o
    8Va                     @   s  d Z ddlmZ edZer ddlmZmZmZmZ ddlm	Z	m
Z
mZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZm Z m!Z!m"Z"m#Z#m$Z$m%Z%m&Z&m'Z'm(Z(m)Z)m*Z*m+Z+m,Z,m-Z-m.Z.m/Z/m0Z0m1Z1m2Z2m3Z3m4Z4m5Z5m6Z6m7Z7m8Z8m9Z9m:Z:m;Z;m<Z<m=Z=m>Z>m?Z?m@Z@mAZAmBZBmCZCmDZDmEZEmFZFmGZGmHZHmIZImJZJmKZKmLZLmMZMmNZNmOZOmPZPmQZQmRZRmSZSmTZTmUZUmVZVmWZWmXZXmYZYmZZZm[Z[m\Z\m]Z]m^Z^m_Z_m`Z`maZambZbmcZcmdZdmeZemfZfmgZgmhZhmiZimjZjmkZkmlZlmmZmmnZnmoZompZpmqZqmrZrmsZsmtZtmuZumvZvmwZwmxZxmyZymzZzm{Z{m|Z|m}Z}m~Z~mZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZm Z mZmZmZmZmZmZmZmZm	Z	m
Z
mZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZm Z m!Z!m"Z"m#Z#m$Z$m%Z%m&Z&m'Z'm(Z(m)Z)m*Z*m+Z+m,Z,m-Z-m.Z.m/Z/m0Z0m1Z1m2Z2m3Z3m4Z4m5Z5m6Z6m7Z7m8Z8m9Z9m:Z:m;Z;m<Z<m=Z=m>Z>m?Z?m@Z@mAZAmBZBmCZCmDZDmEZEmFZFmGZGmHZHmIZImJZJmKZKmLZLmMZMmNZNmOZOmPZPmQZQmRZRmSZSmTZTmUZUmVZVmWZWmXZXmYZYmZZZm[Z[m\Z\m]Z]m^Z^m_Z_m`Z`maZambZbmcZcmdZdmeZemfZfmgZgmhZhmiZimjZjmkZkmlZlmmZmmnZnmoZompZpmqZqmrZrmsZsmtZtmuZumvZvmwZwmxZxmyZymzZzm{Z{m|Z|m}Z}m~Z~mZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmHZHmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmÐZÐmĐZĐmŐZŐmƐZƐmǐZǐmZmȐZȐmɐZɐmʐZʐmːZːm̐Z̐m͐Z͐mΐZΐmϐZϐmАZАmѐZѐmҐZҐmӐZӐmԐZԐmՐZՐm֐Z֐mאZאmؐZؐmِZِmڐZڐmېZېmܐZݐmސZߐmZmZmZ ddlmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZmZ ddlmZ ddlmZmZ dd	lmZmZmZmZmZmZmZm Z  dd
lmZmZmZmZmZmZmZm	Z	m
Z
mZmZmZ ddlmZmZmZmZmZmZmZ ddlmZ dd dD \ZZZZZZZZZ Z!Z"Z#Z$Z%Z&Z'Z(Z)Z*Z+Z,Z-Z.Z/Z0Z1Z2Z3Z4Z5Z6dd dD \Z7Z8Z9Z:Z;Z<Z=Z>Z?Z@ZAZBZCZDZEZFZ?Z@ZGZHZIZJZKZLZMZNZOZPZQZRed\	ZSZTZUZVZWZXZYZZZ[dZ\dZ]dZ^dd Z_dd Z`dd Zadd Zbdd Zcdd  Zdd!d" Zed#d$ Zfd%d& Zgd'd( Zhd)d* Zid+d, Zjd-d. Zkd/d0 Zld1d2 Zmd3d4 Znd5d6 Zod7d8 Zpd9d: Zqd;d< Zrd=d> Zsd?d@ ZtdAdB ZudCdD ZvdEdF ZwdGdH ZxdIdJ ZydKdL ZzdMdN Z{dOdP Z|dQdR Z}dSdT Z~dUdV ZdWdX ZdYdZ Zd[d\ Zd]d^ Zd_d` Zdadb Zdcdd Zdedf Zdgdh Zdidj Zdkdl Zdmdn Zdodp Zdqdr Zdsdt Zdudv Zdwdx Zdydz Zd{d| Zd}d~ Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd ZddĄ ZddƄ ZddȄ Zddʄ Zdd̄ Zdd΄ ZddЄ Zdd҄ ZddԄ Zddք Zdd؄ Zddڄ Zdd܄ Zddބ Zdd Zdd Zdd Zdd Zdd Zdd Zdd Zdd ZdS )z
This code is automatically generated. Never edit it manually.
For details of generating the code see `rubi_parsing_guide.md` in `parsetools`.
    )import_modulematchpy)PatternReplacementRuleCustomConstraintis_match(  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SubstForrG  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Factorr  CosIntegralSinIntegralLogIntegralSinhIntegralCoshIntegralRuleErf	PolyGammaExpIntegralEiExpIntegralELogGammaUtilityOperator	FactorialZeta
ProductLogDerivativeDividesHypergeometricPFQIntHideOneQNullrubi_exprubi_logDiscriminantNegativeQuotient)IntegralSsqrtAndOrIntegerFloatModIAbssimplifyMulAddPowsign
EulerGammaWC)symbolsSymbol)sincostancotcscsecr  erf)acoshasinhatanhacothacschasechcoshsinhtanhcothsechcsch)atanacscasinacotacosasecatan2)pic                 C      g | ]}t |qS  r  .0ir  r  V/usr/lib/python3/dist-packages/sympy/integrals/rubi/rules/miscellaneous_integration.py
<listcomp>       r  ZABCFGHabcdefghijklmnpqrtuvswxyzc                 C   r  r  r  r  r  r  r  r     r  )Za1Za2Zb1Zb2Zc1Zc2Zd1Zd2n1n2Ze1Ze2f1f2Zg1Zg2r  r  Zn3ZPqZPmZPxZQmZQrZQxZjnZmnZnon2ZRFxZRGxzi ii Pqq Q R r C k uFNc                  C   s2  ddl m} m}m}m}m}m}m}m}m	}m
}	m}
m}m}m}m}m}m}m}m}m}m}m}m}m}m}m}m}m}m}m}m}m }m!} m"}!m#}"m$}#m%}$m&}%m'}&m(}'m)}(m*})m+}*m,}+m-},m.}-m/}.m0}/m1}0m2}1m3}2m4}3m5}4m6}5m7}6m8}7m9}8m:}9m;}:m<};m=}<m>}=m?}>m@}?mA}@mB}AmC}BmD}CmE}DmF}EmG}FmH}GmI}HmJ}ImK}JmL}KmM}LmN}MmO}NmP}OmQ}PmR}QmS}RmT}SmU}T tVtWtXtYtZdt[d tZdt[d t\ tZdt[d t]  tY|||||| |}Ut^|Ut_}VtVtWt`tYtZdt[d tZdt[d  t] tZdt[d ta tZdt[d tY||||||	| |	}Wt^|Wtb}XtVtWtYtZdt[d tZdt[d t\ tZdt[d t] tZdt[d ta tZdt[d tY|||||||	| |
}Yt^|Ytc}ZtVtWtdtetYtZd	t[d tZdt[d  tZdt[d tZdt[d tetYtZd
t[d tZdt[d  tZdt[d tZdt[d tYt[d tZdt[d tYtZdt[d  tZdt[d  tY||||||||||||
|||}[t^|[tf}\tVtWtdtetYtZd	t[d t[d  tZdt[d tZdt[d tetYtZd
t[d t[d  tZdt[d tZdt[d tYt[d tZdt[d tZdt[d  tY|||||||||
|}]t^|]tg}^tVtWtdtetYtZd	t[d tZdt[d tZdt[d tetYtZd
t[d tZdt[d   tZdt[d tZdt[d tZdt[d tYt[d tZdt[d tYtZdt[d  tZdt[d  tY||||||||||||
|||}_t^|_th}`tVtWtdtetYtZd	t[d t[d tZdt[d tetYtZd
t[d t[d   tZdt[d tZdt[d tZdt[d tYt[d tZdt[d tZdt[d  tY|||||||||
|}at^|ati}btVtWtXtj tYtktl}ct^|ctm}dtVtWtXtntj  tYtkto}et^|etp}ftVtWtXtjtZdt[d  tY||tktq}gt^|gtr}htVtWtXtjtZdt[d  tstZdt[d  tY|||tktt}it^|itu}jtVtWtXtYtktv}kt^|ktw}ltVtWtetYtZdt[d tZdt[d tZdt[d tZd	t[d tetYtZdt[d tZdt[d tZdt[d tZdt[d  tx tZdt[d tY|||||||||
}mt^|mty}ntVtWtetYtZdt[d tZdt[d tZdt[d tZd	t[d tetYtZdt[d tZdt[d tZdt[d tZdt[d  tx tZdt[d tY|||||||||
}ot^|otz}ptVtWtXtZdt[d tn tXt\ tZdt[d t{ tZdt[d  tY||||||}qt^|qt|}rtVtWtXt{tZdt[d tZdt[d tZdt[d  tjtZdt[d tZdt[d tZdt[d  tY|||||||tkt}	}st^|st~}ttVtWtXt{tZdt[d tZdt[d tZdt[d  tntZdt[d tZd	t[d tZdt[d  tjtZdt[d tZdt[d tZdt[d  tY|||||||||||tkt}ut^|ut}vtVtWtXt{tZdt[d tZdt[d tZdt[d  tntZdt[d tZd	t[d tZdt[d  tjtZdt[d tZdt[d tZdt[d  tstZdt[d tZd
t[d tZdt[d  tY||||||||!||||	||| tkt}wt^|wt}xtVtWttjt\ tZdt[d  tZdt[d tY||||"tkt}yt^|yt}ztVtWtjt\ tZdt[d tZdt[d t] tZdt[d tY|||||#tkt}{t^|{t}|tVtWt{tZdt[d tjt\ tZdt[d tZdt[d tZdt[d  tZdt[d tY||||||$tkt}}t^|}t}~tVtWt{tZdt[d tZdt[d tjt\ tZdt[d  tZdt[d t] tZdt[d tY||||||%|tkt	}t^|t}tVtWttjt\ tZdt[d  t{t\ tZdt[d tntZdt[d tZdt[d  tZdt[d tZdt[d  tZdt[d tY||||||||%||tkt}t^|t}tVtWttjt\ tZdt[d  tntZdt[d tZdt[d tZdt[d tZdt[d  tZdt[d tY|||||||%|tkt
}t^|t}tVtWt{tZdt[d tntZdt[d tZdt[d tjt\ tZdt[d  tZdt[d tZdt[d  tZdt[d tY|||||||%|tkt
}t^|t}tVtWtstZdt[d ttjt\ tZdt[d   t{t\ tZdt[d tntZdt[d tZdt[d  tZdt[d tZdt[d  tZdt[d tY|||||||||%||tkt}t^|t}tVtWtstZdt[d ttjt\ tZdt[d   tntZdt[d tZdt[d tZdt[d tZdt[d  tZdt[d tY||||||||%|tkt}t^|t}tVtWt{t\ tZdt[d tZdt[d tZdt[d tjt\ tZdt[d tZdt[d tZdt[d  tZdt[d tY||||||||tkt
}t^|t}tVtWt{t\ tZdt[d tZdt[d tZdt[d tnt\ tZdt[d tZd	t[d tZdt[d  tjt\ tZdt[d tZdt[d tZdt[d  tZdt[d tY||||||||||	||tkt}t^|t}tVtWtdt{ tX tY||tkt}t^|t}tVtWtdt{ tX tntZdt[d  tY|||&tkt}t^|t}tVtWtXtt{tZdt[d tntZdt[d  tZdt[d  tZdt[d  tY|||||tkt}t^|t}tVtWtXt{tZdt[d  tt{tZdt[d tntZdt[d  tZdt[d  tZdt[d  tY|||||	|*|'|(|)tkt}t^|t}tVtWtXt{tZdt[d  tntZdt[d  tt{tZdt[d tntZdt[d  tZdt[d  tZdt[d  tY|||||	|*|,|+|(|)tkt}t^|t}tVtWtXt{tZdt[d tZdt[d tntZdt[d tZdt[d  tZdt[d  tY|||||	|-||.tkt
}t^|t}tVtWtXt{tZdt[d  t{tZdt[d tZdt[d tntZdt[d tZdt[d  tZdt[d  tY|||||	|*|/|0|.tkt}t^|t}tVtWtXtntZdt[d  t{tZdt[d tZdt[d tntZdt[d tZdt[d  tZdt[d  tY|||||	|,|1|2|3|.tkt}t^|t}tVtWtXt{tZdt[d  tntZdt[d  t{tZdt[d tZdt[d tntZdt[d tZdt[d  tZdt[d  tY|||||	|*|,|4|2|3|.tkt}t^|t}tVtWtXtYtZdt[d  tY|||5}t^|t}tVtWtXtYtkt}t^|t}tVtWtXtYtkt}t^|t}tVtWt{tZdt[d tntZdt[d  tstZdt[d  tZdt[d t] tZdt[d tY|||||	| ||6|7
}t^|t}tVtWt{tZdt[d tntZdt[d  tZdt[d t] tZdt[d tY||||| ||6}t^|t}tVtWt{tZdt[d tZdt[d t] tZdt[d tY|||| ||8|9}t^|t}tVtWtYt\ tZdt[d tZdt[d t] tZdt[d tY||||:|;|<}t^|t}tVtWt{t\ tZdt[d tZdt[d t] tZdt[d tY|||| |;|=|>}t^|t}tVtWt{t\ tYtZdt[d  tZdt[d tZdt[d t] tZdt[d tY||||| |;|=}t^|t}tVtWtYtZdt[d tZdt[d tYtZdt[d tZdt[d  tx tZdt[d tY||||*|,|?|@tkt	}t^|t}tVtWtXttYt\ tZdt[d   tY|||Atkt}t^|t}tVtWtXtYtZdt[d tZdt[d tYtZdt[d tZdt[d  tZdt[d tZdt[d  tY|||||%|B||C	}t^|t}tVtWtXtYtZdt[d tZdt[d tYtZdt[d tZdt[d  tZdt[d t]  tY||||||%|B| |C
}t^|t}tVtWtXtYtZdt[d tZdt[d tYtZdt[d tZdt[d  tZdt[d  tY||||%|Atkt}t^|t}tVtWtZdt[dtYtZdt[d tZdt[d tetYt\ tZdt[d tZdt[d   tY||||||D}t^|t}tVtWtXtYtkt}t^|t}tVtWtXtY tY|E|<tktÃ}t^|tă}tVtWtXtYtZdt[d  tY|.|F|E|GtktŃ}t^|tƃ}tVtWtXtYtx  tY|H}t^|tǃ}tVtWtXtY|Itktȃ}t^|tɃ}tVtWt[dtt{t[d tZdt[d   tY|||J}t^|tʃ}tVtWt[dtt{t\ tZdt[d   tY|||K|L}t^|t˃}tVtWt[dtt{t\ tZdt[d   tY|||M|N}t^|t̃}tVtWt{ttXtZdt[d tZdt[d   tY|||A|O}t^|t̓}tVtWtXtYtkt΃}t^|tσ}tVtWtXtYtktЃ}t^|tу}tVtWtYtZdt[d tZdt[d tZdt[d tZdt[d tYtZdt[d tZdt[d tZdt[d tZdt[d  tZdt[d tY||||||||	|P|Q|R|S}t^|t҃}tVtWtXttYtZdt[d tZdt[d  tYtZdt[d tZdt[d  t]  tY||||||%|B|T	}t^|tӃ}tVtWtXtYtktԃ}t^|tՃ}tVtWtXtY}t^|tփ}g |V|X|Z|\|^|`|b|d|f|h|j|l|n|p|r|t|v|x|z|||~|||||||||||||||||||||||||||||||||||đ|Ƒ|ȑ|ʑ|̑|Α|Б|ґ|ԑ|֑|ؑ|ڑ|ܑS )Nr   )Ucons149cons2004cons2cons3cons8cons4cons5cons388cons29cons52cons2005cons2006cons2007cons2008cons50cons127cons210cons36cons37cons38cons1101cons2009cons68cons19cons86cons1039cons1038cons40cons2010cons10cons2011cons2012cons2013cons211cons1833cons1246cons2014cons48cons2015cons2016cons2017cons2018cons54cons2019cons802cons2020cons20cons2021cons588cons2022cons2023cons2024cons2025cons2026cons2027cons2028cons2029cons2030cons669cons198cons2031cons842cons2032cons21cons2033cons150cons47cons2034cons1856cons1249cons263cons2035cons369cons2036cons69cons1481cons746cons1484cons167cons2037cons2038cons1678cons1257cons2039cons349b   acudegfn   CBAmphqr  rs)Z sympy.integrals.rubi.constraintsr  r  r  r  r  r   r!  r"  r#  r$  r%  r&  r'  r(  r)  r*  r+  r,  r-  r.  r/  r0  r1  r2  r3  r4  r5  r6  r7  r8  r9  r:  r;  r<  r=  r>  r?  r@  rA  rB  rC  rD  rE  rF  rG  rH  rI  rJ  rK  rL  rM  rN  rO  rP  rQ  rR  rS  rT  rU  rV  rW  rX  rY  rZ  r[  r\  r]  r^  r_  r`  ra  rb  rc  rd  re  rf  rg  rh  ri  rj  rk  rl  rm  rn  ro  r   r  u_x_r  r  n_p_r   replacement6934d_q_replacement6935replacement6936F_r  replacement6937replacement6938replacement6939replacement6940y_r   With6941replacement6941w_With6942replacement6942With6943replacement6943z_With6944replacement6944With6945replacement6945m_replacement6946replacement6947v_replacement6948With6949replacement6949With6950replacement6950With6951replacement6951a_With6952replacement6952With6953replacement6953With6954replacement6954With6955replacement6955A_With6956replacement6956With6957replacement6957With6958replacement6958With6959replacement6959With6960replacement6960With6961replacement6961With6962replacement6962With6963replacement6963With6964replacement6964With6965replacement6965With6966replacement6966With6967replacement6967With6968replacement6968With6969replacement6969With6970replacement6970With6971replacement6971replacement6972With6973replacement6973With6974replacement6974replacement6975replacement6976replacement6977replacement6978replacement6979replacement6980With6981replacement6981With6982replacement6982replacement6983replacement6984With6985replacement6985replacement6986With6987replacement6987With6988replacement6988With6989replacement6989With6990With6991replacement6991replacement6992replacement6993replacement6994replacement6995With6996replacement6996With6997replacement6997replacement6998replacement6999With7000replacement7000replacement7001)r  r  r  r  r  r   r!  r"  r#  r$  r%  r&  r'  r(  r)  r*  r+  r,  r-  r.  r/  r0  r1  r2  r3  r4  r5  r6  r7  r8  r9  r:  r;  r<  r=  r>  r?  r@  rA  rB  rC  rD  rE  rF  rG  rH  rI  rJ  rK  rL  rM  rN  rO  rP  rQ  rR  rS  rT  rU  rV  rW  rX  rY  rZ  r[  r\  r]  r^  r_  r`  ra  rb  rc  rd  re  rf  rg  rh  ri  rj  rk  rl  rm  rn  ro  Zpattern6934Zrule6934Zpattern6935Zrule6935Zpattern6936Zrule6936Zpattern6937Zrule6937Zpattern6938Zrule6938Zpattern6939Zrule6939Zpattern6940Zrule6940Zpattern6941Zrule6941Zpattern6942Zrule6942Zpattern6943Zrule6943Zpattern6944Zrule6944Zpattern6945Zrule6945Zpattern6946Zrule6946Zpattern6947Zrule6947Zpattern6948Zrule6948Zpattern6949Zrule6949Zpattern6950Zrule6950Zpattern6951Zrule6951Zpattern6952Zrule6952Zpattern6953Zrule6953Zpattern6954Zrule6954Zpattern6955Zrule6955Zpattern6956Zrule6956Zpattern6957Zrule6957Zpattern6958Zrule6958Zpattern6959Zrule6959Zpattern6960Zrule6960Zpattern6961Zrule6961Zpattern6962Zrule6962Zpattern6963Zrule6963Zpattern6964Zrule6964Zpattern6965Zrule6965Zpattern6966Zrule6966Zpattern6967Zrule6967Zpattern6968Zrule6968Zpattern6969Zrule6969Zpattern6970Zrule6970Zpattern6971Zrule6971Zpattern6972Zrule6972Zpattern6973Zrule6973Zpattern6974Zrule6974Zpattern6975Zrule6975Zpattern6976Zrule6976Zpattern6977Zrule6977Zpattern6978Zrule6978Zpattern6979Zrule6979Zpattern6980Zrule6980Zpattern6981Zrule6981Zpattern6982Zrule6982Zpattern6983Zrule6983Zpattern6984Zrule6984Zpattern6985Zrule6985Zpattern6986Zrule6986Zpattern6987Zrule6987Zpattern6988Zrule6988Zpattern6989Zrule6989Zpattern6990Zrule6990Zpattern6991Zrule6991Zpattern6992Zrule6992Zpattern6993Zrule6993Zpattern6994Zrule6994Zpattern6995Zrule6995Zpattern6996Zrule6996Zpattern6997Zrule6997Zpattern6998Zrule6998Zpattern6999Zrule6999Zpattern7000Zrule7000Zpattern7001Zrule7001r  r  r  miscellaneous_integration   s  ^ R
b
r






*
>



T
~


@
P
n
t







 
4
b
|

r



&


r
\
J
L
N
`
p
4
z
r
l
l


.


6
4
4
:



f


 r  c                 C   sd   t |t| || ||  |  t|  | ||  | t|   t|| ||  ||   ||S Nr  r%   r&   r   )rr  rp  rs  ry  r  rt  xr  r  r  r  _  s   dr  c                 C   sT   t ||| ||   |  | | ||  | |   t|| ||  ||   ||S r  r  r   )rr  rp  rs  ru  r  r  rt  r  r  r  r  r  c  s   Tr  c	           	      C   s`   t ||| ||  |  |  | | ||  | | |   t|| ||  || |   ||S r  r  )	rr  rp  rs  ru  ry  r  r  rt  r  r  r  r  r  g  s   `r  c              
   C   sR   t |
| tt|||||   | | ||t|||  t|	|
|   |S r  r  r+   r   r  r}  r|  r{  Frr  rp  rs  ru  rv  rx  rw  ry  r  r  r  r  r  k     Rr  c
           
   
   C   sZ   t || tt|||||	   | |	 |	|	t||	 td t||	 td  |	S Nrq  r  r+   r   r  r  
r}  r{  r  rr  rp  rs  rv  rw  ry  r  r  r  r  r  o     Zr  c              
   C   sR   t |
| tt|||  | | | | ||t|||  t|	|
|   |S r  r  r  r  r  r  r  s  r  r  c
           
   
   C   sZ   t || tt|||	  | | | |	 |	|	t||	 td t||	 td  |	S r  r  r	  r  r  r  r  w  r
  r  c              	   C   sX   t |ttttfrdS zt|| |}tt|}W n tt	fy%   Y dS w |r*dS dS NFT

isinstanceintr  floatr  r  r6   r   	TypeErrorAttributeError)rt  r  yr  resr  r  r  r  {     r  c                 C   s$   t || |}t|tt|| |S r  r  r  logr   )rt  r  r  r  r  r  r  r    s   r  c              	   C   s\   t |ttttfrdS zt|| | |}tt|}W n tt	fy'   Y dS w |r,dS dS r  r  )rt  wr  r  r  r  r  r  r  r    s   r  c                 C   s,   t || | |}t|tt|| | |S r  r  )rt  r  r  r  r  r  r  r  r    s   r  c              	   C   sX   t |ttttfrdS zt|||}tt|}W n tt	fy%   Y dS w |r*dS dS r  r  )r~  rt  r  r  r  r  r  r  r  r    r  r  c                 C   s2   t |||}t||| td   | td  |S r  r  r  r  )r~  rt  r  r  r  r  r  r  r       &r  c              	   C   sj   t |ttttfrdS zt|| |||  |   |}tt|}W n tt	fy.   Y dS w |r3dS dS r  r  )r~  ry  rt  r  r  zr  r  r  r  r  r    s   r  c                 C   sT   t || |||  |   |}t||| td   || td   | td  |S r  r  )r~  ry  rt  r  r  r  r  r  r  r  r    s   6r  c                 C   s4   t |ttttfrdS t| |}t|| |rdS dS r  )r  r  r  r  r  r-  r   rt  r  vr  r  r  r    s   
r  c                 C      t | |}t||S r  )r-  r   r  r  r  r  r       

r  c
           
      C   sn   t | |td  ||td   | tt||t| ||	|    |t|||	|     |   |	|	|	S Nrz  r  r  r   r   r  
rr  rp  rs  ru  rv  rx  r~  ry  rt  r  r  r  r  r       nr  c
           
      C   sz   t ||td  ||td   | tt||	||   |t| ||	|    |t|||	|     |   |	|	|	S r  r   r!  r  r  r  r    s   zr  c                 C   s0   t ||||   | | ||  |  |  |S r  r   rr  r~  ry  r  rt  r  r  r  r  r  r  r    s   0r  c
              	   C   X   t |ttttfrdS zt|	||}
tt|
}W n tt	fy%   Y dS w |r*dS dS r  r  )rr  rp  rs  ru  r~  ry  rt  r  r  r  r  r  r  r  r  r    r  r  c
                 C   sB   t |	||}
t|
tt| ||  | |||  |  |||	|S r  r  r  r+   r   )rr  rp  rs  ru  r~  ry  rt  r  r  r  r  r  r  r  r    s   6r  c              	   C   sX   t |ttttfrdS zt||	|}tt|}W n tt	fy%   Y dS w |r*dS dS r  r  )rr  rp  rs  ru  rv  rx  r~  ry  r  rt  r  r  r  r  r  r  r  r  r  r    r  r  c                 C   sR   t ||	|}t|tt| ||  | |||  |  |||  |  ||||S r  r&  )rr  rp  rs  ru  rv  rx  r~  ry  r  rt  r  r  r  r  r  r  r  r  r    s   Fr  c              	   C   sX   t |ttttfrdS zt|||}tt|}W n tt	fy%   Y dS w |r*dS dS r  r  )rr  rp  rs  ru  rv  rx  rw  r  r~  ry  r  r  rt  r  r  r  r  r  r  r  r  r  r  r    r  r  c                 C   sb   t |||}t|tt| ||  | |||  |	  |||  |
  |||  |  ||||S r  r&  )rr  rp  rs  ru  rv  rx  rw  r  r~  ry  r  r  rt  r  r  r  r  r  r  r  r  r  r    s   Vr  c              	   C   sX   t |ttttfrdS zt|||}tt|}W n tt	fy%   Y dS w |r*dS dS r  r  )rr  rp  ry  rt  r  r  r  r  r  r  r  r    r  r  c                 C   s@   t |||}t| t|||t|| tt|| |||| S r  )r  r  r   r+   )rr  rp  ry  rt  r  r  r  r  r  r  r  (  s   4r  c           	   	   C   sX   t |ttttfrdS zt|||}tt|}W n tt	fy%   Y dS w |r*dS dS r  r  )	rr  rp  ry  r  rt  r  r  r  r  r  r  r  r  .  r  r  c                 C   s6   t |||}t|tt| |||   | ||||S r  r&  )rr  rp  ry  r  rt  r  r  r  r  r  r  r  ;  s   *r  c	              
   C   s   t |ttttfrdS z9td}	td}
t|| || |}
t|||}	tt	t
t|
t|| || |t	t
t|	t|||}W n ttfyP   Y dS w |rUdS dS NFr  r  Tr  r  r  r  r  r  r  r  r  r6   r   r
   r  r  )rr  rp  r~  ry  r  rt  r  r  r  r  r  r  r  r  r  r  A     Br  c	              	   C   sf   t d}	t d}
t|| || |}
t|||}	t|	|
 tt|| | |||   |  ||||S )Nr  r  )r  r  r  r  r+   r   )rr  rp  r~  ry  r  rt  r  r  r  r  r  r  r  r  r  Q  s
   6r  c
              	   C   r%  r  r  )rr  rp  rs  ry  r  r  rt  r  r  r  r  r  r  r  r  r  Z  r  r  c
              	   C   sJ   t |	||}
t|
tt| |||   ||td|    | |||	|S r  r  r  r+   r   r  )rr  rp  rs  ry  r  r  rt  r  r  r  r  r  r  r  r  g     >r  c              	   C   sX   t |ttttfrdS zt|||}tt|}W n tt	fy%   Y dS w |r*dS dS r  r  )r}  r|  rr  rp  rs  ry  r  r  rt  r  r  r  r  r  r  r  r  r  r  m  r  r  c              
   C   sZ   t |||}t|tt| |||   ||||   ||td|    |  ||||S r  r*  )r}  r|  rr  rp  rs  ry  r  r  rt  r  r  r  r  r  r  r  r  r  z  s   Nr  c              	   C   X   t |	ttttfrdS zt|
||	}tt|}W n tt	fy%   Y dS w |r*dS dS r  r  )r}  r|  rr  rs  ry  r  r  rt  r  r  r  r  r  r  r  r  r    r  r  c              
   C   sN   t |
||	}t|tt| ||	|   |||	td|    |  |	|	|
|	S r  r*  )r}  r|  rr  rs  ry  r  r  rt  r  r  r  r  r  r  r  r    s   Br  c              
   C   s   t |
ttttfrdS z9td}td}t|| || |
}t|||
}tt	t
t|t|| || |
t	t
t|t|||
}W n ttfyP   Y dS w |rUdS dS r'  r(  )rr  rp  rs  r~  ry  r  r  rt  r  r  r  r  r  r  r  r  r  r  r    r)  r  c              
   C   sz   t d}t d}t|| || |
}t|||
}t|| tt|
| | ||
|   ||
td|    |  |
|
||
S Nr  r  rz  r  r  r  r  r+   r   r  )rr  rp  rs  r~  ry  r  r  rt  r  r  r  r  r  r  r  r  r  r    s
   Jr  c              
   C   s   t |ttttfrdS z9td}td}t|| || |}t||	|}tt	t
t|t|| || |t	t
t|t||	|}W n ttfyP   Y dS w |rUdS dS r'  r(  )r}  r|  rr  rp  rs  r~  ry  r  r  rt  r  r  r  r  r  r  r  r  r  r  r  r    r)  r  c              
   C   s   t d}t d}t|| || |}t||	|}t|| tt|| | |||    ||||   ||td|    |  ||||S r-  r.  )r}  r|  rr  rp  rs  r~  ry  r  r  rt  r  r  r  r  r  r  r  r  r  r  r    s
   Zr  c              
   C   s   t |
ttttfrdS z9td}td}t|| || |
}t|||
}tt	t
t|t|| || |
t	t
t|t|||
}W n ttfyP   Y dS w |rUdS dS r'  r(  )r}  r|  rr  rs  r~  ry  r  r  rt  r  r  r  r  r  r  r  r  r  r  r    r)  r  c              
   C   s~   t d}t d}t|| || |
}t|||
}t|| tt|
| | ||
|    |||
td|    |  |
|
||
S r-  r.  )r}  r|  rr  rs  r~  ry  r  r  rt  r  r  r  r  r  r  r  r  r  r    s
   Nr  c              	   C   r,  r  r  )rr  rp  rs  ru  r~  ry  r  rt  r  r  r  r  r  r  r  r  r    r  r  c              	   C   sJ   t |
||	}t|tt| ||	|   | |||	|   |  |	|	|
|	S r  r&  )rr  rp  rs  ru  r~  ry  r  rt  r  r  r  r  r  r  r  r    r+  r  c              	   C   sX   t |ttttfrdS zt||
|}tt|}W n tt	fy%   Y dS w |r*dS dS r  r  )rr  rp  rs  ru  rv  rx  r~  ry  r  r  rt  r  r  r  r  r  r  r  r  r  r    r  r  c              	   C   s^   t ||
|}t|tt| |||   | ||||   |  ||||   |	  ||||S r  r&  )rr  rp  rs  ru  rv  rx  r~  ry  r  r  rt  r  r  r  r  r  r  r  r  r    s   Rr  c              	   C   sX   t |ttttfrdS zt|||}tt|}W n tt	fy%   Y dS w |r*dS dS r  r  )r  rt  r  r  r  r  r  r  r  r    r  r  c                 C   s&   t |||}t| | | t|  |S r  )r  r  r  )r  rt  r  r  r  r  r  r  r    s   r  c              	   C   sX   t |ttttfrdS zt|||}tt|}W n tt	fy%   Y dS w |r*dS dS r  r  )r  r~  rt  r  r  r  r  r  r  r  r  r    r  r  c                 C   s2   t |||}t|tt| | ||  ||||S r  r&  )r  r~  rt  r  r  r  r  r  r  r  r  $  r  r  c           	      C   sH   t |ttttfrdS ||t|| |t||   }t||r"dS dS r  r  r  r  r  r  r  r   	rr  rp  r~  r  rt  r  r  r  rs  r  r  r  r  *  s    
r  c           	      C   sN   ||t || |t ||   }t|tt| |||   | |||| |S r  )r  r  r+   r   r0  r  r  r  r  3  s    .r  c
                 C   sP   t |	ttttfrdS ||| t||	 || t||	   }
t|
|	r&dS dS r  r/  rr  rp  r~  r  r  r  rt  r  r  r  rs  r  r  r  r  9     (
r  c
                 C   s~   ||| t ||	 || t ||	   }
t|
| |td  tt| ||	||td     | |	|	||td  | |	S r  r  r  r  r+   r   r1  r  r  r  r  B  s   (Vr  c                 C   sP   t |
ttttfrdS |||	 t||
 || t|	|
   }t||
r&dS dS r  r/  rr  rp  r~  r  r  r  r  rt  r  r  r  rs  r  r  r  r  H  r2  r  c                 C   s   |||	 t ||
 || t |	|
   }t|| |td  tt| ||
||td     | |
|
||td  |	|td   |
S r  r3  r4  r  r  r  r  Q  s   (br  c	           
      C   sP   t |ttttfrdS ||| t|| || t||   }	t|	|r&dS dS r  r/  
rr  rp  r~  r  r  rt  r  r  r  rs  r  r  r  r  W  r2  r  c	           
   
   C   sj   ||| t || || t ||   }	t|	| tt| ||  | | |||||| td   |S r  r  r  r+   r   r  r5  r  r  r  r  `  s   (Br  c
                 C   P   t |	ttttfrdS ||| t||	 || t||	   }
t|
|	r&dS dS r  r/  r1  r  r  r  r  f  r2  r  c
              	   C   sp   ||| t ||	 || t ||	   }
t|
| tt| ||	|   | |	|	||| | td  | |	 S r  r6  r1  r  r  r  r  o  s   (Hr  c
                 C   r7  r  r/  rr  rp  r~  r  r  r  rt  r  r  r  rs  r  r  r  r  u  r2  r  c
                 C   s   ||| t ||	 || t ||	   }
t|
| |td  tt| ||	||td     | |	|	||| td  ||td   |	 S r  r3  r8  r  r  r  r  ~  s   (hr  c                 C   sP   t |
ttttfrdS |||	 t||
 || t|	|
   }t||
r&dS dS r  r/  r4  r  r  r  r    r2  r  c                 C   s   |||	 t ||
 || t |	|
   }t|| |td  tt| ||
||td     | |
|
||| | td  |	|td   |
 S r  r3  r4  r  r  r  r    s   (lr  c              	   C   sJ   t td| td  ttt|| td  |||||| td  |S r  )r  r  r+   r   r  )r~  rt  r  r  r  r  r       Jr  c              	   C   sn   t |ttttfrdS zt| |}ttt|t	| t
|td|}W n ttfy0   Y dS w |r5dS dS )NF   T)r  r  r  r  r  rO  r  r6   r   r  r  r  r  r  rt  r  lstr  r  r  r  r    s   
(r  c                 C   f   t | |}tt|tdt|td ttt|td||t|tdtdt|td  |S Nrz     rq  r:  rO  r  r  r  r+   r   rt  r  r<  r  r  r  r       
\r  c              	   C   V   t |ttttfrdS zt| |}tt|}W n tt	fy$   Y dS w |r)dS dS r  )
r  r  r  r  r  r   r6   r   r  r  r;  r  r  r  r       
r  c                 C   r=  r>  )r   r  r  r  r+   r   rA  r  r  r  r    rB  r  c
           
      C   s   t | t| || t|   || t|   |	| t|   | ||  ||  |	|  t|  t||||   |||   |	||   ||S r  r   )
rr  r~  ry  r  r  rt  r  r  r  r  r  r  r  r    s   r  c                 C   sr   t | t| || t|   || t|   | ||  ||  t|  t||||   |||   ||S r  r   r$  r  r  r  r    s   rr  c                 C   sL   t | t| || t|   | ||  t|  t||||   ||S r  r   )rr  r~  r  rt  r  r  r  r  r  r    s   Lr  c                 C   sp   t t|| td  t| |||    t| ||   |  t||||   | ||   | |  ||S r  )r  r   r  r  r   )rr  rp  ry  r  rt  r  r  r  r  r    s   pr  c                 C   sv   t || t|  | |||   t|  | ||   | t|   t||||   | ||   | |  ||S r  r  r&   r   )rr  rp  ry  r  rt  r  r  r  r  r  r    s   vr  c                 C   s   t || t|  | |||  ||   t|  | ||   |||   t|   t||||   | ||   |||   |  ||S r  rE  )rr  rp  r~  ry  r  rt  r  r  r  r  r  r       r  c                 C   s   t |ttttfrdS || t|  | ||| |    t|   | ||  |||   t|  }tt|tdr?dS dS )NFrq  T)	r  r  r  r  r  r&   r6   r   r  rr  rp  r~  r  r  rt  r  r  r  r  r  r    s   Rr  c              	   C   s   || t |  | ||| |    t |   | ||  |||   t |  }t|t||||   | ||| |    |  ||S r  )r&   r  r   rG  r  r  r  r    s   R8r  c                 C   s@   t |ttttfrdS t|| |||    |}t|rdS dS r  )r  r  r  r  r  r   r)   rr  rp  ry  rt  r  r  r  r  r  r    s   r  c                 C   s$   t || |||    |}t||S r  )r   r   rH  r  r  r  r    s   
r  c                 C   sJ   t td|  ||   t||td| ||   td|   ||S )Nr?  rz  r  r  r   rr  rp  rs  ry  r  r  rt  r  r  r  r  r    r9  r  c                 C   s   t |td| ||   td |  | |||   ||td|    |  t||td| ||   td|   ||S r  rI  rJ  r  r  r  r    s   r  c                 C   sT   t |ttttfrdS t|| |||   ||td|     |}t|r(dS dS )NFrz  T)r  r  r  r  r  r   r  r)   rr  rp  rs  ry  r  rt  r  r  r  r  r  r    s   .r  c                 C   s8   t || |||   ||td|     |}t||S r  )r   r  r   rK  r  r  r  r    s   .
r  c                 C   s^   t || ||  |t|||     | td |td|   |td | ||    |S r  )r   r  r  )rr  rp  rs  r~  ry  rt  r  r  r  r  r  
  s   ^r  c              	   C   rC  r  )
r  r  r  r  r  r"  r6   r   r  r  r;  r  r  r  r    rD  r  c              
   C   s\   t | |}ttdt|td ttt|td|||t|td t|td |S )Nrq  r:  rz  )r"  r  r  r  r+   r   rA  r  r  r  r    s   
Rr  c              	   C   sp   t |ttttfrdS zt| td|}ttt	|t
t|td}W n ttfy1   Y dS w |r6dS dS )NFr   rz  Tr  r  r  r  r  r{  r  r  r6   r   r   r  r  r  r;  r  r  r  r  !  s   $r  c              
   C   sl   t | td|}ttdt|td tttt|td| ||||t|td t|td |S )Nr   rq  rz  r:  r{  r  r  r  r+   r   r#  rA  r  r  r  r  .  s   \r  c              	   C   s   t |ttttfrdS z#t|| td |}ttt	|t
|  t|td td }W n ttfy:   Y dS w |r?dS dS )NFrq  rz  TrL  )r~  rt  r  r<  r  r  r  r  r  4  s   2r  c              
   C   sp   t || td |}ttdt|td tttt|td| ||||t|td t|td |S )Nrq  rz  r:  rM  )r~  rt  r  r<  r  r  r  r  A  s   \r  c                 C   sZ   t | }t|tt||| td  td  t|t|||  |||td|  |S )Nrq  rN  )r4   r  r+   r   r  r   r  )r~  rt  r  kr  r  r  r  G  s   Rr  c              	   C   rC  r  )
r  r  r  r  r  r  r6   r   r  r  r;  r  r  r  r  L  rD  r  c              	   C   s<   t | |}ttdttt|td||t|td|S )Nrz  rq  )r  r  r  r+   r   r  rA  r  r  r  r  Y  s   
2r  c                 C   s   t tdtd|   tttd| t|  | td td  ||t tdtd|   tttd|t|  | td td  || S )Nrq  rz  )r  r  r   r   r  )rr  rp  r  r  r  r  r  r  _  s   r  c                 C   s   t td| |  ttttdtdtdtd t |  |td  t|  | |td    |tttd|td |S )Nrz  rq  rN  r?  r  r  r  r   r   rO  r  r   rr  rp  ry  r  r  r  r  r  r  c  s   r  c                 C   sn   t td| |  ttttdtdtdtd t |  | t|  | |   |tttd||S )Nrq  rN  rz  rP  rQ  r  r  r  r  g  r"  r  c                 C   s4   t ttt|||| |||    |t|||S r  )r   r   r   r  r  )rr  rp  ry  rt  r  r  r  r  r  r  k  s   4r  c                 C   s2   t |ttttfrdS t| |}t|| rdS dS r  )r  r  r  r  r  r#  r  r  r  r  r  r  o  s   

r  c                 C   r  r  )r#  r   r  r  r  r  r  x  r  r  c                 C   s0   t |ttttfrdS t| |}t|rdS dS r  )r  r  r  r  r  r   r)   r  r  r  r  r  ~  s   
r  c                 C   r  r  )r   r   r  r  r  r  r    r  r  c
           
      C   sP   t |	| |  | ||	|   |  |||	|   |  t||	||   |	|	S r  r  )
rr  rp  rs  ru  r~  ry  r  r  rt  r  r  r  r  r    s   Pr  c                 C   s   t td| tdd |  t| |||   ||td|     |td| ||    t||td| ||   td|   ||S )Nr?  rq  rz  )r  r  r  r   rJ  r  r  r  r    rF  r  c              	   C   rC  r  )
r  r  r  r  r  rO  r6   r   r  r  r;  r  r  r  r    rD  r  c                 C   r=  r>  r@  rA  r  r  r  r    rB  r  c                 C   s
   t | |S r  r#  )rt  r  r  r  r  r    s   
r  (  __doc__Zsympy.externalr   r   r   r   r   r   Z%sympy.integrals.rubi.utility_functionr   r	   r
   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r    r!   r"   r#   r$   r%   r&   r'   r(   r)   r*   r+   r,   r-   r.   r/   r0   r1   r2   r3   r4   r5   r6   r7   r8   r9   r:   r;   r<   r=   r>   r?   r@   rA   rB   rC   rD   rE   rF   rG   rH   rI   rJ   rK   rL   rM   rN   rO   rP   rQ   rR   rS   rT   rU   rV   rW   rX   rY   rZ   r[   r\   r]   r^   r_   r`   ra   rb   rc   rd   re   rf   rg   rh   ri   rj   rk   rl   rm   rn   ro   rp   rq   rr   rs   rt   ru   rv   rw   rx   ry   rz   r{   r|   r}   r~   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r  r  r  r  r  r  r  r  r	  r
  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r   r!  r"  r#  r$  r%  r&  r'  r(  r)  r*  r+  r,  r-  r.  r/  r0  r1  r2  r3  r4  r5  r6  r7  r8  r9  r:  r;  r<  r=  r>  r?  r@  rA  rB  rC  rD  rE  rF  rG  rH  rI  rJ  rK  rL  rM  rN  rO  rP  rQ  rR  rS  rT  rU  rV  rW  rX  rY  rZ  r[  r\  r]  r^  r_  r`  ra  rb  rc  rd  re  rf  rg  rh  ri  rj  rk  rl  rm  rn  ro  rp  rq  rr  rs  rt  ru  rv  rw  rx  ry  rz  r{  r|  r}  r~  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  Zexpr  r  r  r  r  Zsympyr  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  Zsympy.integrals.rubi.symbolr  Zsympy.core.symbolr  r  Zsympy.functionsr  r  r  r  r  r  r  Z%sympy.functions.elementary.hyperbolicr  r  r  r  r  r   r  r  r  r  r  r  Z(sympy.functions.elementary.trigonometricr  r  r	  r
  r  r  r  r  ZPir  ZB_ZC_r  ZG_ZH_r  Zb_Zc_r  Ze_Zf_Zg_Zh_Zi_Zj_Zk_Zl_r  r  r  r  Zr_Zt_r  r  Zs_r  r  r  r  Za1_Za2_Zb1_Zb2_Zc1_Zc2_Zd1_Zd2_Zn1_Zn2_Ze1_Ze2_Zf1_Zf2_Zg1_Zg2_Zn3_ZPq_ZPm_ZPx_ZQm_ZQr_ZQx_Zjn_Zmn_Znon2_ZRFx_ZRGx_r  iiZPqqQRr  r{  rO  rt  Z	_UseGammaZ	ShowStepsZStepCounterr  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  <module>   s              nJjB.
 
V















	















	







	

	

	








	

	

	

	

	

	

	












	

	



	















	

	



