1""" 2This code is automatically generated. Never edit it manually. 3For details of generating the code see `rubi_parsing_guide.md` in `parsetools`. 4""" 5 6from sympy.external import import_module 7matchpy = import_module("matchpy") 8 9if matchpy: 10 from matchpy import Pattern, ReplacementRule, CustomConstraint, is_match 11 from sympy.integrals.rubi.utility_function import ( 12 Int, Sum, Set, With, Module, Scan, MapAnd, FalseQ, 13 ZeroQ, NegativeQ, NonzeroQ, FreeQ, NFreeQ, List, Log, PositiveQ, 14 PositiveIntegerQ, NegativeIntegerQ, IntegerQ, IntegersQ, 15 ComplexNumberQ, PureComplexNumberQ, RealNumericQ, PositiveOrZeroQ, 16 NegativeOrZeroQ, FractionOrNegativeQ, NegQ, Equal, Unequal, IntPart, 17 FracPart, RationalQ, ProductQ, SumQ, NonsumQ, Subst, First, Rest, 18 SqrtNumberQ, SqrtNumberSumQ, LinearQ, Sqrt, ArcCosh, Coefficient, 19 Denominator, Hypergeometric2F1, Not, Simplify, FractionalPart, 20 IntegerPart, AppellF1, EllipticPi, EllipticE, EllipticF, ArcTan, 21 ArcCot, ArcCoth, ArcTanh, ArcSin, ArcSinh, ArcCos, ArcCsc, ArcSec, 22 ArcCsch, ArcSech, Sinh, Tanh, Cosh, Sech, Csch, Coth, LessEqual, Less, 23 Greater, GreaterEqual, FractionQ, IntLinearcQ, Expand, IndependentQ, 24 PowerQ, IntegerPowerQ, PositiveIntegerPowerQ, FractionalPowerQ, AtomQ, 25 ExpQ, LogQ, Head, MemberQ, TrigQ, SinQ, CosQ, TanQ, CotQ, SecQ, CscQ, 26 Sin, Cos, Tan, Cot, Sec, Csc, HyperbolicQ, SinhQ, CoshQ, TanhQ, CothQ, 27 SechQ, CschQ, InverseTrigQ, SinCosQ, SinhCoshQ, LeafCount, Numerator, 28 NumberQ, NumericQ, Length, ListQ, Im, Re, InverseHyperbolicQ, 29 InverseFunctionQ, TrigHyperbolicFreeQ, InverseFunctionFreeQ, RealQ, 30 EqQ, FractionalPowerFreeQ, ComplexFreeQ, PolynomialQ, FactorSquareFree, 31 PowerOfLinearQ, Exponent, QuadraticQ, LinearPairQ, BinomialParts, 32 TrinomialParts, PolyQ, EvenQ, OddQ, PerfectSquareQ, NiceSqrtAuxQ, 33 NiceSqrtQ, Together, PosAux, PosQ, CoefficientList, ReplaceAll, 34 ExpandLinearProduct, GCD, ContentFactor, NumericFactor, 35 NonnumericFactors, MakeAssocList, GensymSubst, KernelSubst, 36 ExpandExpression, Apart, SmartApart, MatchQ, 37 PolynomialQuotientRemainder, FreeFactors, NonfreeFactors, 38 RemoveContentAux, RemoveContent, FreeTerms, NonfreeTerms, 39 ExpandAlgebraicFunction, CollectReciprocals, ExpandCleanup, 40 AlgebraicFunctionQ, Coeff, LeadTerm, RemainingTerms, LeadFactor, 41 RemainingFactors, LeadBase, LeadDegree, Numer, Denom, hypergeom, Expon, 42 MergeMonomials, PolynomialDivide, BinomialQ, TrinomialQ, 43 GeneralizedBinomialQ, GeneralizedTrinomialQ, FactorSquareFreeList, 44 PerfectPowerTest, SquareFreeFactorTest, RationalFunctionQ, 45 RationalFunctionFactors, NonrationalFunctionFactors, Reverse, 46 RationalFunctionExponents, RationalFunctionExpand, ExpandIntegrand, 47 SimplerQ, SimplerSqrtQ, SumSimplerQ, BinomialDegree, TrinomialDegree, 48 CancelCommonFactors, SimplerIntegrandQ, GeneralizedBinomialDegree, 49 GeneralizedBinomialParts, GeneralizedTrinomialDegree, 50 GeneralizedTrinomialParts, MonomialQ, MonomialSumQ, 51 MinimumMonomialExponent, MonomialExponent, LinearMatchQ, 52 PowerOfLinearMatchQ, QuadraticMatchQ, CubicMatchQ, BinomialMatchQ, 53 TrinomialMatchQ, GeneralizedBinomialMatchQ, GeneralizedTrinomialMatchQ, 54 QuotientOfLinearsMatchQ, PolynomialTermQ, PolynomialTerms, 55 NonpolynomialTerms, PseudoBinomialParts, NormalizePseudoBinomial, 56 PseudoBinomialPairQ, PseudoBinomialQ, PolynomialGCD, PolyGCD, 57 AlgebraicFunctionFactors, NonalgebraicFunctionFactors, 58 QuotientOfLinearsP, QuotientOfLinearsParts, QuotientOfLinearsQ, 59 Flatten, Sort, AbsurdNumberQ, AbsurdNumberFactors, 60 NonabsurdNumberFactors, SumSimplerAuxQ, Prepend, Drop, 61 CombineExponents, FactorInteger, FactorAbsurdNumber, 62 SubstForInverseFunction, SubstForFractionalPower, 63 SubstForFractionalPowerOfQuotientOfLinears, 64 FractionalPowerOfQuotientOfLinears, SubstForFractionalPowerQ, 65 SubstForFractionalPowerAuxQ, FractionalPowerOfSquareQ, 66 FractionalPowerSubexpressionQ, Apply, FactorNumericGcd, 67 MergeableFactorQ, MergeFactor, MergeFactors, TrigSimplifyQ, 68 TrigSimplify, TrigSimplifyRecur, Order, FactorOrder, Smallest, 69 OrderedQ, MinimumDegree, PositiveFactors, Sign, NonpositiveFactors, 70 PolynomialInAuxQ, PolynomialInQ, ExponentInAux, ExponentIn, 71 PolynomialInSubstAux, PolynomialInSubst, Distrib, DistributeDegree, 72 FunctionOfPower, DivideDegreesOfFactors, MonomialFactor, FullSimplify, 73 FunctionOfLinearSubst, FunctionOfLinear, NormalizeIntegrand, 74 NormalizeIntegrandAux, NormalizeIntegrandFactor, 75 NormalizeIntegrandFactorBase, NormalizeTogether, 76 NormalizeLeadTermSigns, AbsorbMinusSign, NormalizeSumFactors, 77 SignOfFactor, NormalizePowerOfLinear, SimplifyIntegrand, SimplifyTerm, 78 TogetherSimplify, SmartSimplify, SubstForExpn, ExpandToSum, UnifySum, 79 UnifyTerms, UnifyTerm, CalculusQ, FunctionOfInverseLinear, 80 PureFunctionOfSinhQ, PureFunctionOfTanhQ, PureFunctionOfCoshQ, 81 IntegerQuotientQ, OddQuotientQ, EvenQuotientQ, FindTrigFactor, 82 FunctionOfSinhQ, FunctionOfCoshQ, OddHyperbolicPowerQ, FunctionOfTanhQ, 83 FunctionOfTanhWeight, FunctionOfHyperbolicQ, SmartNumerator, 84 SmartDenominator, SubstForAux, ActivateTrig, ExpandTrig, TrigExpand, 85 SubstForTrig, SubstForHyperbolic, InertTrigFreeQ, LCM, 86 SubstForFractionalPowerOfLinear, FractionalPowerOfLinear, 87 InverseFunctionOfLinear, InertTrigQ, InertReciprocalQ, DeactivateTrig, 88 FixInertTrigFunction, DeactivateTrigAux, PowerOfInertTrigSumQ, 89 PiecewiseLinearQ, KnownTrigIntegrandQ, KnownSineIntegrandQ, 90 KnownTangentIntegrandQ, KnownCotangentIntegrandQ, 91 KnownSecantIntegrandQ, TryPureTanSubst, TryTanhSubst, TryPureTanhSubst, 92 AbsurdNumberGCD, AbsurdNumberGCDList, ExpandTrigExpand, 93 ExpandTrigReduce, ExpandTrigReduceAux, NormalizeTrig, TrigToExp, 94 ExpandTrigToExp, TrigReduce, FunctionOfTrig, AlgebraicTrigFunctionQ, 95 FunctionOfHyperbolic, FunctionOfQ, FunctionOfExpnQ, PureFunctionOfSinQ, 96 PureFunctionOfCosQ, PureFunctionOfTanQ, PureFunctionOfCotQ, 97 FunctionOfCosQ, FunctionOfSinQ, OddTrigPowerQ, FunctionOfTanQ, 98 FunctionOfTanWeight, FunctionOfTrigQ, FunctionOfDensePolynomialsQ, 99 FunctionOfLog, PowerVariableExpn, PowerVariableDegree, 100 PowerVariableSubst, EulerIntegrandQ, FunctionOfSquareRootOfQuadratic, 101 SquareRootOfQuadraticSubst, Divides, EasyDQ, ProductOfLinearPowersQ, 102 Rt, NthRoot, AtomBaseQ, SumBaseQ, NegSumBaseQ, AllNegTermQ, 103 SomeNegTermQ, TrigSquareQ, RtAux, TrigSquare, IntSum, IntTerm, Map2, 104 ConstantFactor, SameQ, ReplacePart, CommonFactors, 105 MostMainFactorPosition, FunctionOfExponentialQ, FunctionOfExponential, 106 FunctionOfExponentialFunction, FunctionOfExponentialFunctionAux, 107 FunctionOfExponentialTest, FunctionOfExponentialTestAux, stdev, 108 rubi_test, If, IntQuadraticQ, IntBinomialQ, RectifyTangent, 109 RectifyCotangent, Inequality, Condition, Simp, SimpHelp, SplitProduct, 110 SplitSum, SubstFor, SubstForAux, FresnelS, FresnelC, Erfc, Erfi, Gamma, 111 FunctionOfTrigOfLinearQ, ElementaryFunctionQ, Complex, UnsameQ, 112 _SimpFixFactor, SimpFixFactor, _FixSimplify, FixSimplify, 113 _SimplifyAntiderivativeSum, SimplifyAntiderivativeSum, 114 _SimplifyAntiderivative, SimplifyAntiderivative, _TrigSimplifyAux, 115 TrigSimplifyAux, Cancel, Part, PolyLog, D, Dist, Sum_doit, PolynomialQuotient, Floor, 116 PolynomialRemainder, Factor, PolyLog, CosIntegral, SinIntegral, LogIntegral, SinhIntegral, 117 CoshIntegral, Rule, Erf, PolyGamma, ExpIntegralEi, ExpIntegralE, LogGamma , UtilityOperator, Factorial, 118 Zeta, ProductLog, DerivativeDivides, HypergeometricPFQ, IntHide, OneQ, Null, rubi_exp as exp, rubi_log as log, Discriminant, 119 Negative, Quotient 120 ) 121 from sympy import (Integral, S, sqrt, And, Or, Integer, Float, Mod, I, Abs, simplify, Mul, 122 Add, Pow, sign, EulerGamma) 123 from sympy.integrals.rubi.symbol import WC 124 from sympy.core.symbol import symbols, Symbol 125 from sympy.functions import (sin, cos, tan, cot, csc, sec, sqrt, erf) 126 from sympy.functions.elementary.hyperbolic import (acosh, asinh, atanh, acoth, acsch, asech, cosh, sinh, tanh, coth, sech, csch) 127 from sympy.functions.elementary.trigonometric import (atan, acsc, asin, acot, acos, asec, atan2) 128 from sympy import pi as Pi 129 130 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'] 131 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']] 132 i, ii, Pqq, Q, R, r, C, k, u = symbols('i ii Pqq Q R r C k u') 133 _UseGamma = False 134 ShowSteps = False 135 StepCounter = None 136 137 138def special_functions(): 139 from sympy.integrals.rubi.constraints import cons69, cons2, cons3, cons68, cons19, cons1266, cons8, cons29, cons20, cons168, cons1959, cons1960, cons96, cons263, cons1961, cons1834, cons64, cons1962, cons1963, cons1964, cons249, cons1965, cons1966, cons1967, cons1833, cons4, cons1257, cons21, cons1361, cons1968, cons1969, cons170, cons1970, cons1971, cons33, cons1972, cons1973, cons1974, cons802, cons89, cons90, cons5, cons52, cons91, cons385, cons50, cons1975, cons1976, cons1977, cons54, cons1978, cons1101, cons127, cons1245, cons13, cons139, cons1381, cons1979, cons1980, cons198, cons1981, cons1982, cons1983, cons152, cons465, cons1767, cons165, cons950, cons951, cons1984, cons1985, cons805, cons1986, cons1987, cons1988, cons1989, cons340, cons1990, cons1991, cons1992, cons1993, cons1994, cons1995, cons40, cons1996, cons349, cons1997, cons1998, cons1999, cons2000, cons2001, cons2002, cons2003 140 141 142 pattern6742 = Pattern(Integral(Erf(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons69) 143 rule6742 = ReplacementRule(pattern6742, replacement6742) 144 145 pattern6743 = Pattern(Integral(Erfc(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons69) 146 rule6743 = ReplacementRule(pattern6743, replacement6743) 147 148 pattern6744 = Pattern(Integral(Erfi(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons69) 149 rule6744 = ReplacementRule(pattern6744, replacement6744) 150 151 pattern6745 = Pattern(Integral(Erf(x_*WC('b', S(1)))/x_, x_), cons3, cons3) 152 rule6745 = ReplacementRule(pattern6745, replacement6745) 153 154 pattern6746 = Pattern(Integral(Erfc(x_*WC('b', S(1)))/x_, x_), cons3, cons3) 155 rule6746 = ReplacementRule(pattern6746, replacement6746) 156 157 pattern6747 = Pattern(Integral(Erfi(x_*WC('b', S(1)))/x_, x_), cons3, cons3) 158 rule6747 = ReplacementRule(pattern6747, replacement6747) 159 160 pattern6748 = Pattern(Integral(x_**WC('m', S(1))*Erf(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons19, cons68) 161 rule6748 = ReplacementRule(pattern6748, replacement6748) 162 163 pattern6749 = Pattern(Integral(x_**WC('m', S(1))*Erfc(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons19, cons68) 164 rule6749 = ReplacementRule(pattern6749, replacement6749) 165 166 pattern6750 = Pattern(Integral(x_**WC('m', S(1))*Erfi(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons19, cons68) 167 rule6750 = ReplacementRule(pattern6750, replacement6750) 168 169 pattern6751 = Pattern(Integral(x_*Erf(x_*WC('b', S(1)) + WC('a', S(0)))*exp(x_**S(2)*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons1266) 170 rule6751 = ReplacementRule(pattern6751, replacement6751) 171 172 pattern6752 = Pattern(Integral(x_*Erfc(x_*WC('b', S(1)) + WC('a', S(0)))*exp(x_**S(2)*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons1266) 173 rule6752 = ReplacementRule(pattern6752, replacement6752) 174 175 pattern6753 = Pattern(Integral(x_*Erfi(x_*WC('b', S(1)) + WC('a', S(0)))*exp(x_**S(2)*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons1266) 176 rule6753 = ReplacementRule(pattern6753, replacement6753) 177 178 pattern6754 = Pattern(Integral(x_**m_*Erf(x_*WC('b', S(1)) + WC('a', S(0)))*exp(x_**S(2)*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons20, cons168) 179 rule6754 = ReplacementRule(pattern6754, replacement6754) 180 181 pattern6755 = Pattern(Integral(x_**m_*Erfc(x_*WC('b', S(1)) + WC('a', S(0)))*exp(x_**S(2)*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons20, cons168) 182 rule6755 = ReplacementRule(pattern6755, replacement6755) 183 184 pattern6756 = Pattern(Integral(x_**m_*Erfi(x_*WC('b', S(1)) + WC('a', S(0)))*exp(x_**S(2)*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons20, cons168) 185 rule6756 = ReplacementRule(pattern6756, replacement6756) 186 187 pattern6757 = Pattern(Integral(Erf(x_*WC('b', S(1)))*exp(x_**S(2)*WC('d', S(1)) + WC('c', S(0)))/x_, x_), cons3, cons1959) 188 rule6757 = ReplacementRule(pattern6757, replacement6757) 189 190 pattern6758 = Pattern(Integral(Erfc(x_*WC('b', S(1)))*exp(x_**S(2)*WC('d', S(1)) + WC('c', S(0)))/x_, x_), cons3, cons1959) 191 rule6758 = ReplacementRule(pattern6758, replacement6758) 192 193 pattern6759 = Pattern(Integral(Erfi(x_*WC('b', S(1)))*exp(x_**S(2)*WC('d', S(1)) + WC('c', S(0)))/x_, x_), cons3, cons1960) 194 rule6759 = ReplacementRule(pattern6759, replacement6759) 195 196 pattern6760 = Pattern(Integral(x_**m_*Erf(x_*WC('b', S(1)) + WC('a', S(0)))*exp(x_**S(2)*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons20, cons96) 197 rule6760 = ReplacementRule(pattern6760, replacement6760) 198 199 pattern6761 = Pattern(Integral(x_**m_*Erfc(x_*WC('b', S(1)) + WC('a', S(0)))*exp(x_**S(2)*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons20, cons96) 200 rule6761 = ReplacementRule(pattern6761, replacement6761) 201 202 pattern6762 = Pattern(Integral(x_**m_*Erfi(x_*WC('b', S(1)) + WC('a', S(0)))*exp(x_**S(2)*WC('d', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons29, cons20, cons96) 203 rule6762 = ReplacementRule(pattern6762, replacement6762) 204 205 pattern6763 = Pattern(Integral(Erf(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons69) 206 rule6763 = ReplacementRule(pattern6763, replacement6763) 207 208 pattern6764 = Pattern(Integral(Erfc(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons69) 209 rule6764 = ReplacementRule(pattern6764, replacement6764) 210 211 pattern6765 = Pattern(Integral(Erfi(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons69) 212 rule6765 = ReplacementRule(pattern6765, replacement6765) 213 214 pattern6766 = Pattern(Integral(x_**WC('m', S(1))*Erf(x_*WC('b', S(1)))**S(2), x_), cons3, cons20, cons263, cons1961) 215 rule6766 = ReplacementRule(pattern6766, replacement6766) 216 217 pattern6767 = Pattern(Integral(x_**WC('m', S(1))*Erfc(x_*WC('b', S(1)))**S(2), x_), cons3, cons20, cons1834, cons1961) 218 rule6767 = ReplacementRule(pattern6767, replacement6767) 219 220 pattern6768 = Pattern(Integral(x_**WC('m', S(1))*Erfi(x_*WC('b', S(1)))**S(2), x_), cons3, cons20, cons1834, cons1961) 221 rule6768 = ReplacementRule(pattern6768, replacement6768) 222 223 pattern6769 = Pattern(Integral(x_**WC('m', S(1))*Erf(a_ + x_*WC('b', S(1)))**S(2), x_), cons2, cons3, cons64) 224 rule6769 = ReplacementRule(pattern6769, replacement6769) 225 226 pattern6770 = Pattern(Integral(x_**WC('m', S(1))*Erfc(a_ + x_*WC('b', S(1)))**S(2), x_), cons2, cons3, cons64) 227 rule6770 = ReplacementRule(pattern6770, replacement6770) 228 229 pattern6771 = Pattern(Integral(x_**WC('m', S(1))*Erfi(a_ + x_*WC('b', S(1)))**S(2), x_), cons2, cons3, cons64) 230 rule6771 = ReplacementRule(pattern6771, replacement6771) 231 232 pattern6772 = Pattern(Integral(FresnelS(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons69) 233 rule6772 = ReplacementRule(pattern6772, replacement6772) 234 235 pattern6773 = Pattern(Integral(FresnelC(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons69) 236 rule6773 = ReplacementRule(pattern6773, replacement6773) 237 238 pattern6774 = Pattern(Integral(FresnelS(x_*WC('b', S(1)))/x_, x_), cons3, cons3) 239 rule6774 = ReplacementRule(pattern6774, replacement6774) 240 241 pattern6775 = Pattern(Integral(FresnelC(x_*WC('b', S(1)))/x_, x_), cons3, cons3) 242 rule6775 = ReplacementRule(pattern6775, replacement6775) 243 244 pattern6776 = Pattern(Integral(x_**WC('m', S(1))*FresnelS(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons19, cons68) 245 rule6776 = ReplacementRule(pattern6776, replacement6776) 246 247 pattern6777 = Pattern(Integral(x_**WC('m', S(1))*FresnelC(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons19, cons68) 248 rule6777 = ReplacementRule(pattern6777, replacement6777) 249 250 pattern6778 = Pattern(Integral(FresnelS(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons69) 251 rule6778 = ReplacementRule(pattern6778, replacement6778) 252 253 pattern6779 = Pattern(Integral(FresnelC(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons69) 254 rule6779 = ReplacementRule(pattern6779, replacement6779) 255 256 pattern6780 = Pattern(Integral(x_**m_*FresnelS(x_*WC('b', S(1)))**S(2), x_), cons3, cons20, cons1834, cons1962) 257 rule6780 = ReplacementRule(pattern6780, replacement6780) 258 259 pattern6781 = Pattern(Integral(x_**m_*FresnelC(x_*WC('b', S(1)))**S(2), x_), cons3, cons20, cons1834, cons1962) 260 rule6781 = ReplacementRule(pattern6781, replacement6781) 261 262 pattern6782 = Pattern(Integral(x_*FresnelS(x_*WC('b', S(1)))*sin(x_**S(2)*WC('c', S(1))), x_), cons3, cons8, cons1963) 263 rule6782 = ReplacementRule(pattern6782, replacement6782) 264 265 pattern6783 = Pattern(Integral(x_*FresnelC(x_*WC('b', S(1)))*cos(x_**S(2)*WC('c', S(1))), x_), cons3, cons8, cons1963) 266 rule6783 = ReplacementRule(pattern6783, replacement6783) 267 268 pattern6784 = Pattern(Integral(x_**m_*FresnelS(x_*WC('b', S(1)))*sin(x_**S(2)*WC('c', S(1))), x_), cons3, cons8, cons1963, cons20, cons168, cons1964) 269 rule6784 = ReplacementRule(pattern6784, replacement6784) 270 271 pattern6785 = Pattern(Integral(x_**m_*FresnelC(x_*WC('b', S(1)))*cos(x_**S(2)*WC('c', S(1))), x_), cons3, cons8, cons1963, cons20, cons168, cons1964) 272 rule6785 = ReplacementRule(pattern6785, replacement6785) 273 274 pattern6786 = Pattern(Integral(x_**m_*FresnelS(x_*WC('b', S(1)))*sin(x_**S(2)*WC('c', S(1))), x_), cons3, cons8, cons1963, cons20, cons249, cons1965) 275 rule6786 = ReplacementRule(pattern6786, replacement6786) 276 277 pattern6787 = Pattern(Integral(x_**m_*FresnelC(x_*WC('b', S(1)))*cos(x_**S(2)*WC('c', S(1))), x_), cons3, cons8, cons1963, cons20, cons249, cons1965) 278 rule6787 = ReplacementRule(pattern6787, replacement6787) 279 280 pattern6788 = Pattern(Integral(x_*FresnelS(x_*WC('b', S(1)))*cos(x_**S(2)*WC('c', S(1))), x_), cons3, cons8, cons1963) 281 rule6788 = ReplacementRule(pattern6788, replacement6788) 282 283 pattern6789 = Pattern(Integral(x_*FresnelC(x_*WC('b', S(1)))*sin(x_**S(2)*WC('c', S(1))), x_), cons3, cons8, cons1963) 284 rule6789 = ReplacementRule(pattern6789, replacement6789) 285 286 pattern6790 = Pattern(Integral(x_**m_*FresnelS(x_*WC('b', S(1)))*cos(x_**S(2)*WC('c', S(1))), x_), cons3, cons8, cons1963, cons20, cons168, cons1966) 287 rule6790 = ReplacementRule(pattern6790, replacement6790) 288 289 pattern6791 = Pattern(Integral(x_**m_*FresnelC(x_*WC('b', S(1)))*sin(x_**S(2)*WC('c', S(1))), x_), cons3, cons8, cons1963, cons20, cons168, cons1966) 290 rule6791 = ReplacementRule(pattern6791, replacement6791) 291 292 pattern6792 = Pattern(Integral(x_**m_*FresnelS(x_*WC('b', S(1)))*cos(x_**S(2)*WC('c', S(1))), x_), cons3, cons8, cons1963, cons20, cons96, cons1967) 293 rule6792 = ReplacementRule(pattern6792, replacement6792) 294 295 pattern6793 = Pattern(Integral(x_**m_*FresnelC(x_*WC('b', S(1)))*sin(x_**S(2)*WC('c', S(1))), x_), cons3, cons8, cons1963, cons20, cons96, cons1967) 296 rule6793 = ReplacementRule(pattern6793, replacement6793) 297 298 pattern6794 = Pattern(Integral(ExpIntegralE(n_, x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons4, cons1833) 299 rule6794 = ReplacementRule(pattern6794, replacement6794) 300 301 pattern6795 = Pattern(Integral(x_**WC('m', S(1))*ExpIntegralE(n_, x_*WC('b', S(1))), x_), cons3, cons1257, cons64) 302 rule6795 = ReplacementRule(pattern6795, replacement6795) 303 304 pattern6796 = Pattern(Integral(ExpIntegralE(S(1), x_*WC('b', S(1)))/x_, x_), cons3, cons3) 305 rule6796 = ReplacementRule(pattern6796, replacement6796) 306 307 pattern6797 = Pattern(Integral(x_**m_*ExpIntegralE(n_, x_*WC('b', S(1))), x_), cons3, cons1257, cons20, cons96) 308 rule6797 = ReplacementRule(pattern6797, replacement6797) 309 310 pattern6798 = Pattern(Integral(x_**m_*ExpIntegralE(n_, x_*WC('b', S(1))), x_), cons3, cons19, cons4, cons1257, cons21) 311 rule6798 = ReplacementRule(pattern6798, replacement6798) 312 313 pattern6799 = Pattern(Integral(x_**WC('m', S(1))*ExpIntegralE(n_, x_*WC('b', S(1))), x_), cons3, cons19, cons4, cons1361) 314 rule6799 = ReplacementRule(pattern6799, replacement6799) 315 316 pattern6800 = Pattern(Integral(x_**WC('m', S(1))*ExpIntegralE(n_, a_ + x_*WC('b', S(1))), x_), cons2, cons3, cons19, cons4, cons1968) 317 rule6800 = ReplacementRule(pattern6800, replacement6800) 318 319 pattern6801 = Pattern(Integral(x_**WC('m', S(1))*ExpIntegralE(n_, a_ + x_*WC('b', S(1))), x_), cons2, cons3, cons19, cons1969, cons68) 320 rule6801 = ReplacementRule(pattern6801, replacement6801) 321 322 pattern6802 = Pattern(Integral(ExpIntegralEi(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons69) 323 rule6802 = ReplacementRule(pattern6802, replacement6802) 324 325 pattern6803 = Pattern(Integral(x_**WC('m', S(1))*ExpIntegralEi(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons19, cons68) 326 rule6803 = ReplacementRule(pattern6803, replacement6803) 327 328 pattern6804 = Pattern(Integral(ExpIntegralEi(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons69) 329 rule6804 = ReplacementRule(pattern6804, replacement6804) 330 331 pattern6805 = Pattern(Integral(x_**WC('m', S(1))*ExpIntegralEi(x_*WC('b', S(1)))**S(2), x_), cons3, cons64) 332 rule6805 = ReplacementRule(pattern6805, replacement6805) 333 334 pattern6806 = Pattern(Integral(x_**WC('m', S(1))*ExpIntegralEi(a_ + x_*WC('b', S(1)))**S(2), x_), cons2, cons3, cons64) 335 rule6806 = ReplacementRule(pattern6806, replacement6806) 336 337 pattern6807 = Pattern(Integral(ExpIntegralEi(x_*WC('d', S(1)) + WC('c', S(0)))*exp(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons1266) 338 rule6807 = ReplacementRule(pattern6807, replacement6807) 339 340 pattern6808 = Pattern(Integral(x_**WC('m', S(1))*ExpIntegralEi(x_*WC('d', S(1)) + WC('c', S(0)))*exp(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons64) 341 rule6808 = ReplacementRule(pattern6808, replacement6808) 342 343 pattern6809 = Pattern(Integral(x_**m_*ExpIntegralEi(x_*WC('d', S(1)) + WC('c', S(0)))*exp(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons20, cons96) 344 rule6809 = ReplacementRule(pattern6809, replacement6809) 345 346 pattern6810 = Pattern(Integral(LogIntegral(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons69) 347 rule6810 = ReplacementRule(pattern6810, replacement6810) 348 349 pattern6811 = Pattern(Integral(LogIntegral(x_*WC('b', S(1)))/x_, x_), cons3, cons3) 350 rule6811 = ReplacementRule(pattern6811, replacement6811) 351 352 pattern6812 = Pattern(Integral(x_**WC('m', S(1))*LogIntegral(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons19, cons68) 353 rule6812 = ReplacementRule(pattern6812, replacement6812) 354 355 pattern6813 = Pattern(Integral(SinIntegral(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons69) 356 rule6813 = ReplacementRule(pattern6813, replacement6813) 357 358 pattern6814 = Pattern(Integral(CosIntegral(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons69) 359 rule6814 = ReplacementRule(pattern6814, replacement6814) 360 361 pattern6815 = Pattern(Integral(SinIntegral(x_*WC('b', S(1)))/x_, x_), cons3, cons3) 362 rule6815 = ReplacementRule(pattern6815, replacement6815) 363 364 pattern6816 = Pattern(Integral(CosIntegral(x_*WC('b', S(1)))/x_, x_), cons3, cons3) 365 rule6816 = ReplacementRule(pattern6816, replacement6816) 366 367 pattern6817 = Pattern(Integral(x_**WC('m', S(1))*SinIntegral(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons19, cons68) 368 rule6817 = ReplacementRule(pattern6817, replacement6817) 369 370 pattern6818 = Pattern(Integral(x_**WC('m', S(1))*CosIntegral(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons19, cons68) 371 rule6818 = ReplacementRule(pattern6818, replacement6818) 372 373 pattern6819 = Pattern(Integral(SinIntegral(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons69) 374 rule6819 = ReplacementRule(pattern6819, replacement6819) 375 376 pattern6820 = Pattern(Integral(CosIntegral(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons69) 377 rule6820 = ReplacementRule(pattern6820, replacement6820) 378 379 pattern6821 = Pattern(Integral(x_**WC('m', S(1))*SinIntegral(x_*WC('b', S(1)))**S(2), x_), cons3, cons64) 380 rule6821 = ReplacementRule(pattern6821, replacement6821) 381 382 pattern6822 = Pattern(Integral(x_**WC('m', S(1))*CosIntegral(x_*WC('b', S(1)))**S(2), x_), cons3, cons64) 383 rule6822 = ReplacementRule(pattern6822, replacement6822) 384 385 pattern6823 = Pattern(Integral(x_**WC('m', S(1))*SinIntegral(a_ + x_*WC('b', S(1)))**S(2), x_), cons2, cons3, cons64) 386 rule6823 = ReplacementRule(pattern6823, replacement6823) 387 388 pattern6824 = Pattern(Integral(x_**WC('m', S(1))*CosIntegral(a_ + x_*WC('b', S(1)))**S(2), x_), cons2, cons3, cons64) 389 rule6824 = ReplacementRule(pattern6824, replacement6824) 390 391 pattern6825 = Pattern(Integral(SinIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*sin(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons1266) 392 rule6825 = ReplacementRule(pattern6825, replacement6825) 393 394 pattern6826 = Pattern(Integral(CosIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*cos(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons1266) 395 rule6826 = ReplacementRule(pattern6826, replacement6826) 396 397 pattern6827 = Pattern(Integral(x_**WC('m', S(1))*SinIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*sin(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons64) 398 rule6827 = ReplacementRule(pattern6827, replacement6827) 399 400 pattern6828 = Pattern(Integral(x_**WC('m', S(1))*CosIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*cos(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons64) 401 rule6828 = ReplacementRule(pattern6828, replacement6828) 402 403 pattern6829 = Pattern(Integral(x_**m_*SinIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*sin(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons20, cons96) 404 rule6829 = ReplacementRule(pattern6829, replacement6829) 405 406 pattern6830 = Pattern(Integral(x_**WC('m', S(1))*CosIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*cos(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons20, cons96) 407 rule6830 = ReplacementRule(pattern6830, replacement6830) 408 409 pattern6831 = Pattern(Integral(SinIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*cos(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons1266) 410 rule6831 = ReplacementRule(pattern6831, replacement6831) 411 412 pattern6832 = Pattern(Integral(CosIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*sin(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons1266) 413 rule6832 = ReplacementRule(pattern6832, replacement6832) 414 415 pattern6833 = Pattern(Integral(x_**WC('m', S(1))*SinIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*cos(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons64) 416 rule6833 = ReplacementRule(pattern6833, replacement6833) 417 418 pattern6834 = Pattern(Integral(x_**WC('m', S(1))*CosIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*sin(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons64) 419 rule6834 = ReplacementRule(pattern6834, replacement6834) 420 421 pattern6835 = Pattern(Integral(x_**WC('m', S(1))*SinIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*cos(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons20, cons96) 422 rule6835 = ReplacementRule(pattern6835, replacement6835) 423 424 pattern6836 = Pattern(Integral(x_**m_*CosIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*sin(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons20, cons96) 425 rule6836 = ReplacementRule(pattern6836, replacement6836) 426 427 pattern6837 = Pattern(Integral(SinhIntegral(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons69) 428 rule6837 = ReplacementRule(pattern6837, replacement6837) 429 430 pattern6838 = Pattern(Integral(CoshIntegral(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons69) 431 rule6838 = ReplacementRule(pattern6838, replacement6838) 432 433 pattern6839 = Pattern(Integral(SinhIntegral(x_*WC('b', S(1)))/x_, x_), cons3, cons3) 434 rule6839 = ReplacementRule(pattern6839, replacement6839) 435 436 pattern6840 = Pattern(Integral(CoshIntegral(x_*WC('b', S(1)))/x_, x_), cons3, cons3) 437 rule6840 = ReplacementRule(pattern6840, replacement6840) 438 439 pattern6841 = Pattern(Integral(x_**WC('m', S(1))*SinhIntegral(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons19, cons68) 440 rule6841 = ReplacementRule(pattern6841, replacement6841) 441 442 pattern6842 = Pattern(Integral(x_**WC('m', S(1))*CoshIntegral(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons19, cons68) 443 rule6842 = ReplacementRule(pattern6842, replacement6842) 444 445 pattern6843 = Pattern(Integral(SinhIntegral(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons69) 446 rule6843 = ReplacementRule(pattern6843, replacement6843) 447 448 pattern6844 = Pattern(Integral(CoshIntegral(x_*WC('b', S(1)) + WC('a', S(0)))**S(2), x_), cons2, cons3, cons69) 449 rule6844 = ReplacementRule(pattern6844, replacement6844) 450 451 pattern6845 = Pattern(Integral(x_**WC('m', S(1))*SinhIntegral(x_*WC('b', S(1)))**S(2), x_), cons3, cons64) 452 rule6845 = ReplacementRule(pattern6845, replacement6845) 453 454 pattern6846 = Pattern(Integral(x_**WC('m', S(1))*CoshIntegral(x_*WC('b', S(1)))**S(2), x_), cons3, cons64) 455 rule6846 = ReplacementRule(pattern6846, replacement6846) 456 457 pattern6847 = Pattern(Integral(x_**WC('m', S(1))*SinhIntegral(a_ + x_*WC('b', S(1)))**S(2), x_), cons2, cons3, cons64) 458 rule6847 = ReplacementRule(pattern6847, replacement6847) 459 460 pattern6848 = Pattern(Integral(x_**WC('m', S(1))*CoshIntegral(a_ + x_*WC('b', S(1)))**S(2), x_), cons2, cons3, cons64) 461 rule6848 = ReplacementRule(pattern6848, replacement6848) 462 463 pattern6849 = Pattern(Integral(SinhIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*sinh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons1266) 464 rule6849 = ReplacementRule(pattern6849, replacement6849) 465 466 pattern6850 = Pattern(Integral(CoshIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*cosh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons1266) 467 rule6850 = ReplacementRule(pattern6850, replacement6850) 468 469 pattern6851 = Pattern(Integral(x_**WC('m', S(1))*SinhIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*sinh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons20, cons170) 470 rule6851 = ReplacementRule(pattern6851, replacement6851) 471 472 pattern6852 = Pattern(Integral(x_**WC('m', S(1))*CoshIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*cosh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons20, cons170) 473 rule6852 = ReplacementRule(pattern6852, replacement6852) 474 475 pattern6853 = Pattern(Integral(x_**m_*SinhIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*sinh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons20, cons96) 476 rule6853 = ReplacementRule(pattern6853, replacement6853) 477 478 pattern6854 = Pattern(Integral(x_**WC('m', S(1))*CoshIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*cosh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons20, cons96) 479 rule6854 = ReplacementRule(pattern6854, replacement6854) 480 481 pattern6855 = Pattern(Integral(SinhIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*cosh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons1266) 482 rule6855 = ReplacementRule(pattern6855, replacement6855) 483 484 pattern6856 = Pattern(Integral(CoshIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*sinh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons1266) 485 rule6856 = ReplacementRule(pattern6856, replacement6856) 486 487 pattern6857 = Pattern(Integral(x_**WC('m', S(1))*SinhIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*cosh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons64) 488 rule6857 = ReplacementRule(pattern6857, replacement6857) 489 490 pattern6858 = Pattern(Integral(x_**WC('m', S(1))*CoshIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*sinh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons64) 491 rule6858 = ReplacementRule(pattern6858, replacement6858) 492 493 pattern6859 = Pattern(Integral(x_**WC('m', S(1))*SinhIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*cosh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons20, cons96) 494 rule6859 = ReplacementRule(pattern6859, replacement6859) 495 496 pattern6860 = Pattern(Integral(x_**m_*CoshIntegral(x_*WC('d', S(1)) + WC('c', S(0)))*sinh(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons8, cons29, cons20, cons96) 497 rule6860 = ReplacementRule(pattern6860, replacement6860) 498 499 pattern6861 = Pattern(Integral(Gamma(n_, x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons69) 500 rule6861 = ReplacementRule(pattern6861, replacement6861) 501 502 pattern6862 = Pattern(Integral(Gamma(n_, b_*x_)/x_, x_), cons3, cons4, cons1970) 503 rule6862 = ReplacementRule(pattern6862, replacement6862) 504 505 pattern6863 = Pattern(Integral(x_**WC('m', S(1))*Gamma(n_, b_*x_), x_), cons3, cons19, cons4, cons68) 506 rule6863 = ReplacementRule(pattern6863, replacement6863) 507 508 pattern6864 = Pattern(Integral(x_**WC('m', S(1))*Gamma(n_, a_ + x_*WC('b', S(1))), x_), cons2, cons3, cons19, cons4, cons1971, cons68) 509 rule6864 = ReplacementRule(pattern6864, With6864) 510 511 pattern6865 = Pattern(Integral(LogGamma(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons69) 512 rule6865 = ReplacementRule(pattern6865, replacement6865) 513 514 pattern6866 = Pattern(Integral(x_**WC('m', S(1))*LogGamma(x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons33, cons170) 515 rule6866 = ReplacementRule(pattern6866, replacement6866) 516 517 pattern6867 = Pattern(Integral(PolyGamma(n_, x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons4, cons1833) 518 rule6867 = ReplacementRule(pattern6867, replacement6867) 519 520 pattern6868 = Pattern(Integral(x_**WC('m', S(1))*PolyGamma(n_, x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons4, cons33, cons170) 521 rule6868 = ReplacementRule(pattern6868, replacement6868) 522 523 pattern6869 = Pattern(Integral(x_**WC('m', S(1))*PolyGamma(n_, x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons4, cons33, cons96) 524 rule6869 = ReplacementRule(pattern6869, replacement6869) 525 526 pattern6870 = Pattern(Integral(Gamma(x_*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))*PolyGamma(S(0), x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons4, cons1833) 527 rule6870 = ReplacementRule(pattern6870, replacement6870) 528 529 pattern6871 = Pattern(Integral(Factorial(x_*WC('b', S(1)) + WC('a', S(0)))**WC('n', S(1))*PolyGamma(S(0), x_*WC('b', S(1)) + WC('c', S(0))), x_), cons2, cons3, cons8, cons4, cons1972) 530 rule6871 = ReplacementRule(pattern6871, replacement6871) 531 532 pattern6872 = Pattern(Integral(Zeta(S(2), x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons69) 533 rule6872 = ReplacementRule(pattern6872, replacement6872) 534 535 pattern6873 = Pattern(Integral(Zeta(s_, x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons802, cons1973, cons1974) 536 rule6873 = ReplacementRule(pattern6873, replacement6873) 537 538 pattern6874 = Pattern(Integral(x_**WC('m', S(1))*Zeta(S(2), x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons33) 539 rule6874 = ReplacementRule(pattern6874, replacement6874) 540 541 pattern6875 = Pattern(Integral(x_**WC('m', S(1))*Zeta(s_, x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons802, cons1973, cons1974, cons33, cons170) 542 rule6875 = ReplacementRule(pattern6875, replacement6875) 543 544 pattern6876 = Pattern(Integral(x_**WC('m', S(1))*Zeta(s_, x_*WC('b', S(1)) + WC('a', S(0))), x_), cons2, cons3, cons802, cons1973, cons1974, cons33, cons96) 545 rule6876 = ReplacementRule(pattern6876, replacement6876) 546 547 pattern6877 = Pattern(Integral(PolyLog(n_, (x_**WC('p', S(1))*WC('b', S(1)))**WC('q', S(1))*WC('a', S(1))), x_), cons2, cons3, cons5, cons52, cons89, cons90) 548 rule6877 = ReplacementRule(pattern6877, replacement6877) 549 550 pattern6878 = Pattern(Integral(PolyLog(n_, (x_**WC('p', S(1))*WC('b', S(1)))**WC('q', S(1))*WC('a', S(1))), x_), cons2, cons3, cons5, cons52, cons89, cons91) 551 rule6878 = ReplacementRule(pattern6878, replacement6878) 552 553 pattern6879 = Pattern(Integral(PolyLog(n_, (x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*WC('c', S(1)))/(x_*WC('e', S(1)) + WC('d', S(0))), x_), cons2, cons3, cons8, cons29, cons50, cons4, cons5, cons385) 554 rule6879 = ReplacementRule(pattern6879, replacement6879) 555 556 pattern6880 = Pattern(Integral(PolyLog(n_, (x_**WC('p', S(1))*WC('b', S(1)))**WC('q', S(1))*WC('a', S(1)))/x_, x_), cons2, cons3, cons4, cons5, cons52, cons1975) 557 rule6880 = ReplacementRule(pattern6880, replacement6880) 558 559 pattern6881 = Pattern(Integral(x_**WC('m', S(1))*PolyLog(n_, (x_**WC('p', S(1))*WC('b', S(1)))**WC('q', S(1))*WC('a', S(1))), x_), cons2, cons3, cons19, cons5, cons52, cons68, cons89, cons90) 560 rule6881 = ReplacementRule(pattern6881, replacement6881) 561 562 pattern6882 = Pattern(Integral(x_**WC('m', S(1))*PolyLog(n_, (x_**WC('p', S(1))*WC('b', S(1)))**WC('q', S(1))*WC('a', S(1))), x_), cons2, cons3, cons19, cons5, cons52, cons68, cons89, cons91) 563 rule6882 = ReplacementRule(pattern6882, replacement6882) 564 565 pattern6883 = Pattern(Integral(PolyLog(n_, (x_**WC('p', S(1))*WC('b', S(1)))**WC('q', S(1))*WC('a', S(1)))*log(x_**WC('m', S(1))*WC('c', S(1)))**WC('r', S(1))/x_, x_), cons2, cons3, cons8, cons19, cons4, cons52, cons54, cons1976, cons1977) 566 rule6883 = ReplacementRule(pattern6883, replacement6883) 567 568 pattern6884 = Pattern(Integral(PolyLog(n_, (x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*WC('c', S(1))), x_), cons2, cons3, cons8, cons5, cons89, cons90) 569 rule6884 = ReplacementRule(pattern6884, replacement6884) 570 571 pattern6885 = Pattern(Integral(x_**WC('m', S(1))*PolyLog(n_, (x_*WC('b', S(1)) + WC('a', S(0)))**WC('p', S(1))*WC('c', S(1))), x_), cons2, cons3, cons8, cons19, cons5, cons89, cons90, cons64) 572 rule6885 = ReplacementRule(pattern6885, replacement6885) 573 574 pattern6886 = Pattern(Integral(PolyLog(n_, (F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1))))**WC('p', S(1))*WC('d', S(1))), x_), cons1101, cons2, cons3, cons8, cons29, cons4, cons5, cons1978) 575 rule6886 = ReplacementRule(pattern6886, replacement6886) 576 577 pattern6887 = Pattern(Integral((x_*WC('f', S(1)) + WC('e', S(0)))**WC('m', S(1))*PolyLog(n_, (F_**((x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1))))**WC('p', S(1))*WC('d', S(1))), x_), cons1101, cons2, cons3, cons8, cons29, cons50, cons127, cons4, cons5, cons33, cons170) 578 rule6887 = ReplacementRule(pattern6887, replacement6887) 579 580 pattern6888 = Pattern(Integral(u_*PolyLog(n_, v_), x_), cons4, cons4, CustomConstraint(With6888)) 581 rule6888 = ReplacementRule(pattern6888, replacement6888) 582 583 pattern6889 = Pattern(Integral(u_*PolyLog(n_, v_)*log(w_), x_), cons4, cons1245, CustomConstraint(With6889)) 584 rule6889 = ReplacementRule(pattern6889, replacement6889) 585 586 pattern6890 = Pattern(Integral((ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))**p_, x_), cons2, cons3, cons8, cons13, cons139) 587 rule6890 = ReplacementRule(pattern6890, replacement6890) 588 589 pattern6891 = Pattern(Integral((ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons1381) 590 rule6891 = ReplacementRule(pattern6891, replacement6891) 591 592 pattern6892 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(a_ + x_*WC('b', S(1)))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons3, cons8, cons5, cons64) 593 rule6892 = ReplacementRule(pattern6892, replacement6892) 594 595 pattern6893 = Pattern(Integral((ProductLog(x_**n_*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons8, cons4, cons5, cons1979) 596 rule6893 = ReplacementRule(pattern6893, replacement6893) 597 598 pattern6894 = Pattern(Integral((ProductLog(x_**n_*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons8, cons4, cons1980) 599 rule6894 = ReplacementRule(pattern6894, replacement6894) 600 601 pattern6895 = Pattern(Integral((ProductLog(x_**n_*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons8, cons5, cons198) 602 rule6895 = ReplacementRule(pattern6895, replacement6895) 603 604 pattern6896 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons8, cons19, cons4, cons5, cons68, cons1981) 605 rule6896 = ReplacementRule(pattern6896, replacement6896) 606 607 pattern6897 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons8, cons19, cons4, cons5, cons1982) 608 rule6897 = ReplacementRule(pattern6897, replacement6897) 609 610 pattern6898 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(x_*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons8, cons19, cons1983) 611 rule6898 = ReplacementRule(pattern6898, replacement6898) 612 613 pattern6899 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(x_**n_*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1)), x_), cons2, cons8, cons5, cons152, cons465, cons68) 614 rule6899 = ReplacementRule(pattern6899, replacement6899) 615 616 pattern6900 = Pattern(Integral(S(1)/(d_ + ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('d', S(1))), x_), cons2, cons3, cons29, cons1767) 617 rule6900 = ReplacementRule(pattern6900, replacement6900) 618 619 pattern6901 = Pattern(Integral(ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))/(d_ + ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('d', S(1))), x_), cons2, cons3, cons29, cons1767) 620 rule6901 = ReplacementRule(pattern6901, replacement6901) 621 622 pattern6902 = Pattern(Integral((ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))**p_/(d_ + ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons13, cons165) 623 rule6902 = ReplacementRule(pattern6902, replacement6902) 624 625 pattern6903 = Pattern(Integral(S(1)/((d_ + ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('d', S(1)))*ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))), x_), cons2, cons3, cons29, cons1767) 626 rule6903 = ReplacementRule(pattern6903, replacement6903) 627 628 pattern6904 = Pattern(Integral(S(1)/(sqrt(ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(d_ + ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons950) 629 rule6904 = ReplacementRule(pattern6904, replacement6904) 630 631 pattern6905 = Pattern(Integral(S(1)/(sqrt(ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))*(d_ + ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('d', S(1)))), x_), cons2, cons3, cons8, cons29, cons951) 632 rule6905 = ReplacementRule(pattern6905, replacement6905) 633 634 pattern6906 = Pattern(Integral((ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))**p_/(d_ + ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons13, cons139) 635 rule6906 = ReplacementRule(pattern6906, replacement6906) 636 637 pattern6907 = Pattern(Integral((ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('c', S(1)))**WC('p', S(1))/(d_ + ProductLog(x_*WC('b', S(1)) + WC('a', S(0)))*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons5, cons1984) 638 rule6907 = ReplacementRule(pattern6907, replacement6907) 639 640 pattern6908 = Pattern(Integral(x_**WC('m', S(1))/(d_ + ProductLog(a_ + x_*WC('b', S(1)))*WC('d', S(1))), x_), cons2, cons3, cons29, cons64) 641 rule6908 = ReplacementRule(pattern6908, replacement6908) 642 643 pattern6909 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(a_ + x_*WC('b', S(1)))*WC('c', S(1)))**WC('p', S(1))/(d_ + ProductLog(a_ + x_*WC('b', S(1)))*WC('d', S(1))), x_), cons2, cons3, cons8, cons29, cons5, cons64) 644 rule6909 = ReplacementRule(pattern6909, replacement6909) 645 646 pattern6910 = Pattern(Integral(S(1)/(d_ + ProductLog(x_**n_*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons29, cons198) 647 rule6910 = ReplacementRule(pattern6910, replacement6910) 648 649 pattern6911 = Pattern(Integral((ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1))/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons8, cons29, cons4, cons5, cons1985) 650 rule6911 = ReplacementRule(pattern6911, replacement6911) 651 652 pattern6912 = Pattern(Integral(ProductLog(x_**WC('n', S(1))*WC('a', S(1)))**WC('p', S(1))/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons29, cons805, cons1986) 653 rule6912 = ReplacementRule(pattern6912, replacement6912) 654 655 pattern6913 = Pattern(Integral((ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**p_/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons8, cons29, cons805, cons1987, cons1988) 656 rule6913 = ReplacementRule(pattern6913, replacement6913) 657 658 pattern6914 = Pattern(Integral((ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**p_/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons8, cons29, cons805, cons1987, cons1989) 659 rule6914 = ReplacementRule(pattern6914, replacement6914) 660 661 pattern6915 = Pattern(Integral((ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1))/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons8, cons29, cons340, cons90, cons1990) 662 rule6915 = ReplacementRule(pattern6915, replacement6915) 663 664 pattern6916 = Pattern(Integral((ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1))/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons8, cons29, cons340, cons90, cons1991) 665 rule6916 = ReplacementRule(pattern6916, replacement6916) 666 667 pattern6917 = Pattern(Integral((ProductLog(x_**n_*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1))/(d_ + ProductLog(x_**n_*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons8, cons29, cons5, cons198) 668 rule6917 = ReplacementRule(pattern6917, replacement6917) 669 670 pattern6918 = Pattern(Integral(x_**WC('m', S(1))/(d_ + ProductLog(x_*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons29, cons33, cons170) 671 rule6918 = ReplacementRule(pattern6918, replacement6918) 672 673 pattern6919 = Pattern(Integral(S(1)/(x_*(d_ + ProductLog(x_*WC('a', S(1)))*WC('d', S(1)))), x_), cons2, cons29, cons1992) 674 rule6919 = ReplacementRule(pattern6919, replacement6919) 675 676 pattern6920 = Pattern(Integral(x_**WC('m', S(1))/(d_ + ProductLog(x_*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons29, cons33, cons96) 677 rule6920 = ReplacementRule(pattern6920, replacement6920) 678 679 pattern6921 = Pattern(Integral(x_**WC('m', S(1))/(d_ + ProductLog(x_*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons29, cons19, cons21) 680 rule6921 = ReplacementRule(pattern6921, replacement6921) 681 682 pattern6922 = Pattern(Integral(S(1)/(x_*(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1)))), x_), cons2, cons29, cons4, cons1993) 683 rule6922 = ReplacementRule(pattern6922, replacement6922) 684 685 pattern6923 = Pattern(Integral(x_**WC('m', S(1))/(d_ + ProductLog(x_**n_*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons29, cons152, cons465, cons68) 686 rule6923 = ReplacementRule(pattern6923, replacement6923) 687 688 pattern6924 = Pattern(Integral((ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1))/(x_*(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1)))), x_), cons2, cons8, cons29, cons4, cons5, cons1994) 689 rule6924 = ReplacementRule(pattern6924, replacement6924) 690 691 pattern6925 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1))/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons8, cons29, cons19, cons4, cons5, cons68, cons1995) 692 rule6925 = ReplacementRule(pattern6925, replacement6925) 693 694 pattern6926 = Pattern(Integral(x_**WC('m', S(1))*ProductLog(x_**WC('n', S(1))*WC('a', S(1)))**WC('p', S(1))/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons29, cons19, cons4, cons40, cons1996) 695 rule6926 = ReplacementRule(pattern6926, replacement6926) 696 697 pattern6927 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**p_/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons8, cons29, cons19, cons4, cons68, cons349, cons1997, cons1998) 698 rule6927 = ReplacementRule(pattern6927, replacement6927) 699 700 pattern6928 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**p_/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons8, cons29, cons19, cons4, cons68, cons349, cons1997, cons1999) 701 rule6928 = ReplacementRule(pattern6928, replacement6928) 702 703 pattern6929 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1))/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons8, cons29, cons19, cons4, cons5, cons68, cons2000, cons2001) 704 rule6929 = ReplacementRule(pattern6929, replacement6929) 705 706 pattern6930 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1))/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons8, cons29, cons19, cons4, cons5, cons68, cons2000, cons2002) 707 rule6930 = ReplacementRule(pattern6930, replacement6930) 708 709 pattern6931 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(x_*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1))/(d_ + ProductLog(x_*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons8, cons29, cons19, cons5, cons68) 710 rule6931 = ReplacementRule(pattern6931, replacement6931) 711 712 pattern6932 = Pattern(Integral(x_**WC('m', S(1))*(ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('c', S(1)))**WC('p', S(1))/(d_ + ProductLog(x_**WC('n', S(1))*WC('a', S(1)))*WC('d', S(1))), x_), cons2, cons8, cons29, cons5, cons68, cons152, cons465) 713 rule6932 = ReplacementRule(pattern6932, replacement6932) 714 715 pattern6933 = Pattern(Integral(u_, x_), cons2003) 716 rule6933 = ReplacementRule(pattern6933, replacement6933) 717 return [rule6742, rule6743, rule6744, rule6745, rule6746, rule6747, rule6748, rule6749, rule6750, rule6751, rule6752, rule6753, rule6754, rule6755, rule6756, rule6757, rule6758, rule6759, rule6760, rule6761, rule6762, rule6763, rule6764, rule6765, rule6766, rule6767, rule6768, rule6769, rule6770, rule6771, rule6772, rule6773, rule6774, rule6775, rule6776, rule6777, rule6778, rule6779, rule6780, rule6781, rule6782, rule6783, rule6784, rule6785, rule6786, rule6787, rule6788, rule6789, rule6790, rule6791, rule6792, rule6793, rule6794, rule6795, rule6796, rule6797, rule6798, rule6799, rule6800, rule6801, rule6802, rule6803, rule6804, rule6805, rule6806, rule6807, rule6808, rule6809, rule6810, rule6811, rule6812, rule6813, rule6814, rule6815, rule6816, rule6817, rule6818, rule6819, rule6820, rule6821, rule6822, rule6823, rule6824, rule6825, rule6826, rule6827, rule6828, rule6829, rule6830, rule6831, rule6832, rule6833, rule6834, rule6835, rule6836, rule6837, rule6838, rule6839, rule6840, rule6841, rule6842, rule6843, rule6844, rule6845, rule6846, rule6847, rule6848, rule6849, rule6850, rule6851, rule6852, rule6853, rule6854, rule6855, rule6856, rule6857, rule6858, rule6859, rule6860, rule6861, rule6862, rule6863, rule6864, rule6865, rule6866, rule6867, rule6868, rule6869, rule6870, rule6871, rule6872, rule6873, rule6874, rule6875, rule6876, rule6877, rule6878, rule6879, rule6880, rule6881, rule6882, rule6883, rule6884, rule6885, rule6886, rule6887, rule6888, rule6889, rule6890, rule6891, rule6892, rule6893, rule6894, rule6895, rule6896, rule6897, rule6898, rule6899, rule6900, rule6901, rule6902, rule6903, rule6904, rule6905, rule6906, rule6907, rule6908, rule6909, rule6910, rule6911, rule6912, rule6913, rule6914, rule6915, rule6916, rule6917, rule6918, rule6919, rule6920, rule6921, rule6922, rule6923, rule6924, rule6925, rule6926, rule6927, rule6928, rule6929, rule6930, rule6931, rule6932, rule6933, ] 718 719 720 721 722 723def replacement6742(a, b, x): 724 return Simp(exp(-(a + b*x)**S(2))/(sqrt(Pi)*b), x) + Simp((a + b*x)*Erf(a + b*x)/b, x) 725 726 727def replacement6743(a, b, x): 728 return -Simp(exp(-(a + b*x)**S(2))/(sqrt(Pi)*b), x) + Simp((a + b*x)*Erfc(a + b*x)/b, x) 729 730 731def replacement6744(a, b, x): 732 return -Simp(exp((a + b*x)**S(2))/(sqrt(Pi)*b), x) + Simp((a + b*x)*Erfi(a + b*x)/b, x) 733 734 735def replacement6745(b, x): 736 return Simp(S(2)*b*x*HypergeometricPFQ(List(S(1)/2, S(1)/2), List(S(3)/2, S(3)/2), -b**S(2)*x**S(2))/sqrt(Pi), x) 737 738 739def replacement6746(b, x): 740 return -Int(Erf(b*x)/x, x) + Simp(log(x), x) 741 742 743def replacement6747(b, x): 744 return Simp(S(2)*b*x*HypergeometricPFQ(List(S(1)/2, S(1)/2), List(S(3)/2, S(3)/2), b**S(2)*x**S(2))/sqrt(Pi), x) 745 746 747def replacement6748(a, b, m, x): 748 return -Dist(S(2)*b/(sqrt(Pi)*(m + S(1))), Int(x**(m + S(1))*exp(-(a + b*x)**S(2)), x), x) + Simp(x**(m + S(1))*Erf(a + b*x)/(m + S(1)), x) 749 750 751def replacement6749(a, b, m, x): 752 return Dist(S(2)*b/(sqrt(Pi)*(m + S(1))), Int(x**(m + S(1))*exp(-(a + b*x)**S(2)), x), x) + Simp(x**(m + S(1))*Erfc(a + b*x)/(m + S(1)), x) 753 754 755def replacement6750(a, b, m, x): 756 return -Dist(S(2)*b/(sqrt(Pi)*(m + S(1))), Int(x**(m + S(1))*exp((a + b*x)**S(2)), x), x) + Simp(x**(m + S(1))*Erfi(a + b*x)/(m + S(1)), x) 757 758 759def replacement6751(a, b, c, d, x): 760 return -Dist(b/(sqrt(Pi)*d), Int(exp(-a**S(2) - S(2)*a*b*x + c - x**S(2)*(b**S(2) - d)), x), x) + Simp(Erf(a + b*x)*exp(c + d*x**S(2))/(S(2)*d), x) 761 762 763def replacement6752(a, b, c, d, x): 764 return Dist(b/(sqrt(Pi)*d), Int(exp(-a**S(2) - S(2)*a*b*x + c - x**S(2)*(b**S(2) - d)), x), x) + Simp(Erfc(a + b*x)*exp(c + d*x**S(2))/(S(2)*d), x) 765 766 767def replacement6753(a, b, c, d, x): 768 return -Dist(b/(sqrt(Pi)*d), Int(exp(a**S(2) + S(2)*a*b*x + c + x**S(2)*(b**S(2) + d)), x), x) + Simp(Erfi(a + b*x)*exp(c + d*x**S(2))/(S(2)*d), x) 769 770 771def replacement6754(a, b, c, d, m, x): 772 return -Dist((m + S(-1))/(S(2)*d), Int(x**(m + S(-2))*Erf(a + b*x)*exp(c + d*x**S(2)), x), x) - Dist(b/(sqrt(Pi)*d), Int(x**(m + S(-1))*exp(-a**S(2) - S(2)*a*b*x + c - x**S(2)*(b**S(2) - d)), x), x) + Simp(x**(m + S(-1))*Erf(a + b*x)*exp(c + d*x**S(2))/(S(2)*d), x) 773 774 775def replacement6755(a, b, c, d, m, x): 776 return -Dist((m + S(-1))/(S(2)*d), Int(x**(m + S(-2))*Erfc(a + b*x)*exp(c + d*x**S(2)), x), x) + Dist(b/(sqrt(Pi)*d), Int(x**(m + S(-1))*exp(-a**S(2) - S(2)*a*b*x + c - x**S(2)*(b**S(2) - d)), x), x) + Simp(x**(m + S(-1))*Erfc(a + b*x)*exp(c + d*x**S(2))/(S(2)*d), x) 777 778 779def replacement6756(a, b, c, d, m, x): 780 return -Dist((m + S(-1))/(S(2)*d), Int(x**(m + S(-2))*Erfi(a + b*x)*exp(c + d*x**S(2)), x), x) - Dist(b/(sqrt(Pi)*d), Int(x**(m + S(-1))*exp(a**S(2) + S(2)*a*b*x + c + x**S(2)*(b**S(2) + d)), x), x) + Simp(x**(m + S(-1))*Erfi(a + b*x)*exp(c + d*x**S(2))/(S(2)*d), x) 781 782 783def replacement6757(b, c, d, x): 784 return Simp(S(2)*b*x*HypergeometricPFQ(List(S(1)/2, S(1)), List(S(3)/2, S(3)/2), d*x**S(2))*exp(c)/sqrt(Pi), x) 785 786 787def replacement6758(b, c, d, x): 788 return Int(exp(c + d*x**S(2))/x, x) - Int(Erf(b*x)*exp(c + d*x**S(2))/x, x) 789 790 791def replacement6759(b, c, d, x): 792 return Simp(S(2)*b*x*HypergeometricPFQ(List(S(1)/2, S(1)), List(S(3)/2, S(3)/2), d*x**S(2))*exp(c)/sqrt(Pi), x) 793 794 795def replacement6760(a, b, c, d, m, x): 796 return -Dist(S(2)*d/(m + S(1)), Int(x**(m + S(2))*Erf(a + b*x)*exp(c + d*x**S(2)), x), x) - Dist(S(2)*b/(sqrt(Pi)*(m + S(1))), Int(x**(m + S(1))*exp(-a**S(2) - S(2)*a*b*x + c - x**S(2)*(b**S(2) - d)), x), x) + Simp(x**(m + S(1))*Erf(a + b*x)*exp(c + d*x**S(2))/(m + S(1)), x) 797 798 799def replacement6761(a, b, c, d, m, x): 800 return -Dist(S(2)*d/(m + S(1)), Int(x**(m + S(2))*Erfc(a + b*x)*exp(c + d*x**S(2)), x), x) + Dist(S(2)*b/(sqrt(Pi)*(m + S(1))), Int(x**(m + S(1))*exp(-a**S(2) - S(2)*a*b*x + c - x**S(2)*(b**S(2) - d)), x), x) + Simp(x**(m + S(1))*Erfc(a + b*x)*exp(c + d*x**S(2))/(m + S(1)), x) 801 802 803def replacement6762(a, b, c, d, m, x): 804 return -Dist(S(2)*d/(m + S(1)), Int(x**(m + S(2))*Erfi(a + b*x)*exp(c + d*x**S(2)), x), x) - Dist(S(2)*b/(sqrt(Pi)*(m + S(1))), Int(x**(m + S(1))*exp(a**S(2) + S(2)*a*b*x + c + x**S(2)*(b**S(2) + d)), x), x) + Simp(x**(m + S(1))*Erfi(a + b*x)*exp(c + d*x**S(2))/(m + S(1)), x) 805 806 807def replacement6763(a, b, x): 808 return -Dist(S(4)/sqrt(Pi), Int((a + b*x)*Erf(a + b*x)*exp(-(a + b*x)**S(2)), x), x) + Simp((a + b*x)*Erf(a + b*x)**S(2)/b, x) 809 810 811def replacement6764(a, b, x): 812 return Dist(S(4)/sqrt(Pi), Int((a + b*x)*Erfc(a + b*x)*exp(-(a + b*x)**S(2)), x), x) + Simp((a + b*x)*Erfc(a + b*x)**S(2)/b, x) 813 814 815def replacement6765(a, b, x): 816 return -Dist(S(4)/sqrt(Pi), Int((a + b*x)*Erfi(a + b*x)*exp((a + b*x)**S(2)), x), x) + Simp((a + b*x)*Erfi(a + b*x)**S(2)/b, x) 817 818 819def replacement6766(b, m, x): 820 return -Dist(S(4)*b/(sqrt(Pi)*(m + S(1))), Int(x**(m + S(1))*Erf(b*x)*exp(-b**S(2)*x**S(2)), x), x) + Simp(x**(m + S(1))*Erf(b*x)**S(2)/(m + S(1)), x) 821 822 823def replacement6767(b, m, x): 824 return Dist(S(4)*b/(sqrt(Pi)*(m + S(1))), Int(x**(m + S(1))*Erfc(b*x)*exp(-b**S(2)*x**S(2)), x), x) + Simp(x**(m + S(1))*Erfc(b*x)**S(2)/(m + S(1)), x) 825 826 827def replacement6768(b, m, x): 828 return -Dist(S(4)*b/(sqrt(Pi)*(m + S(1))), Int(x**(m + S(1))*Erfi(b*x)*exp(b**S(2)*x**S(2)), x), x) + Simp(x**(m + S(1))*Erfi(b*x)**S(2)/(m + S(1)), x) 829 830 831def replacement6769(a, b, m, x): 832 return Dist(S(1)/b, Subst(Int((-a/b + x/b)**m*Erf(x)**S(2), x), x, a + b*x), x) 833 834 835def replacement6770(a, b, m, x): 836 return Dist(S(1)/b, Subst(Int((-a/b + x/b)**m*Erfc(x)**S(2), x), x, a + b*x), x) 837 838 839def replacement6771(a, b, m, x): 840 return Dist(S(1)/b, Subst(Int((-a/b + x/b)**m*Erfi(x)**S(2), x), x, a + b*x), x) 841 842 843def replacement6772(a, b, x): 844 return Simp(cos(Pi*(a + b*x)**S(2)/S(2))/(Pi*b), x) + Simp((a + b*x)*FresnelS(a + b*x)/b, x) 845 846 847def replacement6773(a, b, x): 848 return -Simp(sin(Pi*(a + b*x)**S(2)/S(2))/(Pi*b), x) + Simp((a + b*x)*FresnelC(a + b*x)/b, x) 849 850 851def replacement6774(b, x): 852 return Simp(I*b*x*HypergeometricPFQ(List(S(1)/2, S(1)/2), List(S(3)/2, S(3)/2), -I*Pi*b**S(2)*x**S(2)/S(2))/S(2), x) - Simp(I*b*x*HypergeometricPFQ(List(S(1)/2, S(1)/2), List(S(3)/2, S(3)/2), I*Pi*b**S(2)*x**S(2)/S(2))/S(2), x) 853 854 855def replacement6775(b, x): 856 return Simp(b*x*HypergeometricPFQ(List(S(1)/2, S(1)/2), List(S(3)/2, S(3)/2), -I*Pi*b**S(2)*x**S(2)/S(2))/S(2), x) + Simp(b*x*HypergeometricPFQ(List(S(1)/2, S(1)/2), List(S(3)/2, S(3)/2), I*Pi*b**S(2)*x**S(2)/S(2))/S(2), x) 857 858 859def replacement6776(a, b, m, x): 860 return -Dist(b/(m + S(1)), Int(x**(m + S(1))*sin(Pi*(a + b*x)**S(2)/S(2)), x), x) + Simp(x**(m + S(1))*FresnelS(a + b*x)/(m + S(1)), x) 861 862 863def replacement6777(a, b, m, x): 864 return -Dist(b/(m + S(1)), Int(x**(m + S(1))*cos(Pi*(a + b*x)**S(2)/S(2)), x), x) + Simp(x**(m + S(1))*FresnelC(a + b*x)/(m + S(1)), x) 865 866 867def replacement6778(a, b, x): 868 return -Dist(S(2), Int((a + b*x)*FresnelS(a + b*x)*sin(Pi*(a + b*x)**S(2)/S(2)), x), x) + Simp((a + b*x)*FresnelS(a + b*x)**S(2)/b, x) 869 870 871def replacement6779(a, b, x): 872 return -Dist(S(2), Int((a + b*x)*FresnelC(a + b*x)*cos(Pi*(a + b*x)**S(2)/S(2)), x), x) + Simp((a + b*x)*FresnelC(a + b*x)**S(2)/b, x) 873 874 875def replacement6780(b, m, x): 876 return -Dist(S(2)*b/(m + S(1)), Int(x**(m + S(1))*FresnelS(b*x)*sin(Pi*b**S(2)*x**S(2)/S(2)), x), x) + Simp(x**(m + S(1))*FresnelS(b*x)**S(2)/(m + S(1)), x) 877 878 879def replacement6781(b, m, x): 880 return -Dist(S(2)*b/(m + S(1)), Int(x**(m + S(1))*FresnelC(b*x)*cos(Pi*b**S(2)*x**S(2)/S(2)), x), x) + Simp(x**(m + S(1))*FresnelC(b*x)**S(2)/(m + S(1)), x) 881 882 883def replacement6782(b, c, x): 884 return Dist(S(1)/(S(2)*Pi*b), Int(sin(Pi*b**S(2)*x**S(2)), x), x) - Simp(FresnelS(b*x)*cos(Pi*b**S(2)*x**S(2)/S(2))/(Pi*b**S(2)), x) 885 886 887def replacement6783(b, c, x): 888 return -Dist(S(1)/(S(2)*Pi*b), Int(sin(Pi*b**S(2)*x**S(2)), x), x) + Simp(FresnelC(b*x)*sin(Pi*b**S(2)*x**S(2)/S(2))/(Pi*b**S(2)), x) 889 890 891def replacement6784(b, c, m, x): 892 return Dist(S(1)/(S(2)*Pi*b), Int(x**(m + S(-1))*sin(Pi*b**S(2)*x**S(2)), x), x) + Dist((m + S(-1))/(Pi*b**S(2)), Int(x**(m + S(-2))*FresnelS(b*x)*cos(Pi*b**S(2)*x**S(2)/S(2)), x), x) - Simp(x**(m + S(-1))*FresnelS(b*x)*cos(Pi*b**S(2)*x**S(2)/S(2))/(Pi*b**S(2)), x) 893 894 895def replacement6785(b, c, m, x): 896 return -Dist(S(1)/(S(2)*Pi*b), Int(x**(m + S(-1))*sin(Pi*b**S(2)*x**S(2)), x), x) - Dist((m + S(-1))/(Pi*b**S(2)), Int(x**(m + S(-2))*FresnelC(b*x)*sin(Pi*b**S(2)*x**S(2)/S(2)), x), x) + Simp(x**(m + S(-1))*FresnelC(b*x)*sin(Pi*b**S(2)*x**S(2)/S(2))/(Pi*b**S(2)), x) 897 898 899def replacement6786(b, c, m, x): 900 return Dist(b/(S(2)*m + S(2)), Int(x**(m + S(1))*cos(Pi*b**S(2)*x**S(2)), x), x) - Dist(Pi*b**S(2)/(m + S(1)), Int(x**(m + S(2))*FresnelS(b*x)*cos(Pi*b**S(2)*x**S(2)/S(2)), x), x) - Simp(b*x**(m + S(2))/(S(2)*(m + S(1))*(m + S(2))), x) + Simp(x**(m + S(1))*FresnelS(b*x)*sin(Pi*b**S(2)*x**S(2)/S(2))/(m + S(1)), x) 901 902 903def replacement6787(b, c, m, x): 904 return -Dist(b/(S(2)*m + S(2)), Int(x**(m + S(1))*cos(Pi*b**S(2)*x**S(2)), x), x) + Dist(Pi*b**S(2)/(m + S(1)), Int(x**(m + S(2))*FresnelC(b*x)*sin(Pi*b**S(2)*x**S(2)/S(2)), x), x) - Simp(b*x**(m + S(2))/(S(2)*(m + S(1))*(m + S(2))), x) + Simp(x**(m + S(1))*FresnelC(b*x)*cos(Pi*b**S(2)*x**S(2)/S(2))/(m + S(1)), x) 905 906 907def replacement6788(b, c, x): 908 return Dist(S(1)/(S(2)*Pi*b), Int(cos(Pi*b**S(2)*x**S(2)), x), x) - Simp(x/(S(2)*Pi*b), x) + Simp(FresnelS(b*x)*sin(Pi*b**S(2)*x**S(2)/S(2))/(Pi*b**S(2)), x) 909 910 911def replacement6789(b, c, x): 912 return Dist(S(1)/(S(2)*Pi*b), Int(cos(Pi*b**S(2)*x**S(2)), x), x) + Simp(x/(S(2)*Pi*b), x) - Simp(FresnelC(b*x)*cos(Pi*b**S(2)*x**S(2)/S(2))/(Pi*b**S(2)), x) 913 914 915def replacement6790(b, c, m, x): 916 return Dist(S(1)/(S(2)*Pi*b), Int(x**(m + S(-1))*cos(Pi*b**S(2)*x**S(2)), x), x) - Dist((m + S(-1))/(Pi*b**S(2)), Int(x**(m + S(-2))*FresnelS(b*x)*sin(Pi*b**S(2)*x**S(2)/S(2)), x), x) - Simp(x**m/(S(2)*Pi*b*m), x) + Simp(x**(m + S(-1))*FresnelS(b*x)*sin(Pi*b**S(2)*x**S(2)/S(2))/(Pi*b**S(2)), x) 917 918 919def replacement6791(b, c, m, x): 920 return Dist(S(1)/(S(2)*Pi*b), Int(x**(m + S(-1))*cos(Pi*b**S(2)*x**S(2)), x), x) + Dist((m + S(-1))/(Pi*b**S(2)), Int(x**(m + S(-2))*FresnelC(b*x)*cos(Pi*b**S(2)*x**S(2)/S(2)), x), x) + Simp(x**m/(S(2)*Pi*b*m), x) - Simp(x**(m + S(-1))*FresnelC(b*x)*cos(Pi*b**S(2)*x**S(2)/S(2))/(Pi*b**S(2)), x) 921 922 923def replacement6792(b, c, m, x): 924 return -Dist(b/(S(2)*m + S(2)), Int(x**(m + S(1))*sin(Pi*b**S(2)*x**S(2)), x), x) + Dist(Pi*b**S(2)/(m + S(1)), Int(x**(m + S(2))*FresnelS(b*x)*sin(Pi*b**S(2)*x**S(2)/S(2)), x), x) + Simp(x**(m + S(1))*FresnelS(b*x)*cos(Pi*b**S(2)*x**S(2)/S(2))/(m + S(1)), x) 925 926 927def replacement6793(b, c, m, x): 928 return -Dist(b/(S(2)*m + S(2)), Int(x**(m + S(1))*sin(Pi*b**S(2)*x**S(2)), x), x) - Dist(Pi*b**S(2)/(m + S(1)), Int(x**(m + S(2))*FresnelC(b*x)*cos(Pi*b**S(2)*x**S(2)/S(2)), x), x) + Simp(x**(m + S(1))*FresnelC(b*x)*sin(Pi*b**S(2)*x**S(2)/S(2))/(m + S(1)), x) 929 930 931def replacement6794(a, b, n, x): 932 return -Simp(ExpIntegralE(n + S(1), a + b*x)/b, x) 933 934 935def replacement6795(b, m, n, x): 936 return Dist(m/b, Int(x**(m + S(-1))*ExpIntegralE(n + S(1), b*x), x), x) - Simp(x**m*ExpIntegralE(n + S(1), b*x)/b, x) 937 938 939def replacement6796(b, x): 940 return -Simp(EulerGamma*log(x), x) + Simp(b*x*HypergeometricPFQ(List(S(1), S(1), S(1)), List(S(2), S(2), S(2)), -b*x), x) - Simp(log(b*x)**S(2)/S(2), x) 941 942 943def replacement6797(b, m, n, x): 944 return Dist(b/(m + S(1)), Int(x**(m + S(1))*ExpIntegralE(n + S(-1), b*x), x), x) + Simp(x**(m + S(1))*ExpIntegralE(n, b*x)/(m + S(1)), x) 945 946 947def replacement6798(b, m, n, x): 948 return -Simp(x**(m + S(1))*HypergeometricPFQ(List(m + S(1), m + S(1)), List(m + S(2), m + S(2)), -b*x)/(m + S(1))**S(2), x) + Simp(x**m*(b*x)**(-m)*Gamma(m + S(1))*log(x)/b, x) 949 950 951def replacement6799(b, m, n, x): 952 return -Simp(x**(m + S(1))*ExpIntegralE(-m, b*x)/(m + n), x) + Simp(x**(m + S(1))*ExpIntegralE(n, b*x)/(m + n), x) 953 954 955def replacement6800(a, b, m, n, x): 956 return Dist(m/b, Int(x**(m + S(-1))*ExpIntegralE(n + S(1), a + b*x), x), x) - Simp(x**m*ExpIntegralE(n + S(1), a + b*x)/b, x) 957 958 959def replacement6801(a, b, m, n, x): 960 return Dist(b/(m + S(1)), Int(x**(m + S(1))*ExpIntegralE(n + S(-1), a + b*x), x), x) + Simp(x**(m + S(1))*ExpIntegralE(n, a + b*x)/(m + S(1)), x) 961 962 963def replacement6802(a, b, x): 964 return -Simp(exp(a + b*x)/b, x) + Simp((a + b*x)*ExpIntegralEi(a + b*x)/b, x) 965 966 967def replacement6803(a, b, m, x): 968 return -Dist(b/(m + S(1)), Int(x**(m + S(1))*exp(a + b*x)/(a + b*x), x), x) + Simp(x**(m + S(1))*ExpIntegralEi(a + b*x)/(m + S(1)), x) 969 970 971def replacement6804(a, b, x): 972 return -Dist(S(2), Int(ExpIntegralEi(a + b*x)*exp(a + b*x), x), x) + Simp((a + b*x)*ExpIntegralEi(a + b*x)**S(2)/b, x) 973 974 975def replacement6805(b, m, x): 976 return -Dist(S(2)/(m + S(1)), Int(x**m*ExpIntegralEi(b*x)*exp(b*x), x), x) + Simp(x**(m + S(1))*ExpIntegralEi(b*x)**S(2)/(m + S(1)), x) 977 978 979def replacement6806(a, b, m, x): 980 return -Dist(a*m/(b*(m + S(1))), Int(x**(m + S(-1))*ExpIntegralEi(a + b*x)**S(2), x), x) - Dist(S(2)/(m + S(1)), Int(x**m*ExpIntegralEi(a + b*x)*exp(a + b*x), x), x) + Simp(x**(m + S(1))*ExpIntegralEi(a + b*x)**S(2)/(m + S(1)), x) + Simp(a*x**m*ExpIntegralEi(a + b*x)**S(2)/(b*(m + S(1))), x) 981 982 983def replacement6807(a, b, c, d, x): 984 return -Dist(d/b, Int(exp(a + c + x*(b + d))/(c + d*x), x), x) + Simp(ExpIntegralEi(c + d*x)*exp(a + b*x)/b, x) 985 986 987def replacement6808(a, b, c, d, m, x): 988 return -Dist(d/b, Int(x**m*exp(a + c + x*(b + d))/(c + d*x), x), x) - Dist(m/b, Int(x**(m + S(-1))*ExpIntegralEi(c + d*x)*exp(a + b*x), x), x) + Simp(x**m*ExpIntegralEi(c + d*x)*exp(a + b*x)/b, x) 989 990 991def replacement6809(a, b, c, d, m, x): 992 return -Dist(b/(m + S(1)), Int(x**(m + S(1))*ExpIntegralEi(c + d*x)*exp(a + b*x), x), x) - Dist(d/(m + S(1)), Int(x**(m + S(1))*exp(a + c + x*(b + d))/(c + d*x), x), x) + Simp(x**(m + S(1))*ExpIntegralEi(c + d*x)*exp(a + b*x)/(m + S(1)), x) 993 994 995def replacement6810(a, b, x): 996 return -Simp(ExpIntegralEi(S(2)*log(a + b*x))/b, x) + Simp((a + b*x)*LogIntegral(a + b*x)/b, x) 997 998 999def replacement6811(b, x): 1000 return -Simp(b*x, x) + Simp(LogIntegral(b*x)*log(b*x), x) 1001 1002 1003def replacement6812(a, b, m, x): 1004 return -Dist(b/(m + S(1)), Int(x**(m + S(1))/log(a + b*x), x), x) + Simp(x**(m + S(1))*LogIntegral(a + b*x)/(m + S(1)), x) 1005 1006 1007def replacement6813(a, b, x): 1008 return Simp(cos(a + b*x)/b, x) + Simp((a + b*x)*SinIntegral(a + b*x)/b, x) 1009 1010 1011def replacement6814(a, b, x): 1012 return -Simp(sin(a + b*x)/b, x) + Simp((a + b*x)*CosIntegral(a + b*x)/b, x) 1013 1014 1015def replacement6815(b, x): 1016 return Simp(b*x*HypergeometricPFQ(List(S(1), S(1), S(1)), List(S(2), S(2), S(2)), -I*b*x)/S(2), x) + Simp(b*x*HypergeometricPFQ(List(S(1), S(1), S(1)), List(S(2), S(2), S(2)), I*b*x)/S(2), x) 1017 1018 1019def replacement6816(b, x): 1020 return Simp(EulerGamma*log(x), x) - Simp(I*b*x*HypergeometricPFQ(List(S(1), S(1), S(1)), List(S(2), S(2), S(2)), -I*b*x)/S(2), x) + Simp(I*b*x*HypergeometricPFQ(List(S(1), S(1), S(1)), List(S(2), S(2), S(2)), I*b*x)/S(2), x) + Simp(log(b*x)**S(2)/S(2), x) 1021 1022 1023def replacement6817(a, b, m, x): 1024 return -Dist(b/(m + S(1)), Int(x**(m + S(1))*sin(a + b*x)/(a + b*x), x), x) + Simp(x**(m + S(1))*SinIntegral(a + b*x)/(m + S(1)), x) 1025 1026 1027def replacement6818(a, b, m, x): 1028 return -Dist(b/(m + S(1)), Int(x**(m + S(1))*cos(a + b*x)/(a + b*x), x), x) + Simp(x**(m + S(1))*CosIntegral(a + b*x)/(m + S(1)), x) 1029 1030 1031def replacement6819(a, b, x): 1032 return -Dist(S(2), Int(SinIntegral(a + b*x)*sin(a + b*x), x), x) + Simp((a + b*x)*SinIntegral(a + b*x)**S(2)/b, x) 1033 1034 1035def replacement6820(a, b, x): 1036 return -Dist(S(2), Int(CosIntegral(a + b*x)*cos(a + b*x), x), x) + Simp((a + b*x)*CosIntegral(a + b*x)**S(2)/b, x) 1037 1038 1039def replacement6821(b, m, x): 1040 return -Dist(S(2)/(m + S(1)), Int(x**m*SinIntegral(b*x)*sin(b*x), x), x) + Simp(x**(m + S(1))*SinIntegral(b*x)**S(2)/(m + S(1)), x) 1041 1042 1043def replacement6822(b, m, x): 1044 return -Dist(S(2)/(m + S(1)), Int(x**m*CosIntegral(b*x)*cos(b*x), x), x) + Simp(x**(m + S(1))*CosIntegral(b*x)**S(2)/(m + S(1)), x) 1045 1046 1047def replacement6823(a, b, m, x): 1048 return -Dist(a*m/(b*(m + S(1))), Int(x**(m + S(-1))*SinIntegral(a + b*x)**S(2), x), x) - Dist(S(2)/(m + S(1)), Int(x**m*SinIntegral(a + b*x)*sin(a + b*x), x), x) + Simp(x**(m + S(1))*SinIntegral(a + b*x)**S(2)/(m + S(1)), x) + Simp(a*x**m*SinIntegral(a + b*x)**S(2)/(b*(m + S(1))), x) 1049 1050 1051def replacement6824(a, b, m, x): 1052 return -Dist(a*m/(b*(m + S(1))), Int(x**(m + S(-1))*CosIntegral(a + b*x)**S(2), x), x) - Dist(S(2)/(m + S(1)), Int(x**m*CosIntegral(a + b*x)*cos(a + b*x), x), x) + Simp(x**(m + S(1))*CosIntegral(a + b*x)**S(2)/(m + S(1)), x) + Simp(a*x**m*CosIntegral(a + b*x)**S(2)/(b*(m + S(1))), x) 1053 1054 1055def replacement6825(a, b, c, d, x): 1056 return Dist(d/b, Int(sin(c + d*x)*cos(a + b*x)/(c + d*x), x), x) - Simp(SinIntegral(c + d*x)*cos(a + b*x)/b, x) 1057 1058 1059def replacement6826(a, b, c, d, x): 1060 return -Dist(d/b, Int(sin(a + b*x)*cos(c + d*x)/(c + d*x), x), x) + Simp(CosIntegral(c + d*x)*sin(a + b*x)/b, x) 1061 1062 1063def replacement6827(a, b, c, d, m, x): 1064 return Dist(d/b, Int(x**m*sin(c + d*x)*cos(a + b*x)/(c + d*x), x), x) + Dist(m/b, Int(x**(m + S(-1))*SinIntegral(c + d*x)*cos(a + b*x), x), x) - Simp(x**m*SinIntegral(c + d*x)*cos(a + b*x)/b, x) 1065 1066 1067def replacement6828(a, b, c, d, m, x): 1068 return -Dist(d/b, Int(x**m*sin(a + b*x)*cos(c + d*x)/(c + d*x), x), x) - Dist(m/b, Int(x**(m + S(-1))*CosIntegral(c + d*x)*sin(a + b*x), x), x) + Simp(x**m*CosIntegral(c + d*x)*sin(a + b*x)/b, x) 1069 1070 1071def replacement6829(a, b, c, d, m, x): 1072 return -Dist(b/(m + S(1)), Int(x**(m + S(1))*SinIntegral(c + d*x)*cos(a + b*x), x), x) - Dist(d/(m + S(1)), Int(x**(m + S(1))*sin(a + b*x)*sin(c + d*x)/(c + d*x), x), x) + Simp(x**(m + S(1))*SinIntegral(c + d*x)*sin(a + b*x)/(m + S(1)), x) 1073 1074 1075def replacement6830(a, b, c, d, m, x): 1076 return Dist(b/(m + S(1)), Int(x**(m + S(1))*CosIntegral(c + d*x)*sin(a + b*x), x), x) - Dist(d/(m + S(1)), Int(x**(m + S(1))*cos(a + b*x)*cos(c + d*x)/(c + d*x), x), x) + Simp(x**(m + S(1))*CosIntegral(c + d*x)*cos(a + b*x)/(m + S(1)), x) 1077 1078 1079def replacement6831(a, b, c, d, x): 1080 return -Dist(d/b, Int(sin(a + b*x)*sin(c + d*x)/(c + d*x), x), x) + Simp(SinIntegral(c + d*x)*sin(a + b*x)/b, x) 1081 1082 1083def replacement6832(a, b, c, d, x): 1084 return Dist(d/b, Int(cos(a + b*x)*cos(c + d*x)/(c + d*x), x), x) - Simp(CosIntegral(c + d*x)*cos(a + b*x)/b, x) 1085 1086 1087def replacement6833(a, b, c, d, m, x): 1088 return -Dist(d/b, Int(x**m*sin(a + b*x)*sin(c + d*x)/(c + d*x), x), x) - Dist(m/b, Int(x**(m + S(-1))*SinIntegral(c + d*x)*sin(a + b*x), x), x) + Simp(x**m*SinIntegral(c + d*x)*sin(a + b*x)/b, x) 1089 1090 1091def replacement6834(a, b, c, d, m, x): 1092 return Dist(d/b, Int(x**m*cos(a + b*x)*cos(c + d*x)/(c + d*x), x), x) + Dist(m/b, Int(x**(m + S(-1))*CosIntegral(c + d*x)*cos(a + b*x), x), x) - Simp(x**m*CosIntegral(c + d*x)*cos(a + b*x)/b, x) 1093 1094 1095def replacement6835(a, b, c, d, m, x): 1096 return Dist(b/(m + S(1)), Int(x**(m + S(1))*SinIntegral(c + d*x)*sin(a + b*x), x), x) - Dist(d/(m + S(1)), Int(x**(m + S(1))*sin(c + d*x)*cos(a + b*x)/(c + d*x), x), x) + Simp(x**(m + S(1))*SinIntegral(c + d*x)*cos(a + b*x)/(m + S(1)), x) 1097 1098 1099def replacement6836(a, b, c, d, m, x): 1100 return -Dist(b/(m + S(1)), Int(x**(m + S(1))*CosIntegral(c + d*x)*cos(a + b*x), x), x) - Dist(d/(m + S(1)), Int(x**(m + S(1))*sin(a + b*x)*cos(c + d*x)/(c + d*x), x), x) + Simp(x**(m + S(1))*CosIntegral(c + d*x)*sin(a + b*x)/(m + S(1)), x) 1101 1102 1103def replacement6837(a, b, x): 1104 return -Simp(cosh(a + b*x)/b, x) + Simp((a + b*x)*SinhIntegral(a + b*x)/b, x) 1105 1106 1107def replacement6838(a, b, x): 1108 return -Simp(sinh(a + b*x)/b, x) + Simp((a + b*x)*CoshIntegral(a + b*x)/b, x) 1109 1110 1111def replacement6839(b, x): 1112 return Simp(b*x*HypergeometricPFQ(List(S(1), S(1), S(1)), List(S(2), S(2), S(2)), -b*x)/S(2), x) + Simp(b*x*HypergeometricPFQ(List(S(1), S(1), S(1)), List(S(2), S(2), S(2)), b*x)/S(2), x) 1113 1114 1115def replacement6840(b, x): 1116 return Simp(EulerGamma*log(x), x) - Simp(b*x*HypergeometricPFQ(List(S(1), S(1), S(1)), List(S(2), S(2), S(2)), -b*x)/S(2), x) + Simp(b*x*HypergeometricPFQ(List(S(1), S(1), S(1)), List(S(2), S(2), S(2)), b*x)/S(2), x) + Simp(log(b*x)**S(2)/S(2), x) 1117 1118 1119def replacement6841(a, b, m, x): 1120 return -Dist(b/(m + S(1)), Int(x**(m + S(1))*sinh(a + b*x)/(a + b*x), x), x) + Simp(x**(m + S(1))*SinhIntegral(a + b*x)/(m + S(1)), x) 1121 1122 1123def replacement6842(a, b, m, x): 1124 return -Dist(b/(m + S(1)), Int(x**(m + S(1))*cosh(a + b*x)/(a + b*x), x), x) + Simp(x**(m + S(1))*CoshIntegral(a + b*x)/(m + S(1)), x) 1125 1126 1127def replacement6843(a, b, x): 1128 return -Dist(S(2), Int(SinhIntegral(a + b*x)*sinh(a + b*x), x), x) + Simp((a + b*x)*SinhIntegral(a + b*x)**S(2)/b, x) 1129 1130 1131def replacement6844(a, b, x): 1132 return -Dist(S(2), Int(CoshIntegral(a + b*x)*cosh(a + b*x), x), x) + Simp((a + b*x)*CoshIntegral(a + b*x)**S(2)/b, x) 1133 1134 1135def replacement6845(b, m, x): 1136 return -Dist(S(2)/(m + S(1)), Int(x**m*SinhIntegral(b*x)*sinh(b*x), x), x) + Simp(x**(m + S(1))*SinhIntegral(b*x)**S(2)/(m + S(1)), x) 1137 1138 1139def replacement6846(b, m, x): 1140 return -Dist(S(2)/(m + S(1)), Int(x**m*CoshIntegral(b*x)*cosh(b*x), x), x) + Simp(x**(m + S(1))*CoshIntegral(b*x)**S(2)/(m + S(1)), x) 1141 1142 1143def replacement6847(a, b, m, x): 1144 return -Dist(a*m/(b*(m + S(1))), Int(x**(m + S(-1))*SinhIntegral(a + b*x)**S(2), x), x) - Dist(S(2)/(m + S(1)), Int(x**m*SinhIntegral(a + b*x)*sinh(a + b*x), x), x) + Simp(x**(m + S(1))*SinhIntegral(a + b*x)**S(2)/(m + S(1)), x) + Simp(a*x**m*SinhIntegral(a + b*x)**S(2)/(b*(m + S(1))), x) 1145 1146 1147def replacement6848(a, b, m, x): 1148 return -Dist(a*m/(b*(m + S(1))), Int(x**(m + S(-1))*CoshIntegral(a + b*x)**S(2), x), x) - Dist(S(2)/(m + S(1)), Int(x**m*CoshIntegral(a + b*x)*cosh(a + b*x), x), x) + Simp(x**(m + S(1))*CoshIntegral(a + b*x)**S(2)/(m + S(1)), x) + Simp(a*x**m*CoshIntegral(a + b*x)**S(2)/(b*(m + S(1))), x) 1149 1150 1151def replacement6849(a, b, c, d, x): 1152 return -Dist(d/b, Int(sinh(c + d*x)*cosh(a + b*x)/(c + d*x), x), x) + Simp(SinhIntegral(c + d*x)*cosh(a + b*x)/b, x) 1153 1154 1155def replacement6850(a, b, c, d, x): 1156 return -Dist(d/b, Int(sinh(a + b*x)*cosh(c + d*x)/(c + d*x), x), x) + Simp(CoshIntegral(c + d*x)*sinh(a + b*x)/b, x) 1157 1158 1159def replacement6851(a, b, c, d, m, x): 1160 return -Dist(d/b, Int(x**m*sinh(c + d*x)*cosh(a + b*x)/(c + d*x), x), x) - Dist(m/b, Int(x**(m + S(-1))*SinhIntegral(c + d*x)*cosh(a + b*x), x), x) + Simp(x**m*SinhIntegral(c + d*x)*cosh(a + b*x)/b, x) 1161 1162 1163def replacement6852(a, b, c, d, m, x): 1164 return -Dist(d/b, Int(x**m*sinh(a + b*x)*cosh(c + d*x)/(c + d*x), x), x) - Dist(m/b, Int(x**(m + S(-1))*CoshIntegral(c + d*x)*sinh(a + b*x), x), x) + Simp(x**m*CoshIntegral(c + d*x)*sinh(a + b*x)/b, x) 1165 1166 1167def replacement6853(a, b, c, d, m, x): 1168 return -Dist(b/(m + S(1)), Int(x**(m + S(1))*SinhIntegral(c + d*x)*cosh(a + b*x), x), x) - Dist(d/(m + S(1)), Int(x**(m + S(1))*sinh(a + b*x)*sinh(c + d*x)/(c + d*x), x), x) + Simp(x**(m + S(1))*SinhIntegral(c + d*x)*sinh(a + b*x)/(m + S(1)), x) 1169 1170 1171def replacement6854(a, b, c, d, m, x): 1172 return -Dist(b/(m + S(1)), Int(x**(m + S(1))*CoshIntegral(c + d*x)*sinh(a + b*x), x), x) - Dist(d/(m + S(1)), Int(x**(m + S(1))*cosh(a + b*x)*cosh(c + d*x)/(c + d*x), x), x) + Simp(x**(m + S(1))*CoshIntegral(c + d*x)*cosh(a + b*x)/(m + S(1)), x) 1173 1174 1175def replacement6855(a, b, c, d, x): 1176 return -Dist(d/b, Int(sinh(a + b*x)*sinh(c + d*x)/(c + d*x), x), x) + Simp(SinhIntegral(c + d*x)*sinh(a + b*x)/b, x) 1177 1178 1179def replacement6856(a, b, c, d, x): 1180 return -Dist(d/b, Int(cosh(a + b*x)*cosh(c + d*x)/(c + d*x), x), x) + Simp(CoshIntegral(c + d*x)*cosh(a + b*x)/b, x) 1181 1182 1183def replacement6857(a, b, c, d, m, x): 1184 return -Dist(d/b, Int(x**m*sinh(a + b*x)*sinh(c + d*x)/(c + d*x), x), x) - Dist(m/b, Int(x**(m + S(-1))*SinhIntegral(c + d*x)*sinh(a + b*x), x), x) + Simp(x**m*SinhIntegral(c + d*x)*sinh(a + b*x)/b, x) 1185 1186 1187def replacement6858(a, b, c, d, m, x): 1188 return -Dist(d/b, Int(x**m*cosh(a + b*x)*cosh(c + d*x)/(c + d*x), x), x) - Dist(m/b, Int(x**(m + S(-1))*CoshIntegral(c + d*x)*cosh(a + b*x), x), x) + Simp(x**m*CoshIntegral(c + d*x)*cosh(a + b*x)/b, x) 1189 1190 1191def replacement6859(a, b, c, d, m, x): 1192 return -Dist(b/(m + S(1)), Int(x**(m + S(1))*SinhIntegral(c + d*x)*sinh(a + b*x), x), x) - Dist(d/(m + S(1)), Int(x**(m + S(1))*sinh(c + d*x)*cosh(a + b*x)/(c + d*x), x), x) + Simp(x**(m + S(1))*SinhIntegral(c + d*x)*cosh(a + b*x)/(m + S(1)), x) 1193 1194 1195def replacement6860(a, b, c, d, m, x): 1196 return -Dist(b/(m + S(1)), Int(x**(m + S(1))*CoshIntegral(c + d*x)*cosh(a + b*x), x), x) - Dist(d/(m + S(1)), Int(x**(m + S(1))*sinh(a + b*x)*cosh(c + d*x)/(c + d*x), x), x) + Simp(x**(m + S(1))*CoshIntegral(c + d*x)*sinh(a + b*x)/(m + S(1)), x) 1197 1198 1199def replacement6861(a, b, n, x): 1200 return -Simp(Gamma(n + S(1), a + b*x)/b, x) + Simp((a + b*x)*Gamma(n, a + b*x)/b, x) 1201 1202 1203def replacement6862(b, n, x): 1204 return Simp(Gamma(n)*log(x), x) - Simp((b*x)**n*HypergeometricPFQ(List(n, n), List(n + S(1), n + S(1)), -b*x)/n**S(2), x) 1205 1206 1207def replacement6863(b, m, n, x): 1208 return Simp(x**(m + S(1))*Gamma(n, b*x)/(m + S(1)), x) - Simp(x**m*(b*x)**(-m)*Gamma(m + n + S(1), b*x)/(b*(m + S(1))), x) 1209 1210 1211def With6864(a, b, m, n, x): 1212 _UseGamma = True 1213 return Dist(b/(m + S(1)), Int(x**(m + S(1))*(a + b*x)**(n + S(-1))*exp(-a - b*x), x), x) + Simp(x**(m + S(1))*Gamma(n, a + b*x)/(m + S(1)), x) 1214 1215 1216def replacement6865(a, b, x): 1217 return Simp(PolyGamma(S(-2), a + b*x)/b, x) 1218 1219 1220def replacement6866(a, b, m, x): 1221 return -Dist(m/b, Int(x**(m + S(-1))*PolyGamma(S(-2), a + b*x), x), x) + Simp(x**m*PolyGamma(S(-2), a + b*x)/b, x) 1222 1223 1224def replacement6867(a, b, n, x): 1225 return Simp(PolyGamma(n + S(-1), a + b*x)/b, x) 1226 1227 1228def replacement6868(a, b, m, n, x): 1229 return -Dist(m/b, Int(x**(m + S(-1))*PolyGamma(n + S(-1), a + b*x), x), x) + Simp(x**m*PolyGamma(n + S(-1), a + b*x)/b, x) 1230 1231 1232def replacement6869(a, b, m, n, x): 1233 return -Dist(b/(m + S(1)), Int(x**(m + S(1))*PolyGamma(n + S(1), a + b*x), x), x) + Simp(x**(m + S(1))*PolyGamma(n, a + b*x)/(m + S(1)), x) 1234 1235 1236def replacement6870(a, b, n, x): 1237 return Simp(Gamma(a + b*x)**n/(b*n), x) 1238 1239 1240def replacement6871(a, b, c, n, x): 1241 return Simp(Factorial(a + b*x)**n/(b*n), x) 1242 1243 1244def replacement6872(a, b, x): 1245 return Int(PolyGamma(S(1), a + b*x), x) 1246 1247 1248def replacement6873(a, b, s, x): 1249 return -Simp(Zeta(s + S(-1), a + b*x)/(b*(s + S(-1))), x) 1250 1251 1252def replacement6874(a, b, m, x): 1253 return Int(x**m*PolyGamma(S(1), a + b*x), x) 1254 1255 1256def replacement6875(a, b, m, s, x): 1257 return Dist(m/(b*(s + S(-1))), Int(x**(m + S(-1))*Zeta(s + S(-1), a + b*x), x), x) - Simp(x**m*Zeta(s + S(-1), a + b*x)/(b*(s + S(-1))), x) 1258 1259 1260def replacement6876(a, b, m, s, x): 1261 return Dist(b*s/(m + S(1)), Int(x**(m + S(1))*Zeta(s + S(1), a + b*x), x), x) + Simp(x**(m + S(1))*Zeta(s, a + b*x)/(m + S(1)), x) 1262 1263 1264def replacement6877(a, b, n, p, q, x): 1265 return -Dist(p*q, Int(PolyLog(n + S(-1), a*(b*x**p)**q), x), x) + Simp(x*PolyLog(n, a*(b*x**p)**q), x) 1266 1267 1268def replacement6878(a, b, n, p, q, x): 1269 return -Dist(S(1)/(p*q), Int(PolyLog(n + S(1), a*(b*x**p)**q), x), x) + Simp(x*PolyLog(n + S(1), a*(b*x**p)**q)/(p*q), x) 1270 1271 1272def replacement6879(a, b, c, d, e, n, p, x): 1273 return Simp(PolyLog(n + S(1), c*(a + b*x)**p)/(e*p), x) 1274 1275 1276def replacement6880(a, b, n, p, q, x): 1277 return Simp(PolyLog(n + S(1), a*(b*x**p)**q)/(p*q), x) 1278 1279 1280def replacement6881(a, b, m, n, p, q, x): 1281 return -Dist(p*q/(m + S(1)), Int(x**m*PolyLog(n + S(-1), a*(b*x**p)**q), x), x) + Simp(x**(m + S(1))*PolyLog(n, a*(b*x**p)**q)/(m + S(1)), x) 1282 1283 1284def replacement6882(a, b, m, n, p, q, x): 1285 return -Dist((m + S(1))/(p*q), Int(x**m*PolyLog(n + S(1), a*(b*x**p)**q), x), x) + Simp(x**(m + S(1))*PolyLog(n + S(1), a*(b*x**p)**q)/(p*q), x) 1286 1287 1288def replacement6883(a, b, c, m, n, p, q, r, x): 1289 return -Dist(m*r/(p*q), Int(PolyLog(n + S(1), a*(b*x**p)**q)*log(c*x**m)**(r + S(-1))/x, x), x) + Simp(PolyLog(n + S(1), a*(b*x**p)**q)*log(c*x**m)**r/(p*q), x) 1290 1291 1292def replacement6884(a, b, c, n, p, x): 1293 return -Dist(p, Int(PolyLog(n + S(-1), c*(a + b*x)**p), x), x) + Dist(a*p, Int(PolyLog(n + S(-1), c*(a + b*x)**p)/(a + b*x), x), x) + Simp(x*PolyLog(n, c*(a + b*x)**p), x) 1294 1295 1296def replacement6885(a, b, c, m, n, p, x): 1297 return -Dist(b*p/(m + S(1)), Int(x**(m + S(1))*PolyLog(n + S(-1), c*(a + b*x)**p)/(a + b*x), x), x) + Simp(x**(m + S(1))*PolyLog(n, c*(a + b*x)**p)/(m + S(1)), x) 1298 1299 1300def replacement6886(F, a, b, c, d, n, p, x): 1301 return Simp(PolyLog(n + S(1), d*(F**(c*(a + b*x)))**p)/(b*c*p*log(F)), x) 1302 1303 1304def replacement6887(F, a, b, c, d, e, f, m, n, p, x): 1305 return -Dist(f*m/(b*c*p*log(F)), Int((e + f*x)**(m + S(-1))*PolyLog(n + S(1), d*(F**(c*(a + b*x)))**p), x), x) + Simp((e + f*x)**m*PolyLog(n + S(1), d*(F**(c*(a + b*x)))**p)/(b*c*p*log(F)), x) 1306 1307 1308def With6888(n, u, v, x): 1309 if isinstance(x, (int, Integer, float, Float)): 1310 return False 1311 try: 1312 w = DerivativeDivides(v, u*v, x) 1313 res = Not(FalseQ(w)) 1314 except (TypeError, AttributeError): 1315 return False 1316 if res: 1317 return True 1318 return False 1319 1320 1321def replacement6888(n, u, v, x): 1322 1323 w = DerivativeDivides(v, u*v, x) 1324 return Simp(w*PolyLog(n + S(1), v), x) 1325 1326 1327def With6889(n, u, v, w, x): 1328 if isinstance(x, (int, Integer, float, Float)): 1329 return False 1330 try: 1331 z = DerivativeDivides(v, u*v, x) 1332 res = Not(FalseQ(z)) 1333 except (TypeError, AttributeError): 1334 return False 1335 if res: 1336 return True 1337 return False 1338 1339 1340def replacement6889(n, u, v, w, x): 1341 1342 z = DerivativeDivides(v, u*v, x) 1343 return -Int(SimplifyIntegrand(z*D(w, x)*PolyLog(n + S(1), v)/w, x), x) + Simp(z*PolyLog(n + S(1), v)*log(w), x) 1344 1345 1346def replacement6890(a, b, c, p, x): 1347 return Dist(p/(c*(p + S(1))), Int((c*ProductLog(a + b*x))**(p + S(1))/(ProductLog(a + b*x) + S(1)), x), x) + Simp((c*ProductLog(a + b*x))**p*(a + b*x)/(b*(p + S(1))), x) 1348 1349 1350def replacement6891(a, b, c, p, x): 1351 return -Dist(p, Int((c*ProductLog(a + b*x))**p/(ProductLog(a + b*x) + S(1)), x), x) + Simp((c*ProductLog(a + b*x))**p*(a + b*x)/b, x) 1352 1353 1354def replacement6892(a, b, c, m, p, x): 1355 return Dist(S(1)/b, Subst(Int(ExpandIntegrand((c*ProductLog(x))**p, (-a/b + x/b)**m, x), x), x, a + b*x), x) 1356 1357 1358def replacement6893(a, c, n, p, x): 1359 return -Dist(n*p, Int((c*ProductLog(a*x**n))**p/(ProductLog(a*x**n) + S(1)), x), x) + Simp(x*(c*ProductLog(a*x**n))**p, x) 1360 1361 1362def replacement6894(a, c, n, p, x): 1363 return Dist(n*p/(c*(n*p + S(1))), Int((c*ProductLog(a*x**n))**(p + S(1))/(ProductLog(a*x**n) + S(1)), x), x) + Simp(x*(c*ProductLog(a*x**n))**p/(n*p + S(1)), x) 1364 1365 1366def replacement6895(a, c, n, p, x): 1367 return -Subst(Int((c*ProductLog(a*x**(-n)))**p/x**S(2), x), x, S(1)/x) 1368 1369 1370def replacement6896(a, c, m, n, p, x): 1371 return -Dist(n*p/(m + S(1)), Int(x**m*(c*ProductLog(a*x**n))**p/(ProductLog(a*x**n) + S(1)), x), x) + Simp(x**(m + S(1))*(c*ProductLog(a*x**n))**p/(m + S(1)), x) 1372 1373 1374def replacement6897(a, c, m, n, p, x): 1375 return Dist(n*p/(c*(m + n*p + S(1))), Int(x**m*(c*ProductLog(a*x**n))**(p + S(1))/(ProductLog(a*x**n) + S(1)), x), x) + Simp(x**(m + S(1))*(c*ProductLog(a*x**n))**p/(m + n*p + S(1)), x) 1376 1377 1378def replacement6898(a, c, m, p, x): 1379 return Dist(S(1)/c, Int(x**m*(c*ProductLog(a*x))**(p + S(1))/(ProductLog(a*x) + S(1)), x), x) + Int(x**m*(c*ProductLog(a*x))**p/(ProductLog(a*x) + S(1)), x) 1380 1381 1382def replacement6899(a, c, m, n, p, x): 1383 return -Subst(Int(x**(-m + S(-2))*(c*ProductLog(a*x**(-n)))**p, x), x, S(1)/x) 1384 1385 1386def replacement6900(a, b, d, x): 1387 return Simp((a + b*x)/(b*d*ProductLog(a + b*x)), x) 1388 1389 1390def replacement6901(a, b, d, x): 1391 return -Int(S(1)/(d*ProductLog(a + b*x) + d), x) + Simp(d*x, x) 1392 1393 1394def replacement6902(a, b, c, d, p, x): 1395 return -Dist(c*p, Int((c*ProductLog(a + b*x))**(p + S(-1))/(d*ProductLog(a + b*x) + d), x), x) + Simp(c*(c*ProductLog(a + b*x))**(p + S(-1))*(a + b*x)/(b*d), x) 1396 1397 1398def replacement6903(a, b, d, x): 1399 return Simp(ExpIntegralEi(ProductLog(a + b*x))/(b*d), x) 1400 1401 1402def replacement6904(a, b, c, d, x): 1403 return Simp(Erfi(sqrt(c*ProductLog(a + b*x))/Rt(c, S(2)))*Rt(Pi*c, S(2))/(b*c*d), x) 1404 1405 1406def replacement6905(a, b, c, d, x): 1407 return Simp(Erf(sqrt(c*ProductLog(a + b*x))/Rt(-c, S(2)))*Rt(-Pi*c, S(2))/(b*c*d), x) 1408 1409 1410def replacement6906(a, b, c, d, p, x): 1411 return -Dist(S(1)/(c*(p + S(1))), Int((c*ProductLog(a + b*x))**(p + S(1))/(d*ProductLog(a + b*x) + d), x), x) + Simp((c*ProductLog(a + b*x))**p*(a + b*x)/(b*d*(p + S(1))), x) 1412 1413 1414def replacement6907(a, b, c, d, p, x): 1415 return Simp((-ProductLog(a + b*x))**(-p)*(c*ProductLog(a + b*x))**p*Gamma(p + S(1), -ProductLog(a + b*x))/(b*d), x) 1416 1417 1418def replacement6908(a, b, d, m, x): 1419 return Dist(S(1)/b, Subst(Int(ExpandIntegrand(S(1)/(d*ProductLog(x) + d), (-a/b + x/b)**m, x), x), x, a + b*x), x) 1420 1421 1422def replacement6909(a, b, c, d, m, p, x): 1423 return Dist(S(1)/b, Subst(Int(ExpandIntegrand((c*ProductLog(x))**p/(d*ProductLog(x) + d), (-a/b + x/b)**m, x), x), x, a + b*x), x) 1424 1425 1426def replacement6910(a, d, n, x): 1427 return -Subst(Int(S(1)/(x**S(2)*(d*ProductLog(a*x**(-n)) + d)), x), x, S(1)/x) 1428 1429 1430def replacement6911(a, c, d, n, p, x): 1431 return Simp(c*x*(c*ProductLog(a*x**n))**(p + S(-1))/d, x) 1432 1433 1434def replacement6912(a, d, n, p, x): 1435 return Simp(a**p*ExpIntegralEi(-p*ProductLog(a*x**n))/(d*n), x) 1436 1437 1438def replacement6913(a, c, d, n, p, x): 1439 return Simp(a**(-S(1)/n)*c**(-S(1)/n)*Erfi(sqrt(c*ProductLog(a*x**n))/Rt(c*n, S(2)))*Rt(Pi*c*n, S(2))/(d*n), x) 1440 1441 1442def replacement6914(a, c, d, n, p, x): 1443 return Simp(a**(-S(1)/n)*c**(-S(1)/n)*Erf(sqrt(c*ProductLog(a*x**n))/Rt(-c*n, S(2)))*Rt(-Pi*c*n, S(2))/(d*n), x) 1444 1445 1446def replacement6915(a, c, d, n, p, x): 1447 return -Dist(c*(n*(p + S(-1)) + S(1)), Int((c*ProductLog(a*x**n))**(p + S(-1))/(d*ProductLog(a*x**n) + d), x), x) + Simp(c*x*(c*ProductLog(a*x**n))**(p + S(-1))/d, x) 1448 1449 1450def replacement6916(a, c, d, n, p, x): 1451 return -Dist(S(1)/(c*(n*p + S(1))), Int((c*ProductLog(a*x**n))**(p + S(1))/(d*ProductLog(a*x**n) + d), x), x) + Simp(x*(c*ProductLog(a*x**n))**p/(d*(n*p + S(1))), x) 1452 1453 1454def replacement6917(a, c, d, n, p, x): 1455 return -Subst(Int((c*ProductLog(a*x**(-n)))**p/(x**S(2)*(d*ProductLog(a*x**(-n)) + d)), x), x, S(1)/x) 1456 1457 1458def replacement6918(a, d, m, x): 1459 return -Dist(m/(m + S(1)), Int(x**m/((d*ProductLog(a*x) + d)*ProductLog(a*x)), x), x) + Simp(x**(m + S(1))/(d*(m + S(1))*ProductLog(a*x)), x) 1460 1461 1462def replacement6919(a, d, x): 1463 return Simp(log(ProductLog(a*x))/d, x) 1464 1465 1466def replacement6920(a, d, m, x): 1467 return -Int(x**m*ProductLog(a*x)/(d*ProductLog(a*x) + d), x) + Simp(x**(m + S(1))/(d*(m + S(1))), x) 1468 1469 1470def replacement6921(a, d, m, x): 1471 return Simp(x**m*(-(m + S(1))*ProductLog(a*x))**(-m)*Gamma(m + S(1), -(m + S(1))*ProductLog(a*x))*exp(-m*ProductLog(a*x))/(a*d*(m + S(1))), x) 1472 1473 1474def replacement6922(a, d, n, x): 1475 return Simp(log(ProductLog(a*x**n))/(d*n), x) 1476 1477 1478def replacement6923(a, d, m, n, x): 1479 return -Subst(Int(x**(-m + S(-2))/(d*ProductLog(a*x**(-n)) + d), x), x, S(1)/x) 1480 1481 1482def replacement6924(a, c, d, n, p, x): 1483 return Simp((c*ProductLog(a*x**n))**p/(d*n*p), x) 1484 1485 1486def replacement6925(a, c, d, m, n, p, x): 1487 return Simp(c*x**(m + S(1))*(c*ProductLog(a*x**n))**(p + S(-1))/(d*(m + S(1))), x) 1488 1489 1490def replacement6926(a, d, m, n, p, x): 1491 return Simp(a**p*ExpIntegralEi(-p*ProductLog(a*x**n))/(d*n), x) 1492 1493 1494def replacement6927(a, c, d, m, n, p, x): 1495 return Simp(a**(p + S(-1)/2)*c**(p + S(-1)/2)*Erf(sqrt(c*ProductLog(a*x**n))/Rt(c/(p + S(-1)/2), S(2)))*Rt(Pi*c/(p + S(-1)/2), S(2))/(d*n), x) 1496 1497 1498def replacement6928(a, c, d, m, n, p, x): 1499 return Simp(a**(p + S(-1)/2)*c**(p + S(-1)/2)*Erfi(sqrt(c*ProductLog(a*x**n))/Rt(-c/(p + S(-1)/2), S(2)))*Rt(-Pi*c/(p + S(-1)/2), S(2))/(d*n), x) 1500 1501 1502def replacement6929(a, c, d, m, n, p, x): 1503 return -Dist(c*(m + n*(p + S(-1)) + S(1))/(m + S(1)), Int(x**m*(c*ProductLog(a*x**n))**(p + S(-1))/(d*ProductLog(a*x**n) + d), x), x) + Simp(c*x**(m + S(1))*(c*ProductLog(a*x**n))**(p + S(-1))/(d*(m + S(1))), x) 1504 1505 1506def replacement6930(a, c, d, m, n, p, x): 1507 return -Dist((m + S(1))/(c*(m + n*p + S(1))), Int(x**m*(c*ProductLog(a*x**n))**(p + S(1))/(d*ProductLog(a*x**n) + d), x), x) + Simp(x**(m + S(1))*(c*ProductLog(a*x**n))**p/(d*(m + n*p + S(1))), x) 1508 1509 1510def replacement6931(a, c, d, m, p, x): 1511 return Simp(x**m*(c*ProductLog(a*x))**p*(-(m + S(1))*ProductLog(a*x))**(-m - p)*Gamma(m + p + S(1), -(m + S(1))*ProductLog(a*x))*exp(-m*ProductLog(a*x))/(a*d*(m + S(1))), x) 1512 1513 1514def replacement6932(a, c, d, m, n, p, x): 1515 return -Subst(Int(x**(-m + S(-2))*(c*ProductLog(a*x**(-n)))**p/(d*ProductLog(a*x**(-n)) + d), x), x, S(1)/x) 1516 1517 1518def replacement6933(u, x): 1519 return Subst(Int(SimplifyIntegrand((x + S(1))*SubstFor(ProductLog(x), u, x)*exp(x), x), x), x, ProductLog(x)) 1520