))**FracPart(n)*cos(e/S(2) + f*x/S(2))**(-S(2)*FracPart(n)), Int((c + d*x)**m*cos(e/S(2) + f*x/S(2))**(S(2)*n), x), x) def replacement3260(a, b, c, d, e, f, m, n, x): return Dist((S(2)*a)**IntPart(n)*(a + b*cos(e + f*x))**FracPart(n)*sin(e/S(2) + f*x/S(2))**(-S(2)*FracPart(n)), Int((c + d*x)**m*sin(e/S(2) + f*x/S(2))**(S(2)*n), x), x) def replacement3261(a, b, c, d, e, f, m, x): return Dist(S(2), Int((c + d*x)**m*exp(I*(e + f*x))/(S(2)*a*exp(I*(e + f*x)) - I*b*exp(S(2)*I*(e + f*x)) + I*b), x), x) def replacement3262(a, b, c, d, e, f, m, x): return Dist(S(2), Int((c + d*x)**m*exp(I*(e + f*x))/(S(2)*a*exp(I*(e + f*x)) + b*exp(S(2)*I*(e + f*x)) + b), x), x) def replacement3263(a, b, c, d, e, f, m, x): return Dist(a/(a**S(2) - b**S(2)), Int((c + d*x)**m/(a + b*sin(e + f*x)), x), x) - Dist(b*d*m/(f*(a**S(2) - b**S(2))), Int((c + d*x)**(m + S(-1))*cos(e + f*x)/(a + b*sin(e + f*x)), x), x) + Simp(b*(c + d*x)**m*cos(e + f*x)/(f*(a + b*sin(e + f*x))*(a**S(2) - b**S(2))), x) def replacement3264(a, b, c, d, e, f, m, x): return Dist(a/(a**S(2) - b**S(2)), Int((c + d*x)**m/(a + b*cos(e + f*x)), x), x) + Dist(b*d*m/(f*(a**S(2) - b**S(2))), Int((c + d*x)**(m + S(-1))*sin(e + f*x)/(a + b*cos(e + f*x)), x), x) - Simp(b*(c + d*x)**m*sin(e + f*x)/(f*(a + b*cos(e + f*x))*(a**S(2) - b**S(2))), x) def replacement3265(a, b, c, d, e, f, m, n, x): return Dist(a/(a**S(2) - b**S(2)), Int((a + b*sin(e + f*x))**(n + S(1))*(c + d*x)**m, x), x) - Dist(b*(n + S(2))/((a**S(2) - b**S(2))*(n + S(1))), Int((a + b*sin(e + f*x))**(n + S(1))*(c + d*x)**m*sin(e + f*x), x), x) + Dist(b*d*m/(f*(a**S(2) - b**S(2))*(n + S(1))), Int((a + b*sin(e + f*x))**(n + S(1))*(c + d*x)**(m + S(-1))*cos(e + f*x), x), x) - Simp(b*(a + b*sin(e + f*x))**(n + S(1))*(c + d*x)**m*cos(e + f*x)/(f*(a**S(2) - b**S(2))*(n + S(1))), x) def replacement3266(a, b, c, d, e, f, m, n, x): return Dist(a/(a**S(2) - b**S(2)), Int((a + b*cos(e + f*x))**(n + S(1))*(c + d*x)**m, x), x) - Dist(b*(n + S(2))/((a**S(2) - b**S(2))*(n + S(1))), Int((a + b*cos(e + f*x))**(n + S(1))*(c + d*x)**m*cos(e + f*x), x), x) - Dist(b*d*m/(f*(a**S(2) - b**S(2))*(n + S(1))), Int((a + b*cos(e + f*x))**(n + S(1))*(c + d*x)**(m + S(-1))*sin(e + f*x), x), x) + Simp(b*(a + b*cos(e + f*x))**(n + S(1))*(c + d*x)**m*sin(e + f*x)/(f*(a**S(2) - b**S(2))*(n + S(1))), x) def replacement3267(a, b, m, n, u, v, x): return Int((a + b*sin(ExpandToSum(v, x)))**n*ExpandToSum(u, x)**m, x) def replacement3268(a, b, m, n, u, v, x): return Int((a + b*cos(ExpandToSum(v, x)))**n*ExpandToSum(u, x)**m, x) def replacement3269(a, b, c, d, e, f, m, n, x): return Int((a + b*sin(e + f*x))**n*(c + d*x)**m, x) def replacement3270(a, b, c, d, e, f, m, n, x): return Int((a + b*cos(e + f*x))**n*(c + d*x)**m, x) def replacement3271(a, b, c, d, n, p, x): return Int(ExpandIntegrand(sin(c + d*x), (a + b*x**n)**p, x), x) def replacement3272(a, b, c, d, n, p, x): return Int(ExpandIntegrand(cos(c + d*x), (a + b*x**n)**p, x), x) def replacement3273(a, b, c, d, n, p, x): return -Dist(d/(b*n*(p + S(1))), Int(x**(S(1) - n)*(a + b*x**n)**(p + S(1))*cos(c + d*x), x), x) - Dist((S(1) - n)/(b*n*(p + S(1))), Int(x**(-n)*(a + b*x**n)**(p + S(1))*sin(c + d*x), x), x) + Simp(x**(S(1) - n)*(a + b*x**n)**(p + S(1))*sin(c + d*x)/(b*n*(p + S(1))), x) def replacement3274(a, b, c, d, n, p, x): return Dist(d/(b*n*(p + S(1))), Int(x**(S(1) - n)*(a + b*x**n)**(p + S(1))*sin(c + d*x), x), x) - Dist((S(1) - n)/(b*n*(p + S(1))), Int(x**(-n)*(a + b*x**n)**(p + S(1))*cos(c + d*x), x), x) + Simp(x**(S(1) - n)*(a + b*x**n)**(p + S(1))*cos(c + d*x)/(b*n*(p + S(1))), x) def replacement3275(a, b, c, d, n, p, x): return Int(ExpandIntegrand(sin(c + d*x), (a + b*x**n)**p, x), x) def replacement3276(a, b, c, d, n, p, x): return Int(ExpandIntegrand(cos(c + d*x), (a + b*x**n)**p, x), x) def replacement3277(a, b, c, d, n, p, x): return Int(x**(n*p)*(a*x**(-n) + b)**p*sin(c + d*x), x) def replacement3278(a, b, c, d, n, p, x): return Int(x**(n*p)*(a*x**(-n) + b)**p*cos(c + d*x), x) def replacement3279(a, b, c, d, n, p, x): return Int((a + b*x**n)**p*sin(c + d*x), x) def replacement3280(a, b, c, d, n, p, x): return Int((a + b*x**n)**p*cos(c + d*x), x) def replacement3281(a, b, c, d, e, m, n, p, x): return Int(ExpandIntegrand(sin(c + d*x), (e*x)**m*(a + b*x**n)**p, x), x) def replacement3282(a, b, c, d, e, m, n, p, x): return Int(ExpandIntegrand(cos(c + d*x), (e*x)**m*(a + b*x**n)**p, x), x) def replacement3283(a, b, c, d, e, m, n, p, x): return -Dist(d*e**m/(b*n*(p + S(1))), Int((a + b*x**n)**(p + S(1))*cos(c + d*x), x), x) + Simp(e**m*(a + b*x**n)**(p + S(1))*sin(c + d*x)/(b*n*(p + S(1))), x) def replacement3284(a, b, c, d, e, m, n, p, x): return Dist(d*e**m/(b*n*(p + S(1))), Int((a + b*x**n)**(p + S(1))*sin(c + d*x), x), x) + Simp(e**m*(a + b*x**n)**(p + S(1))*cos(c + d*x)/(b*n*(p + S(1))), x) def replacement3285(a, b, c, d, m, n, p, x): return -Dist(d/(b*n*(p + S(1))), Int(x**(m - n + S(1))*(a + b*x**n)**(p + S(1))*cos(c + d*x), x), x) - Dist((m - n + S(1))/(b*n*(p + S(1))), Int(x**(m - n)*(a + b*x**n)**(p + S(1))*sin(c + d*x), x), x) + Simp(x**(m - n + S(1))*(a + b*x**n)**(p + S(1))*sin(c + d*x)/(b*n*(p + S(1))), x) def replacement3286(a, b, c, d, m, n, p, x): return Dist(d/(b*n*(p + S(1))), Int(x**(m - n + S(1))*(a + b*x**n)**(p + S(1))*sin(c + d*x), x), x) - Dist((m - n + S(1))/(b*n*(p + S(1))), Int(x**(m - n)*(a + b*x**n)**(p + S(1))*cos(c + d*x), x), x) + Simp(x**(m - n + S(1))*(a + b*x**n)**(p + S(1))*cos(c + d*x)/(b*n*(p + S(1))), x) def replacement3287(a, b, c, d, m, n, p, x): return Int(ExpandIntegrand(sin(c + d*x), x**m*(a + b*x**n)**p, x), x) def replacement3288(a, b, c, d, m, n, p, x): return Int(ExpandIntegrand(cos(c + d*x), x**m*(a + b*x**n)**p, x), x) def replacement3289(a, b, c, d, m, n, p, x): return Int(x**(m + n*p)*(a*x**(-n) + b)**p*sin(c + d*x), x) def replacement3290(a, b, c, d, m, n, p, x): return Int(x**(m + n*p)*(a*x**(-n) + b)**p*cos(c + d*x), x) def replacement3291(a, b, c, d, e, m, n, p, x): return Int((e*x)**m*(a + b*x**n)**p*sin(c + d*x), x) def replacement3292(a, b, c, d, e, m, n, p, x): return Int((e*x)**m*(a + b*x**n)**p*cos(c + d*x), x) def replacement3293(d, x): return Simp(sqrt(S(2))*sqrt(Pi)*FresnelS(sqrt(S(2))*x*sqrt(S(1)/Pi)*Rt(d, S(2)))/(S(2)*Rt(d, S(2))), x) def replacement3294(d, x): return Simp(sqrt(S(2))*sqrt(Pi)*FresnelC(sqrt(S(2))*x*sqrt(S(1)/Pi)*Rt(d, S(2)))/(S(2)*Rt(d, S(2))), x) def replacement3295(c, d, x): return Dist(sin(c), Int(cos(d*x**S(2)), x), x) + Dist(cos(c), Int(sin(d*x**S(2)), x), x) def replacement3296(c, d, x): return -Dist(sin(c), Int(sin(d*x**S(2)), x), x) + Dist(cos(c), Int(cos(d*x**S(2)), x), x) def replacement3297(c, d, n, x): return Dist(I/S(2), Int(exp(-I*c - I*d*x**n), x), x) - Dist(I/S(2), Int(exp(I*c + I*d*x**n), x), x) def replacement3298(c, d, n, x): return Dist(S(1)/2, Int(exp(-I*c - I*d*x**n), x), x) + Dist(S(1)/2, Int(exp(I*c + I*d*x**n), x), x) def replacement3299(a, b, c, d, n, p, x): return Int(ExpandTrigReduce((a + b*sin(c + d*x**n))**p, x), x) def replacement3300(a, b, c, d, n, p, x): return Int(ExpandTrigReduce((a + b*cos(c + d*x**n))**p, x), x) def replacement3301(a, b, c, d, n, p, x): return -Subst(Int((a + b*sin(c + d*x**(-n)))**p/x**S(2), x), x, S(1)/x) def replacement3302(a, b, c, d, n, p, x): return -Subst(Int((a + b*cos(c + d*x**(-n)))**p/x**S(2), x), x, S(1)/x) def With3303(a, b, c, d, n, p, x): k = Denominator(n) return Dist(k, Subst(Int(x**(k + S(-1))*(a + b*sin(c + d*x**(k*n)))**p, x), x, x**(S(1)/k)), x) def With3304(a, b, c, d, n, p, x): k = Denominator(n) return Dist(k, Subst(Int(x**(k + S(-1))*(a + b*cos(c + d*x**(k*n)))**p, x), x, x**(S(1)/k)), x) def replacement3305(c, d, n, x): return Dist(I/S(2), Int(exp(-I*c - I*d*x**n), x), x) - Dist(I/S(2), Int(exp(I*c + I*d*x**n), x), x) def replacement3306(c, d, n, x): return Dist(S(1)/2, Int(exp(-I*c - I*d*x**n), x), x) + Dist(S(1)/2, Int(exp(I*c + I*d*x**n), x), x) def replacement3307(a, b, c, d, n, p, x): return Int(ExpandTrigReduce((a + b*sin(c + d*x**n))**p, x), x) def replacement3308(a, b, c, d, n, p, x): return Int(ExpandTrigReduce((a + b*cos(c + d*x**n))**p, x), x) def replacement3309(a, b, c, d, n, p, u, x): return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a + b*sin(c + d*x**n))**p, x), x, u), x) def replacement3310(a, b, c, d, n, p, u, x): return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a + b*cos(c + d*x**n))**p, x), x, u), x) def replacement3311(a, b, c, d, n, p, u, x): return Int((a + b*sin(c + d*u**n))**p, x) def replacement3312(a, b, c, d, n, p, u, x): return Int((a + b*cos(c + d*u**n))**p, x) def replacement3313(a, b, p, u, x): return Int((a + b*sin(ExpandToSum(u, x)))**p, x) def replacement3314(a, b, p, u, x): return Int((a + b*cos(ExpandToSum(u, x)))**p, x) def replacement3315(d, n, x): return Simp(SinIntegral(d*x**n)/n, x) def replacement3316(d, n, x): return Simp(CosIntegral(d*x**n)/n, x) def replacement3317(c, d, n, x): return Dist(sin(c), Int(cos(d*x**n)/x, x), x) + Dist(cos(c), Int(sin(d*x**n)/x, x), x) def replacement3318(c, d, n, x): return -Dist(sin(c), Int(sin(d*x**n)/x, x), x) + Dist(cos(c), Int(cos(d*x**n)/x, x), x) def With3319(a, b, c, d, m, n, p, x): if isinstance(x, (int, Integer, float, Float)): return False mn = (m + S(1))/n if And(IntegerQ(mn), Or(Equal(p, S(1)), Greater(mn, S(0)))): return True return False def replacement3319(a, b, c, d, m, n, p, x): mn = (m + S(1))/n return Dist(S(1)/n, Subst(Int(x**(mn + S(-1))*(a + b*sin(c + d*x))**p, x), x, x**n), x) def With3320(a, b, c, d, m, n, p, x): if isinstance(x, (int, Integer, float, Float)): return False mn = (m + S(1))/n if And(IntegerQ(mn), Or(Equal(p, S(1)), Greater(mn, S(0)))): return True return False def replacement3320(a, b, c, d, m, n, p, x): mn = (m + S(1))/n return Dist(S(1)/n, Subst(Int(x**(mn + S(-1))*(a + b*cos(c + d*x))**p, x), x, x**n), x) def With3321(a, b, c, d, e, m, n, p, x): if isinstance(x, (int, Integer, float, Float)): return False mn = (m + S(1))/n if And(IntegerQ(mn), Or(Equal(p, S(1)), Greater(mn, S(0)))): return True return False def replacement3321(a, b, c, d, e, m, n, p, x): mn = (m + S(1))/n return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(a + b*sin(c + d*x**n))**p, x), x) def With3322(a, b, c, d, e, m, n, p, x): if isinstance(x, (int, Integer, float, Float)): return False mn = (m + S(1))/n if And(IntegerQ(mn), Or(Equal(p, S(1)), Greater(mn, S(0)))): return True return False def replacement3322(a, b, c, d, e, m, n, p, x): mn = (m + S(1))/n return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(a + b*cos(c + d*x**n))**p, x), x) def replacement3323(a, b, m, n, x): return Dist(S(2)/n, Subst(Int(sin(a + b*x**S(2)), x), x, x**(n/S(2))), x) def replacement3324(a, b, m, n, x): return Dist(S(2)/n, Subst(Int(cos(a + b*x**S(2)), x), x, x**(n/S(2))), x) def replacement3325(c, d, e, m, n, x): return Dist(e**n*(m - n + S(1))/(d*n), Int((e*x)**(m - n)*cos(c + d*x**n), x), x) - Simp(e**(n + S(-1))*(e*x)**(m - n + S(1))*cos(c + d*x**n)/(d*n), x) def replacement3326(c, d, e, m, n, x): return -Dist(e**n*(m - n + S(1))/(d*n), Int((e*x)**(m - n)*sin(c + d*x**n), x), x) + Simp(e**(n + S(-1))*(e*x)**(m - n + S(1))*sin(c + d*x**n)/(d*n), x) def replacement3327(c, d, e, m, n, x): return -Dist(d*e**(-n)*n/(m + S(1)), Int((e*x)**(m + n)*cos(c + d*x**n), x), x) + Simp((e*x)**(m + S(1))*sin(c + d*x**n)/(e*(m + S(1))), x) def replacement3328(c, d, e, m, n, x): return Dist(d*e**(-n)*n/(m + S(1)), Int((e*x)**(m + n)*sin(c + d*x**n), x), x) + Simp((e*x)**(m + S(1))*cos(c + d*x**n)/(e*(m + S(1))), x) def replacement3329(c, d, e, m, n, x): return Dist(I/S(2), Int((e*x)**m*exp(-I*c - I*d*x**n), x), x) - Dist(I/S(2), Int((e*x)**m*exp(I*c + I*d*x**n), x), x) def replacement3330(c, d, e, m, n, x): return Dist(S(1)/2, Int((e*x)**m*exp(-I*c - I*d*x**n), x), x) + Dist(S(1)/2, Int((e*x)**m*exp(I*c + I*d*x**n), x), x) def replacement3331(a, b, m, n, p, x): return Dist(b*n*p/(n + S(-1)), Int(sin(a + b*x**n)**(p + S(-1))*cos(a + b*x**n), x), x) - Simp(x**(S(1) - n)*sin(a + b*x**n)**p/(n + S(-1)), x) def replacement3332(a, b, m, n, p, x): return -Dist(b*n*p/(n + S(-1)), Int(sin(a + b*x**n)*cos(a + b*x**n)**(p + S(-1)), x), x) - Simp(x**(S(1) - n)*cos(a + b*x**n)**p/(n + S(-1)), x) def replacement3333(a, b, m, n, p, x): return Dist((p + S(-1))/p, Int(x**m*sin(a + b*x**n)**(p + S(-2)), x), x) + Simp(sin(a + b*x**n)**p/(b**S(2)*n*p**S(2)), x) - Simp(x**n*sin(a + b*x**n)**(p + S(-1))*cos(a + b*x**n)/(b*n*p), x) def replacement3334(a, b, m, n, p, x): return Dist((p + S(-1))/p, Int(x**m*cos(a + b*x**n)**(p + S(-2)), x), x) + Simp(cos(a + b*x**n)**p/(b**S(2)*n*p**S(2)), x) + Simp(x**n*sin(a + b*x**n)*cos(a + b*x**n)**(p + S(-1))/(b*n*p), x) def replacement3335(a, b, m, n, p, x): return Dist((p + S(-1))/p, Int(x**m*sin(a + b*x**n)**(p + S(-2)), x), x) - Dist((m - S(2)*n + S(1))*(m - n + S(1))/(b**S(2)*n**S(2)*p**S(2)), Int(x**(m - S(2)*n)*sin(a + b*x**n)**p, x), x) + Simp(x**(m - S(2)*n + S(1))*(m - n + S(1))*sin(a + b*x**n)**p/(b**S(2)*n**S(2)*p**S(2)), x) - Simp(x**(m - n + S(1))*sin(a + b*x**n)**(p + S(-1))*cos(a + b*x**n)/(b*n*p), x) def replacement3336(a, b, m, n, p, x): return Dist((p + S(-1))/p, Int(x**m*cos(a + b*x**n)**(p + S(-2)), x), x) - Dist((m - S(2)*n + S(1))*(m - n + S(1))/(b**S(2)*n**S(2)*p**S(2)), Int(x**(m - S(2)*n)*cos(a + b*x**n)**p, x), x) + Simp(x**(m - S(2)*n + S(1))*(m - n + S(1))*cos(a + b*x**n)**p/(b**S(2)*n**S(2)*p**S(2)), x) + Simp(x**(m - n + S(1))*sin(a + b*x**n)*cos(a + b*x**n)**(p + S(-1))/(b*n*p), x) def replacement3337(a, b, m, n, p, x): return -Dist(b**S(2)*n**S(2)*p**S(2)/((m + S(1))*(m + n + S(1))), Int(x**(m + S(2)*n)*sin(a + b*x**n)**p, x), x) + Dist(b**S(2)*n**S(2)*p*(p + S(-1))/((m + S(1))*(m + n + S(1))), Int(x**(m + S(2)*n)*sin(a + b*x**n)**(p + S(-2)), x), x) + Simp(x**(m + S(1))*sin(a + b*x**n)**p/(m + S(1)), x) - Simp(b*n*p*x**(m + n + S(1))*sin(a + b*x**n)**(p + S(-1))*cos(a + b*x**n)/((m + S(1))*(m + n + S(1))), x) def replacement3338(a, b, m, n, p, x): return -Dist(b**S(2)*n**S(2)*p**S(2)/((m + S(1))*(m + n + S(1))), Int(x**(m + S(2)*n)*cos(a + b*x**n)**p, x), x) + Dist(b**S(2)*n**S(2)*p*(p + S(-1))/((m + S(1))*(m + n + S(1))), Int(x**(m + S(2)*n)*cos(a + b*x**n)**(p + S(-2)), x), x) + Simp(x**(m + S(1))*cos(a + b*x**n)**p/(m + S(1)), x) + Simp(b*n*p*x**(m + n + S(1))*sin(a + b*x**n)*cos(a + b*x**n)**(p + S(-1))/((m + S(1))*(m + n + S(1))), x) def With3339(a, b, c, d, e, m, n, p, x): k = Denominator(m) return Dist(k/e, Subst(Int(x**(k*(m + S(1)) + S(-1))*(a + b*sin(c + d*e**(-n)*x**(k*n)))**p, x), x, (e*x)**(S(1)/k)), x) def With3340(a, b, c, d, e, m, n, p, x): k = Denominator(m) return Dist(k/e, Subst(Int(x**(k*(m + S(1)) + S(-1))*(a + b*cos(c + d*e**(-n)*x**(k*n)))**p, x), x, (e*x)**(S(1)/k)), x) def replacement3341(a, b, c, d, e, m, n, p, x): return Int(ExpandTrigReduce((e*x)**m, (a + b*sin(c + d*x**n))**p, x), x) def replacement3342(a, b, c, d, e, m, n, p, x): return Int(ExpandTrigReduce((e*x)**m, (a + b*cos(c + d*x**n))**p, x), x) def replacement3343(a, b, m, n, p, x): return Dist((p + S(2))/(p + S(1)), Int(x**m*sin(a + b*x**n)**(p + S(2)), x), x) - Simp(sin(a + b*x**n)**(p + S(2))/(b**S(2)*n*(p + S(1))*(p + S(2))), x) + Simp(x**n*sin(a + b*x**n)**(p + S(1))*cos(a + b*x**n)/(b*n*(p + S(1))), x) def replacement3344(a, b, m, n, p, x): return Dist((p + S(2))/(p + S(1)), Int(x**m*cos(a + b*x**n)**(p + S(2)), x), x) - Simp(cos(a + b*x**n)**(p + S(2))/(b**S(2)*n*(p + S(1))*(p + S(2))), x) - Simp(x**n*sin(a + b*x**n)*cos(a + b*x**n)**(p + S(1))/(b*n*(p + S(1))), x) def replacement3345(a, b, m, n, p, x): return Dist((p + S(2))/(p + S(1)), Int(x**m*sin(a + b*x**n)**(p + S(2)), x), x) + Dist((m - S(2)*n + S(1))*(m - n + S(1))/(b**S(2)*n**S(2)*(p + S(1))*(p + S(2))), Int(x**(m - S(2)*n)*sin(a + b*x**n)**(p + S(2)), x), x) + Simp(x**(m - n + S(1))*sin(a + b*x**n)**(p + S(1))*cos(a + b*x**n)/(b*n*(p + S(1))), x) - Simp(x**(m - S(2)*n + S(1))*(m - n + S(1))*sin(a + b*x**n)**(p + S(2))/(b**S(2)*n**S(2)*(p + S(1))*(p + S(2))), x) def replacement3346(a, b, m, n, p, x): return Dist((p + S(2))/(p + S(1)), Int(x**m*cos(a + b*x**n)**(p + S(2)), x), x) + Dist((m - S(2)*n + S(1))*(m - n + S(1))/(b**S(2)*n**S(2)*(p + S(1))*(p + S(2))), Int(x**(m - S(2)*n)*cos(a + b*x**n)**(p + S(2)), x), x) - Simp(x**(m - n + S(1))*sin(a + b*x**n)*cos(a + b*x**n)**(p + S(1))/(b*n*(p + S(1))), x) - Simp(x**(m - S(2)*n + S(1))*(m - n + S(1))*cos(a + b*x**n)**(p + S(2))/(b**S(2)*n**S(2)*(p + S(1))*(p + S(2))), x) def replacement3347(a, b, c, d, m, n, p, x): return -Subst(Int(x**(-m + S(-2))*(a + b*sin(c + d*x**(-n)))**p, x), x, S(1)/x) def replacement3348(a, b, c, d, m, n, p, x): return -Subst(Int(x**(-m + S(-2))*(a + b*cos(c + d*x**(-n)))**p, x), x, S(1)/x) def With3349(a, b, c, d, e, m, n, p, x): k = Denominator(m) return -Dist(k/e, Subst(Int(x**(-k*(m + S(1)) + S(-1))*(a + b*sin(c + d*e**(-n)*x**(-k*n)))**p, x), x, (e*x)**(-S(1)/k)), x) def With3350(a, b, c, d, e, m, n, p, x): k = Denominator(m) return -Dist(k/e, Subst(Int(x**(-k*(m + S(1)) + S(-1))*(a + b*cos(c + d*e**(-n)*x**(-k*n)))**p, x), x, (e*x)**(-S(1)/k)), x) def replacement3351(a, b, c, d, e, m, n, p, x): return -Dist((e*x)**m*(S(1)/x)**m, Subst(Int(x**(-m + S(-2))*(a + b*sin(c + d*x**(-n)))**p, x), x, S(1)/x), x) def replacement3352(a, b, c, d, e, m, n, p, x): return -Dist((e*x)**m*(S(1)/x)**m, Subst(Int(x**(-m + S(-2))*(a + b*cos(c + d*x**(-n)))**p, x), x, S(1)/x), x) def With3353(a, b, c, d, m, n, p, x): k = Denominator(n) return Dist(k, Subst(Int(x**(k*(m + S(1)) + S(-1))*(a + b*sin(c + d*x**(k*n)))**p, x), x, x**(S(1)/k)), x) def With3354(a, b, c, d, m, n, p, x): k = Denominator(n) return Dist(k, Subst(Int(x**(k*(m + S(1)) + S(-1))*(a + b*cos(c + d*x**(k*n)))**p, x), x, x**(S(1)/k)), x) def replacement3355(a, b, c, d, e, m, n, p, x): return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(a + b*sin(c + d*x**n))**p, x), x) def replacement3356(a, b, c, d, e, m, n, p, x): return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(a + b*cos(c + d*x**n))**p, x), x) def replacement3357(a, b, c, d, m, n, p, x): return Dist(S(1)/(m + S(1)), Subst(Int((a + b*sin(c + d*x**(n/(m + S(1)))))**p, x), x, x**(m + S(1))), x) def replacement3358(a, b, c, d, m, n, p, x): return Dist(S(1)/(m + S(1)), Subst(Int((a + b*cos(c + d*x**(n/(m + S(1)))))**p, x), x, x**(m + S(1))), x) def replacement3359(a, b, c, d, e, m, n, p, x): return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(a + b*sin(c + d*x**n))**p, x), x) def replacement3360(a, b, c, d, e, m, n, p, x): return Dist(e**IntPart(m)*x**(-FracPart(m))*(e*x)**FracPart(m), Int(x**m*(a + b*cos(c + d*x**n))**p, x), x) def replacement3361(c, d, e, m, n, x): return Dist(I/S(2), Int((e*x)**m*exp(-I*c - I*d*x**n), x), x) - Dist(I/S(2), Int((e*x)**m*exp(I*c + I*d*x**n), x), x) def replacement3362(c, d, e, m, n, x): return Dist(S(1)/2, Int((e*x)**m*exp(-I*c - I*d*x**n), x), x) + Dist(S(1)/2, Int((e*x)**m*exp(I*c + I*d*x**n), x), x) def replacement3363(a, b, c, d, e, m, n, p, x): return Int(ExpandTrigReduce((e*x)**m, (a + b*sin(c + d*x**n))**p, x), x) def replacement3364(a, b, c, d, e, m, n, p, x): return Int(ExpandTrigReduce((e*x)**m, (a + b*cos(c + d*x**n))**p, x), x) def replacement3365(a, b, c, d, m, n, p, u, x): return Dist(Coefficient(u, x, S(1))**(-m + S(-1)), Subst(Int((a + b*sin(c + d*x**n))**p*(x - Coefficient(u, x, S(0)))**m, x), x, u), x) def replacement3366(a, b, c, d, m, n, p, u, x): return Dist(Coefficient(u, x, S(1))**(-m + S(-1)), Subst(Int((a + b*cos(c + d*x**n))**p*(x - Coefficient(u, x, S(0)))**m, x), x, u), x) def replacement3367(a, b, c, d, e, m, n, p, u, x): return Int((e*x)**m*(a + b*sin(c + d*u**n))**p, x) def replacement3368(a, b, c, d, e, m, n, p, u, x): return Int((e*x)**m*(a + b*cos(c + d*u**n))**p, x) def replacement3369(a, b, e, m, p, u, x): return Int((e*x)**m*(a + b*sin(ExpandToSum(u, x)))**p, x) def replacement3370(a, b, e, m, p, u, x): return Int((e*x)**m*(a + b*cos(ExpandToSum(u, x)))**p, x) def replacement3371(a, b, m, n, p, x): return Simp(sin(a + b*x**n)**(p + S(1))/(b*n*(p + S(1))), x) def replacement3372(a, b, m, n, p, x): return -Simp(cos(a + b*x**n)**(p + S(1))/(b*n*(p + S(1))), x) def replacement3373(a, b, m, n, p, x): return -Dist((m - n + S(1))/(b*n*(p + S(1))), Int(x**(m - n)*sin(a + b*x**n)**(p + S(1)), x), x) + Simp(x**(m - n + S(1))*sin(a + b*x**n)**(p + S(1))/(b*n*(p + S(1))), x) def replacement3374(a, b, m, n, p, x): return Dist((m - n + S(1))/(b*n*(p + S(1))), Int(x**(m - n)*cos(a + b*x**n)**(p + S(1)), x), x) - Simp(x**(m - n + S(1))*cos(a + b*x**n)**(p + S(1))/(b*n*(p + S(1))), x) def replacement3375(a, b, c, x): return Int(sin((b + S(2)*c*x)**S(2)/(S(4)*c)), x) def replacement3376(a, b, c, x): return Int(cos((b + S(2)*c*x)**S(2)/(S(4)*c)), x) def replacement3377(a, b, c, x): return -Dist(sin((-S(4)*a*c + b**S(2))/(S(4)*c)), Int(cos((b + S(2)*c*x)**S(2)/(S(4)*c)), x), x) + Dist(cos((-S(4)*a*c + b**S(2))/(S(4)*c)), Int(sin((b + S(2)*c*x)**S(2)/(S(4)*c)), x), x) def replacement3378(a, b, c, x): return Dist(sin((-S(4)*a*c + b**S(2))/(S(4)*c)), Int(sin((b + S(2)*c*x)**S(2)/(S(4)*c)), x), x) + Dist(cos((-S(4)*a*c + b**S(2))/(S(4)*c)), Int(cos((b + S(2)*c*x)**S(2)/(S(4)*c)), x), x) def replacement3379(a, b, c, n, x): return Int(ExpandTrigReduce(sin(a + b*x + c*x**S(2))**n, x), x) def replacement3380(a, b, c, n, x): return Int(ExpandTrigReduce(cos(a + b*x + c*x**S(2))**n, x), x) def replacement3381(n, v, x): return Int(sin(ExpandToSum(v, x))**n, x) def replacement3382(n, v, x): return Int(cos(ExpandToSum(v, x))**n, x) def replacement3383(a, b, c, d, e, x): return -Simp(e*cos(a + b*x + c*x**S(2))/(S(2)*c), x) def replacement3384(a, b, c, d, e, x): return Simp(e*sin(a + b*x + c*x**S(2))/(S(2)*c), x) def replacement3385(a, b, c, d, e, x): return Dist((-b*e + S(2)*c*d)/(S(2)*c), Int(sin(a + b*x + c*x**S(2)), x), x) - Simp(e*cos(a + b*x + c*x**S(2))/(S(2)*c), x) def replacement3386(a, b, c, d, e, x): return Dist((-b*e + S(2)*c*d)/(S(2)*c), Int(cos(a + b*x + c*x**S(2)), x), x) + Simp(e*sin(a + b*x + c*x**S(2))/(S(2)*c), x) def replacement3387(a, b, c, d, e, m, x): return Dist(e**S(2)*(m + S(-1))/(S(2)*c), Int((d + e*x)**(m + S(-2))*cos(a + b*x + c*x**S(2)), x), x) - Simp(e*(d + e*x)**(m + S(-1))*cos(a + b*x + c*x**S(2))/(S(2)*c), x) def replacement3388(a, b, c, d, e, m, x): return -Dist(e**S(2)*(m + S(-1))/(S(2)*c), Int((d + e*x)**(m + S(-2))*sin(a + b*x + c*x**S(2)), x), x) + Simp(e*(d + e*x)**(m + S(-1))*sin(a + b*x + c*x**S(2))/(S(2)*c), x) def replacement3389(a, b, c, d, e, m, x): return -Dist((b*e - S(2)*c*d)/(S(2)*c), Int((d + e*x)**(m + S(-1))*sin(a + b*x + c*x**S(2)), x), x) + Dist(e**S(2)*(m + S(-1))/(S(2)*c), Int((d + e*x)**(m + S(-2))*cos(a + b*x + c*x**S(2)), x), x) - Simp(e*(d + e*x)**(m + S(-1))*cos(a + b*x + c*x**S(2))/(S(2)*c), x) def replacement3390(a, b, c, d, e, m, x): return -Dist((b*e - S(2)*c*d)/(S(2)*c), Int((d + e*x)**(m + S(-1))*cos(a + b*x + c*x**S(2)), x), x) - Dist(e**S(2)*(m + S(-1))/(S(2)*c), Int((d + e*x)**(m + S(-2))*sin(a + b*x + c*x**S(2)), x), x) + Simp(e*(d + e*x)**(m + S(-1))*sin(a + b*x + c*x**S(2))/(S(2)*c), x) def replacement3391(a, b, c, d, e, m, x): return -Dist(S(2)*c/(e**S(2)*(m + S(1))), Int((d + e*x)**(m + S(2))*cos(a + b*x + c*x**S(2)), x), x) + Simp((d + e*x)**(m + S(1))*sin(a + b*x + c*x**S(2))/(e*(m + S(1))), x) def replacement3392(a, b, c, d, e, m, x): return Dist(S(2)*c/(e**S(2)*(m + S(1))), Int((d + e*x)**(m + S(2))*sin(a + b*x + c*x**S(2)), x), x) + Simp((d + e*x)**(m + S(1))*cos(a + b*x + c*x**S(2))/(e*(m + S(1))), x) def replacement3393(a, b, c, d, e, m, x): return -Dist(S(2)*c/(e**S(2)*(m + S(1))), Int((d + e*x)**(m + S(2))*cos(a + b*x + c*x**S(2)), x), x) - Dist((b*e - S(2)*c*d)/(e**S(2)*(m + S(1))), Int((d + e*x)**(m + S(1))*cos(a + b*x + c*x**S(2)), x), x) + Simp((d + e*x)**(m + S(1))*sin(a + b*x + c*x**S(2))/(e*(m + S(1))), x) def replacement3394(a, b, c, d, e, m, x): return Dist(S(2)*c/(e**S(2)*(m + S(1))), Int((d + e*x)**(m + S(2))*sin(a + b*x + c*x**S(2)), x), x) + Dist((b*e - S(2)*c*d)/(e**S(2)*(m + S(1))), Int((d + e*x)**(m + S(1))*sin(a + b*x + c*x**S(2)), x), x) + Simp((d + e*x)**(m + S(1))*cos(a + b*x + c*x**S(2))/(e*(m + S(1))), x) def replacement3395(a, b, c, d, e, m, x): return Int((d + e*x)**m*sin(a + b*x + c*x**S(2)), x) def replacement3396(a, b, c, d, e, m, x): return Int((d + e*x)**m*cos(a + b*x + c*x**S(2)), x) def replacement3397(a, b, c, d, e, m, n, x): return Int(ExpandTrigReduce((d + e*x)**m, sin(a + b*x + c*x**S(2))**n, x), x) def replacement3398(a, b, c, d, e, m, n, x): return Int(ExpandTrigReduce((d + e*x)**m, cos(a + b*x + c*x**S(2))**n, x), x) def replacement3399(m, n, u, v, x): return Int(ExpandToSum(u, x)**m*sin(ExpandToSum(v, x))**n, x) def replacement3400(m, n, u, v, x): return Int(ExpandToSum(u, x)**m*cos(ExpandToSum(v, x))**n, x)