""" 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 tangent(): from sympy.integrals.rubi.constraints import cons1510, cons2, cons3, cons50, cons127, cons19, cons4, cons1511, cons20, cons25, cons95, cons167, cons96, cons1512, cons1172, cons89, cons1513, cons91, cons168, cons685, cons33, cons268, cons34, cons1514, cons1515, cons21, cons157, cons1252, cons1516, cons1517, cons1518, cons1519, cons1520, cons1521, cons1522, cons1523, cons1524, cons82, cons81, cons1525, cons1526, cons1527, cons1528, cons29, cons5, cons1529, cons8, cons1263, cons1266, cons1441, cons465, cons1442, cons1530, cons1257, cons1531, cons90, cons1532, cons1533, cons1534, cons517, cons1535, cons1536, cons1537, cons1538, cons1539, cons1540, cons68, cons1541, cons86, cons1230, cons150, cons198, cons1542, cons87, cons72, cons1543, cons73, cons1414, cons269, cons1305, cons170, cons1544, cons1545, cons1324, cons1546, cons1325, cons1547, cons1548, cons1549, cons1550, cons79, cons1551, cons1552, cons1553, cons1425, cons1387, cons113, cons274, cons1327, cons1329, cons1336, cons1554, cons1555, cons1335, cons1556, cons1557, cons1338, cons1558, cons1559, cons810, cons382, cons210, cons149, cons1419, cons52, cons40, cons36, cons37, cons348, cons1420, cons1428, cons1560, cons1561, cons1562, cons1258, cons1563, cons38, cons35, cons1435, cons1564, cons1565, cons1566, cons1567, cons1568, cons1569, cons1570, cons1571, cons1494, cons1458, cons1572, cons1483, cons1481, cons745, cons1499, cons48, cons47, cons228, cons1482, cons64, cons530, cons1573, cons1574, cons812, cons813, cons1362, cons1575, cons1497, cons70, cons71, cons825, cons826, cons1576, cons1577, cons1578, cons1579, cons1580, cons1581, cons49, cons241, cons1582 pattern3401 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons19, cons4, cons1510) rule3401 = ReplacementRule(pattern3401, replacement3401) pattern3402 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons19, cons4, cons1510) rule3402 = ReplacementRule(pattern3402, replacement3402) pattern3403 = Pattern(Integral(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('n', S(1)), x_), cons50, cons127, cons1511) rule3403 = ReplacementRule(pattern3403, replacement3403) pattern3404 = Pattern(Integral((S(1)/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1)), x_), cons50, cons127, cons1511) rule3404 = ReplacementRule(pattern3404, replacement3404) pattern3405 = Pattern(Integral((WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1)), x_), cons3, cons50, cons127, cons4, cons20, cons25) rule3405 = ReplacementRule(pattern3405, replacement3405) pattern3406 = Pattern(Integral((WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1)), x_), cons3, cons50, cons127, cons4, cons20, cons25) rule3406 = ReplacementRule(pattern3406, replacement3406) pattern3407 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons95, cons167, cons96, cons1512, cons1172) rule3407 = ReplacementRule(pattern3407, replacement3407) pattern3408 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons95, cons167, cons96, cons1512, cons1172) rule3408 = ReplacementRule(pattern3408, replacement3408) pattern3409 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons19, cons89, cons167, cons1172, cons1513) rule3409 = ReplacementRule(pattern3409, replacement3409) pattern3410 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons19, cons89, cons167, cons1172, cons1513) rule3410 = ReplacementRule(pattern3410, replacement3410) pattern3411 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons95, cons91, cons168, cons1172) rule3411 = ReplacementRule(pattern3411, replacement3411) pattern3412 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons95, cons91, cons168, cons1172) rule3412 = ReplacementRule(pattern3412, replacement3412) pattern3413 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons19, cons89, cons91, cons685, cons1172) rule3413 = ReplacementRule(pattern3413, replacement3413) pattern3414 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons19, cons89, cons91, cons685, cons1172) rule3414 = ReplacementRule(pattern3414, replacement3414) pattern3415 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons50, cons127, cons4, cons33, cons268, cons1172) rule3415 = ReplacementRule(pattern3415, replacement3415) pattern3416 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons50, cons127, cons4, cons33, cons268, cons1172) rule3416 = ReplacementRule(pattern3416, replacement3416) pattern3417 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons50, cons127, cons4, cons33, cons34, cons685, cons1172) rule3417 = ReplacementRule(pattern3417, replacement3417) pattern3418 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons50, cons127, cons4, cons33, cons34, cons685, cons1172) rule3418 = ReplacementRule(pattern3418, replacement3418) pattern3419 = Pattern(Integral((WC('a', 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('n', S(1)), x_), cons2, cons50, cons127, cons19, cons1514) rule3419 = ReplacementRule(pattern3419, replacement3419) pattern3420 = Pattern(Integral((WC('a', 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('n', S(1)), x_), cons2, cons50, cons127, cons19, cons1514) rule3420 = ReplacementRule(pattern3420, replacement3420) pattern3421 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons50, cons127, cons19, cons4, cons1515) rule3421 = ReplacementRule(pattern3421, replacement3421) pattern3422 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons50, cons127, cons19, cons4, cons1515) rule3422 = ReplacementRule(pattern3422, replacement3422) pattern3423 = Pattern(Integral((WC('a', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons19, cons4, cons21, cons25) rule3423 = ReplacementRule(pattern3423, replacement3423) pattern3424 = Pattern(Integral((WC('a', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons19, cons4, cons21, cons25) rule3424 = ReplacementRule(pattern3424, replacement3424) pattern3425 = Pattern(Integral((WC('a', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons19, cons4, cons21, cons25) rule3425 = ReplacementRule(pattern3425, replacement3425) pattern3426 = Pattern(Integral((WC('a', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons50, cons127, cons19, cons4, cons157) rule3426 = ReplacementRule(pattern3426, replacement3426) pattern3427 = Pattern(Integral((WC('a', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons50, cons127, cons19, cons4, cons157) rule3427 = ReplacementRule(pattern3427, replacement3427) pattern3428 = Pattern(Integral((WC('a', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons50, cons127, cons19, cons1252, cons1516) rule3428 = ReplacementRule(pattern3428, replacement3428) pattern3429 = Pattern(Integral((WC('a', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons50, cons127, cons19, cons1252, cons1516) rule3429 = ReplacementRule(pattern3429, replacement3429) pattern3430 = Pattern(Integral((WC('b', S(1))*tan(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))))**m_, x_), cons3, cons50, cons127, cons4, cons1517, cons1518) rule3430 = ReplacementRule(pattern3430, replacement3430) pattern3431 = Pattern(Integral((WC('b', S(1))/tan(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))))**m_, x_), cons3, cons50, cons127, cons4, cons1517, cons1518) rule3431 = ReplacementRule(pattern3431, replacement3431) pattern3432 = Pattern(Integral((WC('a', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons95, cons91, cons1519, cons1172) rule3432 = ReplacementRule(pattern3432, replacement3432) pattern3433 = Pattern(Integral((WC('a', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons95, cons91, cons1519, cons1172) rule3433 = ReplacementRule(pattern3433, replacement3433) pattern3434 = Pattern(Integral((WC('a', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons19, cons89, cons91, cons1172) rule3434 = ReplacementRule(pattern3434, replacement3434) pattern3435 = Pattern(Integral((WC('a', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons19, cons89, cons91, cons1172) rule3435 = ReplacementRule(pattern3435, replacement3435) pattern3436 = Pattern(Integral((WC('a', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons95, cons167, cons1520, cons1172) rule3436 = ReplacementRule(pattern3436, replacement3436) pattern3437 = Pattern(Integral((WC('a', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons95, cons167, cons1520, cons1172) rule3437 = ReplacementRule(pattern3437, replacement3437) pattern3438 = Pattern(Integral((WC('a', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons19, cons89, cons167, cons1521, cons1172) rule3438 = ReplacementRule(pattern3438, replacement3438) pattern3439 = Pattern(Integral((WC('a', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons19, cons89, cons167, cons1521, cons1172) rule3439 = ReplacementRule(pattern3439, replacement3439) pattern3440 = Pattern(Integral((WC('a', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons4, cons33, cons1522, cons1172) rule3440 = ReplacementRule(pattern3440, replacement3440) pattern3441 = Pattern(Integral((WC('a', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons4, cons33, cons1522, cons1172) rule3441 = ReplacementRule(pattern3441, replacement3441) pattern3442 = Pattern(Integral((WC('a', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons4, cons33, cons1523, cons1521, cons1172) rule3442 = ReplacementRule(pattern3442, replacement3442) pattern3443 = Pattern(Integral((WC('a', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons4, cons33, cons1523, cons1521, cons1172) rule3443 = ReplacementRule(pattern3443, replacement3443) pattern3444 = Pattern(Integral(S(1)/(sqrt(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons3, cons50, cons127, cons1524) rule3444 = ReplacementRule(pattern3444, replacement3444) pattern3445 = Pattern(Integral(S(1)/(sqrt(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons3, cons50, cons127, cons1524) rule3445 = ReplacementRule(pattern3445, replacement3445) pattern3446 = Pattern(Integral(sqrt(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons3, cons50, cons127, cons1524) rule3446 = ReplacementRule(pattern3446, replacement3446) pattern3447 = Pattern(Integral(sqrt(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons3, cons50, cons127, cons1524) rule3447 = ReplacementRule(pattern3447, replacement3447) pattern3448 = Pattern(Integral((WC('a', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons19, cons4, cons82, cons81) rule3448 = ReplacementRule(pattern3448, replacement3448) pattern3449 = Pattern(Integral((WC('a', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons19, cons4, cons82, cons81) rule3449 = ReplacementRule(pattern3449, replacement3449) pattern3450 = Pattern(Integral((WC('a', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons19, cons4, cons1525, cons1526) rule3450 = ReplacementRule(pattern3450, replacement3450) pattern3451 = Pattern(Integral((WC('a', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons19, cons4, cons1525, cons1526) rule3451 = ReplacementRule(pattern3451, replacement3451) pattern3452 = Pattern(Integral((WC('a', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons19, cons4, cons21, cons25) rule3452 = ReplacementRule(pattern3452, replacement3452) pattern3453 = Pattern(Integral((WC('a', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons50, cons127, cons19, cons4, cons21, cons25) rule3453 = ReplacementRule(pattern3453, replacement3453) pattern3454 = Pattern(Integral(((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*WC('a', S(1)))**WC('m', S(1))*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons5, cons89, cons167, cons1527, cons1528) rule3454 = ReplacementRule(pattern3454, replacement3454) pattern3455 = Pattern(Integral(((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*WC('a', S(1)))**WC('m', S(1))*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons5, cons89, cons167, cons1527, cons1528) rule3455 = ReplacementRule(pattern3455, replacement3455) pattern3456 = Pattern(Integral(((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**p_*WC('a', S(1)))**WC('m', S(1))*(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons5, cons89, cons91, cons1529, cons1528) rule3456 = ReplacementRule(pattern3456, replacement3456) pattern3457 = Pattern(Integral(((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**p_*WC('a', S(1)))**WC('m', S(1))*(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons5, cons89, cons91, cons1529, cons1528) rule3457 = ReplacementRule(pattern3457, replacement3457) pattern3458 = Pattern(Integral((WC('b', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons3, cons8, cons29, cons89, cons167) rule3458 = ReplacementRule(pattern3458, replacement3458) pattern3459 = Pattern(Integral((WC('b', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons3, cons8, cons29, cons89, cons167) rule3459 = ReplacementRule(pattern3459, replacement3459) pattern3460 = Pattern(Integral((WC('b', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons3, cons8, cons29, cons89, cons91) rule3460 = ReplacementRule(pattern3460, replacement3460) pattern3461 = Pattern(Integral((WC('b', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons3, cons8, cons29, cons89, cons91) rule3461 = ReplacementRule(pattern3461, replacement3461) pattern3462 = Pattern(Integral(tan(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons8, cons29, cons1263) rule3462 = ReplacementRule(pattern3462, replacement3462) pattern3463 = Pattern(Integral(S(1)/tan(x_*WC('d', S(1)) + WC('c', S(0))), x_), cons8, cons29, cons1263) rule3463 = ReplacementRule(pattern3463, replacement3463) pattern3464 = Pattern(Integral((WC('b', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons3, cons8, cons29, cons4, cons25) rule3464 = ReplacementRule(pattern3464, replacement3464) pattern3465 = Pattern(Integral((WC('b', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons3, cons8, cons29, cons4, cons25) rule3465 = ReplacementRule(pattern3465, replacement3465) pattern3466 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**S(2), x_), cons2, cons3, cons8, cons29, cons1266) rule3466 = ReplacementRule(pattern3466, replacement3466) pattern3467 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**S(2), x_), cons2, cons3, cons8, cons29, cons1266) rule3467 = ReplacementRule(pattern3467, replacement3467) pattern3468 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons1441, cons89, cons167) rule3468 = ReplacementRule(pattern3468, replacement3468) pattern3469 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons1441, cons89, cons167) rule3469 = ReplacementRule(pattern3469, replacement3469) pattern3470 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons1441, cons89, cons465) rule3470 = ReplacementRule(pattern3470, replacement3470) pattern3471 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons1441, cons89, cons465) rule3471 = ReplacementRule(pattern3471, replacement3471) pattern3472 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1441) rule3472 = ReplacementRule(pattern3472, replacement3472) pattern3473 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1441) rule3473 = ReplacementRule(pattern3473, replacement3473) pattern3474 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons4, cons1441) rule3474 = ReplacementRule(pattern3474, replacement3474) pattern3475 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons4, cons1441) rule3475 = ReplacementRule(pattern3475, replacement3475) pattern3476 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons1442, cons89, cons167) rule3476 = ReplacementRule(pattern3476, replacement3476) pattern3477 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons1442, cons89, cons167) rule3477 = ReplacementRule(pattern3477, replacement3477) pattern3478 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons1442, cons89, cons91) rule3478 = ReplacementRule(pattern3478, replacement3478) pattern3479 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons1442, cons89, cons91) rule3479 = ReplacementRule(pattern3479, replacement3479) pattern3480 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1442) rule3480 = ReplacementRule(pattern3480, replacement3480) pattern3481 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons1442) rule3481 = ReplacementRule(pattern3481, replacement3481) pattern3482 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons4, cons1442) rule3482 = ReplacementRule(pattern3482, replacement3482) pattern3483 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons4, cons1442) rule3483 = ReplacementRule(pattern3483, replacement3483) pattern3484 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons19, cons1530) rule3484 = ReplacementRule(pattern3484, replacement3484) pattern3485 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons19, cons1530) rule3485 = ReplacementRule(pattern3485, replacement3485) pattern3486 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(S(1)/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons4, cons1441, cons1517) rule3486 = ReplacementRule(pattern3486, replacement3486) pattern3487 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(S(1)/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons4, cons1441, cons1517) rule3487 = ReplacementRule(pattern3487, replacement3487) pattern3488 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons1441, cons1257) rule3488 = ReplacementRule(pattern3488, replacement3488) pattern3489 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons1441, cons1257) rule3489 = ReplacementRule(pattern3489, replacement3489) pattern3490 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons1441) rule3490 = ReplacementRule(pattern3490, replacement3490) pattern3491 = Pattern(Integral(S(1)/(sqrt(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons1441) rule3491 = ReplacementRule(pattern3491, replacement3491) pattern3492 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons29, cons50, cons127, cons1441, cons1531, cons95, cons90) rule3492 = ReplacementRule(pattern3492, replacement3492) pattern3493 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons29, cons50, cons127, cons1441, cons1531, cons95, cons90) rule3493 = ReplacementRule(pattern3493, replacement3493) pattern3494 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons29, cons50, cons127, cons1441, cons1531, cons95, cons91) rule3494 = ReplacementRule(pattern3494, replacement3494) pattern3495 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons29, cons50, cons127, cons1441, cons1531, cons95, cons91) rule3495 = ReplacementRule(pattern3495, replacement3495) pattern3496 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons1441, cons1531) rule3496 = ReplacementRule(pattern3496, replacement3496) pattern3497 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons1441, cons1531) rule3497 = ReplacementRule(pattern3497, replacement3497) pattern3498 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons1441, cons1532) rule3498 = ReplacementRule(pattern3498, replacement3498) pattern3499 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons1441, cons1532) rule3499 = ReplacementRule(pattern3499, replacement3499) pattern3500 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons1441, cons1533, cons25) rule3500 = ReplacementRule(pattern3500, replacement3500) pattern3501 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons1441, cons1533, cons25) rule3501 = ReplacementRule(pattern3501, replacement3501) pattern3502 = Pattern(Integral(sqrt(WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons1441) rule3502 = ReplacementRule(pattern3502, replacement3502) pattern3503 = Pattern(Integral(sqrt(WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons1441) rule3503 = ReplacementRule(pattern3503, replacement3503) pattern3504 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons29, cons50, cons127, cons1441, cons95, cons167, cons1534, cons517) rule3504 = ReplacementRule(pattern3504, replacement3504) pattern3505 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons29, cons50, cons127, cons1441, cons95, cons167, cons1535, cons517) rule3505 = ReplacementRule(pattern3505, replacement3505) pattern3506 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons29, cons50, cons127, cons1441, cons95, cons90, cons96, cons1172) rule3506 = ReplacementRule(pattern3506, replacement3506) pattern3507 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons29, cons50, cons127, cons1441, cons95, cons90, cons96, cons1172) rule3507 = ReplacementRule(pattern3507, replacement3507) pattern3508 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons1441, cons89, cons90, cons1521, cons1172) rule3508 = ReplacementRule(pattern3508, replacement3508) pattern3509 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons1441, cons89, cons90, cons1521, cons1172) rule3509 = ReplacementRule(pattern3509, replacement3509) pattern3510 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/sqrt(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons1441) rule3510 = ReplacementRule(pattern3510, replacement3510) pattern3511 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/sqrt(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons1441) rule3511 = ReplacementRule(pattern3511, replacement3511) pattern3512 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons1441, cons89, cons91, cons1536, cons517) rule3512 = ReplacementRule(pattern3512, replacement3512) pattern3513 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons1441, cons89, cons91, cons1536, cons517) rule3513 = ReplacementRule(pattern3513, replacement3513) pattern3514 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons29, cons50, cons127, cons1441, cons95, cons465, cons168, cons1537, cons1521, cons1172) rule3514 = ReplacementRule(pattern3514, replacement3514) pattern3515 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons29, cons50, cons127, cons1441, cons95, cons465, cons168, cons1537, cons1521, cons1172) rule3515 = ReplacementRule(pattern3515, replacement3515) pattern3516 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons1441, cons89, cons465, cons1538, cons1172) rule3516 = ReplacementRule(pattern3516, replacement3516) pattern3517 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons1441, cons89, cons465, cons1538, cons1172) rule3517 = ReplacementRule(pattern3517, replacement3517) pattern3518 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons1441, cons1539, cons89) rule3518 = ReplacementRule(pattern3518, replacement3518) pattern3519 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons1441, cons1539, cons89) rule3519 = ReplacementRule(pattern3519, replacement3519) pattern3520 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons1441, cons1540, cons1538) rule3520 = ReplacementRule(pattern3520, replacement3520) pattern3521 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons1441, cons1540, cons1538) rule3521 = ReplacementRule(pattern3521, replacement3521) pattern3522 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons1441) rule3522 = ReplacementRule(pattern3522, replacement3522) pattern3523 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons1441) rule3523 = ReplacementRule(pattern3523, replacement3523) pattern3524 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(S(1)/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons4, cons1442, cons1517) rule3524 = ReplacementRule(pattern3524, replacement3524) pattern3525 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(S(1)/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons4, cons1442, cons1517) rule3525 = ReplacementRule(pattern3525, replacement3525) pattern3526 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**S(2)*cos(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons50, cons127, cons1442) rule3526 = ReplacementRule(pattern3526, replacement3526) pattern3527 = Pattern(Integral((a_ + WC('b', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**S(2)*tan(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons2, cons3, cons50, cons127, cons1442) rule3527 = ReplacementRule(pattern3527, replacement3527) pattern3528 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons29, cons50, cons127, cons19, cons1442, cons68) rule3528 = ReplacementRule(pattern3528, replacement3528) pattern3529 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons29, cons50, cons127, cons19, cons1442, cons68) rule3529 = ReplacementRule(pattern3529, replacement3529) pattern3530 = Pattern(Integral(S(1)/((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*cos(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons1442) rule3530 = ReplacementRule(pattern3530, replacement3530) pattern3531 = Pattern(Integral(S(1)/((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*sin(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons1442) rule3531 = ReplacementRule(pattern3531, replacement3531) pattern3532 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons1442, cons1541) rule3532 = ReplacementRule(pattern3532, replacement3532) pattern3533 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons1442, cons1541) rule3533 = ReplacementRule(pattern3533, replacement3533) pattern3534 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_/(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons1442, cons86) rule3534 = ReplacementRule(pattern3534, replacement3534) pattern3535 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_/(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons29, cons50, cons127, cons1442, cons86) rule3535 = ReplacementRule(pattern3535, replacement3535) pattern3536 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons1442, cons1526) rule3536 = ReplacementRule(pattern3536, replacement3536) pattern3537 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons1442, cons1526) rule3537 = ReplacementRule(pattern3537, replacement3537) pattern3538 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*tan(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, cons1441) rule3538 = ReplacementRule(pattern3538, replacement3538) pattern3539 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/tan(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, cons1441) rule3539 = ReplacementRule(pattern3539, replacement3539) pattern3540 = Pattern(Integral(S(1)/((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sqrt(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons29, cons50, cons127, cons1441) rule3540 = ReplacementRule(pattern3540, replacement3540) pattern3541 = Pattern(Integral(S(1)/((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)*sqrt(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons29, cons50, cons127, cons1441) rule3541 = ReplacementRule(pattern3541, replacement3541) pattern3542 = Pattern(Integral((WC('d', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons21) rule3542 = ReplacementRule(pattern3542, replacement3542) pattern3543 = Pattern(Integral((WC('d', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons21) rule3543 = ReplacementRule(pattern3543, replacement3543) pattern3544 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*sin(x_*WC('f', S(1)) + WC('e', S(0)))**m_, x_), cons2, cons3, cons50, cons127, cons4, cons1517) rule3544 = ReplacementRule(pattern3544, replacement3544) pattern3545 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*cos(x_*WC('f', S(1)) + WC('e', S(0)))**m_, x_), cons2, cons3, cons50, cons127, cons4, cons1517) rule3545 = ReplacementRule(pattern3545, replacement3545) pattern3546 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons50, cons127, cons1230, cons150) rule3546 = ReplacementRule(pattern3546, replacement3546) pattern3547 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons50, cons127, cons1230, cons150) rule3547 = ReplacementRule(pattern3547, replacement3547) pattern3548 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons50, cons127, cons1230, cons198, cons1542) rule3548 = ReplacementRule(pattern3548, replacement3548) pattern3549 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons50, cons127, cons1230, cons198, cons1542) rule3549 = ReplacementRule(pattern3549, replacement3549) pattern3550 = Pattern(Integral((WC('d', S(1))/sin(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons21) rule3550 = ReplacementRule(pattern3550, replacement3550) pattern3551 = Pattern(Integral((WC('d', S(1))/cos(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons29, cons50, cons127, cons19, cons4, cons21) rule3551 = ReplacementRule(pattern3551, replacement3551) pattern3552 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*sin(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1))*cos(x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons50, cons127, cons19, cons5, cons87) rule3552 = ReplacementRule(pattern3552, replacement3552) pattern3553 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', 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, cons5, cons87) rule3553 = ReplacementRule(pattern3553, replacement3553) pattern3554 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons72, cons1441, cons20, cons1543) rule3554 = ReplacementRule(pattern3554, replacement3554) pattern3555 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons72, cons1441, cons20, cons1543) rule3555 = ReplacementRule(pattern3555, replacement3555) pattern3556 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons72, cons1441) rule3556 = ReplacementRule(pattern3556, replacement3556) pattern3557 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons72, cons1441) rule3557 = ReplacementRule(pattern3557, replacement3557) pattern3558 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons72) rule3558 = ReplacementRule(pattern3558, replacement3558) pattern3559 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(c_ + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons72) rule3559 = ReplacementRule(pattern3559, replacement3559) pattern3560 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1414) rule3560 = ReplacementRule(pattern3560, replacement3560) pattern3561 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1414) rule3561 = ReplacementRule(pattern3561, replacement3561) pattern3562 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1441, cons33, cons269) rule3562 = ReplacementRule(pattern3562, replacement3562) pattern3563 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1441, cons33, cons269) rule3563 = ReplacementRule(pattern3563, replacement3563) pattern3564 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons73, cons1441, cons1305) rule3564 = ReplacementRule(pattern3564, replacement3564) pattern3565 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons73, cons1441, cons1305) rule3565 = ReplacementRule(pattern3565, replacement3565) pattern3566 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1442, cons33, cons170) rule3566 = ReplacementRule(pattern3566, replacement3566) pattern3567 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1442, cons33, cons170) rule3567 = ReplacementRule(pattern3567, replacement3567) pattern3568 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1442, cons33, cons96) rule3568 = ReplacementRule(pattern3568, replacement3568) pattern3569 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1442, cons33, cons96) rule3569 = ReplacementRule(pattern3569, replacement3569) pattern3570 = Pattern(Integral((c_ + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))/(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1442, cons1544) rule3570 = ReplacementRule(pattern3570, replacement3570) pattern3571 = Pattern(Integral((c_ + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))/(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1442, cons1544) rule3571 = ReplacementRule(pattern3571, replacement3571) pattern3572 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1442, cons1545) rule3572 = ReplacementRule(pattern3572, replacement3572) pattern3573 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1442, cons1545) rule3573 = ReplacementRule(pattern3573, replacement3573) pattern3574 = Pattern(Integral((c_ + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons3, cons8, cons29, cons50, cons127, cons1324) rule3574 = ReplacementRule(pattern3574, replacement3574) pattern3575 = Pattern(Integral((c_ + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons3, cons8, cons29, cons50, cons127, cons1324) rule3575 = ReplacementRule(pattern3575, replacement3575) pattern3576 = Pattern(Integral((c_ + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons3, cons8, cons29, cons50, cons127, cons1546) rule3576 = ReplacementRule(pattern3576, replacement3576) pattern3577 = Pattern(Integral((c_ + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons3, cons8, cons29, cons50, cons127, cons1546) rule3577 = ReplacementRule(pattern3577, replacement3577) pattern3578 = Pattern(Integral((c_ + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons3, cons8, cons29, cons50, cons127, cons1325, cons1547) rule3578 = ReplacementRule(pattern3578, replacement3578) pattern3579 = Pattern(Integral((c_ + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons3, cons8, cons29, cons50, cons127, cons1325, cons1547) rule3579 = ReplacementRule(pattern3579, replacement3579) pattern3580 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1442, cons1547, cons1548) rule3580 = ReplacementRule(pattern3580, replacement3580) pattern3581 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1442, cons1547, cons1548) rule3581 = ReplacementRule(pattern3581, replacement3581) pattern3582 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1442, cons1547, cons1549, cons1550) rule3582 = ReplacementRule(pattern3582, With3582) pattern3583 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1442, cons1547, cons1549, cons1550) rule3583 = ReplacementRule(pattern3583, With3583) pattern3584 = Pattern(Integral((c_ + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons73, cons1442, cons1546) rule3584 = ReplacementRule(pattern3584, replacement3584) pattern3585 = Pattern(Integral((c_ + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons73, cons1442, cons1546) rule3585 = ReplacementRule(pattern3585, replacement3585) pattern3586 = Pattern(Integral((WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons3, cons8, cons29, cons50, cons127, cons19, cons1547, cons79) rule3586 = ReplacementRule(pattern3586, replacement3586) pattern3587 = Pattern(Integral((WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons3, cons8, cons29, cons50, cons127, cons19, cons1547, cons79) rule3587 = ReplacementRule(pattern3587, replacement3587) pattern3588 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons73, cons1442, cons1547, cons21) rule3588 = ReplacementRule(pattern3588, replacement3588) pattern3589 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons73, cons1442, cons1547, cons21) rule3589 = ReplacementRule(pattern3589, replacement3589) pattern3590 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons33, cons34, cons1441) rule3590 = ReplacementRule(pattern3590, replacement3590) pattern3591 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons33, cons34, cons1441) rule3591 = ReplacementRule(pattern3591, replacement3591) pattern3592 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**S(2)/(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1442) rule3592 = ReplacementRule(pattern3592, replacement3592) pattern3593 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**S(2)/(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1442) rule3593 = ReplacementRule(pattern3593, replacement3593) pattern3594 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons33, cons96, cons1442) rule3594 = ReplacementRule(pattern3594, replacement3594) pattern3595 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons33, cons96, cons1442) rule3595 = ReplacementRule(pattern3595, replacement3595) pattern3596 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons73, cons1551, cons1552) rule3596 = ReplacementRule(pattern3596, replacement3596) pattern3597 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons73, cons1551, cons1552) rule3597 = ReplacementRule(pattern3597, replacement3597) pattern3598 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1441, cons1547) rule3598 = ReplacementRule(pattern3598, replacement3598) pattern3599 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1441, cons1547) rule3599 = ReplacementRule(pattern3599, replacement3599) pattern3600 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1441, cons1547, cons95, cons1553, cons1425) rule3600 = ReplacementRule(pattern3600, replacement3600) pattern3601 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1441, cons1547, cons95, cons1553, cons1425) rule3601 = ReplacementRule(pattern3601, replacement3601) pattern3602 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1441, cons1547, cons95, cons1553, cons1387) rule3602 = ReplacementRule(pattern3602, replacement3602) pattern3603 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1441, cons1547, cons95, cons1553, cons1387) rule3603 = ReplacementRule(pattern3603, replacement3603) pattern3604 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1441, cons1547, cons95, cons113, cons96) rule3604 = ReplacementRule(pattern3604, replacement3604) pattern3605 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1441, cons1547, cons95, cons113, cons96) rule3605 = ReplacementRule(pattern3605, replacement3605) pattern3606 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons73, cons1441, cons1547, cons157, cons274) rule3606 = ReplacementRule(pattern3606, replacement3606) pattern3607 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons73, cons1441, cons1547, cons157, cons274) rule3607 = ReplacementRule(pattern3607, replacement3607) pattern3608 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1441, cons1547, cons89, cons1327) rule3608 = ReplacementRule(pattern3608, replacement3608) pattern3609 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1441, cons1547, cons89, cons1327) rule3609 = ReplacementRule(pattern3609, replacement3609) pattern3610 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1441, cons1547, cons89, cons167) rule3610 = ReplacementRule(pattern3610, replacement3610) pattern3611 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1441, cons1547, cons89, cons167) rule3611 = ReplacementRule(pattern3611, replacement3611) pattern3612 = Pattern(Integral(S(1)/((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1441, cons1547) rule3612 = ReplacementRule(pattern3612, replacement3612) pattern3613 = Pattern(Integral(S(1)/((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1441, cons1547) rule3613 = ReplacementRule(pattern3613, replacement3613) pattern3614 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons73, cons1441, cons1547, cons1329) rule3614 = ReplacementRule(pattern3614, replacement3614) pattern3615 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons73, cons1441, cons1547, cons1329) rule3615 = ReplacementRule(pattern3615, replacement3615) pattern3616 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1441, cons1547, cons95, cons168, cons91, cons1336) rule3616 = ReplacementRule(pattern3616, replacement3616) pattern3617 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1441, cons1547, cons95, cons168, cons91, cons1336) rule3617 = ReplacementRule(pattern3617, replacement3617) pattern3618 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1441, cons1547) rule3618 = ReplacementRule(pattern3618, replacement3618) pattern3619 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1441, cons1547) rule3619 = ReplacementRule(pattern3619, replacement3619) pattern3620 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/sqrt(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1441, cons1547) rule3620 = ReplacementRule(pattern3620, replacement3620) pattern3621 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/sqrt(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1441, cons1547) rule3621 = ReplacementRule(pattern3621, replacement3621) pattern3622 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons73, cons1441, cons1547, cons517, cons168, cons1521, cons1336) rule3622 = ReplacementRule(pattern3622, replacement3622) pattern3623 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons73, cons1441, cons1547, cons517, cons168, cons1521, cons1336) rule3623 = ReplacementRule(pattern3623, replacement3623) pattern3624 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sqrt(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1441, cons1547, cons33, cons269, cons1554) rule3624 = ReplacementRule(pattern3624, replacement3624) pattern3625 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*sqrt(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1441, cons1547, cons33, cons269, cons1554) rule3625 = ReplacementRule(pattern3625, replacement3625) pattern3626 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1441, cons1547, cons95, cons269, cons167, cons1336) rule3626 = ReplacementRule(pattern3626, replacement3626) pattern3627 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1441, cons1547, cons95, cons269, cons167, cons1336) rule3627 = ReplacementRule(pattern3627, replacement3627) pattern3628 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons73, cons1441, cons1547, cons33, cons269, cons1336) rule3628 = ReplacementRule(pattern3628, replacement3628) pattern3629 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons73, cons1441, cons1547, cons33, cons269, cons1336) rule3629 = ReplacementRule(pattern3629, replacement3629) pattern3630 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons73, cons1441, cons1547, cons89, cons167, cons1521, cons1555) rule3630 = ReplacementRule(pattern3630, replacement3630) pattern3631 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons73, cons1441, cons1547, cons89, cons167, cons1521, cons1555) rule3631 = ReplacementRule(pattern3631, replacement3631) pattern3632 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons73, cons1441, cons1547, cons89, cons91, cons1555) rule3632 = ReplacementRule(pattern3632, replacement3632) pattern3633 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons73, cons1441, cons1547, cons89, cons91, cons1555) rule3633 = ReplacementRule(pattern3633, replacement3633) pattern3634 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_/(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons73, cons1441, cons1547) rule3634 = ReplacementRule(pattern3634, replacement3634) pattern3635 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_/(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons73, cons1441, cons1547) rule3635 = ReplacementRule(pattern3635, replacement3635) pattern3636 = Pattern(Integral(sqrt(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1441, cons1547) rule3636 = ReplacementRule(pattern3636, replacement3636) pattern3637 = Pattern(Integral(sqrt(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1441, cons1547) rule3637 = ReplacementRule(pattern3637, replacement3637) pattern3638 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons73, cons1441, cons1547) rule3638 = ReplacementRule(pattern3638, replacement3638) pattern3639 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons73, cons1441, cons1547) rule3639 = ReplacementRule(pattern3639, replacement3639) pattern3640 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1442, cons1547, cons95, cons1335, cons91, cons517) rule3640 = ReplacementRule(pattern3640, replacement3640) pattern3641 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1442, cons1547, cons95, cons1335, cons91, cons517) rule3641 = ReplacementRule(pattern3641, replacement3641) pattern3642 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons73, cons1442, cons1547, cons517, cons1335, cons1556, cons1557) rule3642 = ReplacementRule(pattern3642, replacement3642) pattern3643 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons73, cons1442, cons1547, cons517, cons1335, cons1556, cons1557) rule3643 = ReplacementRule(pattern3643, replacement3643) pattern3644 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1442, cons1547, cons95, cons96, cons1338, cons517) rule3644 = ReplacementRule(pattern3644, replacement3644) pattern3645 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1442, cons1547, cons95, cons96, cons1338, cons517) rule3645 = ReplacementRule(pattern3645, replacement3645) pattern3646 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1442, cons1547, cons95, cons96, cons90, cons517) rule3646 = ReplacementRule(pattern3646, replacement3646) pattern3647 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1442, cons1547, cons95, cons96, cons90, cons517) rule3647 = ReplacementRule(pattern3647, replacement3647) pattern3648 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons73, cons1442, cons1547, cons517, cons96, cons1558, cons1559) rule3648 = ReplacementRule(pattern3648, replacement3648) pattern3649 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons73, cons1442, cons1547, cons517, cons96, cons1558, cons1559) rule3649 = ReplacementRule(pattern3649, replacement3649) pattern3650 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1442, cons1547, cons95, cons168, cons90, cons810) rule3650 = ReplacementRule(pattern3650, replacement3650) pattern3651 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1442, cons1547, cons95, cons168, cons90, cons810) rule3651 = ReplacementRule(pattern3651, replacement3651) pattern3652 = Pattern(Integral(S(1)/((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1442, cons1547) rule3652 = ReplacementRule(pattern3652, replacement3652) pattern3653 = Pattern(Integral(S(1)/((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1442, cons1547) rule3653 = ReplacementRule(pattern3653, replacement3653) pattern3654 = Pattern(Integral(sqrt(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons73, cons1442, cons1547) rule3654 = ReplacementRule(pattern3654, replacement3654) pattern3655 = Pattern(Integral(sqrt(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons73, cons1442, cons1547) rule3655 = ReplacementRule(pattern3655, replacement3655) pattern3656 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1442, cons1547) rule3656 = ReplacementRule(pattern3656, replacement3656) pattern3657 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**(S(3)/2)/(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons73, cons1442, cons1547) rule3657 = ReplacementRule(pattern3657, replacement3657) pattern3658 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_/(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons73, cons1442, cons1547, cons21) rule3658 = ReplacementRule(pattern3658, replacement3658) pattern3659 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_/(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons73, cons1442, cons1547, cons21) rule3659 = ReplacementRule(pattern3659, replacement3659) pattern3660 = Pattern(Integral((c_ + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons73, cons1442, cons1547) rule3660 = ReplacementRule(pattern3660, replacement3660) pattern3661 = Pattern(Integral((c_ + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons73, cons1442, cons1547) rule3661 = ReplacementRule(pattern3661, replacement3661) pattern3662 = Pattern(Integral((WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons29, cons50, cons127, cons4, cons25, cons20) rule3662 = ReplacementRule(pattern3662, replacement3662) pattern3663 = Pattern(Integral((WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons29, cons50, cons127, cons4, cons25, cons20) rule3663 = ReplacementRule(pattern3663, replacement3663) pattern3664 = Pattern(Integral(((WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**p_*WC('c', S(1)))**n_*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons5, cons25, cons21) rule3664 = ReplacementRule(pattern3664, replacement3664) pattern3665 = Pattern(Integral(((WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**p_*WC('c', S(1)))**n_*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons5, cons25, cons21) rule3665 = ReplacementRule(pattern3665, replacement3665) pattern3666 = 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))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons382) rule3666 = ReplacementRule(pattern3666, replacement3666) pattern3667 = 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))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons382) rule3667 = ReplacementRule(pattern3667, replacement3667) pattern3668 = Pattern(Integral((WC('g', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(c_ + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons5, cons149, cons20, cons87) rule3668 = ReplacementRule(pattern3668, replacement3668) pattern3669 = Pattern(Integral((WC('g', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(c_ + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons5, cons149, cons20, cons87) rule3669 = ReplacementRule(pattern3669, replacement3669) pattern3670 = Pattern(Integral((WC('g', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**q_)**p_*(c_ + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons52, cons149, cons1419) rule3670 = ReplacementRule(pattern3670, replacement3670) pattern3671 = Pattern(Integral(((S(1)/tan(x_*WC('f', S(1)) + WC('e', S(0))))**q_*WC('g', S(1)))**p_*(c_ + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons52, cons149, cons1419) rule3671 = ReplacementRule(pattern3671, replacement3671) pattern3672 = 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))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons5, cons87) rule3672 = ReplacementRule(pattern3672, replacement3672) pattern3673 = 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))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons5, cons87) rule3673 = ReplacementRule(pattern3673, replacement3673) pattern3674 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*tan(x_*WC('f', S(1)) + WC('e', S(0)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons25, cons20, cons40) rule3674 = ReplacementRule(pattern3674, replacement3674) pattern3675 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(S(1)/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons25, cons20, cons40) rule3675 = ReplacementRule(pattern3675, replacement3675) pattern3676 = Pattern(Integral((WC('g', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons4, cons5, cons25, cons20, cons149) rule3676 = ReplacementRule(pattern3676, replacement3676) pattern3677 = Pattern(Integral((WC('g', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**p_*(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons4, cons5, cons25, cons20, cons149) rule3677 = ReplacementRule(pattern3677, replacement3677) pattern3678 = 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))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons25, cons21) rule3678 = ReplacementRule(pattern3678, replacement3678) pattern3679 = 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))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(c_ + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons210, cons19, cons4, cons5, cons25, cons21) rule3679 = ReplacementRule(pattern3679, replacement3679) pattern3680 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons4, cons72, cons1441) rule3680 = ReplacementRule(pattern3680, replacement3680) pattern3681 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(c_ + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons4, cons72, cons1441) rule3681 = ReplacementRule(pattern3681, replacement3681) pattern3682 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73) rule3682 = ReplacementRule(pattern3682, replacement3682) pattern3683 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73) rule3683 = ReplacementRule(pattern3683, replacement3683) pattern3684 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons33, cons96, cons1441) rule3684 = ReplacementRule(pattern3684, replacement3684) pattern3685 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons33, cons96, cons1441) rule3685 = ReplacementRule(pattern3685, replacement3685) pattern3686 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons33, cons96, cons1442) rule3686 = ReplacementRule(pattern3686, replacement3686) pattern3687 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons33, cons96, cons1442) rule3687 = ReplacementRule(pattern3687, replacement3687) pattern3688 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons73, cons1551) rule3688 = ReplacementRule(pattern3688, replacement3688) pattern3689 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons73, cons1551) rule3689 = ReplacementRule(pattern3689, replacement3689) pattern3690 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1441, cons95, cons168, cons91) rule3690 = ReplacementRule(pattern3690, replacement3690) pattern3691 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1441, cons95, cons168, cons91) rule3691 = ReplacementRule(pattern3691, replacement3691) pattern3692 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons4, cons73, cons1441, cons33, cons168, cons348) rule3692 = ReplacementRule(pattern3692, replacement3692) pattern3693 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons4, cons73, cons1441, cons33, cons168, cons348) rule3693 = ReplacementRule(pattern3693, replacement3693) pattern3694 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1441, cons95, cons269, cons90) rule3694 = ReplacementRule(pattern3694, replacement3694) pattern3695 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1441, cons95, cons269, cons90) rule3695 = ReplacementRule(pattern3695, replacement3695) pattern3696 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons4, cons73, cons1441, cons33, cons269, cons1329) rule3696 = ReplacementRule(pattern3696, replacement3696) pattern3697 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons4, cons73, cons1441, cons33, cons269, cons1329) rule3697 = ReplacementRule(pattern3697, replacement3697) pattern3698 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons73, cons1441, cons89, cons90) rule3698 = ReplacementRule(pattern3698, replacement3698) pattern3699 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons73, cons1441, cons89, cons90) rule3699 = ReplacementRule(pattern3699, replacement3699) pattern3700 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons73, cons1441, cons89, cons91) rule3700 = ReplacementRule(pattern3700, replacement3700) pattern3701 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons73, cons1441, cons89, cons91) rule3701 = ReplacementRule(pattern3701, replacement3701) pattern3702 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons4, cons73, cons1441, cons1420) rule3702 = ReplacementRule(pattern3702, replacement3702) pattern3703 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons4, cons73, cons1441, cons1420) rule3703 = ReplacementRule(pattern3703, replacement3703) pattern3704 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons73, cons1441, cons1428) rule3704 = ReplacementRule(pattern3704, replacement3704) pattern3705 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons73, cons1441, cons1428) rule3705 = ReplacementRule(pattern3705, replacement3705) pattern3706 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons4, cons73, cons1441, cons1428) rule3706 = ReplacementRule(pattern3706, replacement3706) pattern3707 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons4, cons73, cons1441, cons1428) rule3707 = ReplacementRule(pattern3707, replacement3707) pattern3708 = Pattern(Integral((A_ + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons4, cons73, cons1442, cons21, cons25, cons1515, cons1560) rule3708 = ReplacementRule(pattern3708, replacement3708) pattern3709 = Pattern(Integral((A_ + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons4, cons73, cons1442, cons21, cons25, cons1515, cons1560) rule3709 = ReplacementRule(pattern3709, replacement3709) pattern3710 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons4, cons73, cons1442, cons21, cons25, cons1515, cons1561) rule3710 = ReplacementRule(pattern3710, replacement3710) pattern3711 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons4, cons73, cons1442, cons21, cons25, cons1515, cons1561) rule3711 = ReplacementRule(pattern3711, replacement3711) pattern3712 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1442, cons1547, cons95, cons168, cons91, cons1336) rule3712 = ReplacementRule(pattern3712, replacement3712) pattern3713 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1442, cons1547, cons95, cons168, cons91, cons1336) rule3713 = ReplacementRule(pattern3713, replacement3713) pattern3714 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons4, cons73, cons1442, cons1547, cons33, cons168, cons1336, cons1562) rule3714 = ReplacementRule(pattern3714, replacement3714) pattern3715 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons4, cons73, cons1442, cons1547, cons33, cons168, cons1336, cons1562) rule3715 = ReplacementRule(pattern3715, replacement3715) pattern3716 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1442, cons1547, cons95, cons96, cons1327, cons1336) rule3716 = ReplacementRule(pattern3716, replacement3716) pattern3717 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1442, cons1547, cons95, cons96, cons1327, cons1336) rule3717 = ReplacementRule(pattern3717, replacement3717) pattern3718 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons4, cons73, cons1442, cons1547, cons33, cons96, cons1336, cons1559) rule3718 = ReplacementRule(pattern3718, replacement3718) pattern3719 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons4, cons73, cons1442, cons1547, cons33, cons96, cons1336, cons1559) rule3719 = ReplacementRule(pattern3719, replacement3719) pattern3720 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1442, cons1547, cons95, cons1258, cons1327) rule3720 = ReplacementRule(pattern3720, replacement3720) pattern3721 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1442, cons1547, cons95, cons1258, cons1327) rule3721 = ReplacementRule(pattern3721, replacement3721) pattern3722 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))/((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1442, cons1547) rule3722 = ReplacementRule(pattern3722, replacement3722) pattern3723 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))/((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1442, cons1547) rule3723 = ReplacementRule(pattern3723, replacement3723) pattern3724 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1442, cons1547) rule3724 = ReplacementRule(pattern3724, replacement3724) pattern3725 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))/(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1442, cons1547) rule3725 = ReplacementRule(pattern3725, replacement3725) pattern3726 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons4, cons73, cons1442, cons1547) rule3726 = ReplacementRule(pattern3726, replacement3726) pattern3727 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons4, cons73, cons1442, cons1547) rule3727 = ReplacementRule(pattern3727, replacement3727) pattern3728 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1442, cons1547) rule3728 = ReplacementRule(pattern3728, replacement3728) pattern3729 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*sqrt(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))/sqrt(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons73, cons1442, cons1547) rule3729 = ReplacementRule(pattern3729, replacement3729) pattern3730 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons4, cons73, cons1442, cons1560) rule3730 = ReplacementRule(pattern3730, replacement3730) pattern3731 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons4, cons73, cons1442, cons1560) rule3731 = ReplacementRule(pattern3731, replacement3731) pattern3732 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons4, cons73, cons1442, cons1561) rule3732 = ReplacementRule(pattern3732, replacement3732) pattern3733 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons19, cons4, cons73, cons1442, cons1561) rule3733 = ReplacementRule(pattern3733, replacement3733) pattern3734 = Pattern(Integral((A_ + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons50, cons127, cons36, cons38, cons19, cons1563) rule3734 = ReplacementRule(pattern3734, replacement3734) pattern3735 = Pattern(Integral((A_ + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons50, cons127, cons36, cons38, cons19, cons1563) rule3735 = ReplacementRule(pattern3735, replacement3735) pattern3736 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons50, cons127, cons36, cons37, cons38, cons19, cons35) rule3736 = ReplacementRule(pattern3736, replacement3736) pattern3737 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons50, cons127, cons36, cons37, cons38, cons19, cons35) rule3737 = ReplacementRule(pattern3737, replacement3737) pattern3738 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons50, cons127, cons36, cons38, cons19, cons1435) rule3738 = ReplacementRule(pattern3738, replacement3738) pattern3739 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons50, cons127, cons36, cons38, cons19, cons1435) rule3739 = ReplacementRule(pattern3739, replacement3739) pattern3740 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons50, cons127, cons36, cons37, cons38, cons1564, cons33, cons34, cons1441) rule3740 = ReplacementRule(pattern3740, replacement3740) pattern3741 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons50, cons127, cons36, cons37, cons38, cons1564, cons33, cons34, cons1441) rule3741 = ReplacementRule(pattern3741, replacement3741) pattern3742 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons50, cons127, cons36, cons38, cons1565, cons33, cons34, cons1441) rule3742 = ReplacementRule(pattern3742, replacement3742) pattern3743 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons50, cons127, cons36, cons38, cons1565, cons33, cons34, cons1441) rule3743 = ReplacementRule(pattern3743, replacement3743) pattern3744 = Pattern(Integral((A_ + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons36, cons37, cons38, cons1442, cons1566) rule3744 = ReplacementRule(pattern3744, replacement3744) pattern3745 = Pattern(Integral((A_ + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons36, cons37, cons38, cons1442, cons1566) rule3745 = ReplacementRule(pattern3745, replacement3745) pattern3746 = Pattern(Integral((A_ + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/tan(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons50, cons127, cons36, cons37, cons38, cons1567) rule3746 = ReplacementRule(pattern3746, replacement3746) pattern3747 = Pattern(Integral((A_ + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*tan(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons50, cons127, cons36, cons37, cons38, cons1567) rule3747 = ReplacementRule(pattern3747, replacement3747) pattern3748 = Pattern(Integral((A_ + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/tan(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons50, cons127, cons36, cons38, cons1567) rule3748 = ReplacementRule(pattern3748, replacement3748) pattern3749 = Pattern(Integral((A_ + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*tan(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons50, cons127, cons36, cons38, cons1567) rule3749 = ReplacementRule(pattern3749, replacement3749) pattern3750 = Pattern(Integral((A_ + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons36, cons37, cons38, cons1564, cons1442, cons1568) rule3750 = ReplacementRule(pattern3750, replacement3750) pattern3751 = Pattern(Integral((A_ + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons36, cons37, cons38, cons1564, cons1442, cons1568) rule3751 = ReplacementRule(pattern3751, replacement3751) pattern3752 = Pattern(Integral((A_ + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons36, cons38, cons1565, cons1442, cons1567) rule3752 = ReplacementRule(pattern3752, replacement3752) pattern3753 = Pattern(Integral((A_ + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons50, cons127, cons36, cons38, cons1565, cons1442, cons1567) rule3753 = ReplacementRule(pattern3753, replacement3753) pattern3754 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons50, cons127, cons36, cons37, cons38, cons1564, cons33, cons96, cons1442) rule3754 = ReplacementRule(pattern3754, replacement3754) pattern3755 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons50, cons127, cons36, cons37, cons38, cons1564, cons33, cons96, cons1442) rule3755 = ReplacementRule(pattern3755, replacement3755) pattern3756 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons36, cons38, cons1565, cons33, cons96, cons1442) rule3756 = ReplacementRule(pattern3756, replacement3756) pattern3757 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_, x_), cons2, cons3, cons50, cons127, cons36, cons38, cons1565, cons33, cons96, cons1442) rule3757 = ReplacementRule(pattern3757, replacement3757) pattern3758 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons50, cons127, cons36, cons37, cons38, cons19, cons1564, cons1551) rule3758 = ReplacementRule(pattern3758, replacement3758) pattern3759 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons50, cons127, cons36, cons37, cons38, cons19, cons1564, cons1551) rule3759 = ReplacementRule(pattern3759, replacement3759) pattern3760 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons50, cons127, cons36, cons38, cons19, cons1565, cons1551) rule3760 = ReplacementRule(pattern3760, replacement3760) pattern3761 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons50, cons127, cons36, cons38, cons19, cons1565, cons1551) rule3761 = ReplacementRule(pattern3761, replacement3761) pattern3762 = Pattern(Integral((A_ + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons19, cons4, cons1563) rule3762 = ReplacementRule(pattern3762, replacement3762) pattern3763 = Pattern(Integral((A_ + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons19, cons4, cons1563) rule3763 = ReplacementRule(pattern3763, replacement3763) pattern3764 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons73, cons1547, cons89, cons91) rule3764 = ReplacementRule(pattern3764, replacement3764) pattern3765 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons73, cons1547, cons89, cons91) rule3765 = ReplacementRule(pattern3765, replacement3765) pattern3766 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons73, cons1547, cons89, cons91) rule3766 = ReplacementRule(pattern3766, replacement3766) pattern3767 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons73, cons1547, cons89, cons91) rule3767 = ReplacementRule(pattern3767, replacement3767) pattern3768 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons4, cons73, cons1547, cons348) rule3768 = ReplacementRule(pattern3768, replacement3768) pattern3769 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons4, cons73, cons1547, cons348) rule3769 = ReplacementRule(pattern3769, replacement3769) pattern3770 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons4, cons73, cons1547, cons348) rule3770 = ReplacementRule(pattern3770, replacement3770) pattern3771 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('A', S(0)) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons4, cons73, cons1547, cons348) rule3771 = ReplacementRule(pattern3771, replacement3771) pattern3772 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons4, cons73, cons1441, cons1569) rule3772 = ReplacementRule(pattern3772, replacement3772) pattern3773 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons4, cons73, cons1441, cons1569) rule3773 = ReplacementRule(pattern3773, replacement3773) pattern3774 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons4, cons73, cons1441, cons1569) rule3774 = ReplacementRule(pattern3774, replacement3774) pattern3775 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('A', S(0)) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons4, cons73, cons1441, cons1569) rule3775 = ReplacementRule(pattern3775, replacement3775) pattern3776 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons19, cons73, cons1441, cons1305, cons89, cons91, cons1547) rule3776 = ReplacementRule(pattern3776, replacement3776) pattern3777 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons19, cons73, cons1441, cons1305, cons89, cons91, cons1547) rule3777 = ReplacementRule(pattern3777, replacement3777) pattern3778 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons19, cons73, cons1441, cons1305, cons89, cons91, cons1547) rule3778 = ReplacementRule(pattern3778, replacement3778) pattern3779 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons19, cons73, cons1441, cons1305, cons89, cons91, cons1547) rule3779 = ReplacementRule(pattern3779, replacement3779) pattern3780 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons19, cons4, cons73, cons1441, cons1305, cons685) rule3780 = ReplacementRule(pattern3780, replacement3780) pattern3781 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1))*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons19, cons4, cons73, cons1441, cons1305, cons685) rule3781 = ReplacementRule(pattern3781, replacement3781) pattern3782 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))*tan(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, cons1441, cons1305, cons685) rule3782 = ReplacementRule(pattern3782, replacement3782) pattern3783 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('A', S(0)) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))/tan(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, cons1441, cons1305, cons685) rule3783 = ReplacementRule(pattern3783, replacement3783) pattern3784 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons73, cons1442, cons1547, cons95, cons170, cons91) rule3784 = ReplacementRule(pattern3784, replacement3784) pattern3785 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons73, cons1442, cons1547, cons95, cons170, cons91) rule3785 = ReplacementRule(pattern3785, replacement3785) pattern3786 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons73, cons1442, cons1547, cons95, cons170, cons91) rule3786 = ReplacementRule(pattern3786, replacement3786) pattern3787 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons73, cons1442, cons1547, cons95, cons170, cons91) rule3787 = ReplacementRule(pattern3787, replacement3787) pattern3788 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons4, cons73, cons1442, cons1547, cons33, cons170, cons1570) rule3788 = ReplacementRule(pattern3788, replacement3788) pattern3789 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons4, cons73, cons1442, cons1547, cons33, cons170, cons1570) rule3789 = ReplacementRule(pattern3789, replacement3789) pattern3790 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons4, cons73, cons1442, cons1547, cons33, cons170, cons1570) rule3790 = ReplacementRule(pattern3790, replacement3790) pattern3791 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('m', S(1))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons4, cons73, cons1442, cons1547, cons33, cons170, cons1570) rule3791 = ReplacementRule(pattern3791, replacement3791) pattern3792 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons4, cons73, cons1442, cons1547, cons33, cons96, cons1559) rule3792 = ReplacementRule(pattern3792, replacement3792) pattern3793 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons4, cons73, cons1442, cons1547, cons33, cons96, cons1559) rule3793 = ReplacementRule(pattern3793, replacement3793) pattern3794 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons4, cons73, cons1442, cons1547, cons33, cons96, cons1559) rule3794 = ReplacementRule(pattern3794, replacement3794) pattern3795 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons4, cons73, cons1442, cons1547, cons33, cons96, cons1559) rule3795 = ReplacementRule(pattern3795, replacement3795) pattern3796 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons73, cons1442, cons1547) rule3796 = ReplacementRule(pattern3796, replacement3796) pattern3797 = Pattern(Integral((WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons73, cons1442, cons1547) rule3797 = ReplacementRule(pattern3797, replacement3797) pattern3798 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons73, cons1442, cons1547) rule3798 = ReplacementRule(pattern3798, replacement3798) pattern3799 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons73, cons1442, cons1547) rule3799 = ReplacementRule(pattern3799, replacement3799) pattern3800 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons4, cons73, cons1442, cons1547, cons1329, cons1571) rule3800 = ReplacementRule(pattern3800, replacement3800) pattern3801 = Pattern(Integral((WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))/(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons4, cons73, cons1442, cons1547, cons1329, cons1571) rule3801 = ReplacementRule(pattern3801, replacement3801) pattern3802 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons4, cons73, cons1442, cons1547, cons1329, cons1571) rule3802 = ReplacementRule(pattern3802, replacement3802) pattern3803 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_/(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons4, cons73, cons1442, cons1547, cons1329, cons1571) rule3803 = ReplacementRule(pattern3803, replacement3803) pattern3804 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons19, cons4, cons73, cons1442, cons1547) rule3804 = ReplacementRule(pattern3804, replacement3804) pattern3805 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(WC('A', S(0)) + WC('B', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons37, cons38, cons19, cons4, cons73, cons1442, cons1547) rule3805 = ReplacementRule(pattern3805, replacement3805) pattern3806 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons19, cons4, cons73, cons1442, cons1547) rule3806 = ReplacementRule(pattern3806, replacement3806) pattern3807 = Pattern(Integral((WC('A', S(0)) + WC('C', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))**S(2))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**m_*(WC('c', S(0)) + WC('d', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons36, cons38, cons19, cons4, cons73, cons1442, cons1547) rule3807 = ReplacementRule(pattern3807, replacement3807) pattern3808 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons1494) rule3808 = ReplacementRule(pattern3808, replacement3808) pattern3809 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons1494) rule3809 = ReplacementRule(pattern3809, replacement3809) pattern3810 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons1458) rule3810 = ReplacementRule(pattern3810, replacement3810) pattern3811 = Pattern(Integral(S(1)/(a_ + WC('b', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons1458) rule3811 = ReplacementRule(pattern3811, replacement3811) pattern3812 = Pattern(Integral((a_ + (WC('e', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons4, cons5, cons1572) rule3812 = ReplacementRule(pattern3812, replacement3812) pattern3813 = Pattern(Integral((a_ + (WC('e', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons4, cons5, cons1572) rule3813 = ReplacementRule(pattern3813, replacement3813) pattern3814 = Pattern(Integral((a_ + (WC('e', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**p_*sin(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons4, cons5, cons1517) rule3814 = ReplacementRule(pattern3814, With3814) pattern3815 = Pattern(Integral((a_ + (WC('e', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**p_*cos(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons4, cons5, cons1517) rule3815 = ReplacementRule(pattern3815, With3815) pattern3816 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**WC('p', S(1))*sin(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons1483, cons1481, cons40) rule3816 = ReplacementRule(pattern3816, With3816) pattern3817 = Pattern(Integral((a_ + (S(1)/tan(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, cons1483, cons1481, cons40) rule3817 = ReplacementRule(pattern3817, With3817) pattern3818 = Pattern(Integral((a_ + (WC('e', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**WC('p', S(1))*(S(1)/cos(x_*WC('d', S(1)) + WC('c', S(0))))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons4, cons5, cons1517) rule3818 = ReplacementRule(pattern3818, With3818) pattern3819 = Pattern(Integral((a_ + (WC('e', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**WC('p', S(1))*(S(1)/sin(x_*WC('d', S(1)) + WC('c', S(0))))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons4, cons5, cons1517) rule3819 = ReplacementRule(pattern3819, With3819) pattern3820 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0)))**n_)**WC('p', S(1))*(S(1)/cos(x_*WC('d', S(1)) + WC('c', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons1230, cons745, cons40) rule3820 = ReplacementRule(pattern3820, With3820) pattern3821 = Pattern(Integral((a_ + (S(1)/tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**WC('p', S(1))*(S(1)/sin(x_*WC('d', S(1)) + WC('c', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons1230, cons745, cons40) rule3821 = ReplacementRule(pattern3821, With3821) pattern3822 = Pattern(Integral((a_ + (WC('e', S(1))*tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**p_*tan(x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons5, cons1499) rule3822 = ReplacementRule(pattern3822, replacement3822) pattern3823 = Pattern(Integral((a_ + (WC('e', S(1))/tan(x_*WC('d', S(1)) + WC('c', S(0))))**n_*WC('b', S(1)))**p_*(S(1)/tan(x_*WC('d', S(1)) + WC('c', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons5, cons1499) rule3823 = ReplacementRule(pattern3823, replacement3823) pattern3824 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*tan(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) rule3824 = ReplacementRule(pattern3824, replacement3824) pattern3825 = Pattern(Integral((a_ + (S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons4, cons48, cons47, cons40) rule3825 = ReplacementRule(pattern3825, replacement3825) pattern3826 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons4, cons48, cons47, cons149) rule3826 = ReplacementRule(pattern3826, replacement3826) pattern3827 = Pattern(Integral((a_ + (S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)))**p_, x_), cons2, cons3, cons8, cons29, cons50, cons4, cons48, cons47, cons149) rule3827 = ReplacementRule(pattern3827, replacement3827) pattern3828 = Pattern(Integral(S(1)/(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1))), x_), cons2, cons3, cons8, cons29, cons50, cons4, cons48, cons228) rule3828 = ReplacementRule(pattern3828, With3828) pattern3829 = Pattern(Integral(S(1)/((S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons50, cons4, cons48, cons228) rule3829 = ReplacementRule(pattern3829, With3829) pattern3830 = Pattern(Integral(((WC('f', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (WC('f', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**p_*sin(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons5, cons48, cons1517) rule3830 = ReplacementRule(pattern3830, replacement3830) pattern3831 = Pattern(Integral(((WC('f', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (WC('f', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**p_*cos(x_*WC('e', S(1)) + WC('d', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons5, cons48, cons1517) rule3831 = ReplacementRule(pattern3831, replacement3831) pattern3832 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*tan(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, cons1483, cons1481, cons40) rule3832 = ReplacementRule(pattern3832, With3832) pattern3833 = Pattern(Integral(((S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)) + WC('c', S(1))*tan(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, cons1483, cons1481, cons40) rule3833 = ReplacementRule(pattern3833, With3833) pattern3834 = Pattern(Integral(((WC('f', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (WC('f', S(1))*tan(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)))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons5, cons48, cons1482) rule3834 = ReplacementRule(pattern3834, replacement3834) pattern3835 = Pattern(Integral(((WC('f', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (WC('f', S(1))/tan(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)))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons5, cons48, cons1482) rule3835 = ReplacementRule(pattern3835, replacement3835) pattern3836 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*tan(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, cons48, cons1483, cons1481, cons40) rule3836 = ReplacementRule(pattern3836, With3836) pattern3837 = Pattern(Integral(((S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (S(1)/tan(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)))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons48, cons1483, cons1481, cons40) rule3837 = ReplacementRule(pattern3837, With3837) pattern3838 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**p_*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons48, cons47, cons40) rule3838 = ReplacementRule(pattern3838, replacement3838) pattern3839 = Pattern(Integral(((S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**p_*(S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons48, cons47, cons40) rule3839 = ReplacementRule(pattern3839, replacement3839) pattern3840 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n2', S(1)))**p_*tan(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) rule3840 = ReplacementRule(pattern3840, replacement3840) pattern3841 = Pattern(Integral(((S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**p_*(S(1)/tan(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) rule3841 = ReplacementRule(pattern3841, replacement3841) pattern3842 = Pattern(Integral(((WC('f', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (WC('f', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**p_*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons5, cons48, cons228) rule3842 = ReplacementRule(pattern3842, replacement3842) pattern3843 = Pattern(Integral(((WC('f', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (WC('f', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**p_*(S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons5, cons48, cons228) rule3843 = ReplacementRule(pattern3843, replacement3843) pattern3844 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*tan(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, cons19, cons4, cons48, cons47, cons40) rule3844 = ReplacementRule(pattern3844, replacement3844) pattern3845 = Pattern(Integral(((S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (S(1)/tan(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))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons48, cons47, cons40) rule3845 = ReplacementRule(pattern3845, replacement3845) pattern3846 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**WC('n', S(1)) + WC('c', S(1))*tan(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))))**WC('m', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons5, cons48, cons47, cons149) rule3846 = ReplacementRule(pattern3846, replacement3846) pattern3847 = Pattern(Integral(((S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n', S(1))*WC('b', S(1)) + (S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**WC('n2', S(1))*WC('c', S(1)) + WC('a', S(0)))**p_*tan(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) rule3847 = ReplacementRule(pattern3847, replacement3847) pattern3848 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**n_ + WC('c', S(1))*tan(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, cons5, cons48, cons228, cons1481) rule3848 = ReplacementRule(pattern3848, With3848) pattern3849 = Pattern(Integral(((S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**n2_*WC('c', S(1)) + (S(1)/tan(x_*WC('e', S(1)) + WC('d', S(0))))**n_*WC('b', S(1)) + WC('a', S(0)))**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, cons5, cons48, cons228, cons1481) rule3849 = ReplacementRule(pattern3849, With3849) pattern3850 = Pattern(Integral((A_ + WC('B', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0))))*(a_ + WC('b', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons47, cons87) rule3850 = ReplacementRule(pattern3850, replacement3850) pattern3851 = Pattern(Integral((A_ + WC('B', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0))))*(a_ + WC('b', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons47, cons87) rule3851 = ReplacementRule(pattern3851, replacement3851) pattern3852 = Pattern(Integral((A_ + WC('B', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0))))*(a_ + WC('b', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons47, cons25) rule3852 = ReplacementRule(pattern3852, replacement3852) pattern3853 = Pattern(Integral((A_ + WC('B', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0))))*(a_ + WC('b', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons47, cons25) rule3853 = ReplacementRule(pattern3853, replacement3853) pattern3854 = Pattern(Integral((A_ + WC('B', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0))))/(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons228) rule3854 = ReplacementRule(pattern3854, With3854) pattern3855 = Pattern(Integral((A_ + WC('B', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0))))/(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons228) rule3855 = ReplacementRule(pattern3855, With3855) pattern3856 = Pattern(Integral((A_ + WC('B', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0))))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))*tan(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons228, cons87) rule3856 = ReplacementRule(pattern3856, replacement3856) pattern3857 = Pattern(Integral((A_ + WC('B', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0))))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0))) + WC('c', S(1))/tan(x_*WC('e', S(1)) + WC('d', S(0)))**S(2))**n_, x_), cons2, cons3, cons8, cons29, cons50, cons36, cons37, cons228, cons87) rule3857 = ReplacementRule(pattern3857, replacement3857) pattern3858 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons8, cons29, cons50, cons127, cons64) rule3858 = ReplacementRule(pattern3858, replacement3858) pattern3859 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))), x_), cons8, cons29, cons50, cons127, cons64) rule3859 = ReplacementRule(pattern3859, replacement3859) pattern3860 = Pattern(Integral((WC('b', S(1))*tan(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, cons167, cons170) rule3860 = ReplacementRule(pattern3860, replacement3860) pattern3861 = Pattern(Integral((WC('b', S(1))/tan(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, cons167, cons170) rule3861 = ReplacementRule(pattern3861, replacement3861) pattern3862 = Pattern(Integral((WC('b', S(1))*tan(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, cons170) rule3862 = ReplacementRule(pattern3862, replacement3862) pattern3863 = Pattern(Integral((WC('b', S(1))/tan(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, cons170) rule3863 = ReplacementRule(pattern3863, replacement3863) pattern3864 = Pattern(Integral((a_ + WC('b', S(1))*tan(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, cons530) rule3864 = ReplacementRule(pattern3864, replacement3864) pattern3865 = Pattern(Integral((a_ + WC('b', S(1))/tan(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, cons530) rule3865 = ReplacementRule(pattern3865, replacement3865) pattern3866 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons1441, cons33, cons170) rule3866 = ReplacementRule(pattern3866, replacement3866) pattern3867 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons1441, cons33, cons170) rule3867 = ReplacementRule(pattern3867, replacement3867) pattern3868 = Pattern(Integral(S(1)/((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(x_*WC('d', S(1)) + WC('c', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons1441) rule3868 = ReplacementRule(pattern3868, replacement3868) pattern3869 = Pattern(Integral(S(1)/((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(x_*WC('d', S(1)) + WC('c', S(0)))**S(2)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons1441) rule3869 = ReplacementRule(pattern3869, replacement3869) pattern3870 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**m_/(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons1441, cons33, cons96, cons1512) rule3870 = ReplacementRule(pattern3870, replacement3870) pattern3871 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**m_/(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons1441, cons33, cons96, cons1512) rule3871 = ReplacementRule(pattern3871, replacement3871) pattern3872 = Pattern(Integral(S(1)/((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))*(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons1441) rule3872 = ReplacementRule(pattern3872, replacement3872) pattern3873 = Pattern(Integral(S(1)/((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))*(x_*WC('d', S(1)) + WC('c', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons1441) rule3873 = ReplacementRule(pattern3873, replacement3873) pattern3874 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**m_/(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons1441, cons21) rule3874 = ReplacementRule(pattern3874, replacement3874) pattern3875 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**m_/(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons1441, cons21) rule3875 = ReplacementRule(pattern3875, replacement3875) pattern3876 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons1441, cons1573) rule3876 = ReplacementRule(pattern3876, replacement3876) pattern3877 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons1441, cons1573) rule3877 = ReplacementRule(pattern3877, replacement3877) pattern3878 = Pattern(Integral((a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons1441, cons198) rule3878 = ReplacementRule(pattern3878, replacement3878) pattern3879 = Pattern(Integral((a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**n_*(x_*WC('d', S(1)) + WC('c', S(0)))**m_, x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons1441, cons198) rule3879 = ReplacementRule(pattern3879, replacement3879) pattern3880 = Pattern(Integral((a_ + WC('b', S(1))*tan(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, cons1441, cons1574, cons33, cons170) rule3880 = ReplacementRule(pattern3880, With3880) pattern3881 = Pattern(Integral((a_ + WC('b', S(1))/tan(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, cons1441, cons1574, cons33, cons170) rule3881 = ReplacementRule(pattern3881, With3881) pattern3882 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons1442, cons64) rule3882 = ReplacementRule(pattern3882, replacement3882) pattern3883 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))/(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0)))), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons1442, cons64) rule3883 = ReplacementRule(pattern3883, replacement3883) pattern3884 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))/(a_ + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons1442) rule3884 = ReplacementRule(pattern3884, replacement3884) pattern3885 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))/(a_ + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**S(2), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons1442) rule3885 = ReplacementRule(pattern3885, replacement3885) pattern3886 = Pattern(Integral((a_ + WC('b', S(1))*tan(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, cons1442, cons198, cons64) rule3886 = ReplacementRule(pattern3886, replacement3886) pattern3887 = Pattern(Integral((a_ + WC('b', S(1))/tan(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, cons1442, cons198, cons64) rule3887 = ReplacementRule(pattern3887, replacement3887) pattern3888 = Pattern(Integral(u_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*tan(v_))**WC('n', S(1)), x_), cons2, cons3, cons19, cons4, cons812, cons813) rule3888 = ReplacementRule(pattern3888, replacement3888) pattern3889 = Pattern(Integral(u_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/tan(v_))**WC('n', S(1)), x_), cons2, cons3, cons19, cons4, cons812, cons813) rule3889 = ReplacementRule(pattern3889, replacement3889) pattern3890 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons1362) rule3890 = ReplacementRule(pattern3890, replacement3890) pattern3891 = Pattern(Integral((x_*WC('d', S(1)) + WC('c', S(0)))**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/tan(x_*WC('f', S(1)) + WC('e', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons127, cons19, cons4, cons1362) rule3891 = ReplacementRule(pattern3891, replacement3891) pattern3892 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons5, cons1575, cons40) rule3892 = ReplacementRule(pattern3892, replacement3892) pattern3893 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons5, cons1575, cons40) rule3893 = ReplacementRule(pattern3893, replacement3893) pattern3894 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons4, cons5, cons1497) rule3894 = ReplacementRule(pattern3894, replacement3894) pattern3895 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons4, cons5, cons1497) rule3895 = ReplacementRule(pattern3895, replacement3895) pattern3896 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(u_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons4, cons5, cons70, cons71) rule3896 = ReplacementRule(pattern3896, replacement3896) pattern3897 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(u_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons4, cons5, cons70, cons71) rule3897 = ReplacementRule(pattern3897, replacement3897) pattern3898 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))*tan(u_))**WC('p', S(1)), x_), cons2, cons3, cons5, cons825, cons826) rule3898 = ReplacementRule(pattern3898, replacement3898) pattern3899 = Pattern(Integral((WC('a', S(0)) + WC('b', S(1))/tan(u_))**WC('p', S(1)), x_), cons2, cons3, cons5, cons825, cons826) rule3899 = ReplacementRule(pattern3899, replacement3899) pattern3900 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*tan(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons19, cons4, cons5, cons1576, cons40) rule3900 = ReplacementRule(pattern3900, replacement3900) pattern3901 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/tan(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons19, cons4, cons5, cons1576, cons40) rule3901 = ReplacementRule(pattern3901, replacement3901) pattern3902 = Pattern(Integral(x_**WC('m', S(1))*tan(x_**n_*WC('d', S(1)) + WC('c', S(0)))**S(2), x_), cons8, cons29, cons19, cons4, cons1577) rule3902 = ReplacementRule(pattern3902, replacement3902) pattern3903 = Pattern(Integral(x_**WC('m', S(1))/tan(x_**n_*WC('d', S(1)) + WC('c', S(0)))**S(2), x_), cons8, cons29, cons19, cons4, cons1577) rule3903 = ReplacementRule(pattern3903, replacement3903) pattern3904 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*tan(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons19, cons4, cons5, cons1578) rule3904 = ReplacementRule(pattern3904, replacement3904) pattern3905 = Pattern(Integral(x_**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/tan(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons19, cons4, cons5, cons1578) rule3905 = ReplacementRule(pattern3905, replacement3905) pattern3906 = Pattern(Integral((e_*x_)**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*tan(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons5, cons1499) rule3906 = ReplacementRule(pattern3906, replacement3906) pattern3907 = Pattern(Integral((e_*x_)**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/tan(x_**n_*WC('d', S(1)) + WC('c', S(0))))**WC('p', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons5, cons1499) rule3907 = ReplacementRule(pattern3907, replacement3907) pattern3908 = Pattern(Integral((e_*x_)**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))*tan(u_))**WC('p', S(1)), x_), cons2, cons3, cons50, cons19, cons5, cons825, cons826) rule3908 = ReplacementRule(pattern3908, replacement3908) pattern3909 = Pattern(Integral((e_*x_)**WC('m', S(1))*(WC('a', S(0)) + WC('b', S(1))/tan(u_))**WC('p', S(1)), x_), cons2, cons3, cons50, cons19, cons5, cons825, cons826) rule3909 = ReplacementRule(pattern3909, replacement3909) pattern3910 = Pattern(Integral(x_**WC('m', S(1))*(S(1)/cos(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0))))**WC('p', S(1))*tan(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0)))**WC('q', S(1)), x_), cons2, cons3, cons5, cons33, cons87, cons1579, cons1580) rule3910 = ReplacementRule(pattern3910, replacement3910) pattern3911 = Pattern(Integral(x_**WC('m', S(1))*(S(1)/sin(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0))))**WC('p', S(1))*(S(1)/tan(x_**WC('n', S(1))*WC('b', S(1)) + WC('a', S(0))))**WC('q', S(1)), x_), cons2, cons3, cons5, cons33, cons87, cons1579, cons1580) rule3911 = ReplacementRule(pattern3911, replacement3911) pattern3912 = Pattern(Integral(tan(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons8, cons4, cons1581) rule3912 = ReplacementRule(pattern3912, replacement3912) pattern3913 = Pattern(Integral((S(1)/tan(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons4, cons1581) rule3913 = ReplacementRule(pattern3913, replacement3913) pattern3914 = Pattern(Integral((d_ + x_*WC('e', S(1)))*tan(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons50, cons49) rule3914 = ReplacementRule(pattern3914, replacement3914) pattern3915 = Pattern(Integral((d_ + x_*WC('e', S(1)))/tan(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons50, cons49) rule3915 = ReplacementRule(pattern3915, replacement3915) pattern3916 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))*tan(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons50, cons241) rule3916 = ReplacementRule(pattern3916, replacement3916) pattern3917 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))/tan(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons50, cons241) rule3917 = ReplacementRule(pattern3917, replacement3917) pattern3918 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*tan(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons1582) rule3918 = ReplacementRule(pattern3918, replacement3918) pattern3919 = Pattern(Integral((x_*WC('e', S(1)) + WC('d', S(0)))**WC('m', S(1))*(S(1)/tan(x_**S(2)*WC('c', S(1)) + x_*WC('b', S(1)) + WC('a', S(0))))**WC('n', S(1)), x_), cons2, cons3, cons8, cons29, cons50, cons19, cons4, cons1582) rule3919 = ReplacementRule(pattern3919, replacement3919) return [rule3401, rule3402, rule3403, rule3404, rule3405, rule3406, rule3407, rule3408, rule3409, rule3410, rule3411, rule3412, rule3413, rule3414, rule3415, rule3416, rule3417, rule3418, rule3419, rule3420, rule3421, rule3422, rule3423, rule3424, rule3425, rule3426, rule3427, rule3428, rule3429, rule3430, rule3431, rule3432, rule3433, rule3434, rule3435, rule3436, rule3437, rule3438, rule3439, rule3440, rule3441, rule3442, rule3443, rule3444, rule3445, rule3446, rule3447, rule3448, rule3449, rule3450, rule3451, rule3452, rule3453, rule3454, rule3455, rule3456, rule3457, rule3458, rule3459, rule3460, rule3461, rule3462, rule3463, rule3464, rule3465, rule3466, rule3467, rule3468, rule3469, rule3470, rule3471, rule3472, rule3473, rule3474, rule3475, rule3476, rule3477, rule3478, rule3479, rule3480, rule3481, rule3482, rule3483, rule3484, rule3485, rule3486, rule3487, rule3488, rule3489, rule3490, rule3491, rule3492, rule3493, rule3494, rule3495, rule3496, rule3497, rule3498, rule3499, rule3500, rule3501, rule3502, rule3503, rule3504, rule3505, rule3506, rule3507, rule3508, rule3509, rule3510, rule3511, rule3512, rule3513, rule3514, rule3515, rule3516, rule3517, rule3518, rule3519, rule3520, rule3521, rule3522, rule3523, rule3524, rule3525, rule3526, rule3527, rule3528, rule3529, rule3530, rule3531, rule3532, rule3533, rule3534, rule3535, rule3536, rule3537, rule3538, rule3539, rule3540, rule3541, rule3542, rule3543, rule3544, rule3545, rule3546, rule3547, rule3548, rule3549, rule3550, rule3551, rule3552, rule3553, rule3554, rule3555, rule3556, rule3557, rule3558, rule3559, rule3560, rule3561, rule3562, rule3563, rule3564, rule3565, rule3566, rule3567, rule3568, rule3569, rule3570, rule3571, rule3572, rule3573, rule3574, rule3575, rule3576, rule3577, rule3578, rule3579, rule3580, rule3581, rule3582, rule3583, rule3584, rule3585, rule3586, rule3587, rule3588, rule3589, rule3590, rule3591, rule3592, rule3593, rule3594, rule3595, rule3596, rule3597, rule3598, rule3599, rule3600, rule3601, rule3602, rule3603, rule3604, rule3605, rule3606, rule3607, rule3608, rule3609, rule3610, rule3611, rule3612, rule3613, rule3614, rule3615, rule3616, rule3617, rule3618, rule3619, rule3620, rule3621, rule3622, rule3623, rule3624, rule3625, rule3626, rule3627, rule3628, rule3629, rule3630, rule3631, rule3632, rule3633, rule3634, rule3635, rule3636, rule3637, rule3638, rule3639, rule3640, rule3641, rule3642, rule3643, rule3644, rule3645, rule3646, rule3647, rule3648, rule3649, rule3650, rule3651, rule3652, rule3653, rule3654, rule3655, rule3656, rule3657, rule3658, rule3659, rule3660, rule3661, rule3662, rule3663, rule3664, rule3665, rule3666, rule3667, rule3668, rule3669, rule3670, rule3671, rule3672, rule3673, rule3674, rule3675, rule3676, rule3677, rule3678, rule3679, rule3680, rule3681, rule3682, rule3683, rule3684, rule3685, rule3686, rule3687, rule3688, rule3689, rule3690, rule3691, rule3692, rule3693, rule3694, rule3695, rule3696, rule3697, rule3698, rule3699, rule3700, rule3701, rule3702, rule3703, rule3704, rule3705, rule3706, rule3707, rule3708, rule3709, rule3710, rule3711, rule3712, rule3713, rule3714, rule3715, rule3716, rule3717, rule3718, rule3719, rule3720, rule3721, rule3722, rule3723, rule3724, rule3725, rule3726, rule3727, rule3728, rule3729, rule3730, rule3731, rule3732, rule3733, rule3734, rule3735, rule3736, rule3737, rule3738, rule3739, rule3740, rule3741, rule3742, rule3743, rule3744, rule3745, rule3746, rule3747, rule3748, rule3749, rule3750, rule3751, rule3752, rule3753, rule3754, rule3755, rule3756, rule3757, rule3758, rule3759, rule3760, rule3761, rule3762, rule3763, rule3764, rule3765, rule3766, rule3767, rule3768, rule3769, rule3770, rule3771, rule3772, rule3773, rule3774, rule3775, rule3776, rule3777, rule3778, rule3779, rule3780, rule3781, rule3782, rule3783, rule3784, rule3785, rule3786, rule3787, rule3788, rule3789, rule3790, rule3791, rule3792, rule3793, rule3794, rule3795, rule3796, rule3797, rule3798, rule3799, rule3800, rule3801, rule3802, rule3803, rule3804, rule3805, rule3806, rule3807, rule3808, rule3809, rule3810, rule3811, rule3812, rule3813, rule3814, rule3815, rule3816, rule3817, rule3818, rule3819, rule3820, rule3821, rule3822, rule3823, rule3824, rule3825, rule3826, rule3827, rule3828, rule3829, rule3830, rule3831, rule3832, rule3833, rule3834, rule3835, rule3836, rule3837, rule3838, rule3839, rule3840, rule3841, rule3842, rule3843, rule3844, rule3845, rule3846, rule3847, rule3848, rule3849, rule3850, rule3851, rule3852, rule3853, rule3854, rule3855, rule3856, rule3857, rule3858, rule3859, rule3860, rule3861, rule3862, rule3863, rule3864, rule3865, rule3866, rule3867, rule3868, rule3869, rule3870, rule3871, rule3872, rule3873, rule3874, rule3875, rule3876, rule3877, rule3878, rule3879, rule3880, rule3881, rule3882, rule3883, rule3884, rule3885, rule3886, rule3887, rule3888, rule3889, rule3890, rule3891, rule3892, rule3893, rule3894, rule3895, rule3896, rule3897, rule3898, rule3899, rule3900, rule3901, rule3902, rule3903, rule3904, rule3905, rule3906, rule3907, rule3908, rule3909, rule3910, rule3911, rule3912, rule3913, rule3914, rule3915, rule3916, rule3917, rule3918, rule3919, ] def replacement3401(a, b, e, f, m, n, x): return -Simp(b*(a*sin(e + f*x))**m*(b*tan(e + f*x))**(n + S(-1))/(f*m), x) def replacement3402(a, b, e, f, m, n, x): return Simp(b*(a*cos(e + f*x))**m*(b/tan(e + f*x))**(n + S(-1))/(f*m), x) def replacement3403(e, f, m, n, x): return -Dist(S(1)/f, Subst(Int(x**(-n)*(S(1) - x**S(2))**(m/S(2) + n/S(2) + S(-1)/2), x), x, cos(e + f*x)), x) def replacement3404(e, f, m, n, x): return Dist(S(1)/f, Subst(Int(x**(-n)*(S(1) - x**S(2))**(m/S(2) + n/S(2) + S(-1)/2), x), x, sin(e + f*x)), x) def replacement3405(b, e, f, m, n, x): return Dist(b**(-m), Int((b*tan(e + f*x))**(m + n)*(S(1)/cos(e + f*x))**(-m), x), x) def replacement3406(b, e, f, m, n, x): return Dist(b**(-m), Int((b/tan(e + f*x))**(m + n)*(S(1)/sin(e + f*x))**(-m), x), x) def replacement3407(a, b, e, f, m, n, x): return -Dist(b**S(2)*(m + S(2))/(a**S(2)*(n + S(-1))), Int((a*sin(e + f*x))**(m + S(2))*(b*tan(e + f*x))**(n + S(-2)), x), x) + Simp(b*(a*sin(e + f*x))**(m + S(2))*(b*tan(e + f*x))**(n + S(-1))/(a**S(2)*f*(n + S(-1))), x) def replacement3408(a, b, e, f, m, n, x): return -Dist(b**S(2)*(m + S(2))/(a**S(2)*(n + S(-1))), Int((a*cos(e + f*x))**(m + S(2))*(b/tan(e + f*x))**(n + S(-2)), x), x) - Simp(b*(a*cos(e + f*x))**(m + S(2))*(b/tan(e + f*x))**(n + S(-1))/(a**S(2)*f*(n + S(-1))), x) def replacement3409(a, b, e, f, m, n, x): return -Dist(b**S(2)*(m + n + S(-1))/(n + S(-1)), Int((a*sin(e + f*x))**m*(b*tan(e + f*x))**(n + S(-2)), x), x) + Simp(b*(a*sin(e + f*x))**m*(b*tan(e + f*x))**(n + S(-1))/(f*(n + S(-1))), x) def replacement3410(a, b, e, f, m, n, x): return -Dist(b**S(2)*(m + n + S(-1))/(n + S(-1)), Int((a*cos(e + f*x))**m*(b/tan(e + f*x))**(n + S(-2)), x), x) - Simp(b*(a*cos(e + f*x))**m*(b/tan(e + f*x))**(n + S(-1))/(f*(n + S(-1))), x) def replacement3411(a, b, e, f, m, n, x): return -Dist(a**S(2)*(n + S(1))/(b**S(2)*m), Int((a*sin(e + f*x))**(m + S(-2))*(b*tan(e + f*x))**(n + S(2)), x), x) + Simp((a*sin(e + f*x))**m*(b*tan(e + f*x))**(n + S(1))/(b*f*m), x) def replacement3412(a, b, e, f, m, n, x): return -Dist(a**S(2)*(n + S(1))/(b**S(2)*m), Int((a*cos(e + f*x))**(m + S(-2))*(b/tan(e + f*x))**(n + S(2)), x), x) - Simp((a*cos(e + f*x))**m*(b/tan(e + f*x))**(n + S(1))/(b*f*m), x) def replacement3413(a, b, e, f, m, n, x): return -Dist((n + S(1))/(b**S(2)*(m + n + S(1))), Int((a*sin(e + f*x))**m*(b*tan(e + f*x))**(n + S(2)), x), x) + Simp((a*sin(e + f*x))**m*(b*tan(e + f*x))**(n + S(1))/(b*f*(m + n + S(1))), x) def replacement3414(a, b, e, f, m, n, x): return -Dist((n + S(1))/(b**S(2)*(m + n + S(1))), Int((a*cos(e + f*x))**m*(b/tan(e + f*x))**(n + S(2)), x), x) - Simp((a*cos(e + f*x))**m*(b/tan(e + f*x))**(n + S(1))/(b*f*(m + n + S(1))), x) def replacement3415(a, b, e, f, m, n, x): return Dist(a**S(2)*(m + n + S(-1))/m, Int((a*sin(e + f*x))**(m + S(-2))*(b*tan(e + f*x))**n, x), x) - Simp(b*(a*sin(e + f*x))**m*(b*tan(e + f*x))**(n + S(-1))/(f*m), x) def replacement3416(a, b, e, f, m, n, x): return Dist(a**S(2)*(m + n + S(-1))/m, Int((a*cos(e + f*x))**(m + S(-2))*(b/tan(e + f*x))**n, x), x) + Simp(b*(a*cos(e + f*x))**m*(b/tan(e + f*x))**(n + S(-1))/(f*m), x) def replacement3417(a, b, e, f, m, n, x): return Dist((m + S(2))/(a**S(2)*(m + n + S(1))), Int((a*sin(e + f*x))**(m + S(2))*(b*tan(e + f*x))**n, x), x) + Simp(b*(a*sin(e + f*x))**(m + S(2))*(b*tan(e + f*x))**(n + S(-1))/(a**S(2)*f*(m + n + S(1))), x) def replacement3418(a, b, e, f, m, n, x): return Dist((m + S(2))/(a**S(2)*(m + n + S(1))), Int((a*cos(e + f*x))**(m + S(2))*(b/tan(e + f*x))**n, x), x) - Simp(b*(a*cos(e + f*x))**(m + S(2))*(b/tan(e + f*x))**(n + S(-1))/(a**S(2)*f*(m + n + S(1))), x) def replacement3419(a, e, f, m, n, x): return Dist(S(1)/f, Subst(Int(x**(m + n)*(a**S(2) - x**S(2))**(-n/S(2) + S(-1)/2), x), x, a*sin(e + f*x)), x) def replacement3420(a, e, f, m, n, x): return -Dist(S(1)/f, Subst(Int(x**(m + n)*(a**S(2) - x**S(2))**(-n/S(2) + S(-1)/2), x), x, a*cos(e + f*x)), x) def replacement3421(a, b, e, f, m, n, x): return Dist(a**(S(1) - S(2)*IntPart(n/S(2) + S(1)/2))*b**(S(2)*IntPart(n/S(2) + S(1)/2) + S(-1))*(a*sin(e + f*x))**(-S(2)*FracPart(n/S(2) + S(1)/2))*(b*tan(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)/f, Subst(Int((a*x)**(m + n)*(S(1) - x**S(2))**(-n/S(2) + S(-1)/2), x), x, sin(e + f*x)), x) def replacement3422(a, b, e, f, m, n, x): return -Dist(a**(S(1) - S(2)*IntPart(n/S(2) + S(1)/2))*b**(S(2)*IntPart(n/S(2) + S(1)/2) + S(-1))*(a*cos(e + f*x))**(-S(2)*FracPart(n/S(2) + S(1)/2))*(b/tan(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)/f, Subst(Int((a*x)**(m + n)*(S(1) - x**S(2))**(-n/S(2) + S(-1)/2), x), x, cos(e + f*x)), x) def replacement3423(a, b, e, f, m, n, x): return Dist((S(1)/(a*cos(e + f*x)))**FracPart(m)*(a*cos(e + f*x))**FracPart(m), Int((S(1)/(a*cos(e + f*x)))**(-m)*(b*tan(e + f*x))**n, x), x) def replacement3424(a, b, e, f, m, n, x): return Dist((S(1)/(a*sin(e + f*x)))**FracPart(m)*(a*sin(e + f*x))**FracPart(m), Int((S(1)/(a*sin(e + f*x)))**(-m)*(b/tan(e + f*x))**n, x), x) def replacement3425(a, b, e, f, m, n, x): return Dist((a/tan(e + f*x))**m*(b*tan(e + f*x))**m, Int((b*tan(e + f*x))**(-m + n), x), x) def replacement3426(a, b, e, f, m, n, x): return -Simp((a/cos(e + f*x))**m*(b*tan(e + f*x))**(n + S(1))/(b*f*m), x) def replacement3427(a, b, e, f, m, n, x): return Simp((a/sin(e + f*x))**m*(b/tan(e + f*x))**(n + S(1))/(b*f*m), x) def replacement3428(a, b, e, f, m, n, x): return Dist(a/f, Subst(Int((a*x)**(m + S(-1))*(x**S(2) + S(-1))**(n/S(2) + S(-1)/2), x), x, S(1)/cos(e + f*x)), x) def replacement3429(a, b, e, f, m, n, x): return -Dist(a/f, Subst(Int((a*x)**(m + S(-1))*(x**S(2) + S(-1))**(n/S(2) + S(-1)/2), x), x, S(1)/sin(e + f*x)), x) def replacement3430(b, e, f, m, n, x): return Dist(S(1)/f, Subst(Int((b*x)**n*(x**S(2) + S(1))**(m/S(2) + S(-1)), x), x, tan(e + f*x)), x) def replacement3431(b, e, f, m, n, x): return -Dist(S(1)/f, Subst(Int((b*x)**n*(x**S(2) + S(1))**(m/S(2) + S(-1)), x), x, S(1)/tan(e + f*x)), x) def replacement3432(a, b, e, f, m, n, x): return -Dist(a**S(2)*(m + S(-2))/(b**S(2)*(n + S(1))), Int((a/cos(e + f*x))**(m + S(-2))*(b*tan(e + f*x))**(n + S(2)), x), x) + Simp(a**S(2)*(a/cos(e + f*x))**(m + S(-2))*(b*tan(e + f*x))**(n + S(1))/(b*f*(n + S(1))), x) def replacement3433(a, b, e, f, m, n, x): return -Dist(a**S(2)*(m + S(-2))/(b**S(2)*(n + S(1))), Int((a/sin(e + f*x))**(m + S(-2))*(b/tan(e + f*x))**(n + S(2)), x), x) - Simp(a**S(2)*(a/sin(e + f*x))**(m + S(-2))*(b/tan(e + f*x))**(n + S(1))/(b*f*(n + S(1))), x) def replacement3434(a, b, e, f, m, n, x): return -Dist((m + n + S(1))/(b**S(2)*(n + S(1))), Int((a/cos(e + f*x))**m*(b*tan(e + f*x))**(n + S(2)), x), x) + Simp((a/cos(e + f*x))**m*(b*tan(e + f*x))**(n + S(1))/(b*f*(n + S(1))), x) def replacement3435(a, b, e, f, m, n, x): return -Dist((m + n + S(1))/(b**S(2)*(n + S(1))), Int((a/sin(e + f*x))**m*(b/tan(e + f*x))**(n + S(2)), x), x) - Simp((a/sin(e + f*x))**m*(b/tan(e + f*x))**(n + S(1))/(b*f*(n + S(1))), x) def replacement3436(a, b, e, f, m, n, x): return -Dist(b**S(2)*(n + S(-1))/(a**S(2)*m), Int((a/cos(e + f*x))**(m + S(2))*(b*tan(e + f*x))**(n + S(-2)), x), x) + Simp(b*(a/cos(e + f*x))**m*(b*tan(e + f*x))**(n + S(-1))/(f*m), x) def replacement3437(a, b, e, f, m, n, x): return -Dist(b**S(2)*(n + S(-1))/(a**S(2)*m), Int((a/sin(e + f*x))**(m + S(2))*(b/tan(e + f*x))**(n + S(-2)), x), x) - Simp(b*(a/sin(e + f*x))**m*(b/tan(e + f*x))**(n + S(-1))/(f*m), x) def replacement3438(a, b, e, f, m, n, x): return -Dist(b**S(2)*(n + S(-1))/(m + n + S(-1)), Int((a/cos(e + f*x))**m*(b*tan(e + f*x))**(n + S(-2)), x), x) + Simp(b*(a/cos(e + f*x))**m*(b*tan(e + f*x))**(n + S(-1))/(f*(m + n + S(-1))), x) def replacement3439(a, b, e, f, m, n, x): return -Dist(b**S(2)*(n + S(-1))/(m + n + S(-1)), Int((a/sin(e + f*x))**m*(b/tan(e + f*x))**(n + S(-2)), x), x) - Simp(b*(a/sin(e + f*x))**m*(b/tan(e + f*x))**(n + S(-1))/(f*(m + n + S(-1))), x) def replacement3440(a, b, e, f, m, n, x): return Dist((m + n + S(1))/(a**S(2)*m), Int((a/cos(e + f*x))**(m + S(2))*(b*tan(e + f*x))**n, x), x) - Simp((a/cos(e + f*x))**m*(b*tan(e + f*x))**(n + S(1))/(b*f*m), x) def replacement3441(a, b, e, f, m, n, x): return Dist((m + n + S(1))/(a**S(2)*m), Int((a/sin(e + f*x))**(m + S(2))*(b/tan(e + f*x))**n, x), x) + Simp((a/sin(e + f*x))**m*(b/tan(e + f*x))**(n + S(1))/(b*f*m), x) def replacement3442(a, b, e, f, m, n, x): return Dist(a**S(2)*(m + S(-2))/(m + n + S(-1)), Int((a/cos(e + f*x))**(m + S(-2))*(b*tan(e + f*x))**n, x), x) + Simp(a**S(2)*(a/cos(e + f*x))**(m + S(-2))*(b*tan(e + f*x))**(n + S(1))/(b*f*(m + n + S(-1))), x) def replacement3443(a, b, e, f, m, n, x): return Dist(a**S(2)*(m + S(-2))/(m + n + S(-1)), Int((a/sin(e + f*x))**(m + S(-2))*(b/tan(e + f*x))**n, x), x) - Simp(a**S(2)*(a/sin(e + f*x))**(m + S(-2))*(b/tan(e + f*x))**(n + S(1))/(b*f*(m + n + S(-1))), x) def replacement3444(b, e, f, x): return Dist(sqrt(sin(e + f*x))/(sqrt(b*tan(e + f*x))*sqrt(cos(e + f*x))), Int(S(1)/(sqrt(sin(e + f*x))*sqrt(cos(e + f*x))), x), x) def replacement3445(b, e, f, x): return Dist(sqrt(cos(e + f*x))/(sqrt(b/tan(e + f*x))*sqrt(sin(e + f*x))), Int(S(1)/(sqrt(sin(e + f*x))*sqrt(cos(e + f*x))), x), x) def replacement3446(b, e, f, x): return Dist(sqrt(b*tan(e + f*x))*sqrt(cos(e + f*x))/sqrt(sin(e + f*x)), Int(sqrt(sin(e + f*x))*sqrt(cos(e + f*x)), x), x) def replacement3447(b, e, f, x): return Dist(sqrt(b/tan(e + f*x))*sqrt(sin(e + f*x))/sqrt(cos(e + f*x)), Int(sqrt(sin(e + f*x))*sqrt(cos(e + f*x)), x), x) def replacement3448(a, b, e, f, m, n, x): return Dist(a**(m + n)*(a/cos(e + f*x))**(-n)*(b*sin(e + f*x))**(-n)*(b*tan(e + f*x))**n, Int((b*sin(e + f*x))**n*cos(e + f*x)**(-m - n), x), x) def replacement3449(a, b, e, f, m, n, x): return Dist(a**(m + n)*(a/sin(e + f*x))**(-n)*(b*cos(e + f*x))**(-n)*(b/tan(e + f*x))**n, Int((b*cos(e + f*x))**n*sin(e + f*x)**(-m - n), x), x) def replacement3450(a, b, e, f, m, n, x): return Simp((a/cos(e + f*x))**m*(b*tan(e + f*x))**(n + S(1))*(cos(e + f*x)**S(2))**(m/S(2) + n/S(2) + S(1)/2)*Hypergeometric2F1(n/S(2) + S(1)/2, m/S(2) + n/S(2) + S(1)/2, n/S(2) + S(3)/2, sin(e + f*x)**S(2))/(b*f*(n + S(1))), x) def replacement3451(a, b, e, f, m, n, x): return -Simp((a/sin(e + f*x))**m*(b/tan(e + f*x))**(n + S(1))*(sin(e + f*x)**S(2))**(m/S(2) + n/S(2) + S(1)/2)*Hypergeometric2F1(n/S(2) + S(1)/2, m/S(2) + n/S(2) + S(1)/2, n/S(2) + S(3)/2, cos(e + f*x)**S(2))/(b*f*(n + S(1))), x) def replacement3452(a, b, e, f, m, n, x): return Dist((sin(e + f*x)/a)**FracPart(m)*(a/sin(e + f*x))**FracPart(m), Int((sin(e + f*x)/a)**(-m)*(b*tan(e + f*x))**n, x), x) def replacement3453(a, b, e, f, m, n, x): return Dist((cos(e + f*x)/a)**FracPart(m)*(a/cos(e + f*x))**FracPart(m), Int((cos(e + f*x)/a)**(-m)*(b/tan(e + f*x))**n, x), x) def replacement3454(a, b, d, e, f, m, n, p, x): return -Dist(b**S(2)*(n + S(-1))/(m*p + n + S(-1)), Int((a*(d/cos(e + f*x))**p)**m*(b*tan(e + f*x))**(n + S(-2)), x), x) + Simp(b*(a*(d/cos(e + f*x))**p)**m*(b*tan(e + f*x))**(n + S(-1))/(f*(m*p + n + S(-1))), x) def replacement3455(a, b, d, e, f, m, n, p, x): return -Dist(b**S(2)*(n + S(-1))/(m*p + n + S(-1)), Int((a*(d/sin(e + f*x))**p)**m*(b/tan(e + f*x))**(n + S(-2)), x), x) - Simp(b*(a*(d/sin(e + f*x))**p)**m*(b/tan(e + f*x))**(n + S(-1))/(f*(m*p + n + S(-1))), x) def replacement3456(a, b, d, e, f, m, n, p, x): return -Dist((m*p + n + S(1))/(b**S(2)*(n + S(1))), Int((a*(d/cos(e + f*x))**p)**m*(b*tan(e + f*x))**(n + S(2)), x), x) + Simp((a*(d/cos(e + f*x))**p)**m*(b*tan(e + f*x))**(n + S(1))/(b*f*(n + S(1))), x) def replacement3457(a, b, d, e, f, m, n, p, x): return -Dist((m*p + n + S(1))/(b**S(2)*(n + S(1))), Int((a*(d/sin(e + f*x))**p)**m*(b/tan(e + f*x))**(n + S(2)), x), x) + Simp((a*(d/sin(e + f*x))**p)**m*(b/tan(e + f*x))**(n + S(1))/(b*f*(n + S(1))), x) def replacement3458(b, c, d, n, x): return -Dist(b**S(2), Int((b*tan(c + d*x))**(n + S(-2)), x), x) + Simp(b*(b*tan(c + d*x))**(n + S(-1))/(d*(n + S(-1))), x) def replacement3459(b, c, d, n, x): return -Dist(b**S(2), Int((b/tan(c + d*x))**(n + S(-2)), x), x) - Simp(b*(b/tan(c + d*x))**(n + S(-1))/(d*(n + S(-1))), x) def replacement3460(b, c, d, n, x): return -Dist(b**(S(-2)), Int((b*tan(c + d*x))**(n + S(2)), x), x) + Simp((b*tan(c + d*x))**(n + S(1))/(b*d*(n + S(1))), x) def replacement3461(b, c, d, n, x): return -Dist(b**(S(-2)), Int((b/tan(c + d*x))**(n + S(2)), x), x) - Simp((b/tan(c + d*x))**(n + S(1))/(b*d*(n + S(1))), x) def replacement3462(c, d, x): return -Simp(log(RemoveContent(cos(c + d*x), x))/d, x) def replacement3463(c, d, x): return Simp(log(RemoveContent(sin(c + d*x), x))/d, x) def replacement3464(b, c, d, n, x): return Dist(b/d, Subst(Int(x**n/(b**S(2) + x**S(2)), x), x, b*tan(c + d*x)), x) def replacement3465(b, c, d, n, x): return -Dist(b/d, Subst(Int(x**n/(b**S(2) + x**S(2)), x), x, b/tan(c + d*x)), x) def replacement3466(a, b, c, d, x): return Dist(S(2)*a*b, Int(tan(c + d*x), x), x) + Simp(x*(a**S(2) - b**S(2)), x) + Simp(b**S(2)*tan(c + d*x)/d, x) def replacement3467(a, b, c, d, x): return Dist(S(2)*a*b, Int(S(1)/tan(c + d*x), x), x) + Simp(x*(a**S(2) - b**S(2)), x) - Simp(b**S(2)/(d*tan(c + d*x)), x) def replacement3468(a, b, c, d, n, x): return Dist(S(2)*a, Int((a + b*tan(c + d*x))**(n + S(-1)), x), x) + Simp(b*(a + b*tan(c + d*x))**(n + S(-1))/(d*(n + S(-1))), x) def replacement3469(a, b, c, d, n, x): return Dist(S(2)*a, Int((a + b/tan(c + d*x))**(n + S(-1)), x), x) - Simp(b*(a + b/tan(c + d*x))**(n + S(-1))/(d*(n + S(-1))), x) def replacement3470(a, b, c, d, n, x): return Dist(S(1)/(S(2)*a), Int((a + b*tan(c + d*x))**(n + S(1)), x), x) + Simp(a*(a + b*tan(c + d*x))**n/(S(2)*b*d*n), x) def replacement3471(a, b, c, d, n, x): return Dist(S(1)/(S(2)*a), Int((a + b/tan(c + d*x))**(n + S(1)), x), x) - Simp(a*(a + b/tan(c + d*x))**n/(S(2)*b*d*n), x) def replacement3472(a, b, c, d, x): return Dist(-S(2)*b/d, Subst(Int(S(1)/(S(2)*a - x**S(2)), x), x, sqrt(a + b*tan(c + d*x))), x) def replacement3473(a, b, c, d, x): return Dist(S(2)*b/d, Subst(Int(S(1)/(S(2)*a - x**S(2)), x), x, sqrt(a + b/tan(c + d*x))), x) def replacement3474(a, b, c, d, n, x): return -Dist(b/d, Subst(Int((a + x)**(n + S(-1))/(a - x), x), x, b*tan(c + d*x)), x) def replacement3475(a, b, c, d, n, x): return Dist(b/d, Subst(Int((a + x)**(n + S(-1))/(a - x), x), x, b/tan(c + d*x)), x) def replacement3476(a, b, c, d, n, x): return Int((a + b*tan(c + d*x))**(n + S(-2))*(a**S(2) + S(2)*a*b*tan(c + d*x) - b**S(2)), x) + Simp(b*(a + b*tan(c + d*x))**(n + S(-1))/(d*(n + S(-1))), x) def replacement3477(a, b, c, d, n, x): return Int((a + b/tan(c + d*x))**(n + S(-2))*(a**S(2) + S(2)*a*b/tan(c + d*x) - b**S(2)), x) - Simp(b*(a + b/tan(c + d*x))**(n + S(-1))/(d*(n + S(-1))), x) def replacement3478(a, b, c, d, n, x): return Dist(S(1)/(a**S(2) + b**S(2)), Int((a - b*tan(c + d*x))*(a + b*tan(c + d*x))**(n + S(1)), x), x) + Simp(b*(a + b*tan(c + d*x))**(n + S(1))/(d*(a**S(2) + b**S(2))*(n + S(1))), x) def replacement3479(a, b, c, d, n, x): return Dist(S(1)/(a**S(2) + b**S(2)), Int((a - b/tan(c + d*x))*(a + b/tan(c + d*x))**(n + S(1)), x), x) - Simp(b*(a + b/tan(c + d*x))**(n + S(1))/(d*(a**S(2) + b**S(2))*(n + S(1))), x) def replacement3480(a, b, c, d, x): return Dist(b/(a**S(2) + b**S(2)), Int((-a*tan(c + d*x) + b)/(a + b*tan(c + d*x)), x), x) + Simp(a*x/(a**S(2) + b**S(2)), x) def replacement3481(a, b, c, d, x): return Dist(b/(a**S(2) + b**S(2)), Int((-a/tan(c + d*x) + b)/(a + b/tan(c + d*x)), x), x) + Simp(a*x/(a**S(2) + b**S(2)), x) def replacement3482(a, b, c, d, n, x): return Dist(b/d, Subst(Int((a + x)**n/(b**S(2) + x**S(2)), x), x, b*tan(c + d*x)), x) def replacement3483(a, b, c, d, n, x): return -Dist(b/d, Subst(Int((a + x)**n/(b**S(2) + x**S(2)), x), x, b/tan(c + d*x)), x) def replacement3484(a, b, d, e, f, m, x): return Dist(a, Int((d/cos(e + f*x))**m, x), x) + Simp(b*(d/cos(e + f*x))**m/(f*m), x) def replacement3485(a, b, d, e, f, m, x): return Dist(a, Int((d/sin(e + f*x))**m, x), x) - Simp(b*(d/sin(e + f*x))**m/(f*m), x) def replacement3486(a, b, e, f, m, n, x): return Dist(a**(S(2) - m)/(b*f), Subst(Int((a - x)**(m/S(2) + S(-1))*(a + x)**(m/S(2) + n + S(-1)), x), x, b*tan(e + f*x)), x) def replacement3487(a, b, e, f, m, n, x): return -Dist(a**(S(2) - m)/(b*f), Subst(Int((a - x)**(m/S(2) + S(-1))*(a + x)**(m/S(2) + n + S(-1)), x), x, b/tan(e + f*x)), x) def replacement3488(a, b, d, e, f, m, n, x): return Simp(b*(d/cos(e + f*x))**m*(a + b*tan(e + f*x))**n/(a*f*m), x) def replacement3489(a, b, d, e, f, m, n, x): return -Simp(b*(d/sin(e + f*x))**m*(a + b/tan(e + f*x))**n/(a*f*m), x) def replacement3490(a, b, e, f, x): return Dist(-S(2)*a/(b*f), Subst(Int(S(1)/(-a*x**S(2) + S(2)), x), x, S(1)/(sqrt(a + b*tan(e + f*x))*cos(e + f*x))), x) def replacement3491(a, b, e, f, x): return Dist(S(2)*a/(b*f), Subst(Int(S(1)/(-a*x**S(2) + S(2)), x), x, S(1)/(sqrt(a + b/tan(e + f*x))*sin(e + f*x))), x) def replacement3492(a, b, d, e, f, m, n, x): return Dist(a/(S(2)*d**S(2)), Int((d/cos(e + f*x))**(m + S(2))*(a + b*tan(e + f*x))**(n + S(-1)), x), x) + Simp(b*(d/cos(e + f*x))**m*(a + b*tan(e + f*x))**n/(a*f*m), x) def replacement3493(a, b, d, e, f, m, n, x): return Dist(a/(S(2)*d**S(2)), Int((d/sin(e + f*x))**(m + S(2))*(a + b/tan(e + f*x))**(n + S(-1)), x), x) - Simp(b*(d/sin(e + f*x))**m*(a + b/tan(e + f*x))**n/(a*f*m), x) def replacement3494(a, b, d, e, f, m, n, x): return Dist(S(2)*d**S(2)/a, Int((d/cos(e + f*x))**(m + S(-2))*(a + b*tan(e + f*x))**(n + S(1)), x), x) + Simp(S(2)*d**S(2)*(d/cos(e + f*x))**(m + S(-2))*(a + b*tan(e + f*x))**(n + S(1))/(b*f*(m + S(-2))), x) def replacement3495(a, b, d, e, f, m, n, x): return Dist(S(2)*d**S(2)/a, Int((d/sin(e + f*x))**(m + S(-2))*(a + b/tan(e + f*x))**(n + S(1)), x), x) + Simp(-S(2)*d**S(2)*(d/sin(e + f*x))**(m + S(-2))*(a + b/tan(e + f*x))**(n + S(1))/(b*f*(m + S(-2))), x) def replacement3496(a, b, d, e, f, m, n, x): return Dist((a/d)**(S(2)*IntPart(n))*(d/cos(e + f*x))**(-S(2)*FracPart(n))*(a - b*tan(e + f*x))**FracPart(n)*(a + b*tan(e + f*x))**FracPart(n), Int((a - b*tan(e + f*x))**(-n), x), x) def replacement3497(a, b, d, e, f, m, n, x): return Dist((a/d)**(S(2)*IntPart(n))*(d/sin(e + f*x))**(-S(2)*FracPart(n))*(a - b/tan(e + f*x))**FracPart(n)*(a + b/tan(e + f*x))**FracPart(n), Int((a - b/tan(e + f*x))**(-n), x), x) def replacement3498(a, b, d, e, f, m, n, x): return Simp(S(2)*b*(d/cos(e + f*x))**m*(a + b*tan(e + f*x))**(n + S(-1))/(f*m), x) def replacement3499(a, b, d, e, f, m, n, x): return Simp(-S(2)*b*(d/sin(e + f*x))**m*(a + b/tan(e + f*x))**(n + S(-1))/(f*m), x) def replacement3500(a, b, d, e, f, m, n, x): return Dist(a*(m + S(2)*n + S(-2))/(m + n + S(-1)), Int((d/cos(e + f*x))**m*(a + b*tan(e + f*x))**(n + S(-1)), x), x) + Simp(b*(d/cos(e + f*x))**m*(a + b*tan(e + f*x))**(n + S(-1))/(f*(m + n + S(-1))), x) def replacement3501(a, b, d, e, f, m, n, x): return Dist(a*(m + S(2)*n + S(-2))/(m + n + S(-1)), Int((d/sin(e + f*x))**m*(a + b/tan(e + f*x))**(n + S(-1)), x), x) - Simp(b*(d/sin(e + f*x))**m*(a + b/tan(e + f*x))**(n + S(-1))/(f*(m + n + S(-1))), x) def replacement3502(a, b, d, e, f, x): return Dist(-S(4)*b*d**S(2)/f, Subst(Int(x**S(2)/(a**S(2) + d**S(2)*x**S(4)), x), x, sqrt(a + b*tan(e + f*x))/sqrt(d/cos(e + f*x))), x) def replacement3503(a, b, d, e, f, x): return Dist(S(4)*b*d**S(2)/f, Subst(Int(x**S(2)/(a**S(2) + d**S(2)*x**S(4)), x), x, sqrt(a + b/tan(e + f*x))/sqrt(d/sin(e + f*x))), x) def replacement3504(a, b, d, e, f, m, n, x): return -Dist(b**S(2)*(m + S(2)*n + S(-2))/(d**S(2)*m), Int((d/cos(e + f*x))**(m + S(2))*(a + b*tan(e + f*x))**(n + S(-2)), x), x) + Simp(S(2)*b*(d/cos(e + f*x))**m*(a + b*tan(e + f*x))**(n + S(-1))/(f*m), x) def replacement3505(a, b, d, e, f, m, n, x): return -Dist(b**S(2)*(m + S(2)*n + S(-2))/(d**S(2)*m), Int((d/sin(e + f*x))**(m + S(2))*(a + b/tan(e + f*x))**(n + S(-2)), x), x) + Simp(-S(2)*b*(d/sin(e + f*x))**m*(a + b/tan(e + f*x))**(n + S(-1))/(f*m), x) def replacement3506(a, b, d, e, f, m, n, x): return Dist(a*(m + n)/(d**S(2)*m), Int((d/cos(e + f*x))**(m + S(2))*(a + b*tan(e + f*x))**(n + S(-1)), x), x) + Simp(b*(d/cos(e + f*x))**m*(a + b*tan(e + f*x))**n/(a*f*m), x) def replacement3507(a, b, d, e, f, m, n, x): return Dist(a*(m + n)/(d**S(2)*m), Int((d/sin(e + f*x))**(m + S(2))*(a + b/tan(e + f*x))**(n + S(-1)), x), x) - Simp(b*(d/sin(e + f*x))**m*(a + b/tan(e + f*x))**n/(a*f*m), x) def replacement3508(a, b, d, e, f, m, n, x): return Dist(a*(m + S(2)*n + S(-2))/(m + n + S(-1)), Int((d/cos(e + f*x))**m*(a + b*tan(e + f*x))**(n + S(-1)), x), x) + Simp(b*(d/cos(e + f*x))**m*(a + b*tan(e + f*x))**(n + S(-1))/(f*(m + n + S(-1))), x) def replacement3509(a, b, d, e, f, m, n, x): return Dist(a*(m + S(2)*n + S(-2))/(m + n + S(-1)), Int((d/sin(e + f*x))**m*(a + b/tan(e + f*x))**(n + S(-1)), x), x) - Simp(b*(d/sin(e + f*x))**m*(a + b/tan(e + f*x))**(n + S(-1))/(f*(m + n + S(-1))), x) def replacement3510(a, b, d, e, f, x): return Dist(d/(sqrt(a - b*tan(e + f*x))*sqrt(a + b*tan(e + f*x))*cos(e + f*x)), Int(sqrt(d/cos(e + f*x))*sqrt(a - b*tan(e + f*x)), x), x) def replacement3511(a, b, d, e, f, x): return Dist(d/(sqrt(a - b/tan(e + f*x))*sqrt(a + b/tan(e + f*x))*sin(e + f*x)), Int(sqrt(d/sin(e + f*x))*sqrt(a - b/tan(e + f*x)), x), x) def replacement3512(a, b, d, e, f, m, n, x): return -Dist(d**S(2)*(m + S(-2))/(b**S(2)*(m + S(2)*n)), Int((d/cos(e + f*x))**(m + S(-2))*(a + b*tan(e + f*x))**(n + S(2)), x), x) + Simp(S(2)*d**S(2)*(d/cos(e + f*x))**(m + S(-2))*(a + b*tan(e + f*x))**(n + S(1))/(b*f*(m + S(2)*n)), x) def replacement3513(a, b, d, e, f, m, n, x): return -Dist(d**S(2)*(m + S(-2))/(b**S(2)*(m + S(2)*n)), Int((d/sin(e + f*x))**(m + S(-2))*(a + b/tan(e + f*x))**(n + S(2)), x), x) + Simp(-S(2)*d**S(2)*(d/sin(e + f*x))**(m + S(-2))*(a + b/tan(e + f*x))**(n + S(1))/(b*f*(m + S(2)*n)), x) def replacement3514(a, b, d, e, f, m, n, x): return Dist(d**S(2)*(m + S(-2))/(a*(m + n + S(-1))), Int((d/cos(e + f*x))**(m + S(-2))*(a + b*tan(e + f*x))**(n + S(1)), x), x) + Simp(d**S(2)*(d/cos(e + f*x))**(m + S(-2))*(a + b*tan(e + f*x))**(n + S(1))/(b*f*(m + n + S(-1))), x) def replacement3515(a, b, d, e, f, m, n, x): return Dist(d**S(2)*(m + S(-2))/(a*(m + n + S(-1))), Int((d/sin(e + f*x))**(m + S(-2))*(a + b/tan(e + f*x))**(n + S(1)), x), x) - Simp(d**S(2)*(d/sin(e + f*x))**(m + S(-2))*(a + b/tan(e + f*x))**(n + S(1))/(b*f*(m + n + S(-1))), x) def replacement3516(a, b, d, e, f, m, n, x): return Dist((m + n)/(a*(m + S(2)*n)), Int((d/cos(e + f*x))**m*(a + b*tan(e + f*x))**(n + S(1)), x), x) + Simp(a*(d/cos(e + f*x))**m*(a + b*tan(e + f*x))**n/(b*f*(m + S(2)*n)), x) def replacement3517(a, b, d, e, f, m, n, x): return Dist((m + n)/(a*(m + S(2)*n)), Int((d/sin(e + f*x))**m*(a + b/tan(e + f*x))**(n + S(1)), x), x) - Simp(a*(d/sin(e + f*x))**m*(a + b/tan(e + f*x))**n/(b*f*(m + S(2)*n)), x) def replacement3518(a, b, d, e, f, m, n, x): return Dist(a*(m + S(2)*n + S(-2))/(m + n + S(-1)), Int((d/cos(e + f*x))**m*(a + b*tan(e + f*x))**(n + S(-1)), x), x) + Simp(b*(d/cos(e + f*x))**m*(a + b*tan(e + f*x))**(n + S(-1))/(f*(m + n + S(-1))), x) def replacement3519(a, b, d, e, f, m, n, x): return Dist(a*(m + S(2)*n + S(-2))/(m + n + S(-1)), Int((d/sin(e + f*x))**m*(a + b/tan(e + f*x))**(n + S(-1)), x), x) - Simp(b*(d/sin(e + f*x))**m*(a + b/tan(e + f*x))**(n + S(-1))/(f*(m + n + S(-1))), x) def replacement3520(a, b, d, e, f, m, n, x): return Dist((m + n)/(a*(m + S(2)*n)), Int((d/cos(e + f*x))**m*(a + b*tan(e + f*x))**(n + S(1)), x), x) + Simp(a*(d/cos(e + f*x))**m*(a + b*tan(e + f*x))**n/(b*f*(m + S(2)*n)), x) def replacement3521(a, b, d, e, f, m, n, x): return Dist((m + n)/(a*(m + S(2)*n)), Int((d/sin(e + f*x))**m*(a + b/tan(e + f*x))**(n + S(1)), x), x) - Simp(a*(d/sin(e + f*x))**m*(a + b/tan(e + f*x))**n/(b*f*(m + S(2)*n)), x) def replacement3522(a, b, d, e, f, m, n, x): return Dist((d/cos(e + f*x))**m*(a - b*tan(e + f*x))**(-m/S(2))*(a + b*tan(e + f*x))**(-m/S(2)), Int((a - b*tan(e + f*x))**(m/S(2))*(a + b*tan(e + f*x))**(m/S(2) + n), x), x) def replacement3523(a, b, d, e, f, m, n, x): return Dist((d/sin(e + f*x))**m*(a - b/tan(e + f*x))**(-m/S(2))*(a + b/tan(e + f*x))**(-m/S(2)), Int((a - b/tan(e + f*x))**(m/S(2))*(a + b/tan(e + f*x))**(m/S(2) + n), x), x) def replacement3524(a, b, e, f, m, n, x): return Dist(S(1)/(b*f), Subst(Int((S(1) + x**S(2)/b**S(2))**(m/S(2) + S(-1))*(a + x)**n, x), x, b*tan(e + f*x)), x) def replacement3525(a, b, e, f, m, n, x): return -Dist(S(1)/(b*f), Subst(Int((S(1) + x**S(2)/b**S(2))**(m/S(2) + S(-1))*(a + x)**n, x), x, b/tan(e + f*x)), x) def replacement3526(a, b, e, f, x): return Simp(b**S(2)*atanh(sin(e + f*x))/f, x) + Simp((a**S(2) - b**S(2))*sin(e + f*x)/f, x) - Simp(S(2)*a*b*cos(e + f*x)/f, x) def replacement3527(a, b, e, f, x): return -Simp(b**S(2)*atanh(cos(e + f*x))/f, x) - Simp((a**S(2) - b**S(2))*cos(e + f*x)/f, x) + Simp(S(2)*a*b*sin(e + f*x)/f, x) def replacement3528(a, b, d, e, f, m, x): return Dist(S(1)/(m + S(1)), Int((d/cos(e + f*x))**m*(a**S(2)*(m + S(1)) + a*b*(m + S(2))*tan(e + f*x) - b**S(2)), x), x) + Simp(b*(d/cos(e + f*x))**m*(a + b*tan(e + f*x))/(f*(m + S(1))), x) def replacement3529(a, b, d, e, f, m, x): return Dist(S(1)/(m + S(1)), Int((d/sin(e + f*x))**m*(a**S(2)*(m + S(1)) + a*b*(m + S(2))/tan(e + f*x) - b**S(2)), x), x) - Simp(b*(d/sin(e + f*x))**m*(a + b/tan(e + f*x))/(f*(m + S(1))), x) def replacement3530(a, b, e, f, x): return -Dist(S(1)/f, Subst(Int(S(1)/(a**S(2) + b**S(2) - x**S(2)), x), x, (-a*tan(e + f*x) + b)*cos(e + f*x)), x) def replacement3531(a, b, e, f, x): return Dist(S(1)/f, Subst(Int(S(1)/(a**S(2) + b**S(2) - x**S(2)), x), x, (-a/tan(e + f*x) + b)*sin(e + f*x)), x) def replacement3532(a, b, d, e, f, m, x): return -Dist(d**S(2)/b**S(2), Int((d/cos(e + f*x))**(m + S(-2))*(a - b*tan(e + f*x)), x), x) + Dist(d**S(2)*(a**S(2) + b**S(2))/b**S(2), Int((d/cos(e + f*x))**(m + S(-2))/(a + b*tan(e + f*x)), x), x) def replacement3533(a, b, d, e, f, m, x): return -Dist(d**S(2)/b**S(2), Int((d/sin(e + f*x))**(m + S(-2))*(a - b/tan(e + f*x)), x), x) + Dist(d**S(2)*(a**S(2) + b**S(2))/b**S(2), Int((d/sin(e + f*x))**(m + S(-2))/(a + b/tan(e + f*x)), x), x) def replacement3534(a, b, d, e, f, m, x): return Dist(b**S(2)/(d**S(2)*(a**S(2) + b**S(2))), Int((d/cos(e + f*x))**(m + S(2))/(a + b*tan(e + f*x)), x), x) + Dist(S(1)/(a**S(2) + b**S(2)), Int((d/cos(e + f*x))**m*(a - b*tan(e + f*x)), x), x) def replacement3535(a, b, d, e, f, m, x): return Dist(b**S(2)/(d**S(2)*(a**S(2) + b**S(2))), Int((d/sin(e + f*x))**(m + S(2))/(a + b/tan(e + f*x)), x), x) + Dist(S(1)/(a**S(2) + b**S(2)), Int((d/sin(e + f*x))**m*(a - b/tan(e + f*x)), x), x) def replacement3536(a, b, d, e, f, m, n, x): return Dist(d**(S(2)*IntPart(m/S(2)))*(d/cos(e + f*x))**(S(2)*FracPart(m/S(2)))*(cos(e + f*x)**(S(-2)))**(-FracPart(m/S(2)))/(b*f), Subst(Int((S(1) + x**S(2)/b**S(2))**(m/S(2) + S(-1))*(a + x)**n, x), x, b*tan(e + f*x)), x) def replacement3537(a, b, d, e, f, m, n, x): return -Dist(d**(S(2)*IntPart(m/S(2)))*(d/sin(e + f*x))**(S(2)*FracPart(m/S(2)))*(sin(e + f*x)**(S(-2)))**(-FracPart(m/S(2)))/(b*f), Subst(Int((S(1) + x**S(2)/b**S(2))**(m/S(2) + S(-1))*(a + x)**n, x), x, b/tan(e + f*x)), x) def replacement3538(a, b, d, e, f, x): return Dist(-S(4)*b/f, Subst(Int(x**S(2)/(a**S(2)*d**S(2) + x**S(4)), x), x, sqrt(d*cos(e + f*x))*sqrt(a + b*tan(e + f*x))), x) def replacement3539(a, b, d, e, f, x): return Dist(S(4)*b/f, Subst(Int(x**S(2)/(a**S(2)*d**S(2) + x**S(4)), x), x, sqrt(d*sin(e + f*x))*sqrt(a + b/tan(e + f*x))), x) def replacement3540(a, b, d, e, f, x): return Dist(S(1)/(d*sqrt(a - b*tan(e + f*x))*sqrt(a + b*tan(e + f*x))*cos(e + f*x)), Int(sqrt(a - b*tan(e + f*x))/sqrt(d*cos(e + f*x)), x), x) def replacement3541(a, b, d, e, f, x): return Dist(S(1)/(d*sqrt(a - b/tan(e + f*x))*sqrt(a + b/tan(e + f*x))*sin(e + f*x)), Int(sqrt(a - b/tan(e + f*x))/sqrt(d*sin(e + f*x)), x), x) def replacement3542(a, b, d, e, f, m, n, x): return Dist((d/cos(e + f*x))**m*(d*cos(e + f*x))**m, Int((d/cos(e + f*x))**(-m)*(a + b*tan(e + f*x))**n, x), x) def replacement3543(a, b, d, e, f, m, n, x): return Dist((d/sin(e + f*x))**m*(d*sin(e + f*x))**m, Int((d/sin(e + f*x))**(-m)*(a + b/tan(e + f*x))**n, x), x) def replacement3544(a, b, e, f, m, n, x): return Dist(b/f, Subst(Int(x**m*(a + x)**n*(b**S(2) + x**S(2))**(-m/S(2) + S(-1)), x), x, b*tan(e + f*x)), x) def replacement3545(a, b, e, f, m, n, x): return -Dist(b/f, Subst(Int(x**m*(a + x)**n*(b**S(2) + x**S(2))**(-m/S(2) + S(-1)), x), x, b/tan(e + f*x)), x) def replacement3546(a, b, e, f, m, n, x): return Int((a + b*tan(e + f*x))**n*sin(e + f*x)**m, x) def replacement3547(a, b, e, f, m, n, x): return Int((a + b/tan(e + f*x))**n*cos(e + f*x)**m, x) def replacement3548(a, b, e, f, m, n, x): return Int((a*cos(e + f*x) + b*sin(e + f*x))**n*sin(e + f*x)**m*cos(e + f*x)**(-n), x) def replacement3549(a, b, e, f, m, n, x): return Int((a*sin(e + f*x) + b*cos(e + f*x))**n*sin(e + f*x)**(-n)*cos(e + f*x)**m, x) def replacement3550(a, b, d, e, f, m, n, x): return Dist((sin(e + f*x)/d)**FracPart(m)*(d/sin(e + f*x))**FracPart(m), Int((sin(e + f*x)/d)**(-m)*(a + b*tan(e + f*x))**n, x), x) def replacement3551(a, b, d, e, f, m, n, x): return Dist((cos(e + f*x)/d)**FracPart(m)*(d/cos(e + f*x))**FracPart(m), Int((cos(e + f*x)/d)**(-m)*(a + b/tan(e + f*x))**n, x), x) def replacement3552(a, b, e, f, m, n, p, x): return Int((a*cos(e + f*x) + b*sin(e + f*x))**n*sin(e + f*x)**p*cos(e + f*x)**(m - n), x) def replacement3553(a, b, e, f, m, n, p, x): return Int((a*sin(e + f*x) + b*cos(e + f*x))**n*sin(e + f*x)**(m - n)*cos(e + f*x)**p, x) def replacement3554(a, b, c, d, e, f, m, n, x): return Dist(a**m*c**m, Int((c + d*tan(e + f*x))**(-m + n)*(S(1)/cos(e + f*x))**(S(2)*m), x), x) def replacement3555(a, b, c, d, e, f, m, n, x): return Dist(a**m*c**m, Int((c + d/tan(e + f*x))**(-m + n)*(S(1)/sin(e + f*x))**(S(2)*m), x), x) def replacement3556(a, b, c, d, e, f, m, n, x): return Dist(a*c/f, Subst(Int((a + b*x)**(m + S(-1))*(c + d*x)**(n + S(-1)), x), x, tan(e + f*x)), x) def replacement3557(a, b, c, d, e, f, m, n, x): return -Dist(a*c/f, Subst(Int((a + b*x)**(m + S(-1))*(c + d*x)**(n + S(-1)), x), x, S(1)/tan(e + f*x)), x) def replacement3558(a, b, c, d, e, f, x): return Simp(x*(a*c - b*d), x) + Simp(b*d*tan(e + f*x)/f, x) def replacement3559(a, b, c, d, e, f, x): return Simp(x*(a*c - b*d), x) - Simp(b*d/(f*tan(e + f*x)), x) def replacement3560(a, b, c, d, e, f, x): return Dist(a*d + b*c, Int(tan(e + f*x), x), x) + Simp(x*(a*c - b*d), x) + Simp(b*d*tan(e + f*x)/f, x) def replacement3561(a, b, c, d, e, f, x): return Dist(a*d + b*c, Int(S(1)/tan(e + f*x), x), x) + Simp(x*(a*c - b*d), x) - Simp(b*d/(f*tan(e + f*x)), x) def replacement3562(a, b, c, d, e, f, m, x): return Dist((a*d + b*c)/(S(2)*a*b), Int((a + b*tan(e + f*x))**(m + S(1)), x), x) - Simp((a + b*tan(e + f*x))**m*(-a*d + b*c)/(S(2)*a*f*m), x) def replacement3563(a, b, c, d, e, f, m, x): return Dist((a*d + b*c)/(S(2)*a*b), Int((a + b/tan(e + f*x))**(m + S(1)), x), x) + Simp((a + b/tan(e + f*x))**m*(-a*d + b*c)/(S(2)*a*f*m), x) def replacement3564(a, b, c, d, e, f, m, x): return Dist((a*d + b*c)/b, Int((a + b*tan(e + f*x))**m, x), x) + Simp(d*(a + b*tan(e + f*x))**m/(f*m), x) def replacement3565(a, b, c, d, e, f, m, x): return Dist((a*d + b*c)/b, Int((a + b/tan(e + f*x))**m, x), x) - Simp(d*(a + b/tan(e + f*x))**m/(f*m), x) def replacement3566(a, b, c, d, e, f, m, x): return Int((a + b*tan(e + f*x))**(m + S(-1))*Simp(a*c - b*d + (a*d + b*c)*tan(e + f*x), x), x) + Simp(d*(a + b*tan(e + f*x))**m/(f*m), x) def replacement3567(a, b, c, d, e, f, m, x): return Int((a + b/tan(e + f*x))**(m + S(-1))*Simp(a*c - b*d + (a*d + b*c)/tan(e + f*x), x), x) - Simp(d*(a + b/tan(e + f*x))**m/(f*m), x) def replacement3568(a, b, c, d, e, f, m, x): return Dist(S(1)/(a**S(2) + b**S(2)), Int((a + b*tan(e + f*x))**(m + S(1))*Simp(a*c + b*d - (-a*d + b*c)*tan(e + f*x), x), x), x) + Simp((a + b*tan(e + f*x))**(m + S(1))*(-a*d + b*c)/(f*(a**S(2) + b**S(2))*(m + S(1))), x) def replacement3569(a, b, c, d, e, f, m, x): return Dist(S(1)/(a**S(2) + b**S(2)), Int((a + b/tan(e + f*x))**(m + S(1))*Simp(a*c + b*d - (-a*d + b*c)/tan(e + f*x), x), x), x) - Simp((a + b/tan(e + f*x))**(m + S(1))*(-a*d + b*c)/(f*(a**S(2) + b**S(2))*(m + S(1))), x) def replacement3570(a, b, c, d, e, f, x): return Simp(c*log(RemoveContent(a*cos(e + f*x) + b*sin(e + f*x), x))/(b*f), x) def replacement3571(a, b, c, d, e, f, x): return -Simp(c*log(RemoveContent(a*sin(e + f*x) + b*cos(e + f*x), x))/(b*f), x) def replacement3572(a, b, c, d, e, f, x): return Dist((-a*d + b*c)/(a**S(2) + b**S(2)), Int((-a*tan(e + f*x) + b)/(a + b*tan(e + f*x)), x), x) + Simp(x*(a*c + b*d)/(a**S(2) + b**S(2)), x) def replacement3573(a, b, c, d, e, f, x): return Dist((-a*d + b*c)/(a**S(2) + b**S(2)), Int((-a/tan(e + f*x) + b)/(a + b/tan(e + f*x)), x), x) + Simp(x*(a*c + b*d)/(a**S(2) + b**S(2)), x) def replacement3574(b, c, d, e, f, x): return Dist(-S(2)*d**S(2)/f, Subst(Int(S(1)/(b*x**S(2) + S(2)*c*d), x), x, (c - d*tan(e + f*x))/sqrt(b*tan(e + f*x))), x) def replacement3575(b, c, d, e, f, x): return Dist(S(2)*d**S(2)/f, Subst(Int(S(1)/(b*x**S(2) + S(2)*c*d), x), x, (c - d/tan(e + f*x))/sqrt(b/tan(e + f*x))), x) def replacement3576(b, c, d, e, f, x): return Dist(S(2)*c**S(2)/f, Subst(Int(S(1)/(b*c - d*x**S(2)), x), x, sqrt(b*tan(e + f*x))), x) def replacement3577(b, c, d, e, f, x): return Dist(-S(2)*c**S(2)/f, Subst(Int(S(1)/(b*c - d*x**S(2)), x), x, sqrt(b/tan(e + f*x))), x) def replacement3578(b, c, d, e, f, x): return Dist(S(2)/f, Subst(Int((b*c + d*x**S(2))/(b**S(2) + x**S(4)), x), x, sqrt(b*tan(e + f*x))), x) def replacement3579(b, c, d, e, f, x): return Dist(-S(2)/f, Subst(Int((b*c + d*x**S(2))/(b**S(2) + x**S(4)), x), x, sqrt(b/tan(e + f*x))), x) def replacement3580(a, b, c, d, e, f, x): return Dist(-S(2)*d**S(2)/f, Subst(Int(S(1)/(-S(4)*a*d**S(2) + S(2)*b*c*d + x**S(2)), x), x, (-S(2)*a*d + b*c - b*d*tan(e + f*x))/sqrt(a + b*tan(e + f*x))), x) def replacement3581(a, b, c, d, e, f, x): return Dist(S(2)*d**S(2)/f, Subst(Int(S(1)/(-S(4)*a*d**S(2) + S(2)*b*c*d + x**S(2)), x), x, (-S(2)*a*d + b*c - b*d/tan(e + f*x))/sqrt(a + b/tan(e + f*x))), x) def With3582(a, b, c, d, e, f, x): q = Rt(a**S(2) + b**S(2), S(2)) return -Dist(S(1)/(S(2)*q), Int((a*c + b*d - c*q + (-a*d + b*c - d*q)*tan(e + f*x))/sqrt(a + b*tan(e + f*x)), x), x) + Dist(S(1)/(S(2)*q), Int((a*c + b*d + c*q + (-a*d + b*c + d*q)*tan(e + f*x))/sqrt(a + b*tan(e + f*x)), x), x) def With3583(a, b, c, d, e, f, x): q = Rt(a**S(2) + b**S(2), S(2)) return -Dist(S(1)/(S(2)*q), Int((a*c + b*d - c*q + (-a*d + b*c - d*q)/tan(e + f*x))/sqrt(a + b/tan(e + f*x)), x), x) + Dist(S(1)/(S(2)*q), Int((a*c + b*d + c*q + (-a*d + b*c + d*q)/tan(e + f*x))/sqrt(a + b/tan(e + f*x)), x), x) def replacement3584(a, b, c, d, e, f, m, x): return Dist(c*d/f, Subst(Int((a + b*x/d)**m/(c*x + d**S(2)), x), x, d*tan(e + f*x)), x) def replacement3585(a, b, c, d, e, f, m, x): return -Dist(c*d/f, Subst(Int((a + b*x/d)**m/(c*x + d**S(2)), x), x, d/tan(e + f*x)), x) def replacement3586(b, c, d, e, f, m, x): return Dist(c, Int((b*tan(e + f*x))**m, x), x) + Dist(d/b, Int((b*tan(e + f*x))**(m + S(1)), x), x) def replacement3587(b, c, d, e, f, m, x): return Dist(c, Int((b/tan(e + f*x))**m, x), x) + Dist(d/b, Int((b/tan(e + f*x))**(m + S(1)), x), x) def replacement3588(a, b, c, d, e, f, m, x): return Dist(c/S(2) - I*d/S(2), Int((a + b*tan(e + f*x))**m*(I*tan(e + f*x) + S(1)), x), x) + Dist(c/S(2) + I*d/S(2), Int((a + b*tan(e + f*x))**m*(-I*tan(e + f*x) + S(1)), x), x) def replacement3589(a, b, c, d, e, f, m, x): return Dist(c/S(2) - I*d/S(2), Int((S(1) + I/tan(e + f*x))*(a + b/tan(e + f*x))**m, x), x) + Dist(c/S(2) + I*d/S(2), Int((S(1) - I/tan(e + f*x))*(a + b/tan(e + f*x))**m, x), x) def replacement3590(a, b, c, d, e, f, m, x): return Dist(S(1)/(S(2)*a**S(2)), Int((a + b*tan(e + f*x))**(m + S(1))*Simp(a*c**S(2) + a*d**S(2) - S(2)*b*c*d - S(2)*b*d**S(2)*tan(e + f*x), x), x), x) - Simp(b*(a + b*tan(e + f*x))**m*(a*c + b*d)**S(2)/(S(2)*a**S(3)*f*m), x) def replacement3591(a, b, c, d, e, f, m, x): return Dist(S(1)/(S(2)*a**S(2)), Int((a + b/tan(e + f*x))**(m + S(1))*Simp(a*c**S(2) + a*d**S(2) - S(2)*b*c*d - S(2)*b*d**S(2)/tan(e + f*x), x), x), x) + Simp(b*(a + b/tan(e + f*x))**m*(a*c + b*d)**S(2)/(S(2)*a**S(3)*f*m), x) def replacement3592(a, b, c, d, e, f, x): return Dist((-a*d + b*c)**S(2)/b**S(2), Int(S(1)/(a + b*tan(e + f*x)), x), x) + Dist(d**S(2)/b, Int(tan(e + f*x), x), x) + Simp(d*x*(-a*d + S(2)*b*c)/b**S(2), x) def replacement3593(a, b, c, d, e, f, x): return Dist((-a*d + b*c)**S(2)/b**S(2), Int(S(1)/(a + b/tan(e + f*x)), x), x) + Dist(d**S(2)/b, Int(S(1)/tan(e + f*x), x), x) + Simp(d*x*(-a*d + S(2)*b*c)/b**S(2), x) def replacement3594(a, b, c, d, e, f, m, x): return Dist(S(1)/(a**S(2) + b**S(2)), Int((a + b*tan(e + f*x))**(m + S(1))*Simp(a*c**S(2) - a*d**S(2) + S(2)*b*c*d - (-S(2)*a*c*d + b*c**S(2) - b*d**S(2))*tan(e + f*x), x), x), x) + Simp((a + b*tan(e + f*x))**(m + S(1))*(-a*d + b*c)**S(2)/(b*f*(a**S(2) + b**S(2))*(m + S(1))), x) def replacement3595(a, b, c, d, e, f, m, x): return Dist(S(1)/(a**S(2) + b**S(2)), Int((a + b/tan(e + f*x))**(m + S(1))*Simp(a*c**S(2) - a*d**S(2) + S(2)*b*c*d - (-S(2)*a*c*d + b*c**S(2) - b*d**S(2))/tan(e + f*x), x), x), x) - Simp((a + b/tan(e + f*x))**(m + S(1))*(-a*d + b*c)**S(2)/(b*f*(a**S(2) + b**S(2))*(m + S(1))), x) def replacement3596(a, b, c, d, e, f, m, x): return Int((a + b*tan(e + f*x))**m*Simp(c**S(2) + S(2)*c*d*tan(e + f*x) - d**S(2), x), x) + Simp(d**S(2)*(a + b*tan(e + f*x))**(m + S(1))/(b*f*(m + S(1))), x) def replacement3597(a, b, c, d, e, f, m, x): return Int((a + b/tan(e + f*x))**m*Simp(c**S(2) + S(2)*c*d/tan(e + f*x) - d**S(2), x), x) - Simp(d**S(2)*(a + b/tan(e + f*x))**(m + S(1))/(b*f*(m + S(1))), x) def replacement3598(a, b, c, d, e, f, x): return Dist(-S(2)*a*b/f, Subst(Int(S(1)/(-S(2)*a**S(2)*x**S(2) + a*c - b*d), x), x, sqrt(c + d*tan(e + f*x))/sqrt(a + b*tan(e + f*x))), x) def replacement3599(a, b, c, d, e, f, x): return Dist(S(2)*a*b/f, Subst(Int(S(1)/(-S(2)*a**S(2)*x**S(2) + a*c - b*d), x), x, sqrt(c + d/tan(e + f*x))/sqrt(a + b/tan(e + f*x))), x) def replacement3600(a, b, c, d, e, f, m, n, x): return Dist(S(2)*a**S(2)/(a*c - b*d), Int((a + b*tan(e + f*x))**(m + S(-1))*(c + d*tan(e + f*x))**(n + S(1)), x), x) + Simp(a*b*(a + b*tan(e + f*x))**(m + S(-1))*(c + d*tan(e + f*x))**(n + S(1))/(f*(m + S(-1))*(a*c - b*d)), x) def replacement3601(a, b, c, d, e, f, m, n, x): return Dist(S(2)*a**S(2)/(a*c - b*d), Int((a + b/tan(e + f*x))**(m + S(-1))*(c + d/tan(e + f*x))**(n + S(1)), x), x) - Simp(a*b*(a + b/tan(e + f*x))**(m + S(-1))*(c + d/tan(e + f*x))**(n + S(1))/(f*(m + S(-1))*(a*c - b*d)), x) def replacement3602(a, b, c, d, e, f, m, n, x): return -Dist((a*c - b*d)/(S(2)*b**S(2)), Int((a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**(n + S(-1)), x), x) + Simp(a*(a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**n/(S(2)*b*f*m), x) def replacement3603(a, b, c, d, e, f, m, n, x): return -Dist((a*c - b*d)/(S(2)*b**S(2)), Int((a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**(n + S(-1)), x), x) - Simp(a*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**n/(S(2)*b*f*m), x) def replacement3604(a, b, c, d, e, f, m, n, x): return Dist(S(1)/(S(2)*a), Int((a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**n, x), x) + Simp(a*(a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))/(S(2)*f*m*(-a*d + b*c)), x) def replacement3605(a, b, c, d, e, f, m, n, x): return Dist(S(1)/(S(2)*a), Int((a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**n, x), x) - Simp(a*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))/(S(2)*f*m*(-a*d + b*c)), x) def replacement3606(a, b, c, d, e, f, m, n, x): return Dist(a/(a*c - b*d), Int((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1)), x), x) - Simp(d*(a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))/(f*m*(c**S(2) + d**S(2))), x) def replacement3607(a, b, c, d, e, f, m, n, x): return Dist(a/(a*c - b*d), Int((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1)), x), x) + Simp(d*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))/(f*m*(c**S(2) + d**S(2))), x) def replacement3608(a, b, c, d, e, f, n, x): return Dist(S(1)/(S(2)*a*(-a*d + b*c)), Int((c + d*tan(e + f*x))**(n + S(-1))*Simp(a*c*d*(n + S(-1)) + b*c**S(2) + b*d**S(2)*n - d*(n + S(-1))*(-a*d + b*c)*tan(e + f*x), x), x), x) - Simp((c + d*tan(e + f*x))**n*(a*c + b*d)/(S(2)*f*(a + b*tan(e + f*x))*(-a*d + b*c)), x) def replacement3609(a, b, c, d, e, f, n, x): return Dist(S(1)/(S(2)*a*(-a*d + b*c)), Int((c + d/tan(e + f*x))**(n + S(-1))*Simp(a*c*d*(n + S(-1)) + b*c**S(2) + b*d**S(2)*n - d*(n + S(-1))*(-a*d + b*c)/tan(e + f*x), x), x), x) + Simp((c + d/tan(e + f*x))**n*(a*c + b*d)/(S(2)*f*(a + b/tan(e + f*x))*(-a*d + b*c)), x) def replacement3610(a, b, c, d, e, f, n, x): return Dist(S(1)/(S(2)*a**S(2)), Int((c + d*tan(e + f*x))**(n + S(-2))*Simp(a*c**S(2) + a*d**S(2)*(n + S(-1)) - b*c*d*n - d*(a*c*(n + S(-2)) + b*d*n)*tan(e + f*x), x), x), x) + Simp((c + d*tan(e + f*x))**(n + S(-1))*(-a*d + b*c)/(S(2)*a*f*(a + b*tan(e + f*x))), x) def replacement3611(a, b, c, d, e, f, n, x): return Dist(S(1)/(S(2)*a**S(2)), Int((c + d/tan(e + f*x))**(n + S(-2))*Simp(a*c**S(2) + a*d**S(2)*(n + S(-1)) - b*c*d*n - d*(a*c*(n + S(-2)) + b*d*n)/tan(e + f*x), x), x), x) - Simp((c + d/tan(e + f*x))**(n + S(-1))*(-a*d + b*c)/(S(2)*a*f*(a + b/tan(e + f*x))), x) def replacement3612(a, b, c, d, e, f, x): return Dist(b/(-a*d + b*c), Int(S(1)/(a + b*tan(e + f*x)), x), x) - Dist(d/(-a*d + b*c), Int(S(1)/(c + d*tan(e + f*x)), x), x) def replacement3613(a, b, c, d, e, f, x): return Dist(b/(-a*d + b*c), Int(S(1)/(a + b/tan(e + f*x)), x), x) - Dist(d/(-a*d + b*c), Int(S(1)/(c + d/tan(e + f*x)), x), x) def replacement3614(a, b, c, d, e, f, n, x): return Dist(S(1)/(S(2)*a*(-a*d + b*c)), Int((c + d*tan(e + f*x))**n*Simp(a*d*(n + S(-1)) + b*c - b*d*n*tan(e + f*x), x), x), x) - Simp(a*(c + d*tan(e + f*x))**(n + S(1))/(S(2)*f*(a + b*tan(e + f*x))*(-a*d + b*c)), x) def replacement3615(a, b, c, d, e, f, n, x): return Dist(S(1)/(S(2)*a*(-a*d + b*c)), Int((c + d/tan(e + f*x))**n*Simp(a*d*(n + S(-1)) + b*c - b*d*n/tan(e + f*x), x), x), x) + Simp(a*(c + d/tan(e + f*x))**(n + S(1))/(S(2)*f*(a + b/tan(e + f*x))*(-a*d + b*c)), x) def replacement3616(a, b, c, d, e, f, m, n, x): return Dist(a/(d*(n + S(1))*(a*d + b*c)), Int((a + b*tan(e + f*x))**(m + S(-2))*(c + d*tan(e + f*x))**(n + S(1))*Simp(b*(-a*d*(m - S(2)*n + S(-4)) + b*c*(m + S(-2))) + (-a**S(2)*d*(m + n + S(-1)) + a*b*c*(m + S(-2)) + b**S(2)*d*(n + S(1)))*tan(e + f*x), x), x), x) - Simp(a**S(2)*(a + b*tan(e + f*x))**(m + S(-2))*(c + d*tan(e + f*x))**(n + S(1))*(-a*d + b*c)/(d*f*(n + S(1))*(a*d + b*c)), x) def replacement3617(a, b, c, d, e, f, m, n, x): return Dist(a/(d*(n + S(1))*(a*d + b*c)), Int((a + b/tan(e + f*x))**(m + S(-2))*(c + d/tan(e + f*x))**(n + S(1))*Simp(b*(-a*d*(m - S(2)*n + S(-4)) + b*c*(m + S(-2))) + (-a**S(2)*d*(m + n + S(-1)) + a*b*c*(m + S(-2)) + b**S(2)*d*(n + S(1)))/tan(e + f*x), x), x), x) + Simp(a**S(2)*(a + b/tan(e + f*x))**(m + S(-2))*(c + d/tan(e + f*x))**(n + S(1))*(-a*d + b*c)/(d*f*(n + S(1))*(a*d + b*c)), x) def replacement3618(a, b, c, d, e, f, x): return Dist(S(2)*a**S(2)/(a*c - b*d), Int(sqrt(a + b*tan(e + f*x)), x), x) - Dist((a*(c**S(2) - d**S(2)) + S(2)*b*c*d)/(a*(c**S(2) + d**S(2))), Int((a - b*tan(e + f*x))*sqrt(a + b*tan(e + f*x))/(c + d*tan(e + f*x)), x), x) def replacement3619(a, b, c, d, e, f, x): return Dist(S(2)*a**S(2)/(a*c - b*d), Int(sqrt(a + b/tan(e + f*x)), x), x) - Dist((a*(c**S(2) - d**S(2)) + S(2)*b*c*d)/(a*(c**S(2) + d**S(2))), Int((a - b/tan(e + f*x))*sqrt(a + b/tan(e + f*x))/(c + d/tan(e + f*x)), x), x) def replacement3620(a, b, c, d, e, f, x): return Dist(S(2)*a, Int(sqrt(a + b*tan(e + f*x))/sqrt(c + d*tan(e + f*x)), x), x) + Dist(b/a, Int(sqrt(a + b*tan(e + f*x))*(a*tan(e + f*x) + b)/sqrt(c + d*tan(e + f*x)), x), x) def replacement3621(a, b, c, d, e, f, x): return Dist(S(2)*a, Int(sqrt(a + b/tan(e + f*x))/sqrt(c + d/tan(e + f*x)), x), x) + Dist(b/a, Int(sqrt(a + b/tan(e + f*x))*(a/tan(e + f*x) + b)/sqrt(c + d/tan(e + f*x)), x), x) def replacement3622(a, b, c, d, e, f, m, n, x): return Dist(a/(d*(m + n + S(-1))), Int((a + b*tan(e + f*x))**(m + S(-2))*(c + d*tan(e + f*x))**n*Simp(a*d*(m + S(2)*n) + b*c*(m + S(-2)) + (a*c*(m + S(-2)) + b*d*(S(3)*m + S(2)*n + S(-4)))*tan(e + f*x), x), x), x) + Simp(b**S(2)*(a + b*tan(e + f*x))**(m + S(-2))*(c + d*tan(e + f*x))**(n + S(1))/(d*f*(m + n + S(-1))), x) def replacement3623(a, b, c, d, e, f, m, n, x): return Dist(a/(d*(m + n + S(-1))), Int((a + b/tan(e + f*x))**(m + S(-2))*(c + d/tan(e + f*x))**n*Simp(a*d*(m + S(2)*n) + b*c*(m + S(-2)) + (a*c*(m + S(-2)) + b*d*(S(3)*m + S(2)*n + S(-4)))/tan(e + f*x), x), x), x) - Simp(b**S(2)*(a + b/tan(e + f*x))**(m + S(-2))*(c + d/tan(e + f*x))**(n + S(1))/(d*f*(m + n + S(-1))), x) def replacement3624(a, b, c, d, e, f, m, x): return Dist(S(1)/(S(4)*a**S(2)*m), Int((a + b*tan(e + f*x))**(m + S(1))*Simp(S(2)*a*c*m + a*d*(S(2)*m + S(1))*tan(e + f*x) + b*d, x)/sqrt(c + d*tan(e + f*x)), x), x) - Simp(b*(a + b*tan(e + f*x))**m*sqrt(c + d*tan(e + f*x))/(S(2)*a*f*m), x) def replacement3625(a, b, c, d, e, f, m, x): return Dist(S(1)/(S(4)*a**S(2)*m), Int((a + b/tan(e + f*x))**(m + S(1))*Simp(S(2)*a*c*m + a*d*(S(2)*m + S(1))/tan(e + f*x) + b*d, x)/sqrt(c + d/tan(e + f*x)), x), x) + Simp(b*(a + b/tan(e + f*x))**m*sqrt(c + d/tan(e + f*x))/(S(2)*a*f*m), x) def replacement3626(a, b, c, d, e, f, m, n, x): return Dist(S(1)/(S(2)*a**S(2)*m), Int((a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**(n + S(-2))*Simp(c*(a*c*m + b*d*(n + S(-1))) - d*(-a*c*(m + n + S(-1)) + b*d*(m - n + S(1)))*tan(e + f*x) - d*(a*d*(n + S(-1)) + b*c*m), x), x), x) - Simp((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(-1))*(-a*d + b*c)/(S(2)*a*f*m), x) def replacement3627(a, b, c, d, e, f, m, n, x): return Dist(S(1)/(S(2)*a**S(2)*m), Int((a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**(n + S(-2))*Simp(c*(a*c*m + b*d*(n + S(-1))) - d*(-a*c*(m + n + S(-1)) + b*d*(m - n + S(1)))/tan(e + f*x) - d*(a*d*(n + S(-1)) + b*c*m), x), x), x) + Simp((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(-1))*(-a*d + b*c)/(S(2)*a*f*m), x) def replacement3628(a, b, c, d, e, f, m, n, x): return Dist(S(1)/(S(2)*a*m*(-a*d + b*c)), Int((a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**n*Simp(-a*d*(S(2)*m + n + S(1)) + b*c*m + b*d*(m + n + S(1))*tan(e + f*x), x), x), x) + Simp(a*(a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))/(S(2)*f*m*(-a*d + b*c)), x) def replacement3629(a, b, c, d, e, f, m, n, x): return Dist(S(1)/(S(2)*a*m*(-a*d + b*c)), Int((a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**n*Simp(-a*d*(S(2)*m + n + S(1)) + b*c*m + b*d*(m + n + S(1))/tan(e + f*x), x), x), x) - Simp(a*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))/(S(2)*f*m*(-a*d + b*c)), x) def replacement3630(a, b, c, d, e, f, m, n, x): return -Dist(S(1)/(a*(m + n + S(-1))), Int((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(-2))*Simp(-a*c**S(2)*(m + n + S(-1)) + d*(-a*c*(m + S(2)*n + S(-2)) + b*d*m)*tan(e + f*x) + d*(a*d*(n + S(-1)) + b*c*m), x), x), x) + Simp(d*(a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(-1))/(f*(m + n + S(-1))), x) def replacement3631(a, b, c, d, e, f, m, n, x): return -Dist(S(1)/(a*(m + n + S(-1))), Int((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(-2))*Simp(-a*c**S(2)*(m + n + S(-1)) + d*(-a*c*(m + S(2)*n + S(-2)) + b*d*m)/tan(e + f*x) + d*(a*d*(n + S(-1)) + b*c*m), x), x), x) - Simp(d*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(-1))/(f*(m + n + S(-1))), x) def replacement3632(a, b, c, d, e, f, m, n, x): return -Dist(S(1)/(a*(c**S(2) + d**S(2))*(n + S(1))), Int((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))*Simp(-a*c*(n + S(1)) + a*d*(m + n + S(1))*tan(e + f*x) + b*d*m, x), x), x) + Simp(d*(a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))/(f*(c**S(2) + d**S(2))*(n + S(1))), x) def replacement3633(a, b, c, d, e, f, m, n, x): return -Dist(S(1)/(a*(c**S(2) + d**S(2))*(n + S(1))), Int((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))*Simp(-a*c*(n + S(1)) + a*d*(m + n + S(1))/tan(e + f*x) + b*d*m, x), x), x) - Simp(d*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))/(f*(c**S(2) + d**S(2))*(n + S(1))), x) def replacement3634(a, b, c, d, e, f, m, x): return Dist(a/(a*c - b*d), Int((a + b*tan(e + f*x))**m, x), x) - Dist(d/(a*c - b*d), Int((a + b*tan(e + f*x))**m*(a*tan(e + f*x) + b)/(c + d*tan(e + f*x)), x), x) def replacement3635(a, b, c, d, e, f, m, x): return Dist(a/(a*c - b*d), Int((a + b/tan(e + f*x))**m, x), x) - Dist(d/(a*c - b*d), Int((a + b/tan(e + f*x))**m*(a/tan(e + f*x) + b)/(c + d/tan(e + f*x)), x), x) def replacement3636(a, b, c, d, e, f, x): return Dist(d/a, Int(sqrt(a + b*tan(e + f*x))*(a*tan(e + f*x) + b)/sqrt(c + d*tan(e + f*x)), x), x) + Dist((a*c - b*d)/a, Int(sqrt(a + b*tan(e + f*x))/sqrt(c + d*tan(e + f*x)), x), x) def replacement3637(a, b, c, d, e, f, x): return Dist(d/a, Int(sqrt(a + b/tan(e + f*x))*(a/tan(e + f*x) + b)/sqrt(c + d/tan(e + f*x)), x), x) + Dist((a*c - b*d)/a, Int(sqrt(a + b/tan(e + f*x))/sqrt(c + d/tan(e + f*x)), x), x) def replacement3638(a, b, c, d, e, f, m, n, x): return Dist(a*b/f, Subst(Int((a + x)**(m + S(-1))*(c + d*x/b)**n/(a*x + b**S(2)), x), x, b*tan(e + f*x)), x) def replacement3639(a, b, c, d, e, f, m, n, x): return -Dist(a*b/f, Subst(Int((a + x)**(m + S(-1))*(c + d*x/b)**n/(a*x + b**S(2)), x), x, b/tan(e + f*x)), x) def replacement3640(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*tan(e + f*x))**(m + S(-3))*(c + d*tan(e + f*x))**(n + S(1))*Simp(a**S(2)*d*(-a*c*(n + S(1)) + b*d*(m + S(-2))) + b*(-S(2)*a*d + b*c)*(a*d*(n + S(1)) + b*c*(m + S(-2))) - b*(a*d*(-a*d + S(2)*b*c)*(m + n + S(-1)) - b**S(2)*(c**S(2)*(m + S(-2)) - d**S(2)*(n + S(1))))*tan(e + f*x)**S(2) - d*(n + S(1))*(-a**S(3)*d + S(3)*a**S(2)*b*c + S(3)*a*b**S(2)*d - b**S(3)*c)*tan(e + f*x), x), x), x) + Simp((a + b*tan(e + f*x))**(m + S(-2))*(c + d*tan(e + f*x))**(n + S(1))*(-a*d + b*c)**S(2)/(d*f*(c**S(2) + d**S(2))*(n + S(1))), x) def replacement3641(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/tan(e + f*x))**(m + S(-3))*(c + d/tan(e + f*x))**(n + S(1))*Simp(a**S(2)*d*(-a*c*(n + S(1)) + b*d*(m + S(-2))) + b*(-S(2)*a*d + b*c)*(a*d*(n + S(1)) + b*c*(m + S(-2))) - b*(a*d*(-a*d + S(2)*b*c)*(m + n + S(-1)) - b**S(2)*(c**S(2)*(m + S(-2)) - d**S(2)*(n + S(1))))/tan(e + f*x)**S(2) - d*(n + S(1))*(-a**S(3)*d + S(3)*a**S(2)*b*c + S(3)*a*b**S(2)*d - b**S(3)*c)/tan(e + f*x), x), x), x) - Simp((a + b/tan(e + f*x))**(m + S(-2))*(c + d/tan(e + f*x))**(n + S(1))*(-a*d + b*c)**S(2)/(d*f*(c**S(2) + d**S(2))*(n + S(1))), x) def replacement3642(a, b, c, d, e, f, m, n, x): return Dist(S(1)/(d*(m + n + S(-1))), Int((a + b*tan(e + f*x))**(m + S(-3))*(c + d*tan(e + f*x))**n*Simp(a**S(3)*d*(m + n + S(-1)) - 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(-4)) + b*c*(m + S(-2)))*tan(e + f*x)**S(2) + b*d*(S(3)*a**S(2) - b**S(2))*(m + n + S(-1))*tan(e + f*x), x), x), x) + Simp(b**S(2)*(a + b*tan(e + f*x))**(m + S(-2))*(c + d*tan(e + f*x))**(n + S(1))/(d*f*(m + n + S(-1))), x) def replacement3643(a, b, c, d, e, f, m, n, x): return Dist(S(1)/(d*(m + n + S(-1))), Int((a + b/tan(e + f*x))**(m + S(-3))*(c + d/tan(e + f*x))**n*Simp(a**S(3)*d*(m + n + S(-1)) - 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(-4)) + b*c*(m + S(-2)))/tan(e + f*x)**S(2) + b*d*(S(3)*a**S(2) - b**S(2))*(m + n + S(-1))/tan(e + f*x), x), x), x) - Simp(b**S(2)*(a + b/tan(e + f*x))**(m + S(-2))*(c + d/tan(e + f*x))**(n + S(1))/(d*f*(m + n + S(-1))), x) def replacement3644(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*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**(n + S(-2))*Simp(a*c**S(2)*(m + S(1)) + a*d**S(2)*(n + S(-1)) + b*c*d*(m - n + S(2)) - d*(m + n)*(-a*d + b*c)*tan(e + f*x)**S(2) - (m + S(1))*(-S(2)*a*c*d + b*c**S(2) - b*d**S(2))*tan(e + f*x), x), x), x) + Simp((a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**(n + S(-1))*(-a*d + b*c)/(f*(a**S(2) + b**S(2))*(m + S(1))), x) def replacement3645(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/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**(n + S(-2))*Simp(a*c**S(2)*(m + S(1)) + a*d**S(2)*(n + S(-1)) + b*c*d*(m - n + S(2)) - d*(m + n)*(-a*d + b*c)/tan(e + f*x)**S(2) - (m + S(1))*(-S(2)*a*c*d + b*c**S(2) - b*d**S(2))/tan(e + f*x), x), x), x) - Simp((a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**(n + S(-1))*(-a*d + b*c)/(f*(a**S(2) + b**S(2))*(m + S(1))), x) def replacement3646(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*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**(n + S(-1))*Simp(a*c*(m + S(1)) - b*d*n - b*d*(m + n + S(1))*tan(e + f*x)**S(2) - (m + S(1))*(-a*d + b*c)*tan(e + f*x), x), x), x) + Simp(b*(a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**n/(f*(a**S(2) + b**S(2))*(m + S(1))), x) def replacement3647(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/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**(n + S(-1))*Simp(a*c*(m + S(1)) - b*d*n - b*d*(m + n + S(1))/tan(e + f*x)**S(2) - (m + S(1))*(-a*d + b*c)/tan(e + f*x), x), x), x) - Simp(b*(a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**n/(f*(a**S(2) + b**S(2))*(m + S(1))), x) def replacement3648(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*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**n*Simp(a*(m + S(1))*(-a*d + b*c) - b**S(2)*d*(m + n + S(2))*tan(e + f*x)**S(2) - b**S(2)*d*(m + n + S(2)) - b*(m + S(1))*(-a*d + b*c)*tan(e + f*x), x), x), x) + Simp(b**S(2)*(a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**(n + S(1))/(f*(a**S(2) + b**S(2))*(m + S(1))*(-a*d + b*c)), x) def replacement3649(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/tan(e + f*x))**(m + S(1))*(c + d/tan(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(2))/tan(e + f*x)**S(2) - b*(m + S(1))*(-a*d + b*c)/tan(e + f*x), x), x), x) - Simp(b**S(2)*(a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**(n + S(1))/(f*(a**S(2) + b**S(2))*(m + S(1))*(-a*d + b*c)), x) def replacement3650(a, b, c, d, e, f, m, n, x): return Dist(S(1)/(m + n + S(-1)), Int((a + b*tan(e + f*x))**(m + S(-2))*(c + d*tan(e + f*x))**(n + S(-1))*Simp(a**S(2)*c*(m + n + S(-1)) - b*(a*d*n + b*c*(m + S(-1))) + b*(a*d*(S(2)*m + n + S(-2)) + b*c*n)*tan(e + f*x)**S(2) + (m + n + S(-1))*(a**S(2)*d + S(2)*a*b*c - b**S(2)*d)*tan(e + f*x), x), x), x) + Simp(b*(a + b*tan(e + f*x))**(m + S(-1))*(c + d*tan(e + f*x))**n/(f*(m + n + S(-1))), x) def replacement3651(a, b, c, d, e, f, m, n, x): return Dist(S(1)/(m + n + S(-1)), Int((a + b/tan(e + f*x))**(m + S(-2))*(c + d/tan(e + f*x))**(n + S(-1))*Simp(a**S(2)*c*(m + n + S(-1)) - b*(a*d*n + b*c*(m + S(-1))) + b*(a*d*(S(2)*m + n + S(-2)) + b*c*n)/tan(e + f*x)**S(2) + (m + n + S(-1))*(a**S(2)*d + S(2)*a*b*c - b**S(2)*d)/tan(e + f*x), x), x), x) - Simp(b*(a + b/tan(e + f*x))**(m + S(-1))*(c + d/tan(e + f*x))**n/(f*(m + n + S(-1))), x) def replacement3652(a, b, c, d, e, f, x): return Dist(b**S(2)/((a**S(2) + b**S(2))*(-a*d + b*c)), Int((-a*tan(e + f*x) + b)/(a + b*tan(e + f*x)), x), x) - Dist(d**S(2)/((c**S(2) + d**S(2))*(-a*d + b*c)), Int((-c*tan(e + f*x) + d)/(c + d*tan(e + f*x)), x), x) + Simp(x*(a*c - b*d)/((a**S(2) + b**S(2))*(c**S(2) + d**S(2))), x) def replacement3653(a, b, c, d, e, f, x): return Dist(b**S(2)/((a**S(2) + b**S(2))*(-a*d + b*c)), Int((-a/tan(e + f*x) + b)/(a + b/tan(e + f*x)), x), x) - Dist(d**S(2)/((c**S(2) + d**S(2))*(-a*d + b*c)), Int((-c/tan(e + f*x) + d)/(c + d/tan(e + f*x)), x), x) + Simp(x*(a*c - b*d)/((a**S(2) + b**S(2))*(c**S(2) + d**S(2))), x) def replacement3654(a, b, c, d, e, f, x): return -Dist(d*(-a*d + b*c)/(c**S(2) + d**S(2)), Int((tan(e + f*x)**S(2) + S(1))/(sqrt(a + b*tan(e + f*x))*(c + d*tan(e + f*x))), x), x) + Dist(S(1)/(c**S(2) + d**S(2)), Int(Simp(a*c + b*d + (-a*d + b*c)*tan(e + f*x), x)/sqrt(a + b*tan(e + f*x)), x), x) def replacement3655(a, b, c, d, e, f, x): return -Dist(d*(-a*d + b*c)/(c**S(2) + d**S(2)), Int((S(1) + tan(e + f*x)**(S(-2)))/(sqrt(a + b/tan(e + f*x))*(c + d/tan(e + f*x))), x), x) + Dist(S(1)/(c**S(2) + d**S(2)), Int(Simp(a*c + b*d + (-a*d + b*c)/tan(e + f*x), x)/sqrt(a + b/tan(e + f*x)), x), x) def replacement3656(a, b, c, d, e, f, x): return Dist((-a*d + b*c)**S(2)/(c**S(2) + d**S(2)), Int((tan(e + f*x)**S(2) + S(1))/(sqrt(a + b*tan(e + f*x))*(c + d*tan(e + f*x))), x), x) + Dist(S(1)/(c**S(2) + d**S(2)), Int(Simp(a**S(2)*c + S(2)*a*b*d - b**S(2)*c + (-a**S(2)*d + S(2)*a*b*c + b**S(2)*d)*tan(e + f*x), x)/sqrt(a + b*tan(e + f*x)), x), x) def replacement3657(a, b, c, d, e, f, x): return Dist((-a*d + b*c)**S(2)/(c**S(2) + d**S(2)), Int((S(1) + tan(e + f*x)**(S(-2)))/(sqrt(a + b/tan(e + f*x))*(c + d/tan(e + f*x))), x), x) + Dist(S(1)/(c**S(2) + d**S(2)), Int(Simp(a**S(2)*c + S(2)*a*b*d - b**S(2)*c + (-a**S(2)*d + S(2)*a*b*c + b**S(2)*d)/tan(e + f*x), x)/sqrt(a + b/tan(e + f*x)), x), x) def replacement3658(a, b, c, d, e, f, m, x): return Dist(d**S(2)/(c**S(2) + d**S(2)), Int((a + b*tan(e + f*x))**m*(tan(e + f*x)**S(2) + S(1))/(c + d*tan(e + f*x)), x), x) + Dist(S(1)/(c**S(2) + d**S(2)), Int((a + b*tan(e + f*x))**m*(c - d*tan(e + f*x)), x), x) def replacement3659(a, b, c, d, e, f, m, x): return Dist(d**S(2)/(c**S(2) + d**S(2)), Int((S(1) + tan(e + f*x)**(S(-2)))*(a + b/tan(e + f*x))**m/(c + d/tan(e + f*x)), x), x) + Dist(S(1)/(c**S(2) + d**S(2)), Int((a + b/tan(e + f*x))**m*(c - d/tan(e + f*x)), x), x) def replacement3660(a, b, c, d, e, f, m, n, x): return Dist(b/f, Subst(Int((a + x)**m*(c + d*x/b)**n/(b**S(2) + x**S(2)), x), x, b*tan(e + f*x)), x) def replacement3661(a, b, c, d, e, f, m, n, x): return -Dist(b/f, Subst(Int((a + x)**m*(c + d*x/b)**n/(b**S(2) + x**S(2)), x), x, b/tan(e + f*x)), x) def replacement3662(a, b, d, e, f, m, n, x): return Dist(d**m, Int((d/tan(e + f*x))**(-m + n)*(a/tan(e + f*x) + b)**m, x), x) def replacement3663(a, b, d, e, f, m, n, x): return Dist(d**m, Int((d*tan(e + f*x))**(-m + n)*(a*tan(e + f*x) + b)**m, x), x) def replacement3664(a, b, c, d, e, f, m, n, p, x): return Dist(c**IntPart(n)*(c*(d*tan(e + f*x))**p)**FracPart(n)*(d*tan(e + f*x))**(-p*FracPart(n)), Int((d*tan(e + f*x))**(n*p)*(a + b*tan(e + f*x))**m, x), x) def replacement3665(a, b, c, d, e, f, m, n, p, x): return Dist(c**IntPart(n)*(c*(d/tan(e + f*x))**p)**FracPart(n)*(d/tan(e + f*x))**(-p*FracPart(n)), Int((d/tan(e + f*x))**(n*p)*(a + b/tan(e + f*x))**m, x), x) def replacement3666(a, b, c, d, e, f, g, m, n, p, x): return Int((g*tan(e + f*x))**p*(a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**n, x) def replacement3667(a, b, c, d, e, f, g, m, n, p, x): return Int((g/tan(e + f*x))**p*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**n, x) def replacement3668(a, b, c, d, e, f, g, m, n, p, x): return Dist(g**(m + n), Int((g/tan(e + f*x))**(-m - n + p)*(a/tan(e + f*x) + b)**m*(c/tan(e + f*x) + d)**n, x), x) def replacement3669(a, b, c, d, e, f, g, m, n, p, x): return Dist(g**(m + n), Int((g*tan(e + f*x))**(-m - n + p)*(a*tan(e + f*x) + b)**m*(c*tan(e + f*x) + d)**n, x), x) def replacement3670(a, b, c, d, e, f, g, m, n, p, q, x): return Dist((g*tan(e + f*x))**(-p*q)*(g*tan(e + f*x)**q)**p, Int((g*tan(e + f*x))**(p*q)*(a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**n, x), x) def replacement3671(a, b, c, d, e, f, g, m, n, p, q, x): return Dist((g*(S(1)/tan(e + f*x))**q)**p*(g/tan(e + f*x))**(-p*q), Int((g/tan(e + f*x))**(p*q)*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**n, x), x) def replacement3672(a, b, c, d, e, f, g, m, n, p, x): return Dist(g**n, Int((g*tan(e + f*x))**(-n + p)*(a + b*tan(e + f*x))**m*(c*tan(e + f*x) + d)**n, x), x) def replacement3673(a, b, c, d, e, f, g, m, n, p, x): return Dist(g**n, Int((g/tan(e + f*x))**(-n + p)*(a + b/tan(e + f*x))**m*(c/tan(e + f*x) + d)**n, x), x) def replacement3674(a, b, c, d, e, f, m, n, p, x): return Int((c + d/tan(e + f*x))**n*(a/tan(e + f*x) + b)**m*(S(1)/tan(e + f*x))**(-m - p), x) def replacement3675(a, b, c, d, e, f, m, n, p, x): return Int((c + d*tan(e + f*x))**n*(a*tan(e + f*x) + b)**m*tan(e + f*x)**(-m - p), x) def replacement3676(a, b, c, d, e, f, g, m, n, p, x): return Dist((g*tan(e + f*x))**p*(S(1)/tan(e + f*x))**p, Int((c + d/tan(e + f*x))**n*(a/tan(e + f*x) + b)**m*(S(1)/tan(e + f*x))**(-m - p), x), x) def replacement3677(a, b, c, d, e, f, g, m, n, p, x): return Dist((g/tan(e + f*x))**p*tan(e + f*x)**p, Int((c + d*tan(e + f*x))**n*(a*tan(e + f*x) + b)**m*tan(e + f*x)**(-m - p), x), x) def replacement3678(a, b, c, d, e, f, g, m, n, p, x): return Dist((g*tan(e + f*x))**n*(c + d/tan(e + f*x))**n*(c*tan(e + f*x) + d)**(-n), Int((g*tan(e + f*x))**(-n + p)*(a + b*tan(e + f*x))**m*(c*tan(e + f*x) + d)**n, x), x) def replacement3679(a, b, c, d, e, f, g, m, n, p, x): return Dist((g/tan(e + f*x))**n*(c + d*tan(e + f*x))**n*(c/tan(e + f*x) + d)**(-n), Int((g/tan(e + f*x))**(-n + p)*(a + b/tan(e + f*x))**m*(c/tan(e + f*x) + d)**n, x), x) def replacement3680(A, B, a, b, c, d, e, f, m, n, x): return Dist(a*c/f, Subst(Int((A + B*x)*(a + b*x)**(m + S(-1))*(c + d*x)**(n + S(-1)), x), x, tan(e + f*x)), x) def replacement3681(A, B, a, b, c, d, e, f, m, n, x): return -Dist(a*c/f, Subst(Int((A + B*x)*(a + b*x)**(m + S(-1))*(c + d*x)**(n + S(-1)), x), x, S(1)/tan(e + f*x)), x) def replacement3682(A, B, a, b, c, d, e, f, x): return Dist(S(1)/b, Int(Simp(A*b*c + (A*b*d + B*(-a*d + b*c))*tan(e + f*x), x)/(a + b*tan(e + f*x)), x), x) + Dist(B*d/b, Int(tan(e + f*x), x), x) def replacement3683(A, B, a, b, c, d, e, f, x): return Dist(S(1)/b, Int(Simp(A*b*c + (A*b*d + B*(-a*d + b*c))/tan(e + f*x), x)/(a + b/tan(e + f*x)), x), x) + Dist(B*d/b, Int(S(1)/tan(e + f*x), x), x) def replacement3684(A, B, a, b, c, d, e, f, m, x): return Dist(S(1)/(S(2)*a*b), Int((a + b*tan(e + f*x))**(m + S(1))*Simp(A*a*d + A*b*c + B*a*c + S(2)*B*a*d*tan(e + f*x) + B*b*d, x), x), x) - Simp((a + b*tan(e + f*x))**m*(A*b - B*a)*(a*c + b*d)/(S(2)*a**S(2)*f*m), x) def replacement3685(A, B, a, b, c, d, e, f, m, x): return Dist(S(1)/(S(2)*a*b), Int((a + b/tan(e + f*x))**(m + S(1))*Simp(A*a*d + A*b*c + B*a*c + S(2)*B*a*d/tan(e + f*x) + B*b*d, x), x), x) + Simp((a + b/tan(e + f*x))**m*(A*b - B*a)*(a*c + b*d)/(S(2)*a**S(2)*f*m), x) def replacement3686(A, B, a, b, c, d, e, f, m, x): return Dist(S(1)/(a**S(2) + b**S(2)), Int((a + b*tan(e + f*x))**(m + S(1))*Simp(A*a*c + A*b*d - B*a*d + B*b*c - (-A*a*d + A*b*c - B*a*c - B*b*d)*tan(e + f*x), x), x), x) + Simp((a + b*tan(e + f*x))**(m + S(1))*(A*b - B*a)*(-a*d + b*c)/(b*f*(a**S(2) + b**S(2))*(m + S(1))), x) def replacement3687(A, B, a, b, c, d, e, f, m, x): return Dist(S(1)/(a**S(2) + b**S(2)), Int((a + b/tan(e + f*x))**(m + S(1))*Simp(A*a*c + A*b*d - B*a*d + B*b*c - (-A*a*d + A*b*c - B*a*c - B*b*d)/tan(e + f*x), x), x), x) - Simp((a + b/tan(e + f*x))**(m + S(1))*(A*b - B*a)*(-a*d + b*c)/(b*f*(a**S(2) + b**S(2))*(m + S(1))), x) def replacement3688(A, B, a, b, c, d, e, f, m, x): return Int((a + b*tan(e + f*x))**m*Simp(A*c - B*d + (A*d + B*c)*tan(e + f*x), x), x) + Simp(B*d*(a + b*tan(e + f*x))**(m + S(1))/(b*f*(m + S(1))), x) def replacement3689(A, B, a, b, c, d, e, f, m, x): return Int((a + b/tan(e + f*x))**m*Simp(A*c - B*d + (A*d + B*c)/tan(e + f*x), x), x) - Simp(B*d*(a + b/tan(e + f*x))**(m + S(1))/(b*f*(m + S(1))), x) def replacement3690(A, B, a, b, c, d, e, f, m, n, x): return -Dist(a/(d*(n + S(1))*(a*d + b*c)), Int((a + b*tan(e + f*x))**(m + S(-1))*(c + d*tan(e + f*x))**(n + S(1))*Simp(A*b*d*(m - n + S(-2)) - B*(a*d*(n + S(1)) + b*c*(m + S(-1))) + (A*a*d*(m + n) - B*(a*c*(m + S(-1)) + b*d*(n + S(1))))*tan(e + f*x), x), x), x) - Simp(a**S(2)*(a + b*tan(e + f*x))**(m + S(-1))*(c + d*tan(e + f*x))**(n + S(1))*(-A*d + B*c)/(d*f*(n + S(1))*(a*d + b*c)), x) def replacement3691(A, B, a, b, c, d, e, f, m, n, x): return -Dist(a/(d*(n + S(1))*(a*d + b*c)), Int((a + b/tan(e + f*x))**(m + S(-1))*(c + d/tan(e + f*x))**(n + S(1))*Simp(A*b*d*(m - n + S(-2)) - B*(a*d*(n + S(1)) + b*c*(m + S(-1))) + (A*a*d*(m + n) - B*(a*c*(m + S(-1)) + b*d*(n + S(1))))/tan(e + f*x), x), x), x) + Simp(a**S(2)*(a + b/tan(e + f*x))**(m + S(-1))*(c + d/tan(e + f*x))**(n + S(1))*(-A*d + B*c)/(d*f*(n + S(1))*(a*d + b*c)), x) def replacement3692(A, B, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(d*(m + n)), Int((a + b*tan(e + f*x))**(m + S(-1))*(c + d*tan(e + f*x))**n*Simp(A*a*d*(m + n) + B*(a*c*(m + S(-1)) - b*d*(n + S(1))) - (B*(m + S(-1))*(-a*d + b*c) - d*(m + n)*(A*b + B*a))*tan(e + f*x), x), x), x) + Simp(B*b*(a + b*tan(e + f*x))**(m + S(-1))*(c + d*tan(e + f*x))**(n + S(1))/(d*f*(m + n)), x) def replacement3693(A, B, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(d*(m + n)), Int((a + b/tan(e + f*x))**(m + S(-1))*(c + d/tan(e + f*x))**n*Simp(A*a*d*(m + n) + B*(a*c*(m + S(-1)) - b*d*(n + S(1))) - (B*(m + S(-1))*(-a*d + b*c) - d*(m + n)*(A*b + B*a))/tan(e + f*x), x), x), x) - Simp(B*b*(a + b/tan(e + f*x))**(m + S(-1))*(c + d/tan(e + f*x))**(n + S(1))/(d*f*(m + n)), x) def replacement3694(A, B, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(S(2)*a**S(2)*m), Int((a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**(n + S(-1))*Simp(A*(a*c*m + b*d*n) - B*(a*d*n + b*c*m) - d*(-A*a*(m + n) + B*b*(m - n))*tan(e + f*x), x), x), x) - Simp((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**n*(A*b - B*a)/(S(2)*a*f*m), x) def replacement3695(A, B, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(S(2)*a**S(2)*m), Int((a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**(n + S(-1))*Simp(A*(a*c*m + b*d*n) - B*(a*d*n + b*c*m) - d*(-A*a*(m + n) + B*b*(m - n))/tan(e + f*x), x), x), x) + Simp((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**n*(A*b - B*a)/(S(2)*a*f*m), x) def replacement3696(A, B, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(S(2)*a*m*(-a*d + b*c)), Int((a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**n*Simp(A*(-a*d*(S(2)*m + n + S(1)) + b*c*m) + B*(a*c*m - b*d*(n + S(1))) + d*(A*b - B*a)*(m + n + S(1))*tan(e + f*x), x), x), x) + Simp((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))*(A*a + B*b)/(S(2)*f*m*(-a*d + b*c)), x) def replacement3697(A, B, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(S(2)*a*m*(-a*d + b*c)), Int((a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**n*Simp(A*(-a*d*(S(2)*m + n + S(1)) + b*c*m) + B*(a*c*m - b*d*(n + S(1))) + d*(A*b - B*a)*(m + n + S(1))/tan(e + f*x), x), x), x) - Simp((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))*(A*a + B*b)/(S(2)*f*m*(-a*d + b*c)), x) def replacement3698(A, B, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(a*(m + n)), Int((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(-1))*Simp(A*a*c*(m + n) - B*(a*d*n + b*c*m) + (A*a*d*(m + n) - B*(-a*c*n + b*d*m))*tan(e + f*x), x), x), x) + Simp(B*(a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**n/(f*(m + n)), x) def replacement3699(A, B, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(a*(m + n)), Int((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(-1))*Simp(A*a*c*(m + n) - B*(a*d*n + b*c*m) + (A*a*d*(m + n) - B*(-a*c*n + b*d*m))/tan(e + f*x), x), x), x) - Simp(B*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**n/(f*(m + n)), x) def replacement3700(A, B, a, b, c, d, e, f, m, n, x): return -Dist(S(1)/(a*(c**S(2) + d**S(2))*(n + S(1))), Int((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))*Simp(A*(-a*c*(n + S(1)) + b*d*m) - B*(a*d*(n + S(1)) + b*c*m) - a*(-A*d + B*c)*(m + n + S(1))*tan(e + f*x), x), x), x) + Simp((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))*(A*d - B*c)/(f*(c**S(2) + d**S(2))*(n + S(1))), x) def replacement3701(A, B, a, b, c, d, e, f, m, n, x): return -Dist(S(1)/(a*(c**S(2) + d**S(2))*(n + S(1))), Int((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))*Simp(A*(-a*c*(n + S(1)) + b*d*m) - B*(a*d*(n + S(1)) + b*c*m) - a*(-A*d + B*c)*(m + n + S(1))/tan(e + f*x), x), x), x) - Simp((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))*(A*d - B*c)/(f*(c**S(2) + d**S(2))*(n + S(1))), x) def replacement3702(A, B, a, b, c, d, e, f, m, n, x): return Dist(B*b/f, Subst(Int((a + b*x)**(m + S(-1))*(c + d*x)**n, x), x, tan(e + f*x)), x) def replacement3703(A, B, a, b, c, d, e, f, m, n, x): return -Dist(B*b/f, Subst(Int((a + b*x)**(m + S(-1))*(c + d*x)**n, x), x, S(1)/tan(e + f*x)), x) def replacement3704(A, B, a, b, c, d, e, f, m, x): return Dist((A*b + B*a)/(a*d + b*c), Int((a + b*tan(e + f*x))**m, x), x) - Dist((-A*d + B*c)/(a*d + b*c), Int((a - b*tan(e + f*x))*(a + b*tan(e + f*x))**m/(c + d*tan(e + f*x)), x), x) def replacement3705(A, B, a, b, c, d, e, f, m, x): return Dist((A*b + B*a)/(a*d + b*c), Int((a + b/tan(e + f*x))**m, x), x) - Dist((-A*d + B*c)/(a*d + b*c), Int((a - b/tan(e + f*x))*(a + b/tan(e + f*x))**m/(c + d/tan(e + f*x)), x), x) def replacement3706(A, B, a, b, c, d, e, f, m, n, x): return -Dist(B/b, Int((a - b*tan(e + f*x))*(a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**n, x), x) + Dist((A*b + B*a)/b, Int((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**n, x), x) def replacement3707(A, B, a, b, c, d, e, f, m, n, x): return -Dist(B/b, Int((a - b/tan(e + f*x))*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**n, x), x) + Dist((A*b + B*a)/b, Int((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**n, x), x) def replacement3708(A, B, a, b, c, d, e, f, m, n, x): return Dist(A**S(2)/f, Subst(Int((a + b*x)**m*(c + d*x)**n/(A - B*x), x), x, tan(e + f*x)), x) def replacement3709(A, B, a, b, c, d, e, f, m, n, x): return -Dist(A**S(2)/f, Subst(Int((a + b*x)**m*(c + d*x)**n/(A - B*x), x), x, S(1)/tan(e + f*x)), x) def replacement3710(A, B, a, b, c, d, e, f, m, n, x): return Dist(A/S(2) - I*B/S(2), Int((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**n*(I*tan(e + f*x) + S(1)), x), x) + Dist(A/S(2) + I*B/S(2), Int((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**n*(-I*tan(e + f*x) + S(1)), x), x) def replacement3711(A, B, a, b, c, d, e, f, m, n, x): return Dist(A/S(2) - I*B/S(2), Int((S(1) + I/tan(e + f*x))*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**n, x), x) + Dist(A/S(2) + I*B/S(2), Int((S(1) - I/tan(e + f*x))*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**n, x), x) def replacement3712(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*tan(e + f*x))**(m + S(-2))*(c + d*tan(e + f*x))**(n + S(1))*Simp(A*a*d*(-a*c*(n + S(1)) + b*d*(m + S(-1))) - b*(-B*b*(c**S(2)*(m + S(-1)) - d**S(2)*(n + S(1))) + d*(m + n)*(-A*a*d + A*b*c + B*a*c))*tan(e + f*x)**S(2) - d*(n + S(1))*((A*a - B*b)*(-a*d + b*c) + (A*b + B*a)*(a*c + b*d))*tan(e + f*x) + (B*b*c - d*(A*b + B*a))*(a*d*(n + S(1)) + b*c*(m + S(-1))), x), x), x) + Simp((a + b*tan(e + f*x))**(m + S(-1))*(c + d*tan(e + f*x))**(n + S(1))*(-A*d + B*c)*(-a*d + b*c)/(d*f*(c**S(2) + d**S(2))*(n + S(1))), x) def replacement3713(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/tan(e + f*x))**(m + S(-2))*(c + d/tan(e + f*x))**(n + S(1))*Simp(A*a*d*(-a*c*(n + S(1)) + b*d*(m + S(-1))) - b*(-B*b*(c**S(2)*(m + S(-1)) - d**S(2)*(n + S(1))) + d*(m + n)*(-A*a*d + A*b*c + B*a*c))/tan(e + f*x)**S(2) - d*(n + S(1))*((A*a - B*b)*(-a*d + b*c) + (A*b + B*a)*(a*c + b*d))/tan(e + f*x) + (B*b*c - d*(A*b + B*a))*(a*d*(n + S(1)) + b*c*(m + S(-1))), x), x), x) - Simp((a + b/tan(e + f*x))**(m + S(-1))*(c + d/tan(e + f*x))**(n + S(1))*(-A*d + B*c)*(-a*d + b*c)/(d*f*(c**S(2) + d**S(2))*(n + S(1))), x) def replacement3714(A, B, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(d*(m + n)), Int((a + b*tan(e + f*x))**(m + S(-2))*(c + d*tan(e + f*x))**n*Simp(A*a**S(2)*d*(m + n) - B*b*(a*d*(n + S(1)) + b*c*(m + S(-1))) + d*(m + n)*(S(2)*A*a*b + B*(a**S(2) - b**S(2)))*tan(e + f*x) - (B*b*(m + S(-1))*(-a*d + b*c) - b*d*(m + n)*(A*b + B*a))*tan(e + f*x)**S(2), x), x), x) + Simp(B*b*(a + b*tan(e + f*x))**(m + S(-1))*(c + d*tan(e + f*x))**(n + S(1))/(d*f*(m + n)), x) def replacement3715(A, B, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(d*(m + n)), Int((a + b/tan(e + f*x))**(m + S(-2))*(c + d/tan(e + f*x))**n*Simp(A*a**S(2)*d*(m + n) - B*b*(a*d*(n + S(1)) + b*c*(m + S(-1))) + d*(m + n)*(S(2)*A*a*b + B*(a**S(2) - b**S(2)))/tan(e + f*x) - (B*b*(m + S(-1))*(-a*d + b*c) - b*d*(m + n)*(A*b + B*a))/tan(e + f*x)**S(2), x), x), x) - Simp(B*b*(a + b/tan(e + f*x))**(m + S(-1))*(c + d/tan(e + f*x))**(n + S(1))/(d*f*(m + n)), x) def replacement3716(A, B, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(b*(a**S(2) + b**S(2))*(m + S(1))), Int((a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**(n + S(-1))*Simp(A*b*(a*c*(m + S(1)) - b*d*n) + B*b*(a*d*n + b*c*(m + S(1))) - b*d*(A*b - B*a)*(m + n + S(1))*tan(e + f*x)**S(2) - b*(m + S(1))*(A*(-a*d + b*c) - B*(a*c + b*d))*tan(e + f*x), x), x), x) + Simp((a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**n*(A*b - B*a)/(f*(a**S(2) + b**S(2))*(m + S(1))), x) def replacement3717(A, B, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(b*(a**S(2) + b**S(2))*(m + S(1))), Int((a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**(n + S(-1))*Simp(A*b*(a*c*(m + S(1)) - b*d*n) + B*b*(a*d*n + b*c*(m + S(1))) - b*d*(A*b - B*a)*(m + n + S(1))/tan(e + f*x)**S(2) - b*(m + S(1))*(A*(-a*d + b*c) - B*(a*c + b*d))/tan(e + f*x), x), x), x) - Simp((a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**n*(A*b - B*a)/(f*(a**S(2) + b**S(2))*(m + S(1))), x) def replacement3718(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*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**n*Simp(A*(a*(m + S(1))*(-a*d + b*c) - b**S(2)*d*(m + n + S(2))) + B*b*(a*d*(n + S(1)) + b*c*(m + S(1))) - b*d*(A*b - B*a)*(m + n + S(2))*tan(e + f*x)**S(2) - (m + S(1))*(A*b - B*a)*(-a*d + b*c)*tan(e + f*x), x), x), x) + Simp(b*(a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**(n + S(1))*(A*b - B*a)/(f*(a**S(2) + b**S(2))*(m + S(1))*(-a*d + b*c)), x) def replacement3719(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/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**n*Simp(A*(a*(m + S(1))*(-a*d + b*c) - b**S(2)*d*(m + n + S(2))) + B*b*(a*d*(n + S(1)) + b*c*(m + S(1))) - b*d*(A*b - B*a)*(m + n + S(2))/tan(e + f*x)**S(2) - (m + S(1))*(A*b - B*a)*(-a*d + b*c)/tan(e + f*x), x), x), x) - Simp(b*(a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**(n + S(1))*(A*b - B*a)/(f*(a**S(2) + b**S(2))*(m + S(1))*(-a*d + b*c)), x) def replacement3720(A, B, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(m + n), Int((a + b*tan(e + f*x))**(m + S(-1))*(c + d*tan(e + f*x))**(n + S(-1))*Simp(A*a*c*(m + n) - B*(a*d*n + b*c*m) + (m + n)*(A*a*d + A*b*c + B*a*c - B*b*d)*tan(e + f*x) + (A*b*d*(m + n) + B*(a*d*m + b*c*n))*tan(e + f*x)**S(2), x), x), x) + Simp(B*(a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**n/(f*(m + n)), x) def replacement3721(A, B, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(m + n), Int((a + b/tan(e + f*x))**(m + S(-1))*(c + d/tan(e + f*x))**(n + S(-1))*Simp(A*a*c*(m + n) - B*(a*d*n + b*c*m) + (m + n)*(A*a*d + A*b*c + B*a*c - B*b*d)/tan(e + f*x) + (A*b*d*(m + n) + B*(a*d*m + b*c*n))/tan(e + f*x)**S(2), x), x), x) - Simp(B*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**n/(f*(m + n)), x) def replacement3722(A, B, a, b, c, d, e, f, x): return Dist(b*(A*b - B*a)/((a**S(2) + b**S(2))*(-a*d + b*c)), Int((-a*tan(e + f*x) + b)/(a + b*tan(e + f*x)), x), x) + Dist(d*(-A*d + B*c)/((c**S(2) + d**S(2))*(-a*d + b*c)), Int((-c*tan(e + f*x) + d)/(c + d*tan(e + f*x)), x), x) + Simp(x*(A*(a*c - b*d) + B*(a*d + b*c))/((a**S(2) + b**S(2))*(c**S(2) + d**S(2))), x) def replacement3723(A, B, a, b, c, d, e, f, x): return Dist(b*(A*b - B*a)/((a**S(2) + b**S(2))*(-a*d + b*c)), Int((-a/tan(e + f*x) + b)/(a + b/tan(e + f*x)), x), x) + Dist(d*(-A*d + B*c)/((c**S(2) + d**S(2))*(-a*d + b*c)), Int((-c/tan(e + f*x) + d)/(c + d/tan(e + f*x)), x), x) + Simp(x*(A*(a*c - b*d) + B*(a*d + b*c))/((a**S(2) + b**S(2))*(c**S(2) + d**S(2))), x) def replacement3724(A, B, a, b, c, d, e, f, x): return -Dist((-A*b + B*a)*(-a*d + b*c)/(a**S(2) + b**S(2)), Int((tan(e + f*x)**S(2) + S(1))/((a + b*tan(e + f*x))*sqrt(c + d*tan(e + f*x))), x), x) + Dist(S(1)/(a**S(2) + b**S(2)), Int(Simp(A*(a*c + b*d) + B*(-a*d + b*c) - (A*(-a*d + b*c) - B*(a*c + b*d))*tan(e + f*x), x)/sqrt(c + d*tan(e + f*x)), x), x) def replacement3725(A, B, a, b, c, d, e, f, x): return -Dist((-A*b + B*a)*(-a*d + b*c)/(a**S(2) + b**S(2)), Int((S(1) + tan(e + f*x)**(S(-2)))/((a + b/tan(e + f*x))*sqrt(c + d/tan(e + f*x))), x), x) + Dist(S(1)/(a**S(2) + b**S(2)), Int(Simp(A*(a*c + b*d) + B*(-a*d + b*c) - (A*(-a*d + b*c) - B*(a*c + b*d))/tan(e + f*x), x)/sqrt(c + d/tan(e + f*x)), x), x) def replacement3726(A, B, a, b, c, d, e, f, n, x): return Dist(b*(A*b - B*a)/(a**S(2) + b**S(2)), Int((c + d*tan(e + f*x))**n*(tan(e + f*x)**S(2) + S(1))/(a + b*tan(e + f*x)), x), x) + Dist(S(1)/(a**S(2) + b**S(2)), Int((c + d*tan(e + f*x))**n*Simp(A*a + B*b - (A*b - B*a)*tan(e + f*x), x), x), x) def replacement3727(A, B, a, b, c, d, e, f, n, x): return Dist(b*(A*b - B*a)/(a**S(2) + b**S(2)), Int((S(1) + tan(e + f*x)**(S(-2)))*(c + d/tan(e + f*x))**n/(a + b/tan(e + f*x)), x), x) + Dist(S(1)/(a**S(2) + b**S(2)), Int((c + d/tan(e + f*x))**n*Simp(A*a + B*b - (A*b - B*a)/tan(e + f*x), x), x), x) def replacement3728(A, B, a, b, c, d, e, f, x): return Dist(B*b, Int((tan(e + f*x)**S(2) + S(1))/(sqrt(a + b*tan(e + f*x))*sqrt(c + d*tan(e + f*x))), x), x) + Int(Simp(A*a - B*b + (A*b + B*a)*tan(e + f*x), x)/(sqrt(a + b*tan(e + f*x))*sqrt(c + d*tan(e + f*x))), x) def replacement3729(A, B, a, b, c, d, e, f, x): return Dist(B*b, Int((S(1) + tan(e + f*x)**(S(-2)))/(sqrt(a + b/tan(e + f*x))*sqrt(c + d/tan(e + f*x))), x), x) + Int(Simp(A*a - B*b + (A*b + B*a)/tan(e + f*x), x)/(sqrt(a + b/tan(e + f*x))*sqrt(c + d/tan(e + f*x))), x) def replacement3730(A, B, a, b, c, d, e, f, m, n, x): return Dist(A**S(2)/f, Subst(Int((a + b*x)**m*(c + d*x)**n/(A - B*x), x), x, tan(e + f*x)), x) def replacement3731(A, B, a, b, c, d, e, f, m, n, x): return -Dist(A**S(2)/f, Subst(Int((a + b*x)**m*(c + d*x)**n/(A - B*x), x), x, S(1)/tan(e + f*x)), x) def replacement3732(A, B, a, b, c, d, e, f, m, n, x): return Dist(A/S(2) - I*B/S(2), Int((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**n*(I*tan(e + f*x) + S(1)), x), x) + Dist(A/S(2) + I*B/S(2), Int((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**n*(-I*tan(e + f*x) + S(1)), x), x) def replacement3733(A, B, a, b, c, d, e, f, m, n, x): return Dist(A/S(2) - I*B/S(2), Int((S(1) + I/tan(e + f*x))*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**n, x), x) + Dist(A/S(2) + I*B/S(2), Int((S(1) - I/tan(e + f*x))*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**n, x), x) def replacement3734(A, C, a, b, e, f, m, x): return Dist(A/(b*f), Subst(Int((a + x)**m, x), x, b*tan(e + f*x)), x) def replacement3735(A, C, a, b, e, f, m, x): return -Dist(A/(b*f), Subst(Int((a + x)**m, x), x, b/tan(e + f*x)), x) def replacement3736(A, B, C, a, b, e, f, m, x): return Dist(b**(S(-2)), Int((a + b*tan(e + f*x))**(m + S(1))*Simp(B*b - C*a + C*b*tan(e + f*x), x), x), x) def replacement3737(A, B, C, a, b, e, f, m, x): return Dist(b**(S(-2)), Int((a + b/tan(e + f*x))**(m + S(1))*Simp(B*b - C*a + C*b/tan(e + f*x), x), x), x) def replacement3738(A, C, a, b, e, f, m, x): return -Dist(C/b**S(2), Int((a - b*tan(e + f*x))*(a + b*tan(e + f*x))**(m + S(1)), x), x) def replacement3739(A, C, a, b, e, f, m, x): return -Dist(C/b**S(2), Int((a - b/tan(e + f*x))*(a + b/tan(e + f*x))**(m + S(1)), x), x) def replacement3740(A, B, C, a, b, e, f, m, x): return Dist(S(1)/(S(2)*a**S(2)*m), Int((a + b*tan(e + f*x))**(m + S(1))*Simp(A*a*(S(2)*m + S(1)) + B*b - C*a - (C*b*(m + S(-1)) + (m + S(1))*(A*b - B*a))*tan(e + f*x), x), x), x) - Simp((a + b*tan(e + f*x))**m*(A*a + B*b - C*a)*tan(e + f*x)/(S(2)*a*f*m), x) def replacement3741(A, B, C, a, b, e, f, m, x): return Dist(S(1)/(S(2)*a**S(2)*m), Int((a + b/tan(e + f*x))**(m + S(1))*Simp(A*a*(S(2)*m + S(1)) + B*b - C*a - (C*b*(m + S(-1)) + (m + S(1))*(A*b - B*a))/tan(e + f*x), x), x), x) + Simp((a + b/tan(e + f*x))**m*(A*a + B*b - C*a)/(S(2)*a*f*m*tan(e + f*x)), x) def replacement3742(A, C, a, b, e, f, m, x): return Dist(S(1)/(S(2)*a**S(2)*m), Int((a + b*tan(e + f*x))**(m + S(1))*Simp(A*a*(S(2)*m + S(1)) - C*a - (A*b*(m + S(1)) + C*b*(m + S(-1)))*tan(e + f*x), x), x), x) - Simp((a + b*tan(e + f*x))**m*(A*a - C*a)*tan(e + f*x)/(S(2)*a*f*m), x) def replacement3743(A, C, a, b, e, f, m, x): return Dist(S(1)/(S(2)*a**S(2)*m), Int((a + b/tan(e + f*x))**(m + S(1))*Simp(A*a*(S(2)*m + S(1)) - C*a - (A*b*(m + S(1)) + C*b*(m + S(-1)))/tan(e + f*x), x), x), x) + Simp((a + b/tan(e + f*x))**m*(A*a - C*a)/(S(2)*a*f*m*tan(e + f*x)), x) def replacement3744(A, B, C, a, b, e, f, x): return Dist((A*b**S(2) - B*a*b + C*a**S(2))/(a**S(2) + b**S(2)), Int((tan(e + f*x)**S(2) + S(1))/(a + b*tan(e + f*x)), x), x) + Simp(x*(A*a + B*b - C*a)/(a**S(2) + b**S(2)), x) def replacement3745(A, B, C, a, b, e, f, x): return Dist((A*b**S(2) - B*a*b + C*a**S(2))/(a**S(2) + b**S(2)), Int((S(1) + tan(e + f*x)**(S(-2)))/(a + b/tan(e + f*x)), x), x) + Simp(x*(A*a + B*b - C*a)/(a**S(2) + b**S(2)), x) def replacement3746(A, B, C, e, f, x): return Dist(A, Int(S(1)/tan(e + f*x), x), x) + Dist(C, Int(tan(e + f*x), x), x) + Simp(B*x, x) def replacement3747(A, B, C, e, f, x): return Dist(A, Int(tan(e + f*x), x), x) + Dist(C, Int(S(1)/tan(e + f*x), x), x) + Simp(B*x, x) def replacement3748(A, C, e, f, x): return Dist(A, Int(S(1)/tan(e + f*x), x), x) + Dist(C, Int(tan(e + f*x), x), x) def replacement3749(A, C, e, f, x): return Dist(A, Int(tan(e + f*x), x), x) + Dist(C, Int(S(1)/tan(e + f*x), x), x) def replacement3750(A, B, C, a, b, e, f, x): return -Dist((A*b - B*a - C*b)/(a**S(2) + b**S(2)), Int(tan(e + f*x), x), x) + Dist((A*b**S(2) - B*a*b + C*a**S(2))/(a**S(2) + b**S(2)), Int((tan(e + f*x)**S(2) + S(1))/(a + b*tan(e + f*x)), x), x) + Simp(x*(A*a + B*b - C*a)/(a**S(2) + b**S(2)), x) def replacement3751(A, B, C, a, b, e, f, x): return -Dist((A*b - B*a - C*b)/(a**S(2) + b**S(2)), Int(S(1)/tan(e + f*x), x), x) + Dist((A*b**S(2) - B*a*b + C*a**S(2))/(a**S(2) + b**S(2)), Int((S(1) + tan(e + f*x)**(S(-2)))/(a + b/tan(e + f*x)), x), x) + Simp(x*(A*a + B*b - C*a)/(a**S(2) + b**S(2)), x) def replacement3752(A, C, a, b, e, f, x): return Dist((A*b**S(2) + C*a**S(2))/(a**S(2) + b**S(2)), Int((tan(e + f*x)**S(2) + S(1))/(a + b*tan(e + f*x)), x), x) - Dist(b*(A - C)/(a**S(2) + b**S(2)), Int(tan(e + f*x), x), x) + Simp(a*x*(A - C)/(a**S(2) + b**S(2)), x) def replacement3753(A, C, a, b, e, f, x): return Dist((A*b**S(2) + C*a**S(2))/(a**S(2) + b**S(2)), Int((S(1) + tan(e + f*x)**(S(-2)))/(a + b/tan(e + f*x)), x), x) - Dist(b*(A - C)/(a**S(2) + b**S(2)), Int(S(1)/tan(e + f*x), x), x) + Simp(a*x*(A - C)/(a**S(2) + b**S(2)), x) def replacement3754(A, B, C, a, b, e, f, m, x): return Dist(S(1)/(a**S(2) + b**S(2)), Int((a + b*tan(e + f*x))**(m + S(1))*Simp(B*b + a*(A - C) - (A*b - B*a - C*b)*tan(e + f*x), x), x), x) + Simp((a + b*tan(e + f*x))**(m + S(1))*(A*b**S(2) - B*a*b + C*a**S(2))/(b*f*(a**S(2) + b**S(2))*(m + S(1))), x) def replacement3755(A, B, C, a, b, e, f, m, x): return Dist(S(1)/(a**S(2) + b**S(2)), Int((a + b/tan(e + f*x))**(m + S(1))*Simp(B*b + a*(A - C) - (A*b - B*a - C*b)/tan(e + f*x), x), x), x) - Simp((a + b/tan(e + f*x))**(m + S(1))*(A*b**S(2) - B*a*b + C*a**S(2))/(b*f*(a**S(2) + b**S(2))*(m + S(1))), x) def replacement3756(A, C, a, b, e, f, m, x): return Dist(S(1)/(a**S(2) + b**S(2)), Int((a + b*tan(e + f*x))**(m + S(1))*Simp(a*(A - C) - (A*b - C*b)*tan(e + f*x), x), x), x) + Simp((a + b*tan(e + f*x))**(m + S(1))*(A*b**S(2) + C*a**S(2))/(b*f*(a**S(2) + b**S(2))*(m + S(1))), x) def replacement3757(A, C, a, b, e, f, m, x): return Dist(S(1)/(a**S(2) + b**S(2)), Int((a + b/tan(e + f*x))**(m + S(1))*Simp(a*(A - C) - (A*b - C*b)/tan(e + f*x), x), x), x) - Simp((a + b/tan(e + f*x))**(m + S(1))*(A*b**S(2) + C*a**S(2))/(b*f*(a**S(2) + b**S(2))*(m + S(1))), x) def replacement3758(A, B, C, a, b, e, f, m, x): return Int((a + b*tan(e + f*x))**m*Simp(A + B*tan(e + f*x) - C, x), x) + Simp(C*(a + b*tan(e + f*x))**(m + S(1))/(b*f*(m + S(1))), x) def replacement3759(A, B, C, a, b, e, f, m, x): return Int((a + b/tan(e + f*x))**m*Simp(A + B/tan(e + f*x) - C, x), x) - Simp(C*(a + b/tan(e + f*x))**(m + S(1))/(b*f*(m + S(1))), x) def replacement3760(A, C, a, b, e, f, m, x): return Dist(A - C, Int((a + b*tan(e + f*x))**m, x), x) + Simp(C*(a + b*tan(e + f*x))**(m + S(1))/(b*f*(m + S(1))), x) def replacement3761(A, C, a, b, e, f, m, x): return Dist(A - C, Int((a + b/tan(e + f*x))**m, x), x) - Simp(C*(a + b/tan(e + f*x))**(m + S(1))/(b*f*(m + S(1))), x) def replacement3762(A, C, a, b, c, d, e, f, m, n, x): return Dist(A/f, Subst(Int((a + b*x)**m*(c + d*x)**n, x), x, tan(e + f*x)), x) def replacement3763(A, C, a, b, c, d, e, f, m, n, x): return -Dist(A/f, Subst(Int((a + b*x)**m*(c + d*x)**n, x), x, S(1)/tan(e + f*x)), x) def replacement3764(A, B, C, a, b, c, d, e, f, n, x): return Dist(S(1)/(d*(c**S(2) + d**S(2))), Int((c + d*tan(e + f*x))**(n + S(1))*Simp(C*b*(c**S(2) + d**S(2))*tan(e + f*x)**S(2) + a*d*(A*c + B*d - C*c) + b*(A*d**S(2) - B*c*d + C*c**S(2)) + d*(-A*a*d + A*b*c + B*a*c + B*b*d + C*a*d - C*b*c)*tan(e + f*x), x), x), x) - Simp((c + d*tan(e + f*x))**(n + S(1))*(-a*d + b*c)*(A*d**S(2) - B*c*d + C*c**S(2))/(d**S(2)*f*(c**S(2) + d**S(2))*(n + S(1))), x) def replacement3765(A, B, C, a, b, c, d, e, f, n, x): return Dist(S(1)/(d*(c**S(2) + d**S(2))), Int((c + d/tan(e + f*x))**(n + S(1))*Simp(C*b*(c**S(2) + d**S(2))/tan(e + f*x)**S(2) + a*d*(A*c + B*d - C*c) + b*(A*d**S(2) - B*c*d + C*c**S(2)) + d*(-A*a*d + A*b*c + B*a*c + B*b*d + C*a*d - C*b*c)/tan(e + f*x), x), x), x) + Simp((c + d/tan(e + f*x))**(n + S(1))*(-a*d + b*c)*(A*d**S(2) - B*c*d + C*c**S(2))/(d**S(2)*f*(c**S(2) + d**S(2))*(n + S(1))), x) def replacement3766(A, C, a, b, c, d, e, f, n, x): return Dist(S(1)/(d*(c**S(2) + d**S(2))), Int((c + d*tan(e + f*x))**(n + S(1))*Simp(C*b*(c**S(2) + d**S(2))*tan(e + f*x)**S(2) + a*d*(A*c - C*c) + b*(A*d**S(2) + C*c**S(2)) + d*(-A*a*d + A*b*c + C*a*d - C*b*c)*tan(e + f*x), x), x), x) - Simp((c + d*tan(e + f*x))**(n + S(1))*(A*d**S(2) + C*c**S(2))*(-a*d + b*c)/(d**S(2)*f*(c**S(2) + d**S(2))*(n + S(1))), x) def replacement3767(A, C, a, b, c, d, e, f, n, x): return Dist(S(1)/(d*(c**S(2) + d**S(2))), Int((c + d/tan(e + f*x))**(n + S(1))*Simp(C*b*(c**S(2) + d**S(2))/tan(e + f*x)**S(2) + a*d*(A*c - C*c) + b*(A*d**S(2) + C*c**S(2)) + d*(-A*a*d + A*b*c + C*a*d - C*b*c)/tan(e + f*x), x), x), x) + Simp((c + d/tan(e + f*x))**(n + S(1))*(A*d**S(2) + C*c**S(2))*(-a*d + b*c)/(d**S(2)*f*(c**S(2) + d**S(2))*(n + S(1))), x) def replacement3768(A, B, C, a, b, c, d, e, f, n, x): return -Dist(S(1)/(d*(n + S(2))), Int((c + d*tan(e + f*x))**n*Simp(-A*a*d*(n + S(2)) + C*b*c - d*(n + S(2))*(A*b + B*a - C*b)*tan(e + f*x) - (C*a*d*(n + S(2)) - b*(-B*d*(n + S(2)) + C*c))*tan(e + f*x)**S(2), x), x), x) + Simp(C*b*(c + d*tan(e + f*x))**(n + S(1))*tan(e + f*x)/(d*f*(n + S(2))), x) def replacement3769(A, B, C, a, b, c, d, e, f, n, x): return -Dist(S(1)/(d*(n + S(2))), Int((c + d/tan(e + f*x))**n*Simp(-A*a*d*(n + S(2)) + C*b*c - d*(n + S(2))*(A*b + B*a - C*b)/tan(e + f*x) - (C*a*d*(n + S(2)) - b*(-B*d*(n + S(2)) + C*c))/tan(e + f*x)**S(2), x), x), x) - Simp(C*b*(c + d/tan(e + f*x))**(n + S(1))/(d*f*(n + S(2))*tan(e + f*x)), x) def replacement3770(A, C, a, b, c, d, e, f, n, x): return -Dist(S(1)/(d*(n + S(2))), Int((c + d*tan(e + f*x))**n*Simp(-A*a*d*(n + S(2)) + C*b*c - d*(n + S(2))*(A*b - C*b)*tan(e + f*x) - (C*a*d*(n + S(2)) - C*b*c)*tan(e + f*x)**S(2), x), x), x) + Simp(C*b*(c + d*tan(e + f*x))**(n + S(1))*tan(e + f*x)/(d*f*(n + S(2))), x) def replacement3771(A, C, a, b, c, d, e, f, n, x): return -Dist(S(1)/(d*(n + S(2))), Int((c + d/tan(e + f*x))**n*Simp(-A*a*d*(n + S(2)) + C*b*c - d*(n + S(2))*(A*b - C*b)/tan(e + f*x) - (C*a*d*(n + S(2)) - C*b*c)/tan(e + f*x)**S(2), x), x), x) - Simp(C*b*(c + d/tan(e + f*x))**(n + S(1))/(d*f*(n + S(2))*tan(e + f*x)), x) def replacement3772(A, B, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(S(2)*a*m*(-a*d + b*c)), Int((a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**n*Simp(a*(-A*d*(S(2)*m + n + S(1)) + B*c*m + C*d*(n + S(1))) + b*(-B*d*(n + S(1)) + c*m*(A + C)) + (A*b*d*(m + n + S(1)) + C*b*d*(m - n + S(-1)) + a*(-B*d*(m + n + S(1)) + S(2)*C*c*m))*tan(e + f*x), x), x), x) + Simp((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))*(A*a + B*b - C*a)/(S(2)*f*m*(-a*d + b*c)), x) def replacement3773(A, B, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(S(2)*a*m*(-a*d + b*c)), Int((a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**n*Simp(a*(-A*d*(S(2)*m + n + S(1)) + B*c*m + C*d*(n + S(1))) + b*(-B*d*(n + S(1)) + c*m*(A + C)) + (A*b*d*(m + n + S(1)) + C*b*d*(m - n + S(-1)) + a*(-B*d*(m + n + S(1)) + S(2)*C*c*m))/tan(e + f*x), x), x), x) - Simp((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))*(A*a + B*b - C*a)/(S(2)*f*m*(-a*d + b*c)), x) def replacement3774(A, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(S(2)*a*m*(-a*d + b*c)), Int((a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**n*Simp(a*(-A*d*(S(2)*m + n + S(1)) + C*d*(n + S(1))) + b*c*m*(A + C) + (A*b*d*(m + n + S(1)) + S(2)*C*a*c*m + C*b*d*(m - n + S(-1)))*tan(e + f*x), x), x), x) + Simp(a*(A - C)*(a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))/(S(2)*f*m*(-a*d + b*c)), x) def replacement3775(A, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(S(2)*a*m*(-a*d + b*c)), Int((a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**n*Simp(a*(-A*d*(S(2)*m + n + S(1)) + C*d*(n + S(1))) + b*c*m*(A + C) + (A*b*d*(m + n + S(1)) + S(2)*C*a*c*m + C*b*d*(m - n + S(-1)))/tan(e + f*x), x), x), x) - Simp(a*(A - C)*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))/(S(2)*f*m*(-a*d + b*c)), x) def replacement3776(A, B, C, a, b, c, d, e, f, m, n, x): return -Dist(S(1)/(a*d*(c**S(2) + d**S(2))*(n + S(1))), Int((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))*Simp(-a*d*(n + S(1))*(A*c + B*d - C*c) - a*(-C*(c**S(2)*m - d**S(2)*(n + S(1))) + d*(-A*d + B*c)*(m + n + S(1)))*tan(e + f*x) + b*m*(A*d**S(2) - B*c*d + C*c**S(2)), x), x), x) + Simp((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))*(A*d**S(2) - B*c*d + C*c**S(2))/(d*f*(c**S(2) + d**S(2))*(n + S(1))), x) def replacement3777(A, B, C, a, b, c, d, e, f, m, n, x): return -Dist(S(1)/(a*d*(c**S(2) + d**S(2))*(n + S(1))), Int((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))*Simp(-a*d*(n + S(1))*(A*c + B*d - C*c) - a*(-C*(c**S(2)*m - d**S(2)*(n + S(1))) + d*(-A*d + B*c)*(m + n + S(1)))/tan(e + f*x) + b*m*(A*d**S(2) - B*c*d + C*c**S(2)), x), x), x) - Simp((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))*(A*d**S(2) - B*c*d + C*c**S(2))/(d*f*(c**S(2) + d**S(2))*(n + S(1))), x) def replacement3778(A, C, a, b, c, d, e, f, m, n, x): return -Dist(S(1)/(a*d*(c**S(2) + d**S(2))*(n + S(1))), Int((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))*Simp(-a*d*(n + S(1))*(A*c - C*c) - a*(-A*d**S(2)*(m + n + S(1)) - C*(c**S(2)*m - d**S(2)*(n + S(1))))*tan(e + f*x) + b*m*(A*d**S(2) + C*c**S(2)), x), x), x) + Simp((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))*(A*d**S(2) + C*c**S(2))/(d*f*(c**S(2) + d**S(2))*(n + S(1))), x) def replacement3779(A, C, a, b, c, d, e, f, m, n, x): return -Dist(S(1)/(a*d*(c**S(2) + d**S(2))*(n + S(1))), Int((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))*Simp(-a*d*(n + S(1))*(A*c - C*c) - a*(-A*d**S(2)*(m + n + S(1)) - C*(c**S(2)*m - d**S(2)*(n + S(1))))/tan(e + f*x) + b*m*(A*d**S(2) + C*c**S(2)), x), x), x) - Simp((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))*(A*d**S(2) + C*c**S(2))/(d*f*(c**S(2) + d**S(2))*(n + S(1))), x) def replacement3780(A, B, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(b*d*(m + n + S(1))), Int((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**n*Simp(A*b*d*(m + n + S(1)) + C*(a*c*m - b*d*(n + S(1))) - (-B*b*d*(m + n + S(1)) + C*m*(-a*d + b*c))*tan(e + f*x), x), x), x) + Simp(C*(a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))/(d*f*(m + n + S(1))), x) def replacement3781(A, B, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(b*d*(m + n + S(1))), Int((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**n*Simp(A*b*d*(m + n + S(1)) + C*(a*c*m - b*d*(n + S(1))) - (-B*b*d*(m + n + S(1)) + C*m*(-a*d + b*c))/tan(e + f*x), x), x), x) - Simp(C*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))/(d*f*(m + n + S(1))), x) def replacement3782(A, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(b*d*(m + n + S(1))), Int((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**n*Simp(A*b*d*(m + n + S(1)) - C*m*(-a*d + b*c)*tan(e + f*x) + C*(a*c*m - b*d*(n + S(1))), x), x), x) + Simp(C*(a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))/(d*f*(m + n + S(1))), x) def replacement3783(A, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(b*d*(m + n + S(1))), Int((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**n*Simp(A*b*d*(m + n + S(1)) - C*m*(-a*d + b*c)/tan(e + f*x) + C*(a*c*m - b*d*(n + S(1))), x), x), x) - Simp(C*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))/(d*f*(m + n + S(1))), x) def replacement3784(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*tan(e + f*x))**(m + S(-1))*(c + d*tan(e + f*x))**(n + S(1))*Simp(A*d*(-a*c*(n + S(1)) + b*d*m) - b*(-C*(c**S(2)*m - d**S(2)*(n + S(1))) + d*(-A*d + B*c)*(m + n + S(1)))*tan(e + f*x)**S(2) - d*(n + S(1))*(B*(a*c + b*d) + (A - C)*(-a*d + b*c))*tan(e + f*x) + (-B*d + C*c)*(a*d*(n + S(1)) + b*c*m), x), x), x) + Simp((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))*(A*d**S(2) + c*(-B*d + C*c))/(d*f*(c**S(2) + d**S(2))*(n + S(1))), x) def replacement3785(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/tan(e + f*x))**(m + S(-1))*(c + d/tan(e + f*x))**(n + S(1))*Simp(A*d*(-a*c*(n + S(1)) + b*d*m) - b*(-C*(c**S(2)*m - d**S(2)*(n + S(1))) + d*(-A*d + B*c)*(m + n + S(1)))/tan(e + f*x)**S(2) - d*(n + S(1))*(B*(a*c + b*d) + (A - C)*(-a*d + b*c))/tan(e + f*x) + (-B*d + C*c)*(a*d*(n + S(1)) + b*c*m), x), x), x) - Simp((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))*(A*d**S(2) + c*(-B*d + C*c))/(d*f*(c**S(2) + d**S(2))*(n + S(1))), x) def replacement3786(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*tan(e + f*x))**(m + S(-1))*(c + d*tan(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(1)) + C*(c**S(2)*m - d**S(2)*(n + S(1))))*tan(e + f*x)**S(2) - d*(A - C)*(n + S(1))*(-a*d + b*c)*tan(e + f*x), x), x), x) + Simp((a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))*(A*d**S(2) + C*c**S(2))/(d*f*(c**S(2) + d**S(2))*(n + S(1))), x) def replacement3787(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/tan(e + f*x))**(m + S(-1))*(c + d/tan(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(1)) + C*(c**S(2)*m - d**S(2)*(n + S(1))))/tan(e + f*x)**S(2) - d*(A - C)*(n + S(1))*(-a*d + b*c)/tan(e + f*x), x), x), x) - Simp((a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))*(A*d**S(2) + C*c**S(2))/(d*f*(c**S(2) + d**S(2))*(n + S(1))), x) def replacement3788(A, B, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(d*(m + n + S(1))), Int((a + b*tan(e + f*x))**(m + S(-1))*(c + d*tan(e + f*x))**n*Simp(A*a*d*(m + n + S(1)) - C*(a*d*(n + S(1)) + b*c*m) + d*(m + n + S(1))*(A*b + B*a - C*b)*tan(e + f*x) - (-B*b*d*(m + n + S(1)) + C*m*(-a*d + b*c))*tan(e + f*x)**S(2), x), x), x) + Simp(C*(a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))/(d*f*(m + n + S(1))), x) def replacement3789(A, B, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(d*(m + n + S(1))), Int((a + b/tan(e + f*x))**(m + S(-1))*(c + d/tan(e + f*x))**n*Simp(A*a*d*(m + n + S(1)) - C*(a*d*(n + S(1)) + b*c*m) + d*(m + n + S(1))*(A*b + B*a - C*b)/tan(e + f*x) - (-B*b*d*(m + n + S(1)) + C*m*(-a*d + b*c))/tan(e + f*x)**S(2), x), x), x) - Simp(C*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))/(d*f*(m + n + S(1))), x) def replacement3790(A, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(d*(m + n + S(1))), Int((a + b*tan(e + f*x))**(m + S(-1))*(c + d*tan(e + f*x))**n*Simp(A*a*d*(m + n + S(1)) - C*m*(-a*d + b*c)*tan(e + f*x)**S(2) - C*(a*d*(n + S(1)) + b*c*m) + d*(A*b - C*b)*(m + n + S(1))*tan(e + f*x), x), x), x) + Simp(C*(a + b*tan(e + f*x))**m*(c + d*tan(e + f*x))**(n + S(1))/(d*f*(m + n + S(1))), x) def replacement3791(A, C, a, b, c, d, e, f, m, n, x): return Dist(S(1)/(d*(m + n + S(1))), Int((a + b/tan(e + f*x))**(m + S(-1))*(c + d/tan(e + f*x))**n*Simp(A*a*d*(m + n + S(1)) - C*m*(-a*d + b*c)/tan(e + f*x)**S(2) - C*(a*d*(n + S(1)) + b*c*m) + d*(A*b - C*b)*(m + n + S(1))/tan(e + f*x), x), x), x) - Simp(C*(a + b/tan(e + f*x))**m*(c + d/tan(e + f*x))**(n + S(1))/(d*f*(m + n + S(1))), x) def replacement3792(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*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**n*Simp(A*(a*(m + S(1))*(-a*d + b*c) - b**S(2)*d*(m + n + S(2))) - d*(A*b**S(2) - a*(B*b - C*a))*(m + n + S(2))*tan(e + f*x)**S(2) - (m + S(1))*(-a*d + b*c)*(A*b - B*a - C*b)*tan(e + f*x) + (B*b - C*a)*(a*d*(n + S(1)) + b*c*(m + S(1))), x), x), x) + Simp((a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**(n + S(1))*(A*b**S(2) - a*(B*b - C*a))/(f*(a**S(2) + b**S(2))*(m + S(1))*(-a*d + b*c)), x) def replacement3793(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/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**n*Simp(A*(a*(m + S(1))*(-a*d + b*c) - b**S(2)*d*(m + n + S(2))) - d*(A*b**S(2) - a*(B*b - C*a))*(m + n + S(2))/tan(e + f*x)**S(2) - (m + S(1))*(-a*d + b*c)*(A*b - B*a - C*b)/tan(e + f*x) + (B*b - C*a)*(a*d*(n + S(1)) + b*c*(m + S(1))), x), x), x) - Simp((a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**(n + S(1))*(A*b**S(2) - a*(B*b - C*a))/(f*(a**S(2) + b**S(2))*(m + S(1))*(-a*d + b*c)), x) def replacement3794(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*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**n*Simp(A*(a*(m + S(1))*(-a*d + b*c) - b**S(2)*d*(m + n + S(2))) - C*a*(a*d*(n + S(1)) + b*c*(m + S(1))) - d*(A*b**S(2) + C*a**S(2))*(m + n + S(2))*tan(e + f*x)**S(2) - (m + S(1))*(A*b - C*b)*(-a*d + b*c)*tan(e + f*x), x), x), x) + Simp((a + b*tan(e + f*x))**(m + S(1))*(c + d*tan(e + f*x))**(n + S(1))*(A*b**S(2) + C*a**S(2))/(f*(a**S(2) + b**S(2))*(m + S(1))*(-a*d + b*c)), x) def replacement3795(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/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**n*Simp(A*(a*(m + S(1))*(-a*d + b*c) - b**S(2)*d*(m + n + S(2))) - C*a*(a*d*(n + S(1)) + b*c*(m + S(1))) - d*(A*b**S(2) + C*a**S(2))*(m + n + S(2))/tan(e + f*x)**S(2) - (m + S(1))*(A*b - C*b)*(-a*d + b*c)/tan(e + f*x), x), x), x) - Simp((a + b/tan(e + f*x))**(m + S(1))*(c + d/tan(e + f*x))**(n + S(1))*(A*b**S(2) + C*a**S(2))/(f*(a**S(2) + b**S(2))*(m + S(1))*(-a*d + b*c)), x) def replacement3796(A, B, C, a, b, c, d, e, f, x): return Dist((A*b**S(2) - B*a*b + C*a**S(2))/((a**S(2) + b**S(2))*(-a*d + b*c)), Int((-a*tan(e + f*x) + b)/(a + b*tan(e + f*x)), x), x) - Dist((A*d**S(2) - B*c*d + C*c**S(2))/((c**S(2) + d**S(2))*(-a*d + b*c)), Int((-c*tan(e + f*x) + d)/(c + d*tan(e + f*x)), x), x) + Simp(x*(a*(A*c + B*d - C*c) + b*(-A*d + B*c + C*d))/((a**S(2) + b**S(2))*(c**S(2) + d**S(2))), x) def replacement3797(A, B, C, a, b, c, d, e, f, x): return Dist((A*b**S(2) - B*a*b + C*a**S(2))/((a**S(2) + b**S(2))*(-a*d + b*c)), Int((-a/tan(e + f*x) + b)/(a + b/tan(e + f*x)), x), x) - Dist((A*d**S(2) - B*c*d + C*c**S(2))/((c**S(2) + d**S(2))*(-a*d + b*c)), Int((-c/tan(e + f*x) + d)/(c + d/tan(e + f*x)), x), x) + Simp(x*(a*(A*c + B*d - C*c) + b*(-A*d + B*c + C*d))/((a**S(2) + b**S(2))*(c**S(2) + d**S(2))), x) def replacement3798(A, C, a, b, c, d, e, f, x): return Dist((A*b**S(2) + C*a**S(2))/((a**S(2) + b**S(2))*(-a*d + b*c)), Int((-a*tan(e + f*x) + b)/(a + b*tan(e + f*x)), x), x) - Dist((A*d**S(2) + C*c**S(2))/((c**S(2) + d**S(2))*(-a*d + b*c)), Int((-c*tan(e + f*x) + d)/(c + d*tan(e + f*x)), x), x) + Simp(x*(a*(A*c - C*c) - b*(A*d - C*d))/((a**S(2) + b**S(2))*(c**S(2) + d**S(2))), x) def replacement3799(A, C, a, b, c, d, e, f, x): return Dist((A*b**S(2) + C*a**S(2))/((a**S(2) + b**S(2))*(-a*d + b*c)), Int((-a/tan(e + f*x) + b)/(a + b/tan(e + f*x)), x), x) - Dist((A*d**S(2) + C*c**S(2))/((c**S(2) + d**S(2))*(-a*d + b*c)), Int((-c/tan(e + f*x) + d)/(c + d/tan(e + f*x)), x), x) + Simp(x*(a*(A*c - C*c) - b*(A*d - C*d))/((a**S(2) + b**S(2))*(c**S(2) + d**S(2))), x) def replacement3800(A, B, C, a, b, c, d, e, f, n, x): return Dist((A*b**S(2) - B*a*b + C*a**S(2))/(a**S(2) + b**S(2)), Int((c + d*tan(e + f*x))**n*(tan(e + f*x)**S(2) + S(1))/(a + b*tan(e + f*x)), x), x) + Dist(S(1)/(a**S(2) + b**S(2)), Int((c + d*tan(e + f*x))**n*Simp(B*b + a*(A - C) + (B*a - b*(A - C))*tan(e + f*x), x), x), x) def replacement3801(A, B, C, a, b, c, d, e, f, n, x): return Dist((A*b**S(2) - B*a*b + C*a**S(2))/(a**S(2) + b**S(2)), Int((S(1) + tan(e + f*x)**(S(-2)))*(c + d/tan(e + f*x))**n/(a + b/tan(e + f*x)), x), x) + Dist(S(1)/(a**S(2) + b**S(2)), Int((c + d/tan(e + f*x))**n*Simp(B*b + a*(A - C) + (B*a - b*(A - C))/tan(e + f*x), x), x), x) def replacement3802(A, C, a, b, c, d, e, f, n, x): return Dist((A*b**S(2) + C*a**S(2))/(a**S(2) + b**S(2)), Int((c + d*tan(e + f*x))**n*(tan(e + f*x)**S(2) + S(1))/(a + b*tan(e + f*x)), x), x) + Dist(S(1)/(a**S(2) + b**S(2)), Int((c + d*tan(e + f*x))**n*Simp(a*(A - C) - (A*b - C*b)*tan(e + f*x), x), x), x) def replacement3803(A, C, a, b, c, d, e, f, n, x): return Dist((A*b**S(2) + C*a**S(2))/(a**S(2) + b**S(2)), Int((S(1) + tan(e + f*x)**(S(-2)))*(c + d/tan(e + f*x))**n/(a + b/tan(e + f*x)), x), x) + Dist(S(1)/(a**S(2) + b**S(2)), Int((c + d/tan(e + f*x))**n*Simp(a*(A - C) - (A*b - C*b)/tan(e + f*x), x), x), x) def replacement3804(A, B, C, 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*(A*b**S(2) + B*b*x + C*x**S(2))/(b**S(2) + x**S(2)), x), x, b*tan(e + f*x)), x) def replacement3805(A, B, C, 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*(A*b**S(2) + B*b*x + C*x**S(2))/(b**S(2) + x**S(2)), x), x, b/tan(e + f*x)), x) def replacement3806(A, C, 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*(A*b**S(2) + C*x**S(2))/(b**S(2) + x**S(2)), x), x, b*tan(e + f*x)), x) def replacement3807(A, C, 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*(A*b**S(2) + C*x**S(2))/(b**S(2) + x**S(2)), x), x, b/tan(e + f*x)), x) def replacement3808(a, b, c, d, x): return Dist(S(1)/a, Int(cos(c + d*x)**S(2), x), x) def replacement3809(a, b, c, d, x): return Dist(S(1)/a, Int(sin(c + d*x)**S(2), x), x) def replacement3810(a, b, c, d, x): return -Dist(b/(a - b), Int(S(1)/((a + b*tan(c + d*x)**S(2))*cos(c + d*x)**S(2)), x), x) + Simp(x/(a - b), x) def replacement3811(a, b, c, d, x): return -Dist(b/(a - b), Int(S(1)/((a + b/tan(c + d*x)**S(2))*sin(c + d*x)**S(2)), x), x) + Simp(x/(a - b), x) def replacement3812(a, b, c, d, e, n, p, x): return Dist(e/d, Subst(Int((a + b*x**n)**p/(e**S(2) + x**S(2)), x), x, e*tan(c + d*x)), x) def replacement3813(a, b, c, d, e, n, p, x): return -Dist(e/d, Subst(Int((a + b*x**n)**p/(e**S(2) + x**S(2)), x), x, e/tan(c + d*x)), x) def With3814(a, b, c, d, e, m, n, p, x): f = FreeFactors(tan(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)), x), x, tan(c + d*x)/f), x) def With3815(a, b, c, d, e, m, n, p, x): f = FreeFactors(S(1)/tan(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)), x), x, S(1)/(f*tan(c + d*x))), x) def With3816(a, b, c, d, m, n, p, x): f = FreeFactors(cos(c + d*x), x) return -Dist(f/d, Subst(Int((f*x)**(-n*p)*(-f**S(2)*x**S(2) + S(1))**(m/S(2) + S(-1)/2)*ExpandToSum(a*(f*x)**n + b*(-f**S(2)*x**S(2) + S(1))**(n/S(2)), x)**p, x), x, cos(c + d*x)/f), x) def With3817(a, b, c, d, m, n, p, x): f = FreeFactors(sin(c + d*x), x) return Dist(f/d, Subst(Int((f*x)**(-n*p)*(-f**S(2)*x**S(2) + S(1))**(m/S(2) + S(-1)/2)*ExpandToSum(a*(f*x)**n + b*(-f**S(2)*x**S(2) + S(1))**(n/S(2)), x)**p, x), x, sin(c + d*x)/f), x) def With3818(a, b, c, d, e, m, n, p, x): f = FreeFactors(tan(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)), x), x, tan(c + d*x)/f), x) def With3819(a, b, c, d, e, m, n, p, x): f = FreeFactors(S(1)/tan(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)), x), x, S(1)/(f*tan(c + d*x))), x) def With3820(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) - n*p/S(2) + S(-1)/2)*ExpandToSum(a*(-f**S(2)*x**S(2) + S(1))**(n/S(2)) + b*(f*x)**n, x)**p, x), x, sin(c + d*x)/f), x) def With3821(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) - n*p/S(2) + S(-1)/2)*ExpandToSum(a*(-f**S(2)*x**S(2) + S(1))**(n/S(2)) + b*(f*x)**n, x)**p, x), x, cos(c + d*x)/f), x) def replacement3822(a, b, c, d, e, m, n, p, x): return Dist(e/d, Subst(Int((x/e)**m*(a + b*x**n)**p/(e**S(2) + x**S(2)), x), x, e*tan(c + d*x)), x) def replacement3823(a, b, c, d, e, m, n, p, x): return -Dist(e/d, Subst(Int((x/e)**m*(a + b*x**n)**p/(e**S(2) + x**S(2)), x), x, e/tan(c + d*x)), x) def replacement3824(a, b, c, d, e, n, n2, p, x): return Dist(S(4)**(-p)*c**(-p), Int((b + S(2)*c*tan(d + e*x)**n)**(S(2)*p), x), x) def replacement3825(a, b, c, d, e, n, n2, p, x): return Dist(S(4)**(-p)*c**(-p), Int((b + S(2)*c*(S(1)/tan(d + e*x))**n)**(S(2)*p), x), x) def replacement3826(a, b, c, d, e, n, n2, p, x): return Dist((b + S(2)*c*tan(d + e*x)**n)**(-S(2)*p)*(a + b*tan(d + e*x)**n + c*tan(d + e*x)**(S(2)*n))**p, Int((b + S(2)*c*tan(d + e*x)**n)**(S(2)*p), x), x) def replacement3827(a, b, c, d, e, n, n2, p, x): return Dist((b + S(2)*c*(S(1)/tan(d + e*x))**n)**(-S(2)*p)*(a + b*(S(1)/tan(d + e*x))**n + c*(S(1)/tan(d + e*x))**(S(2)*n))**p, Int((b + S(2)*c*(S(1)/tan(d + e*x))**n)**(S(2)*p), x), x) def With3828(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*tan(d + e*x)**n - q), x), x) - Dist(S(2)*c/q, Int(S(1)/(b + S(2)*c*tan(d + e*x)**n + q), x), x) def With3829(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*(S(1)/tan(d + e*x))**n - q), x), x) - Dist(S(2)*c/q, Int(S(1)/(b + S(2)*c*(S(1)/tan(d + e*x))**n + q), x), x) def replacement3830(a, b, c, d, e, f, m, n, n2, p, x): return Dist(f/e, Subst(Int(x**m*(f**S(2) + x**S(2))**(-m/S(2) + S(-1))*(a + b*x**n + c*x**(S(2)*n))**p, x), x, f*tan(d + e*x)), x) def replacement3831(a, b, c, d, e, f, m, n, n2, p, x): return -Dist(f/e, Subst(Int(x**m*(f**S(2) + x**S(2))**(-m/S(2) + S(-1))*(a + b*x**n + c*x**(S(2)*n))**p, x), x, f/tan(d + e*x)), x) def With3832(a, b, c, d, e, m, n, n2, p, x): g = FreeFactors(cos(d + e*x), x) return -Dist(g/e, Subst(Int((g*x)**(-S(2)*n*p)*(-g**S(2)*x**S(2) + S(1))**(m/S(2) + S(-1)/2)*ExpandToSum(a*(g*x)**(S(2)*n) + b*(g*x)**n*(-g**S(2)*x**S(2) + S(1))**(n/S(2)) + c*(-g**S(2)*x**S(2) + S(1))**n, x)**p, x), x, cos(d + e*x)/g), x) def With3833(a, b, c, d, e, m, n, n2, p, x): g = FreeFactors(sin(d + e*x), x) return Dist(g/e, Subst(Int((g*x)**(-S(2)*n*p)*(-g**S(2)*x**S(2) + S(1))**(m/S(2) + S(-1)/2)*ExpandToSum(a*(g*x)**(S(2)*n) + b*(g*x)**n*(-g**S(2)*x**S(2) + S(1))**(n/S(2)) + c*(-g**S(2)*x**S(2) + S(1))**n, x)**p, x), x, sin(d + e*x)/g), x) def replacement3834(a, b, c, d, e, f, m, n, n2, p, x): return Dist(f**(m + S(1))/e, Subst(Int((f**S(2) + x**S(2))**(-m/S(2) + S(-1))*(a + b*x**n + c*x**(S(2)*n))**p, x), x, f*tan(d + e*x)), x) def replacement3835(a, b, c, d, e, f, m, n, n2, p, x): return -Dist(f**(m + S(1))/e, Subst(Int((f**S(2) + x**S(2))**(-m/S(2) + S(-1))*(a + b*x**n + c*x**(S(2)*n))**p, x), x, f/tan(d + e*x)), x) def With3836(a, b, c, d, e, 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) - n*p + S(-1)/2)*ExpandToSum(a*(S(1) - x**S(2))**n + b*x**n*(S(1) - x**S(2))**(n/S(2)) + c*x**(S(2)*n), x)**p, x), x, sin(d + e*x)/g), x) def With3837(a, b, c, d, e, 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) - n*p + S(-1)/2)*ExpandToSum(a*(S(1) - x**S(2))**n + b*x**n*(S(1) - x**S(2))**(n/S(2)) + c*x**(S(2)*n), x)**p, x), x, cos(d + e*x)/g), x) def replacement3838(a, b, c, d, e, m, n, n2, p, x): return Dist(S(4)**(-p)*c**(-p), Int((b + S(2)*c*tan(d + e*x)**n)**(S(2)*p)*tan(d + e*x)**m, x), x) def replacement3839(a, b, c, d, e, m, n, n2, p, x): return Dist(S(4)**(-p)*c**(-p), Int((b + S(2)*c*(S(1)/tan(d + e*x))**n)**(S(2)*p)*(S(1)/tan(d + e*x))**m, x), x) def replacement3840(a, b, c, d, e, m, n, n2, p, x): return Dist((b + S(2)*c*tan(d + e*x)**n)**(-S(2)*p)*(a + b*tan(d + e*x)**n + c*tan(d + e*x)**(S(2)*n))**p, Int((b + S(2)*c*tan(d + e*x)**n)**(S(2)*p)*tan(d + e*x)**m, x), x) def replacement3841(a, b, c, d, e, m, n, n2, p, x): return Dist((b + S(2)*c*(S(1)/tan(d + e*x))**n)**(-S(2)*p)*(a + b*(S(1)/tan(d + e*x))**n + c*(S(1)/tan(d + e*x))**(S(2)*n))**p, Int((b + S(2)*c*(S(1)/tan(d + e*x))**n)**(S(2)*p)*(S(1)/tan(d + e*x))**m, x), x) def replacement3842(a, b, c, d, e, f, m, n, n2, p, x): return Dist(f/e, Subst(Int((x/f)**m*(a + b*x**n + c*x**(S(2)*n))**p/(f**S(2) + x**S(2)), x), x, f*tan(d + e*x)), x) def replacement3843(a, b, c, d, e, f, m, n, n2, p, x): return -Dist(f/e, Subst(Int((x/f)**m*(a + b*x**n + c*x**(S(2)*n))**p/(f**S(2) + x**S(2)), x), x, f/tan(d + e*x)), x) def replacement3844(a, b, c, d, e, m, n, n2, p, x): return Dist(S(4)**(-p)*c**(-p), Int((b + S(2)*c*tan(d + e*x)**n)**(S(2)*p)*(S(1)/tan(d + e*x))**m, x), x) def replacement3845(a, b, c, d, e, m, n, n2, p, x): return Dist(S(4)**(-p)*c**(-p), Int((b + S(2)*c*(S(1)/tan(d + e*x))**n)**(S(2)*p)*tan(d + e*x)**m, x), x) def replacement3846(a, b, c, d, e, m, n, n2, p, x): return Dist((b + S(2)*c*tan(d + e*x)**n)**(-S(2)*p)*(a + b*tan(d + e*x)**n + c*tan(d + e*x)**(S(2)*n))**p, Int((b + S(2)*c*tan(d + e*x)**n)**(S(2)*p)*(S(1)/tan(d + e*x))**m, x), x) def replacement3847(a, b, c, d, e, m, n, n2, p, x): return Dist((b + S(2)*c*(S(1)/tan(d + e*x))**n)**(-S(2)*p)*(a + b*(S(1)/tan(d + e*x))**n + c*(S(1)/tan(d + e*x))**(S(2)*n))**p, Int((b + S(2)*c*(S(1)/tan(d + e*x))**n)**(S(2)*p)*tan(d + e*x)**m, x), x) def With3848(a, b, c, d, e, m, n, n2, p, x): g = FreeFactors(S(1)/tan(d + e*x), x) return Dist(g/e, Subst(Int((g*x)**(m - S(2)*n*p)*(a*(g*x)**(S(2)*n) + b*(g*x)**n + c)**p/(g**S(2)*x**S(2) + S(1)), x), x, S(1)/(g*tan(d + e*x))), x) def With3849(a, b, c, d, e, m, n, n2, p, x): g = FreeFactors(tan(d + e*x), x) return -Dist(g/e, Subst(Int((g*x)**(m - S(2)*n*p)*(a*(g*x)**(S(2)*n) + b*(g*x)**n + c)**p/(g**S(2)*x**S(2) + S(1)), x), x, tan(d + e*x)/g), x) def replacement3850(A, B, a, b, c, d, e, n, x): return Dist(S(4)**(-n)*c**(-n), Int((A + B*tan(d + e*x))*(b + S(2)*c*tan(d + e*x))**(S(2)*n), x), x) def replacement3851(A, B, a, b, c, d, e, n, x): return Dist(S(4)**(-n)*c**(-n), Int((A + B/tan(d + e*x))*(b + S(2)*c/tan(d + e*x))**(S(2)*n), x), x) def replacement3852(A, B, a, b, c, d, e, n, x): return Dist((b + S(2)*c*tan(d + e*x))**(-S(2)*n)*(a + b*tan(d + e*x) + c*tan(d + e*x)**S(2))**n, Int((A + B*tan(d + e*x))*(b + S(2)*c*tan(d + e*x))**(S(2)*n), x), x) def replacement3853(A, B, a, b, c, d, e, n, x): return Dist((b + S(2)*c/tan(d + e*x))**(-S(2)*n)*(a + b/tan(d + e*x) + c/tan(d + e*x)**S(2))**n, Int((A + B/tan(d + e*x))*(b + S(2)*c/tan(d + e*x))**(S(2)*n), x), x) def With3854(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)/Simp(b + S(2)*c*tan(d + e*x) - q, x), x), x) + Dist(B + (-S(2)*A*c + B*b)/q, Int(S(1)/Simp(b + S(2)*c*tan(d + e*x) + q, x), x), x) def With3855(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)/Simp(b + S(2)*c/tan(d + e*x) - q, x), x), x) + Dist(B + (-S(2)*A*c + B*b)/q, Int(S(1)/Simp(b + S(2)*c/tan(d + e*x) + q, x), x), x) def replacement3856(A, B, a, b, c, d, e, n, x): return Int(ExpandTrig((A + B*tan(d + e*x))*(a + b*tan(d + e*x) + c*tan(d + e*x)**S(2))**n, x), x) def replacement3857(A, B, a, b, c, d, e, n, x): return Int(ExpandTrig((A + B/tan(d + e*x))*(a + b/tan(d + e*x) + c/tan(d + e*x)**S(2))**n, x), x) def replacement3858(c, d, e, f, m, x): return -Dist(S(2)*I, Int((c + d*x)**m*exp(S(2)*I*(e + f*x))/(exp(S(2)*I*(e + f*x)) + S(1)), x), x) + Simp(I*(c + d*x)**(m + S(1))/(d*(m + S(1))), x) def replacement3859(c, d, e, f, m, x): return -Dist(S(2)*I, Int((c + d*x)**m*exp(S(2)*I*(e + f*x))/(S(1) - exp(S(2)*I*(e + f*x))), x), x) - Simp(I*(c + d*x)**(m + S(1))/(d*(m + S(1))), x) def replacement3860(b, c, d, e, f, m, n, x): return -Dist(b**S(2), Int((b*tan(e + f*x))**(n + S(-2))*(c + d*x)**m, x), x) - Dist(b*d*m/(f*(n + S(-1))), Int((b*tan(e + f*x))**(n + S(-1))*(c + d*x)**(m + S(-1)), x), x) + Simp(b*(b*tan(e + f*x))**(n + S(-1))*(c + d*x)**m/(f*(n + S(-1))), x) def replacement3861(b, c, d, e, f, m, n, x): return -Dist(b**S(2), Int((b/tan(e + f*x))**(n + S(-2))*(c + d*x)**m, x), x) + Dist(b*d*m/(f*(n + S(-1))), Int((b/tan(e + f*x))**(n + S(-1))*(c + d*x)**(m + S(-1)), x), x) - Simp(b*(b/tan(e + f*x))**(n + S(-1))*(c + d*x)**m/(f*(n + S(-1))), x) def replacement3862(b, c, d, e, f, m, n, x): return -Dist(b**(S(-2)), Int((b*tan(e + f*x))**(n + S(2))*(c + d*x)**m, x), x) - Dist(d*m/(b*f*(n + S(1))), Int((b*tan(e + f*x))**(n + S(1))*(c + d*x)**(m + S(-1)), x), x) + Simp((b*tan(e + f*x))**(n + S(1))*(c + d*x)**m/(b*f*(n + S(1))), x) def replacement3863(b, c, d, e, f, m, n, x): return -Dist(b**(S(-2)), Int((b/tan(e + f*x))**(n + S(2))*(c + d*x)**m, x), x) + Dist(d*m/(b*f*(n + S(1))), Int((b/tan(e + f*x))**(n + S(1))*(c + d*x)**(m + S(-1)), x), x) - Simp((b/tan(e + f*x))**(n + S(1))*(c + d*x)**m/(b*f*(n + S(1))), x) def replacement3864(a, b, c, d, e, f, m, n, x): return Int(ExpandIntegrand((c + d*x)**m, (a + b*tan(e + f*x))**n, x), x) def replacement3865(a, b, c, d, e, f, m, n, x): return Int(ExpandIntegrand((c + d*x)**m, (a + b/tan(e + f*x))**n, x), x) def replacement3866(a, b, c, d, e, f, m, x): return Dist(a*d*m/(S(2)*b*f), Int((c + d*x)**(m + S(-1))/(a + b*tan(e + f*x)), x), x) + Simp((c + d*x)**(m + S(1))/(S(2)*a*d*(m + S(1))), x) - Simp(a*(c + d*x)**m/(S(2)*b*f*(a + b*tan(e + f*x))), x) def replacement3867(a, b, c, d, e, f, m, x): return -Dist(a*d*m/(S(2)*b*f), Int((c + d*x)**(m + S(-1))/(a + b/tan(e + f*x)), x), x) + Simp((c + d*x)**(m + S(1))/(S(2)*a*d*(m + S(1))), x) + Simp(a*(c + d*x)**m/(S(2)*b*f*(a + b/tan(e + f*x))), x) def replacement3868(a, b, c, d, e, f, x): return -Dist(f/(a*d), Int(sin(S(2)*e + S(2)*f*x)/(c + d*x), x), x) + Dist(f/(b*d), Int(cos(S(2)*e + S(2)*f*x)/(c + d*x), x), x) - Simp(S(1)/(d*(a + b*tan(e + f*x))*(c + d*x)), x) def replacement3869(a, b, c, d, e, f, x): return Dist(f/(a*d), Int(sin(S(2)*e + S(2)*f*x)/(c + d*x), x), x) + Dist(f/(b*d), Int(cos(S(2)*e + S(2)*f*x)/(c + d*x), x), x) - Simp(S(1)/(d*(a + b/tan(e + f*x))*(c + d*x)), x) def replacement3870(a, b, c, d, e, f, m, x): return Dist(S(2)*b*f/(a*d*(m + S(1))), Int((c + d*x)**(m + S(1))/(a + b*tan(e + f*x)), x), x) + Simp((c + d*x)**(m + S(1))/(d*(a + b*tan(e + f*x))*(m + S(1))), x) + Simp(f*(c + d*x)**(m + S(2))/(b*d**S(2)*(m + S(1))*(m + S(2))), x) def replacement3871(a, b, c, d, e, f, m, x): return -Dist(S(2)*b*f/(a*d*(m + S(1))), Int((c + d*x)**(m + S(1))/(a + b/tan(e + f*x)), x), x) + Simp((c + d*x)**(m + S(1))/(d*(a + b/tan(e + f*x))*(m + S(1))), x) - Simp(f*(c + d*x)**(m + S(2))/(b*d**S(2)*(m + S(1))*(m + S(2))), x) def replacement3872(a, b, c, d, e, f, x): return Dist(S(1)/(S(2)*a), Int(cos(S(2)*e + S(2)*f*x)/(c + d*x), x), x) + Dist(S(1)/(S(2)*b), Int(sin(S(2)*e + S(2)*f*x)/(c + d*x), x), x) + Simp(log(c + d*x)/(S(2)*a*d), x) def replacement3873(a, b, c, d, e, f, x): return -Dist(S(1)/(S(2)*a), Int(cos(S(2)*e + S(2)*f*x)/(c + d*x), x), x) + Dist(S(1)/(S(2)*b), Int(sin(S(2)*e + S(2)*f*x)/(c + d*x), x), x) + Simp(log(c + d*x)/(S(2)*a*d), x) def replacement3874(a, b, c, d, e, f, m, x): return Dist(S(1)/(S(2)*a), Int((c + d*x)**m*exp(S(2)*a*(e + f*x)/b), x), x) + Simp((c + d*x)**(m + S(1))/(S(2)*a*d*(m + S(1))), x) def replacement3875(a, b, c, d, e, f, m, x): return -Dist(S(1)/(S(2)*a), Int((c + d*x)**m*exp(-S(2)*a*(e + f*x)/b), x), x) + Simp((c + d*x)**(m + S(1))/(S(2)*a*d*(m + S(1))), x) def replacement3876(a, b, c, d, e, f, m, n, x): return Int(ExpandIntegrand((c + d*x)**m, (sin(S(2)*e + S(2)*f*x)/(S(2)*b) + cos(S(2)*e + S(2)*f*x)/(S(2)*a) + S(1)/(S(2)*a))**(-n), x), x) def replacement3877(a, b, c, d, e, f, m, n, x): return Int(ExpandIntegrand((c + d*x)**m, (sin(S(2)*e + S(2)*f*x)/(S(2)*b) - cos(S(2)*e + S(2)*f*x)/(S(2)*a) + S(1)/(S(2)*a))**(-n), x), x) def replacement3878(a, b, c, d, e, f, m, n, x): return Int(ExpandIntegrand((c + d*x)**m, (exp(S(2)*a*(e + f*x)/b)/(S(2)*a) + S(1)/(S(2)*a))**(-n), x), x) def replacement3879(a, b, c, d, e, f, m, n, x): return Int(ExpandIntegrand((c + d*x)**m, (S(1)/(S(2)*a) - exp(-S(2)*a*(e + f*x)/b)/(S(2)*a))**(-n), x), x) def With3880(a, b, c, d, e, f, m, n, x): u = IntHide((a + b*tan(e + f*x))**n, x) return -Dist(d*m, Int(Dist((c + d*x)**(m + S(-1)), u, x), x), x) + Dist((c + d*x)**m, u, x) def With3881(a, b, c, d, e, f, m, n, x): u = IntHide((a + b/tan(e + f*x))**n, x) return -Dist(d*m, Int(Dist((c + d*x)**(m + S(-1)), u, x), x), x) + Dist((c + d*x)**m, u, x) def replacement3882(a, b, c, d, e, f, m, x): return -Dist(S(2)*I*b, Int((c + d*x)**m/(a**S(2) + b**S(2) + (a - I*b)**S(2)*exp(S(2)*I*(e + f*x))), x), x) + Simp((c + d*x)**(m + S(1))/(d*(a - I*b)*(m + S(1))), x) def replacement3883(a, b, c, d, e, f, m, x): return Dist(S(2)*I*b, Int((c + d*x)**m/(a**S(2) + b**S(2) - (a + I*b)**S(2)*exp(S(2)*I*(e + f*x))), x), x) + Simp((c + d*x)**(m + S(1))/(d*(a + I*b)*(m + S(1))), x) def replacement3884(a, b, c, d, e, f, x): return Dist(S(1)/(f*(a**S(2) + b**S(2))), Int((S(2)*a*c*f + S(2)*a*d*f*x + b*d)/(a + b*tan(e + f*x)), x), x) - Simp((c + d*x)**S(2)/(S(2)*d*(a**S(2) + b**S(2))), x) - Simp(b*(c + d*x)/(f*(a + b*tan(e + f*x))*(a**S(2) + b**S(2))), x) def replacement3885(a, b, c, d, e, f, x): return -Dist(S(1)/(f*(a**S(2) + b**S(2))), Int((-S(2)*a*c*f - S(2)*a*d*f*x + b*d)/(a + b/tan(e + f*x)), x), x) - Simp((c + d*x)**S(2)/(S(2)*d*(a**S(2) + b**S(2))), x) + Simp(b*(c + d*x)/(f*(a + b/tan(e + f*x))*(a**S(2) + b**S(2))), x) def replacement3886(a, b, c, d, e, f, m, n, x): return Int(ExpandIntegrand((c + d*x)**m, (-S(2)*I*b/(a**S(2) + b**S(2) + (a - I*b)**S(2)*exp(S(2)*I*(e + f*x))) + S(1)/(a - I*b))**(-n), x), x) def replacement3887(a, b, c, d, e, f, m, n, x): return Int(ExpandIntegrand((c + d*x)**m, (S(2)*I*b/(a**S(2) + b**S(2) - (a + I*b)**S(2)*exp(S(2)*I*(e + f*x))) + S(1)/(a + I*b))**(-n), x), x) def replacement3888(a, b, m, n, u, v, x): return Int((a + b*tan(ExpandToSum(v, x)))**n*ExpandToSum(u, x)**m, x) def replacement3889(a, b, m, n, u, v, x): return Int((a + b/tan(ExpandToSum(v, x)))**n*ExpandToSum(u, x)**m, x) def replacement3890(a, b, c, d, e, f, m, n, x): return Int((a + b*tan(e + f*x))**n*(c + d*x)**m, x) def replacement3891(a, b, c, d, e, f, m, n, x): return Int((a + b/tan(e + f*x))**n*(c + d*x)**m, x) def replacement3892(a, b, c, d, n, p, x): return Dist(S(1)/n, Subst(Int(x**(S(-1) + S(1)/n)*(a + b*tan(c + d*x))**p, x), x, x**n), x) def replacement3893(a, b, c, d, n, p, x): return Dist(S(1)/n, Subst(Int(x**(S(-1) + S(1)/n)*(a + b/tan(c + d*x))**p, x), x, x**n), x) def replacement3894(a, b, c, d, n, p, x): return Int((a + b*tan(c + d*x**n))**p, x) def replacement3895(a, b, c, d, n, p, x): return Int((a + b/tan(c + d*x**n))**p, x) def replacement3896(a, b, c, d, n, p, u, x): return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a + b*tan(c + d*x**n))**p, x), x, u), x) def replacement3897(a, b, c, d, n, p, u, x): return Dist(S(1)/Coefficient(u, x, S(1)), Subst(Int((a + b/tan(c + d*x**n))**p, x), x, u), x) def replacement3898(a, b, p, u, x): return Int((a + b*tan(ExpandToSum(u, x)))**p, x) def replacement3899(a, b, p, u, x): return Int((a + b/tan(ExpandToSum(u, x)))**p, x) def replacement3900(a, b, c, d, m, n, p, x): return Dist(S(1)/n, Subst(Int(x**(S(-1) + (m + S(1))/n)*(a + b*tan(c + d*x))**p, x), x, x**n), x) def replacement3901(a, b, c, d, m, n, p, x): return Dist(S(1)/n, Subst(Int(x**(S(-1) + (m + S(1))/n)*(a + b/tan(c + d*x))**p, x), x, x**n), x) def replacement3902(c, d, m, n, x): return -Dist((m - n + S(1))/(d*n), Int(x**(m - n)*tan(c + d*x**n), x), x) - Int(x**m, x) + Simp(x**(m - n + S(1))*tan(c + d*x**n)/(d*n), x) def replacement3903(c, d, m, n, x): return Dist((m - n + S(1))/(d*n), Int(x**(m - n)/tan(c + d*x**n), x), x) - Int(x**m, x) - Simp(x**(m - n + S(1))/(d*n*tan(c + d*x**n)), x) def replacement3904(a, b, c, d, m, n, p, x): return Int(x**m*(a + b*tan(c + d*x**n))**p, x) def replacement3905(a, b, c, d, m, n, p, x): return Int(x**m*(a + b/tan(c + d*x**n))**p, x) def replacement3906(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*tan(c + d*x**n))**p, x), x) def replacement3907(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/tan(c + d*x**n))**p, x), x) def replacement3908(a, b, e, m, p, u, x): return Int((e*x)**m*(a + b*tan(ExpandToSum(u, x)))**p, x) def replacement3909(a, b, e, m, p, u, x): return Int((e*x)**m*(a + b/tan(ExpandToSum(u, x)))**p, x) def replacement3910(a, b, m, n, p, q, x): return -Dist((m - n + S(1))/(b*n*p), Int(x**(m - n)*(S(1)/cos(a + b*x**n))**p, x), x) + Simp(x**(m - n + S(1))*(S(1)/cos(a + b*x**n))**p/(b*n*p), x) def replacement3911(a, b, m, n, p, q, x): return Dist((m - n + S(1))/(b*n*p), Int(x**(m - n)*(S(1)/sin(a + b*x**n))**p, x), x) - Simp(x**(m - n + S(1))*(S(1)/sin(a + b*x**n))**p/(b*n*p), x) def replacement3912(a, b, c, n, x): return Int(tan(a + b*x + c*x**S(2))**n, x) def replacement3913(a, b, c, n, x): return Int((S(1)/tan(a + b*x + c*x**S(2)))**n, x) def replacement3914(a, b, c, d, e, x): return -Simp(e*log(cos(a + b*x + c*x**S(2)))/(S(2)*c), x) def replacement3915(a, b, c, d, e, x): return Simp(e*log(sin(a + b*x + c*x**S(2)))/(S(2)*c), x) def replacement3916(a, b, c, d, e, x): return Dist((-b*e + S(2)*c*d)/(S(2)*c), Int(tan(a + b*x + c*x**S(2)), x), x) - Simp(e*log(cos(a + b*x + c*x**S(2)))/(S(2)*c), x) def replacement3917(a, b, c, d, e, x): return Dist((-b*e + S(2)*c*d)/(S(2)*c), Int(S(1)/tan(a + b*x + c*x**S(2)), x), x) + Simp(e*log(sin(a + b*x + c*x**S(2)))/(S(2)*c), x) def replacement3918(a, b, c, d, e, m, n, x): return Int((d + e*x)**m*tan(a + b*x + c*x**S(2))**n, x) def replacement3919(a, b, c, d, e, m, n, x): return Int((d + e*x)**m*(S(1)/tan(a + b*x + c*x**S(2)))**n, x)