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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  H/usr/lib/python3/dist-packages/sympy/integrals/rubi/rules/exponential.py
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||||}t|t}tytzt|t}d	t~d t}d
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}t|t؃}tytztt|t}dt~d t}dt~d t}dt~d  tt|t}dt~d t}dt~d t}dt~d   t{t|t}d	t~d t}d
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||`|a|b|c|9}t|t܃}tytztt|t}dt~d t}dt~d t}dt~d  tt|t}dt~d t}dt~d t}dt~d   t{t|t}d	t~d t}d
t~d t}dt~d  t}dt~d t t}dt~d  t|||Y|_||||||	|
||`|a|b|d|[}t|t݃}tytztt|t}dt~d t}dt~d t}dt~d  tt|t}dt~d t}dt~d t}dt~d   t{t|t}d	t~d t}d
t~d t}dt~d  t}dt~d t t  t|||Y|_||||||	|
||`|a|b|d|1}t|tރ}tytztt|t}dt~d t}dt~d t}dt~d  tt|t}dt~d t}dt~d t}dt~d   t{t|t}d	t~d t}d
t~d t}dt~d  t}dt~d t t  t|||Y|_||||||	|
||`|a|b||d|2}t|t߃}tytzttt}dt~d  ttt}dt~d   t{tt}dt~d  t}dt~d t t  t|||Y|_|||||b||&|e}t|t}tytzt{t|t}d	t~d t}d
t~d t}dt~d  t{t|t}d	t~d t}d
t~d t}dt~d  t}dt~d t|t}dt~d t}dt~d  t}dt~d  t||||||||||
}t|t}tytzt{t|t}d	t~d t}d
t~d t}dt~d  t|t}dt~d  t{t|t}d	t~d t}d
t~d t}dt~d  t}dt~d t|t}dt~d t}dt~d  t}dt~d  t|||||||||||}t|t}tytzt|t}dt~d t}dt~d t}dt~d t{t t}dt~d t{t t}d
t~d  t}dt~d  t||||||	|
|f|G|g|h}t|t}tytzt{t t|t}dt~d t}dt~d t}dt~d  t{t t}dt~d t{t t}d
t~d  t}dt~d  t||||||	|
|f|G|g|h} t| t}!tytzt{t t}dt~d t t|t}dt~d t}dt~d t}dt~d  t{t t}dt~d t{t t}d
t~d  t}dt~d  t||||||	|
|||f|G|g|h}"t|"t}#tytzt|t}dt~d t{t t}dt~d t{t|t}d	t~d t}d
t~d  t}dt~d   t|||||||i| |	}$t|$t}%tytztt{t t}dt~d t{t t}d
t~d  t  t|||||||j|k|l	}&t|&t}'tytzt{t|t}dt~d t}d	t~d t}dt~d t}dt~d  t|t~d t}d
t~d t|t}dt~d  t}dt~d  t||||||||
||m
}(t|(t})tytzt{t|t}dt~d t}d	t~d t}dt~d t}dt~d  tt|t~d t}d
t~d   t|||||||
||n	}*t|*t}+tytzt{t|t}dt~d t}d	t~d t}dt~d t}dt~d  tt}dt~d  tt|t~d  t|t}dt~d  t}dt~d  t||||||||
||$|},t|,t}-tytzt{t|t}dt~d t}d	t~d t}dt~d t}dt~d  tt}dt~d  ttt|t~d    t|||||||
||$|
}.t|.t}/tytzt{t|t~d t}dt~d t}dt~d t|t~d   t|||||o}0t|0t}1tytzt|t}dt~d t|t}dt~d tt| t  t||@||p|q}2t|2t}3tytzttt{t|t}d	t~d t}d
t~d t}dt~d  t}dt~d t}dt~d  t|||||||||r	}4t|4t}5tytzttt{t|t}d	t~d t}d
t~d t}dt~d  t}dt~d t}dt~d  t|||||||||s	}6t|6t}7tytzt{t t}dt~d t t}dt~d t|||||2}8t|8t}9tytztt||t}:t|:t};tytzt{t t}dt~d t{t t}dt~d  t t}dt~d t||||||1|u}<t|<t}=tytzt{t t}dt~d tt t}dt~d  t t}dt~d t|||Y||||1|u}>t|>t}?tytzt{t t}dt~d t{t t}dt~d  t t}dt~d t||||||2|u}@t|@t}Atytzt{t t}dt~d tt t}dt~d  t t}dt~d t|||Y||||2|u}Bt|Bt}Ctytzt{t tt  t}dt~d t|||Y|v}Dt|Dt}Etytzt{t tt  t}dt~d t|||tt}Ft|Ft}Gtytzt{t tt}dt~d  t t||||$|w|(tt}Ht|Ht}Ig |y|{|}|||||||||||||||||||||||||||||||||||Ñ|ő|Ǒ|ɑ|ˑ|͑|ϑ|ё|ӑ|Ց|ב|ّ|ۑ|ݑ|ߑ|||||||||||||||||||||	||||||||||||!|#|%|'|)|+|-|/|1|3|5|7|9|;|=|?|A|C|E|G|IS )Nr   )xcons33cons170cons517cons1100cons1101cons3cons8cons29cons50cons127cons210cons4cons96cons20cons21cons19cons1102cons130cons2cons246cons139cons554cons1103cons1104cons5cons382cons56cons1105cons1106cons1107cons211cons226cons798cons799cons52cons1108cons806cons1109cons814cons1110cons1111cons1112cons1113cons586cons1114cons1115cons481cons482cons1116cons198cons25cons1117cons55cons1118cons1119cons1120cons1121cons87cons1122cons358cons533cons1123cons1124cons537cons95cons1125cons1126cons178cons369cons168cons746cons70cons842cons1127cons1128cons1129cons27cons73cons1130cons1131cons1132cons820cons1133cons1134cons1135cons1136cons821cons1137cons1138cons1139cons1140cons150cons812cons813cons1141cons1142cons54cons802cons1143cons1144cons1145cons815cons1146cons228cons64cons1147cons1148cons1149cons1150cons1151cons1152cons1153cons465cons1154cons45cons450cons1155cons1156cons1157cons1019f   egbndcmpar  hjkq   usrt   y)Z sympy.integrals.rubi.constraintsr  r  r  r  r  r   r!  r"  r#  r$  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  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  F_x_r  r  r   replacement1904m_replacement1905replacement1906replacement1907r  replacement1908replacement1909n_replacement1910a_replacement1911replacement1912p_With1913u_v_replacement1914With1915replacement1916replacement1917replacement1918replacement1919G_replacement1920replacement1921replacement1922replacement1923w_replacement1924replacement1925replacement1926With1927e_logreplacement1928replacement1929replacement1930replacement1931replacement1932replacement1933With1934replacement1935replacement1936replacement1937replacement1938replacement1939replacement1940replacement1941replacement1942With1943replacement1944replacement1945replacement1946replacement1947replacement1948replacement1949replacement1950replacement1951replacement1952replacement1953replacement1954replacement1955replacement1956replacement1957replacement1958replacement1959replacement1960replacement1961replacement1962replacement1963replacement1964replacement1965replacement1966replacement1967replacement1968replacement1969replacement1970replacement1971With1972r   With1973replacement1973With1974replacement1974replacement1975replacement1976replacement1977replacement1978H_With1979replacement1979replacement1980replacement1981replacement1982replacement1983replacement1984replacement1985replacement1986With1987With1988h_With1989With1990replacement1991replacement1992replacement1993c_replacement1994replacement1995replacement1996expreplacement1997replacement1998replacement1999replacement2000With2001replacement2002replacement2003replacement2004replacement2005replacement2006With2007replacement2007With2008replacement2008(J  r  r  r  r  r  r 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           
      C   s   t || | t|  ||td   t| ||||	    | | |||	  |td   |	|	 t| ||||	    | | |||	  |td   ||td   |	 S Nr  r  r  r  r   r  r+  r  r  r  r       r  c                 C   sH   t | || | | |   t|| |||   t|  |  | |S Nr  r  r  r,  r  r  r  r  r  r-  r  r  r  r       Hr  c                 C   s   t | || | | |   || td   || td   | |  t|td | | |||   t|  |  t| | td   |S Nr)  r  )r  r  r  r  r,  r  r  r  r  r  r  r-  r  r  r  r    s   r  c              
   C   s\   t td| tt| || | | |  || |td  |   ||t|||  |S Nr  )r  r  r+   r   r  r4  r  r  r  r    s   \r  c                 C   s   t | || | | |   | | t|  | t| td   | | |||   t|  | t|   |||  t|  t|td | | |||   t|  |  | | S r6  )r  r  r%   r  r&   r  r7  r  r  r  r       r  c
           
      C   sh   t | | | |||	    | ||||	    | |  t| || |||	    |||	  |  |	|	S r2  r  r   r+  r  r  r  r    s   hr  c              
   C   s>   t t|||  | ||| ||||    |	   |
 ||S r2  )r   r   r,  r  r  r  r  r  r  r  r  r  r  r-  r  r  r  r       >r  c                 C   s   t || || | |	 t|   t|||
  |td  t|| ||||
    |	   | td  |
|
t|||
  | t|| ||||
    |	   | td  || | |	 t|   |
 S r6  r*  r,  r  r  r  r  r  r  r  r  r  r-  r  r  r  r    s   r  c                 C   sr   t ||| ||||    |	   |
 |}t|| t||||  |td   || t|||  | || S r(  r  r  r   r  )r,  r  r  r  r  r  r  r  r  r  r  r-  r  r  r  r  r    s   *Hr  c
           
      C   s6   t ||| |t||	  |   | t||	|  |	S r2  )r   r2  r,  )
r,  r  r  r  r  r  r  r  vr-  r  r  r  r       6r  c
                 C   s   t ||	}
td}ttt|
tt|
td|	t|
td|t|
td  |
| }ttt|
tt|
d|	t|
d|t|
d  |
| }t|
| t	|||| |t
||	  |   |  |	 | |	S Nzr  r  )r,  r  r  r  rW   r   r  r  r  r   r2  )r,  r  r  r  r  r  r  r  r?  r-  uurB  r  r  r  r    s
   
F:>r  c                 C   s:   t ||| ||||    |	   |
 |||  |  |S r2  r   r;  r  r  r  r    s   :r  c                 C   s   t || || | |	 t|   t|||
  |td  ttd|| ||||
    |	  |   |
|
 t|||
  | ttd|| ||||
    |	  |   || | |	 t|   |
 S r6  r*  r=  r  r  r  r    s   r  c              
   C   s   t || || | |	 |
td  t|   t||| ||||    |	   |
td  |||  |td   || t||| ||||    |	   |
td  |||  |  || | |	 |
td  t|   | S Nr  r)  r  r  r  r   r  r;  r  r  r  r    s   r  c                 C   sR   t ||| ||||    |	   |
 |||  |  | ||||    |	  |S r2  rD  r;  r  r  r  r       Rr  c              
   C   s   t |||	|
|    | | | ||||    |   t||| ||||    |   | |||  |  | ||||    |  ||S r2  r:  )r,  Gr  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r-  r  r  r  r    s   r  c                 C   s2   t | ||||    | || | t|   |S r2  r  r  )r,  r  r  r  r  r-  r  r  r  r    s   2r  c                 C   s"   t t| |t||  | ||S r2  r   r   r2  r,  r  r  r?  r-  r  r  r  r       "r  c                 C   s    t t| |t||  |||S r2  rJ  rK  r  r  r  r        r  c                 C   s^   t | ||  ||td   t||td |t||td t||td t|   |S r/  )r  r  r3   r  r,  r  r  r  r?  wr-  r  r  r  r  #     ^r  c                 C   s0   t t| |t||  | t|||  ||S r2  r   r   r2  r,  rN  r  r  r  r  '  s   0r  c                 C   s.   t t| |t||  |t|||  ||S r2  rQ  rN  r  r  r  r  +     .r  c           	   	   C   s   t ||}td}ttt|tt|td|t|td|t|td  || }ttt|tt|d|t|d|t|d  || }t|| t	t
| |t||  | | || | |S rA  )r,  r  r  r  rW   r   r  r  r  r   r   r2  )	r,  r  r  r  r?  rO  r-  rC  rB  r  r  r  r  /  s
   
F:8r  c                 C   sF   t | ||||
    | |
 t||
 |	td   |	td  |
S r/  )r  r  r  )r,  r  r  r  r  r  r  r  r  r  r-  r  r  r  r  7     Fr  c                 C   sR   t | ||||    | ||	td   t|| |
td   |
td  |S r/  r  r  r  )r,  r  r  r  r  r  r  r  r  r  r  r-  r  r  r  r  ;  rG  r  c                 C   s.   t | |||||     || t|   |S r2  rI  r,  r  r  r  r  r-  r  r  r  r  ?  rR  r  c                 C   s^   t | | tt t|||  t|t|  td  td| t|t|  td  |S r8  )r  r  Pir  r  r  r  rU  r  r  r  r  C  rP  r  c                 C   sb   t | | tt t|||  t| t|  td  td| t| t|  td  |S r8  )r  r  rV  r  r  r  r  rU  r  r  r  r  G  s   br  c              	   C   sz   t || t|  t| |||||  |    |||  |  || t| |||||  |    |||   | | S r2  r  r  r   r  r,  r  r  r  r  r  r-  r  r  r  r  K  s   zr  c              
   C   s^   t |}t|| tt| |||||     ||td   |||||  td|  |S r6  r4   r  r+   r   r  )r,  r  r  r  r  r  r-  r  r  r  r  r  O  s   Vr  c                 C   s|   t | | | |||  |  t|  td |   |||   ttd| | |||  |  t|   ||  | S r/  r  r  r  r  rX  r  r  r  r  T  s   |r  c
           
      C   sX   t | |||||	  |    |||	  |   |||	  |  || | t|   |	S r2  rI  
r,  r  r  r  r  r  r  r  r  r-  r  r  r  r  X     Xr  c	           	      C   s6   t | | t||||  |  t|   ||  |S r2  r3  	r,  r  r  r  r  r  r  r  r-  r  r  r  r  \  r@  r  c              
   C   sV   t td||td   tt| |||td    |||||  |td  |S Nr  r  )r  r  r+   r   r,  r  r  r  r  r  r  r-  r  r  r  r  `     Vr  c              	   C      t || td || t|   t| |||||  |    |||  ||   || t| |||||  |    |||  || td   || | t|   | S r/  rF  r_  r  r  r  r  d  r1  r  c              	   C   ra  r/  rF  r_  r  r  r  r  h  r1  r  c              	   C      t || t|  |td  t| |||||  |    |||  ||   || t| |||||  |    |||  |td   ||td   | S r/  r0  r_  r  r  r  r  l  r.  r  c              	   C   rb  r/  r0  r_  r  r  r  r  p  r.  r  c           	   
   C   sj   t |}t|| tt| |||||     |||td  td   |||||  td|  |S rE  rY  )	r,  r  r  r  r  r  r  r-  r  r  r  r  r  t  s   br  c
           
   	   C   sX   t |||	  |  |||	  |  t| |||||	  |    |||	  |  |	|	S r2  r:  r[  r  r  r  r  y  r\  r  c
           
      C   s   t | | | |||	  |  t|  |td  |   |||	  |td   t|td | | |||	  |  t|   ||  |	 S r/  rZ  r[  r  r  r  r  }  s   r  c	           	   
   C   s4  t | | ||  | t| |||||  td    |||  |td   ||t |td |td  td| |td  t|   t| |||||  td    |||  |td   || t| |||||  td    | |||  |td   td| |td  t|   | S Nr  r)  )r  r   r  r  r  	r,  r  r  r  r  r  r  r  r-  r  r  r  r    s   6 r  c	           	   
   C   sN  t td| |td  t|  |td |td   t| |||||  td    |||  |td   || t td| | | | ||   t|  |td |td   t| |||||  td    |||  |td   || t| |||||  td    |||  |td   ||td   | S Nr  r  rF  re  r  r  r  r    s   P r  c
           
   	   C   s   t || | t|  ||td   t| |||||	  |    |||	  |td   |||	  |td   |	|	 t| |||||	  |    |||	  |td   ||td   |	 S rE  r0  r[  r  r  r  r       r  c              
   C   s   t || t| |||||     |||   ||t | | ||  | t| |||||     |||  |||    || S r2  r:  )r,  r  r  r  r  r  r  r-  r  r  r  r       r  c	           	   	   C   s   t || t|  ||td   t| |||||     |||  |td   |||  td  ||t| |||||     |||  |td   ||td   | S r^  r0  re  r  r  r  r    s   r  c	           	      C   s.   t | |||||  |    |||   |S r2  rD  r]  r  r  r  r    rR  r  c                 C   "   t | t|| t|||  |S r2  r   r2  r,  r  r  r?  r-  r  r  r  r    rL  r  c                 C   s.   t t| |||||  |    |||||S r2  )r   r   )r,  r  r  r  r  r  r  r-  r  r  r  r    rR  r  c                 C   s    t | ||t||   | |S r2  )r   r,  )r,  r  r  r  r?  r-  r  r  r  r    rM  r  c
           
   
   C   s   t ||| | ||    tt| ||| |	 | | ||    || | | ||     |	 |	|	|||	  |||	   |	 S r2  r  r+   r   
r,  r  r  r  r  r  r  r  r  r-  r  r  r  r    rh  r  c                 C   s.   t | || | |  t|||
  |	 |
|
S r2  r:  r,  r  r  r  r  r  r  r  r  r  r-  r  r  r  r    rR  r  c                 C   sV   t | | | | ||   ||||
    || ||  |   |||
  |	  |
S r2  rD  rn  r  r  r  r    r`  r  c
           
   
   C   s   t || t| |||||	   |||	     |||	   |	|	t | | ||  | t| |||||	   |||	     |||	  |||	    |	|	 S r2  r:  rm  r  r  r  r    s   r  c              	   C   s   t || | ||   t|  ||	td   t| |||||
   |||
     |||
  |	td   |||
  td  |
|
 t| |||||
   |||
     |||
  |	td   ||	td   |
 S r^  r0  rn  r  r  r  r    s   r  c              
   C   s   t ||| |
 ||	    tt| ||| | | ||   | |
 ||	    || |
 ||	   | |
 ||	     | |||	|
|  |||   | S r2  rl  )r,  r  r  r  r  r  r  r  r  r  r  r-  r  r  r  r    r.  r  c                 C   sV   t | ||td td|    t| |td| |  td td|   ||S Nr  r  )r  r  r   )r,  r  r  r  r-  r  r  r  r    r`  r  c                 C   s   t | t|| |S r2  rj  )r,  r?  r-  r  r  r  r    s   r  c                 C   s>   t | |||  ||td    | td| t|   |S r8  rT  r,  r  r  r  r  r  r-  r  r  r  r    r<  r  c              	   C   s   t |td |td  td| t|   t| |||  ||td    |||  |td   || t| |||  ||td    | |||  |td   td| t|   | S rc  rF  r,  r  r  r  r  r  r  r-  r  r  r  r    rg  r  c                 C   sf   t | ||td td|    t|td| |  td t|  td|   td|  |S ro  )r  r  r  r  rp  r  r  r  r       fr  c              	   C   s   t td| t|  |td |td   t| |||  ||td    |||  |td   || t| |||  ||td    |||  |td   ||td   | S rf  rF  rq  r  r  r  r    r9  r  c              	   C   s   t || td| |  td|  t| |||  ||td    || t| |||  ||td    | td| t|   | S r8  )r  r  r   r  r  rp  r  r  r  r       r  c              
   C   s,  t || td| |  td|  t| |||  ||td    |||  |td   || t |td |td  td| t|   t| |||  ||td    |||  |td   || t| |||  ||td    | |||  |td   td| t|   | S rc  )r  r  r   r  r  rq  r  r  r  r    s   . r  c              
   C   s8  t td| t|  |td |td   t| |||  ||td    |||  |td   || t || td| |  t|  |td |td   t| |||  ||td    |||  |td   || t| |||  ||td    |||  |td   ||td   | S rf  rF  rq  r  r  r  r    s   : r  c                 C   s6   t | |||  ||td    |||  |  |S r8  r   r  rq  r  r  r  r    r@  r  c                 C   ri  r2  rj  rk  r  r  r  r    rL  r  c
                 C   sb   t | ||||	    | | | | |  |	}
t|t|
|	|td   |	|	 t|	| |
|	 S r(  r>  )r,  r  r  r  r  r  r  r  r?  r-  r  r  r  r  r    s   .4r  c                 C   s^   t |ttttfrdS t||	 t| || t|   }tt|t	t
|tdr-dS dS NFr  T)
isinstanceintr  floatr  r   r  r  r'   rR   r  r  r,  rH  r  r  r  r  r  r  r  r  r  r-  r  r  r  r  r       $r  c              
   C   s   t ||	 t| || t|   }t|| | |	 | ||	   t| || t|   tt|t|td  |||t|   |
  ||| ||||   t|  |S r(  r   r  r  r4   r+   r   rx   r  ry  r  r  r  r    s   $r  c                 C   s^   t |ttttfrdS t|| t|  ||	 t|  }tt|t	t
|tdr-dS dS ru  )rv  rw  r  rx  r  r   r  r  r'   rQ   r  r  ry  r  r  r  r    rz  r  c              
   C   s   t || t|  ||	 t|  }tt|||	 t|  tt|t|td  | || || | |   | |t|  | |
  ||||	|||   t|  |S r(  )r   r  r  r4   r+   r   r  rx   ry  r  r  r  r    s   $r  c                 C   sJ   t |||	  |||	 |   | ||  | || |   | | |
  |S r2  rD  r,  rH  r  r  r  r  r  r  r  r  r  r-  r  r  r  r    s   Jr  c                 C   s   t ||	|||    ||
  t|
 ||	 t| || t|   td||	 t| || t|    | ||||     | |  ||	 t|  |S r/  r  r5   r  r  r|  r  r  r  r    s   r  c                 C   s   t ||	|||    | ||||    | | |
td   ttd|
td ||	 t| || t|    td||	 t| || t|    | ||||     | |  || |	 t|  |S r/  )r  r  r5   r  r|  r  r  r  r     s   r   c
           
      C   s6   t ||t||	  | |t||	  | | |  |	S r2  rj  )
r,  rH  r  r  r  r  r  r  r?  r-  r  r  r  r    r@  r  c                 C   sZ   t |ttttfrdS t|	|
 t| || t|  || t|   }t|r+dS dS NFT)rv  rw  r  rx  r  r   r  r'   r,  rH  Hr  r  r  r  r  r  r  r  r  r  r  r  r-  r  r  r  r  r    s   4r  c              
   C   s   t |	|
 t| || t|  || t|   }t|| |	 |
 | ||
   || | | | ||    t| || t|   tt|t|td  |||t|   |  ||| ||||   t|  |S r(  r{  r  r  r  r  r  %  s   4r  c              
   C   sZ   t ||
| |	 | |   t|||||    || | |||    |  |  ||S r2  r:  r,  rH  r  r  r  r  r  r  r  r  r  r  r  r  r  r-  r  r  r  r  +  s   Zr  c                 C   sf   t |||
  ||	|
 |   |||   ||| |   | ||  | || |   | | |  |S r2  rD  r  r  r  r  r  /  rr  r  c                 C   s   t ||
||	|    |||||     ||  t| |	|
 t| || t|  || t|   td|	|
 t| || t|  || t|    | ||||     | |  |	|
 t| || t|   |S r/  r}  r  r  r  r  r  3  s   r  c                 C   s  t ||
||	|    |||||     | ||||    | | | |   | ||||    | | |  t| |	|
 t| || t|  || t|   td|	|
 t| || t|  || t|    | ||||     | |  |	|
 t| || t|   |S r/  r}  r  r  r  r  r  7  s     r  c                 C   sH   t ||t|	|  ||t||   | |t|
|  | | |  |S r2  rj  )r,  rH  r  r  r  r  r  r  r  r  r?  rO  r-  r  r  r  r	  ;  r5  r	  c	           	   	   C   s   t || || | t|   t||td  | ||||    | |||   |  || t| ||||    | |||   |td  || | |td  t|   | S r6  r*  )	r,  r  r  r  r  r  r  r  r-  r  r  r  r
  ?  s   r
  c
           
   
   C   s0  t || || | t|   t|	|| td  | ||||	    | ||	|   |  |	|	 t ||| | |td  t|   t|	|td  | ||||	    | ||	|   |td   |	|	 t|	| | ||||	    | ||	|   |td   || | |td  t|   |	 S r6  r*  )
r,  r  r  r  r  r  r  r  r  r-  r  r  r  r  C  s   2 r  c
                 C   s   t td | | |td  td}
ttd| |
 t|||	  | td| |  | | |
  |	|	ttd| |
 t|||	  | td| |  | | |
  |	|	 S Nr  r  r  r  r  r   r,  r  r  r  r  r  r  r  r?  r-  r  r  r  r  r  G  s   (r  c
                 C   s   t td | | |td  td}
ttd| |
 t| | |||	  |  td| |  | | |
  |	|	ttd| |
 t| | |||	  |  td| |  | | |
  |	|	 S r  r  r  r  r  r  r  L  s   (r  c                 C   s   t td | | |td  td}t| | | td| |  |  t|||  | td| |	  | | |  || t|| | td| |  |  t|||  | td| |	  | | |  || S r  r  )r,  r  r  r  r  r  r  r  r  r  r?  r-  r  r  r  r  r  Q  s   (r  c           	      C   sd   t td| | | | |||   |   |}t|t|||td   || t|||  | S rE  )r  r  r  r   r  )	r,  r  r  r  r  r  r?  r-  r  r  r  r  r  V  s   .6r  c                 C   s6   t | | | | td|  | | | |  |  |S r8  rt  )r,  r  r  r  r  r?  rO  r-  r  r  r  r  [  r@  r  c	           	   	   C   sF   t t| ||||  |   td|||  ||td    ||S r^  r   r   r  )	r,  r  r  r  r  r  r  r  r-  r  r  r  r  _  rS  r  c              	   C   s>   t t| ||||  |   td|||td    ||S r^  r  )r,  r  r  r  r  r  r  r-  r  r  r  r  c  r<  r  c              	   C   sF   t t| ||||
  |   |	| |||
  ||
td    |
|
S r8  r  )r,  r  r  r  r  r  r  r  r  r  r-  r  r  r  r  g  rS  r  c
           
   	   C   s>   t t| ||||	  |   || |||	td    |	|	S r8  r  )
r,  r  r  r  r  r  r  r  r  r-  r  r  r  r  k  r<  r  c              	   C   s&  t ttt|td  t| t|   t| t|   |  ttd t| t|   t| t|    tdt| t|    | t ttt|td t| t|   t| t|   |  ttdt| t|   t| t|    tdt| t|    | S ro  )r  r  rV  r  r  r  r  )r,  r  r  r-  r  r  r  r  o  s   ( r  c                 C   s   t | t|| td  ||  t| |  ||t||  t| |td  | t||  t| |td  |td  | S r6  )r  r   r  r  r  )r  r  r-  r  r  r  r  s  s   r  c              
   C   sV   t td|| | t|   ttt|||  | ||| ||||    | |S r/  )r  r  r  r+   r   r,  r  r  r  r  r  r  r-  r  r  r  r  w  r`  r  c                 C   s   t || | | t|  t|| ||||    |  ||| ||||    |    || t|t||| ||||    |    | S r2  rW  r  r  r  r  r  {  rs  r  c                 C   s8   t | | |  | | | |  t| ||  | ||S r2  r:  )r,  r  r  r  r?  r-  r  r  r  r    s   8r  c                 C   s8   t | |}t|t|| ttt| || ||||S r2  )r  r  r  r+   r   r  )r  r-  r?  r  r  r  r    s   
.r  c                 C   s6   t | ||  | | t| | | | | |  |S r2  rj  r,  r  r  r  r  r?  rO  r-  r  r  r  r    r@  r  c	           	   
   C   sF   t | ||  | ||tt| t|  |t|  |  |  |S r2  )r   r  r2  r  	r,  rH  r  r  r  r  r?  rO  r-  r  r  r  r    rS  r  c                 C   s   t | | |  | | | | | |  |  | t| | | | | |   t| ||  | | t| | | | | |  ||S r2  )r  r2  r   r  r  r  r  r     rh  r   c	           	      C   s   t | | |  ||tt| t|  |t|  |  |   | | | || |  |  t| ||  | ||tt| t|  |t|  |  |  ||S r2  )r  r  r2  r  r   r  r  r  r  r!    s   r!  c                 C   s,   t |tt|t|  |t|  | |S r2  )r   r#  r  r  )r,  rH  r  r?  rO  r-  r  r  r  r"    s   ,r"  c                 C   sP   t |ttttfrdS || t||t|   }t| | t|| r&dS dS r~  )rv  rw  r  rx  r  r  r  r   r,  r  r?  rO  r-  r  rB  r  r  r  r#    s   r#  c                 C   s,   || t ||t|   }t| | | |S r2  )r  r  r  r  r  r  r  r$    s   r$  c                 C   s   t |ttttfrdS |t|| t|  |td t||  }tt	t
||t
||t|t||t
|| |t||t
||  rGdS dS ru  )rv  rw  r  rx  r  r  r  r  r  r#   r   r   r3   r,  r  r  r?  rO  r-  rB  r  r  r  r%    s   ,Hr%  c                 C   sn   |t || t|  |td t ||  }t| | ||td   t||t|| t||t|| |S r/  )r  r  r  r  r3   r   r  r  r  r  r&    s   ,Br&  (  __doc__Zsympy.externalr   r   r   r   r   r   Z%sympy.integrals.rubi.utility_functionr   r	   r
   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r    r!   r"   r#   r$   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  r  r  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
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