Ë
    7^(h‡; ã                  ó  — U d Z ddlmZ ddlZddlmZ ddlmZmZ ddl	m
Z
 ddlmZ ddlmZ dd	lmZ dd
lmZ ddlmZmZmZmZmZ ddlmZ ddlmZ ddlmZmZ ddl m!Z!m"Z"m#Z# ddl$m%Z%m&Z& ddl'm(Z(m)Z)m*Z*m+Z+ ddl,m-Z- ddl.m/Z/ ddl0m1Z1m2Z2m3Z3m4Z4m5Z5m6Z6m7Z7m8Z8m9Z9m:Z:m;Z; ddl<m=Z=m>Z>m?Z? ddl@mAZA ddlBmCZCmDZDmEZEmFZF ddlGmHZH ddlImJZJmKZK ddlLmMZMmNZNmOZOmPZP ddlQmRZRmSZSmTZTmUZU ddlVmWZWmXZX ddlYmZZZm[Z[ ddl\m]Z]m^Z^m_Z_m`Z`maZambZbmcZcmdZdmeZemfZfmgZg ddlhmiZi dd ljmkZkmlZl dd!lmmnZn d"d#lompZp dd$lqmrZrmsZsmtZtmuZumvZv dd%lwmxZxmyZy dd&lzm{Z{ dd'l|m}Z~ dd(l|mZ€  e(d)«      Z�d*„ Z‚d+„ Zƒdd,l„m…Z…  e…d-«      Z†dXd.„Z‡ G d/„ d0eˆ«      Z‰d1„ ZŠd2„ Z‹d3„ ZŒd4„ Z�d5„ ZŽd6„ Z�d7„ Z�d8„ Z‘d9„ Z’d:„ Z“i Z”d;e•d<<   d=„ Z–d>„ Z—d?„ Z˜dYd@„Z™dA„ ZšdZdB„Z›dC„ ZœdZdD„Z�dE„ ZždZdF„ZŸdG„ Z dH„ Z¡dI„ Z¢dJ„ Z£dK„ Z¤da¥ee†dYdL„«       «       Z¦dYdM„Z§dN„ Z¨dO„ Z©dP„ Zªe†dQ„ «       Z«dR„ Z¬dS„ Z­dT„ Z®dU„ Z¯e†dZdV„«       Z°dW„ Z±y)[a¿  
Integrate functions by rewriting them as Meijer G-functions.

There are three user-visible functions that can be used by other parts of the
sympy library to solve various integration problems:

- meijerint_indefinite
- meijerint_definite
- meijerint_inversion

They can be used to compute, respectively, indefinite integrals, definite
integrals over intervals of the real line, and inverse laplace-type integrals
(from c-I*oo to c+I*oo). See the respective docstrings for details.

The main references for this are:

[L] Luke, Y. L. (1969), The Special Functions and Their Approximations,
    Volume 1

[R] Kelly B. Roach.  Meijer G Function Representations.
    In: Proceedings of the 1997 International Symposium on Symbolic and
    Algebraic Computation, pages 205-211, New York, 1997. ACM.

[P] A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev (1990).
    Integrals and Series: More Special Functions, Vol. 3,.
    Gordon and Breach Science Publisher
é    )ÚannotationsN)ÚSYMPY_DEBUG)ÚSÚExpr)ÚAdd)ÚBasic)Úcacheit)ÚTuple)Úfactor_terms)ÚexpandÚ
expand_mulÚexpand_power_baseÚexpand_trigÚFunction)ÚMul)Úilcm)ÚRationalÚpi)ÚEqÚNeÚ_canonical_coeff)Údefault_sort_keyÚordered)ÚDummyÚsymbolsÚWildÚSymbol)Úsympify)Ú	factorial)ÚreÚimÚargÚAbsÚsignÚ
unpolarifyÚpolarifyÚ
polar_liftÚprincipal_branchÚunbranched_argumentÚperiodic_argument)ÚexpÚ	exp_polarÚlog)Úceiling)ÚcoshÚsinhÚ_rewrite_hyperbolics_as_expÚHyperbolicFunction©Úsqrt)Ú	PiecewiseÚpiecewise_fold)ÚcosÚsinÚsincÚTrigonometricFunction)ÚbesseljÚbesselyÚbesseliÚbesselk)Ú
DiracDeltaÚ	Heaviside)Ú
elliptic_kÚ
elliptic_e)ÚerfÚerfcÚerfiÚEiÚexpintÚSiÚCiÚShiÚChiÚfresnelsÚfresnelc)Úgamma)ÚhyperÚmeijerg)ÚSingularityFunctioné   )ÚIntegral)ÚAndÚOrÚBooleanAtomÚNotÚBooleanFunction)ÚcancelÚfactor)Úmultiset_partitions)Údebug)ÚdebugfÚzc                óŽ   ‡— t        | «      } t        | dd«      rt        ˆfd„| j                  D «       «      S  | j                  ‰Ž S )NÚis_PiecewiseFc              3  ó6   •K  — | ]  }t        |g‰¢­Ž –— Œ y ­w©N)Ú_has)Ú.0ÚiÚfs     €úW/var/www/skyplay_api_hub/venv/lib/python3.12/site-packages/sympy/integrals/meijerint.pyú	<genexpr>z_has.<locals>.<genexpr>T   s   øè ø€ Ò1 1”4˜�;˜A–;Ñ1ùs   ƒ)r6   ÚgetattrÚallÚargsÚhas)Úresrf   s    `rg   rc   rc   O   s@   ø€ ô ˜Ó
€CÜˆs�N EÔ*ÜÓ1¨¯©Ô1Ó1Ð1Øˆ3�7‰7�Aˆ;Ðó    c                ó¾  ‡ ‡‡	‡
‡‡‡‡‡‡‡— d„ }t        t        |d«      «      \  ŠŠŠ	Š}t        dd„ g¬«      Š‰t        ‰z  z  Š‰t        j
                  ddfˆ fd„	Š
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<    G d„ dt        «      } ‰
t        ‰‰z
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  z  z  ‰	gg g dg‰‰z  t        ‰	«      ‰‰	dz
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t        ‰‰z
  «      ‰‰z
  ‰	dz
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  z  z  t        ‰dkD  «      «        ‰
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  «      ‰‰z
  ‰	dz
  z  z  ‰	gg g dg‰‰z  t        ‰	«      ‰‰	dz
  z  z  t        ‰dkD  «      «        ‰
t        ‰‰z  d‰z  z  t        z
  «      ‰‰z
  ‰	dz
  z  z  g ‰	gdgg ‰‰z  t        ‰	«      ‰‰	dz
  z  z  t        ‰dkD  «      «        ‰
‰‰z   ‰	 z  d‰	z
  gg dgg ‰‰z  ‰‰	 z  t        ‰	«      z  t         |‰	«      «      ¬«        ‰
t        ‰‰z
  «      ‰	 z  d‰	z
  gd‰	z
  dz  gdgd‰	z
  dz  g‰‰z  dt        t        ‰	z  dz  «      z  t        d‰	z
  «      z  t        ‰«      ‰	 z  z  t        ‰	«      dk  «        ‰
‰‰	z  ‰‰	z  z
  ‰‰z
  z  d‰	gg d‰	gg ‰‰z  ‰‰	dz
  z  t        ‰	t        z  «      z  t        z  «       d„ Šˆˆ	ˆ
ˆˆfd„} |dd«        |dd«        |t        j                  d«        |t        j                  d«       ˆˆ	ˆ
ˆˆˆfd„} |dd«        |dd«        |t        j                  d«        |t        j                  d«        ‰
t!        t#        d«      ‰z  «      g g dgg «        ‰
t%        ‰«      g dgt        j                  gddg‰dz  dz  t        t'        dd«      z  «        ‰
t)        ‰«      g t        j                  gdgt        j                  t        j                  g‰dz  dz  t        t'        dd«      z  «        ‰
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  «      z  ‰d«        |t1        ‰«      ‰z  t        ‰dz
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                  t3        ddgg dgdg‰‰	z  «      fgz   d«        |t1        t        ‰‰	z
  «      «       |t1        t        ‰	«      «      «      t        t3        ddgt        j                  gdgdt        j                  g‰‰	z  «      fgz   d«        |t5        ‰«       |t        j6                   t        z  «      t        j8                  t3        g dgddgg ‰t#        d«      z  «      fgz   d«        ‰
t;        ‰«      dgg t        j                  gddg‰dz  dz  t+        t        «      dz  «        ‰
t=        ‰«      g dgddgt        j                  g‰dz  dz  t+        t        «       dz  «        ‰
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tA        ‰«      g t        j                  dgddgt        j                  t        j                  g‰dz  dz  t        t	        d«      z   dz  «        ‰
tC        ‰	‰«      g ‰	g‰	dz
  dgg ‰«        ‰
tE        ‰«      dgg t        j                  gdg‰dz  dt+        t        «      z  «        ‰
tG        ‰«      g dgdt        j                  gg ‰dz  dt+        t        «      z  «        ‰
tI        ‰«      t        j                  gg dgt'        dd«      g‰dz   ‰t+        t        «      z  «        ‰
tK        ‰«      dgg t'        dd«      gdt'        dd«      gt        dz  ‰dz  z  dz  t        j                  «        ‰
tM        ‰«      dgg t'        dd«      gdt'        dd«      gt        dz  ‰dz  z  dz  t        j                  «        ‰
tO        ‰	‰«      g g ‰	dz  g‰	 dz  g‰dz  dz  «        ‰
tQ        ‰	‰«      g ‰	dz    dz  g‰	dz  ‰	 dz  g‰	dz    dz  g‰dz  dz  «        ‰
tS        ‰	‰«      g d‰	z   dz  g‰	dz  g‰	 dz  d‰	z   dz  g‰dz  dz  t        «        ‰
tU        ‰	‰«      g g ‰	dz  ‰	 dz  gg ‰dz  dz  t        j                  «        ‰
tW        ‰«      t        j                  t        j                  gg dgdg‰ t        j                  «        ‰
tY        ‰«      t        j                  dt        j                  z  gg dgdg‰ t'        dd«      dz  «       y)z8 Add formulae for the function -> meijerg lookup table. c                ó&   — t        | t        g¬«      S )N©Úexclude)r   r^   )Úns    rg   Úwildz"_create_lookup_table.<locals>.wildZ   s   € Ü�A¤˜sÔ#Ð#rn   Úpqabcrs   c                ó(   — | j                   xr | dkD  S ©Nr   )Ú
is_Integer©Úxs    rg   ú<lambda>z&_create_lookup_table.<locals>.<lambda>]   s   € ¨¯©Ò(>¸¸Q¹€ rn   )Ú
propertiesTc	                óŒ   •— ‰	j                  t        | t        «      g «      j                  | |t	        |||||«      fg||f«       y rb   )Ú
setdefaultÚ_mytyper^   ÚappendrP   )
ÚformulaÚanÚapÚbmÚbqr"   ÚfacÚcondÚhintÚtables
            €rg   Úaddz!_create_lookup_table.<locals>.add`   sK   ø€ Ø×Ñœ ¬!Ó,¨bÓ1×8Ñ8¸'Ø'*¬G°B¸¸BÀÀCÓ,HÐ&IÐ%JÈDÐRVð:Xõ 	Yrn   c                ól   •— ‰j                  t        | t        «      g «      j                  | |||f«       y rb   )r~   r   r^   r€   )r�   Úinstr‡   rˆ   r‰   s       €rg   Úaddiz"_create_lookup_table.<locals>.addid   s0   ø€ Ø×ÑÜ�GœQÓ ó	%ß%+¡V¨W°d¸DÀ$Ð,GÕ%Hrn   c           	     ó^   — | t        dgg g dgt        «      f| t        g dgdgg t        «      fgS ©NrR   r   )rP   r^   )Úas    rg   Úconstantz&_create_lookup_table.<locals>.constanth   s>   € Ø”G˜Q˜C  R¨!¨¬aÓ0Ð1Ø”G˜B   a S¨"¬aÓ0Ð1ð3ð 	3rn   © c                  ó   — e Zd Zed„ «       Zy)ú2_create_lookup_table.<locals>.IsNonPositiveIntegerc                ó@   — t        |«      }|j                  du r|dk  S y )NTr   )r%   rx   )Úclsr"   s     rg   Úevalz7_create_lookup_table.<locals>.IsNonPositiveInteger.evalp   s%   € ä˜S“/ˆCØ�~‰~ Ñ%Ø˜a‘x�ð &rn   N)Ú__name__Ú
__module__Ú__qualname__Úclassmethodr—   r’   rn   rg   ÚIsNonPositiveIntegerr”   n   s   „ à	ñ	 ó 
ñ	 rn   rœ   rR   r   )rˆ   é   c                óN   — t         t        dd«      z  | |z  dz  dd| z  z
  z  z  S )Néÿÿÿÿr�   rR   )r   r   )Úrr$   Únus      rg   ÚA1z _create_lookup_table.<locals>.A1ˆ   s/   € Ü”8˜B “?Ñ" T E¨"¡H¨Q¡J°!°a¸±c±'Ñ#:Ñ:Ð:rn   c                óî   •—  ‰t        ‰dz  ‰z   «      |‰z  z   ‰z  ‰dz  ‰z   | z  z  d‰z   dz  dd| z  z
  ‰dz  z   gg ‰|‰z  z
  dz  g‰|‰z  z   dz  g‰‰dz  z  ‰‰d| z  z
  z   ‰| |‰«      z  «       y )Nr�   rR   r3   )r    Úsgnr¢   r�   rŠ   ÚbÚts     €€€€€rg   Útmpaddz$_create_lookup_table.<locals>.tmpadd‹   s¡   ø€ áŒT�!�Q‘$˜‘(‹^˜c !™eÑ# aÑ'¨¨A©°©°A©Ñ5Ø�!‰e�Q‰Y˜˜A˜a™C™ ! A¡#™Ð&¨Ø�#�a‘%‰i˜‰]ˆO˜q 3 q¡5™y¨!™m˜_¨a°°1±©fØ��A�a‘C‘‰L™˜A˜s A›Ñ&õ	(rn   rŸ   c                óH  •—  ‰t        ‰‰t        ‰z  z  z   «      |t        ‰«      z  t        ‰dz  z  z  z   ‰z  ‰‰t        ‰z  z  z   | z  z  d| z
  |‰z  dz  z   gd| z
  |‰z  dz  z
  gdt        j                  gg ‰t        ‰z  z  ‰z  ‰‰dz  | z
  z   ‰| |‰«      z  «       y )Nr�   rR   r   )r4   r^   r   ÚHalf)r    r¤   r¢   r�   rŠ   r¥   ÚpÚqs     €€€€€€rg   r§   z$_create_lookup_table.<locals>.tmpadd—   s³   ø€ ÙŒT�!�aœ˜1™‘f‘*Ó ¤D¨£G¡¬A°°!±©HÑ 4Ñ4°qÑ8¸!¸aÄÀ1Á¹f¹*Àq¹ÑHØ�‰U�S˜‘U˜1‘W‰_Ð  A¡¨¨A©¨a©¡Ð0°1´a·f±f°+¸rØŒa�‰d‰F�1‰H�a˜!˜A™# ™'‘l¡2 a¨¨a£=Ñ0õ	2rn   é   é   c                óˆ   •— | ‰   }t         j                  |z  t        |«      z  t        g dg|dz   z  dg|dz   z  g ‰«      fgS r�   )r   ÚNegativeOner   rP   ©ÚsubsÚNrs   r¦   s     €€rg   Ú	make_log1z'_create_lookup_table.<locals>.make_log1²   sV   ø€ Ø�‰GˆÜ—‘ Ñ!¤)¨A£,Ñ.Ü˜˜a˜S ! a¡%™[¨1¨#¨q°1©u©+°r¸1Ó=ð?ð @ð 	@rn   c           	     ó`   •— | ‰   }t        |«      t        dg|dz   z  g g dg|dz   z  ‰«      fgS r�   )r   rP   r°   s     €€rg   Ú	make_log2z'_create_lookup_table.<locals>.make_log2·   sH   ø€ Ø�‰GˆÜ˜1“Ü˜!˜˜a !™e™ b¨"¨q¨c°1°q±5©k¸1Ó=ð?ð @ð 	@rn   c                ó&   •—  ‰| «       ‰| «      z   S rb   r’   )r±   r³   rµ   s    €€rg   Ú	make_log3z'_create_lookup_table.<locals>.make_log3Á   s   ø€ Ù˜‹¡¨4£Ñ0Ð0rn   z3/2é   N©T)-ÚlistÚmapr   r^   r   ÚOner   r@   rN   rT   rW   r#   r8   r   r    r©   r+   r'   r0   r   r/   r4   r7   r9   r-   rP   rF   ÚImaginaryUnitr¯   rH   rI   rJ   rK   rG   rC   rD   rE   rL   rM   r;   r<   r=   r>   rA   rB   )r‰   rt   Úcr�   r‘   rœ   r§   r·   r¢   r�   rŠ   r¥   r³   rµ   rs   rª   r«   r¦   s   `       @@@@@@@@@@rg   Ú_create_lookup_tabler¿   X   så	  ÿú€ ò$äœ˜T 7Ó+Ó,�M€A€qˆ!ˆQ�ÜˆSÑ>Ð?Ô@€AØ	Œ!ˆQ‰$‰€Aà)*´·±¸DÀtõ YõIò3ð ‘X˜a“[ $¨Ð-Ð.€Eˆ"�Iô œxô  ñ Œ	�!�a‘%Ó˜!˜a™% 1 q¡5Ñ)Ñ)¨A¨3°°B¸¸¸Q¸q¹SÜˆa‹��Q˜‘U‘ÑœS  Q¡›Zô)áŒ	�!�a‘%Ó˜!˜a™% 1 q¡5Ñ)Ñ)¨2°¨s°Q°C¸¸Q¸q¹SÜˆa‹��Q˜‘U‘ÑœS  Q¡›Zô)áŒ	”!�q˜‘s˜a ™c‘lÑ"Ó# Q¨¡U¨a°!©eÑ$4Ñ4°q°c¸2¸rÀAÀ3ÈÈ!ÉÜˆa‹��Q˜‘U‘ÑœS  Q¡›Zô)áŒ	�1�Q‘3˜!˜A™#‘,¤Ñ"Ó# Q¨¡U¨a°!©eÑ$4Ñ4°b¸1¸#À¸sÀBÈÈ!ÉÜˆa‹��Q˜‘U‘ÑœS  Q¡›Zô)áˆˆQ‰�1�"‰˜˜A™�w  Q C¨¨Q¨q©S°!°q°b±'¼%À»(Ñ2BÜÑ% aÓ(Ó)õ+áŒˆA�‰E‹
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Œ1�6‰6�2Ô÷2ñ 2ñ ˆ1ˆa„LÙ
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�1‹œŸ™¤§¡Ð'¨¨a¨S°1°#¸°r¼1¿6¹6ÔBÙŒ
�1‹œŸ™ ¤!§&¡&¡Ð)¨2°¨s°Q°C¸!¸¼XÀbÈ!»_ÈQÑ=NÕOrn   )ÚtimethisrP   c                ó¬   ‡— dd„}‰| j                   vry| j                  rt        | «      fS t        t	        ˆfd„| j
                  D «       |¬«      «      S )z4 Create a hashable entity describing the type of f. c                ó"   — | j                  «       S rb   )Ú	class_keyry   s    rg   Úkeyz_mytype.<locals>.key/  s   € Ø�{‰{‹}Ðrn   r’   c              3  óD   •K  — | ]  }t        |‰«      D ]  }|–— Œ Œ y ­wrb   )r   )rd   r�   r¦   rz   s      €rg   rh   z_mytype.<locals>.<genexpr>6  s#   øè ø€ ÒB˜q´G¸A¸q³MÒB¨qœÐB˜ÑBùs   ƒ ©rÄ   )rz   ztype[Basic]Úreturnztuple[int, int, str])Úfree_symbolsÚis_FunctionÚtypeÚtupleÚsortedrk   )rf   rz   rÄ   s    ` rg   r   r   -  sG   ø€ óð 	�—‘ÑØØ	
�ŠÜ�A‹wˆxˆÜ”ÓB A§F¡FÔBÈÔLÓMÐMrn   c                  ó   — e Zd ZdZy)Ú_CoeffExpValueErrorzD
    Exception raised by _get_coeff_exp, for internal use only.
    N)r˜   r™   rš   Ú__doc__r’   rn   rg   rÎ   rÎ   9  s   „ ñð 	rn   rÎ   c                ó6  — ddl m} t         || «      «      j                  |«      \  }}|s|t        j
                  fS |\  }|j                  r(|j                  |k7  rt        d«      ‚||j                  fS ||k(  r|t        j                  fS t        d| z  «      ‚)aŒ  
    When expr is known to be of the form c*x**b, with c and/or b possibly 1,
    return c, b.

    Examples
    ========

    >>> from sympy.abc import x, a, b
    >>> from sympy.integrals.meijerint import _get_coeff_exp
    >>> _get_coeff_exp(a*x**b, x)
    (a, b)
    >>> _get_coeff_exp(x, x)
    (1, 1)
    >>> _get_coeff_exp(2*x, x)
    (2, 1)
    >>> _get_coeff_exp(x**3, x)
    (1, 3)
    r   )Úpowsimpzexpr not of form a*x**bzexpr not of form a*x**b: %s)Úsympy.simplifyrÑ   r   Úas_coeff_mulr   ÚZeroÚis_PowÚbaserÎ   r+   r¼   )Úexprrz   rÑ   r¾   Úms        rg   Ú_get_coeff_exprÙ   @  sŒ   € õ& 'Ü™w t›}Ó-×:Ñ:¸1Ó=�F€QˆÙØ”!—&‘&ˆyÐØ
�C€QØ‡x‚xØ�6‰6�QŠ;Ü%Ð&?Ó@Ð@Ø�!—%‘%ˆxˆØ	
ˆaŠØ”!—%‘%ˆxˆä!Ð"?À$Ñ"FÓGÐGrn   c                ó:   ‡— ˆfd„Št        «       } ‰| ||«       |S )a�  
    Find the exponents of ``x`` (not including zero) in ``expr``.

    Examples
    ========

    >>> from sympy.integrals.meijerint import _exponents
    >>> from sympy.abc import x, y
    >>> from sympy import sin
    >>> _exponents(x, x)
    {1}
    >>> _exponents(x**2, x)
    {2}
    >>> _exponents(x**2 + x, x)
    {1, 2}
    >>> _exponents(x**3*sin(x + x**y) + 1/x, x)
    {-1, 1, 3, y}
    c                óÜ   •— | |k(  r|j                  dg«       y | j                  r,| j                  |k(  r|j                  | j                  g«       y | j                  D ]  } ‰|||«       Œ y ©NrR   )ÚupdaterÕ   rÖ   r+   rk   )r×   rz   rm   ÚargumentÚ_exponents_s       €rg   rß   z_exponents.<locals>._exponents_u  s_   ø€ Ø�1Š9Ø�J‰J˜�sŒOØØ�;Š;˜4Ÿ9™9¨š>Ø�J‰J˜Ÿ™�zÔ"ØØŸ	™	ò 	*ˆHÙ˜ ! SÕ)ñ	*rn   )Úset)r×   rz   rm   rß   s      @rg   Ú
_exponentsrá   b  s"   ø€ ô&*ô ‹%€CÙ��a˜ÔØ€Jrn   c                ó„   — | j                  t        «      D �ch c]  }||j                  v sŒ|j                  ’Œ c}S c c}w )zB Find the types of functions in expr, to estimate the complexity. )Úatomsr   rÈ   Úfunc)r×   rz   Úes      rg   Ú
_functionsræ   ƒ  s.   € à ŸJ™J¤xÓ0ÖH�q°A¸¿¹Ò4GˆA�F‹FÒHÐHùÒHs   ˜=¬=c                óŒ   ‡‡‡‡— dD �cg c]  }t        |‰g¬«      ‘Œ c}\  ŠŠˆˆˆˆfd„Št        «       } ‰| |«       |S c c}w )ap  
    Find numbers a such that a linear substitution x -> x + a would
    (hopefully) simplify expr.

    Examples
    ========

    >>> from sympy.integrals.meijerint import _find_splitting_points as fsp
    >>> from sympy import sin
    >>> from sympy.abc import x
    >>> fsp(x, x)
    {0}
    >>> fsp((x-1)**3, x)
    {1}
    >>> fsp(sin(x+3)*x, x)
    {-3, 0}
    Úpqrq   c                óð   •— t        | t        «      sy | j                  ‰‰z  ‰z   «      }|r$|‰   dk7  r|j                  |‰    |‰   z  «       y | j                  ry | j
                  D ]  } ‰||«       Œ y rw   )Ú
isinstancer   ÚmatchrŠ   Úis_Atomrk   )r×   rm   rØ   rÞ   Úcompute_innermostrª   r«   rz   s       €€€€rg   rí   z1_find_splitting_points.<locals>.compute_innermostœ  sw   ø€ Ü˜$¤Ô%ØØ�J‰J�q˜‘s˜Q‘wÓˆÙ��1‘˜’Ø�G‰G�Q�q‘T�E˜!˜A™$‘JÔØØ�<Š<ØØŸ	™	ò 	-ˆHÙ˜h¨Õ,ñ	-rn   )r   rà   )r×   rz   rs   Ú	innermostrí   rª   r«   s    `  @@@rg   Ú_find_splitting_pointsrï   ˆ  sE   û€ ð$ +/Ö/ QŒD�˜Q˜CÖ Ò/�D€A€q÷
-ô “€IÙ�d˜IÔ&ØÐùò 0s   ‰Ac                ó^  — t         j                  }t         j                  }t         j                  }t        | «      } t        j                  | «      }|D ]Ô  }||k(  r||z  }Œ||j
                  vr||z  }Œ"|j                  r¢||j                  j
                  vrŠ|j                  j                  |«      \  }}||fk7  r't        |j                  «      j                  |«      \  }}||fk(  r9|||j                  z  z  }|t        t        ||j                  z  d¬«      «      z  }ŒÐ||z  }ŒÖ |||fS )aq  
    Split expression ``f`` into fac, po, g, where fac is a constant factor,
    po = x**s for some s independent of s, and g is "the rest".

    Examples
    ========

    >>> from sympy.integrals.meijerint import _split_mul
    >>> from sympy import sin
    >>> from sympy.abc import s, x
    >>> _split_mul((3*x)**s*sin(x**2)*x, x)
    (3**s, x*x**s, sin(x**2))
    F)r±   )r   r¼   r   r   Ú	make_argsrÈ   rÕ   r+   rÖ   rÓ   r   r%   r&   )	rf   rz   r†   ÚpoÚgrk   r�   r¾   r¦   s	            rg   Ú
_split_mulrô   ¬  s  € ô �%‰%€CÜ	
�‰€BÜ	�‰€AÜ˜!Ó€Aä�=‰=˜Ó€DØò ˆØ�Š6Ø�!‰G‰BØ�a—n‘nÑ$Ø�1‰H‰Cà�xŠx˜A Q§U¡U×%7Ñ%7Ñ7Ø—v‘v×*Ñ*¨1Ó-‘��1Ø˜˜’9Ü% a§f¡fÓ-×:Ñ:¸1Ó=‘D�A�qØ˜˜’9Ø˜!˜QŸU™U™(‘N�BØœ:¤h¨q°!·%±%©x¸eÔ&DÓEÑE�CØØ�‰F‰Aðð  ��Aˆ:Ðrn   c                ó  — t        j                  | «      }g }|D ]d  }|j                  rE|j                  j                  r/|j                  }|j
                  }|dk  r| }d|z  }||g|z  z  }ŒT|j                  |«       Œf |S )a   
    Return a list ``L`` such that ``Mul(*L) == f``.

    If ``f`` is not a ``Mul`` or ``Pow``, ``L=[f]``.
    If ``f=g**n`` for an integer ``n``, ``L=[g]*n``.
    If ``f`` is a ``Mul``, ``L`` comes from applying ``_mul_args`` to all factors of ``f``.
    r   rR   )r   rñ   rÕ   r+   rx   rÖ   r€   )rf   rk   Úgsró   rs   rÖ   s         rg   Ú	_mul_argsr÷   Ó  sƒ   € ô �=‰=˜Ó€DØ	€BØò 	ˆØ�8Š8˜Ÿ™×(Ò(Ø—‘ˆAØ—6‘6ˆDØ�1ŠuØ�B�Ø˜‘v�Ø�4�&˜‘(‰N‰Bà�I‰I�a�Lð	ð €Irn   c                óÐ   — t        | «      }t        |«      dk  ryt        |«      dk(  rt        |«      gS t        |d«      D ��cg c]  \  }}t	        |Ž t	        |Ž f‘Œ c}}S c c}}w )aŸ  
    Find all the ways to split ``f`` into a product of two terms.
    Return None on failure.

    Explanation
    ===========

    Although the order is canonical from multiset_partitions, this is
    not necessarily the best order to process the terms. For example,
    if the case of len(gs) == 2 is removed and multiset is allowed to
    sort the terms, some tests fail.

    Examples
    ========

    >>> from sympy.integrals.meijerint import _mul_as_two_parts
    >>> from sympy import sin, exp, ordered
    >>> from sympy.abc import x
    >>> list(ordered(_mul_as_two_parts(x*sin(x)*exp(x))))
    [(x, exp(x)*sin(x)), (x*exp(x), sin(x)), (x*sin(x), exp(x))]
    r�   N)r÷   ÚlenrË   r[   r   )rf   rö   rz   Úys       rg   Ú_mul_as_two_partsrû   ê  s^   € ô. 
�1‹€BÜ
ˆ2ƒw�‚{ØÜ
ˆ2ƒw�!‚|Ü�b“	ˆ{ÐÜ-@ÀÀQÓ-G×H¡6 A qŒS�!ˆW”c˜1�gÒÓHÐHùÓHs   ÁA"c                ó´  — d„ }t        t        | j                  «      t        | j                  «      z
  «      }|d| j                  z   |dz  z   z  }|dt
        z  |dz
  | j                  z  z  z  }|t         || j                  |«       || j                  |«       || j                  |«       || j                  |«      | j                  |z  |||z  z  z  «      fS )zO Return C, h such that h is a G function of argument z**n and
        g = C*h. c                ó~   — t        j                  | t        |«      «      D ��cg c]  \  }}||z   |z  ‘Œ c}}S c c}}w )z5 (a1, .., ak) -> (a1/n, (a1+1)/n, ..., (ak + n-1)/n) )Ú	itertoolsÚproductÚrange)Úparamsrs   r�   re   s       rg   Úinflatez_inflate_g.<locals>.inflate  s3   € ä&/×&7Ñ&7¸ÄÀaÃÓ&I×J™d˜a ��Q‘˜“	ÓJÐJùÓJs   £9rR   r�   )r   rù   rƒ   r…   r¡   r   ÚdeltarP   r‚   Úaotherr„   ÚbotherrÞ   )ró   rs   r  ÚvÚCs        rg   Ú
_inflate_gr  	  sÃ   € ò
Kô 	
Œ#ˆa�d‰d‹)”c˜!Ÿ$™$“iÑ
Ó €AØ	ˆA�—‘‰H�q˜‘s‰NÑ€AØˆ!ŒB‰$�1�q‘5˜!Ÿ'™'‘/Ñ	"Ñ"€AØŒg‘g˜aŸd™d AÓ&©°·±¸!Ó(<Ù˜aŸd™d AÓ&©°·±¸!Ó(<Ø—j‘j !‘m a¨!¨A©#¡hÑ.ó0ð 0ð 0rn   c                óÀ   — d„ }t         || j                  «       || j                  «       || j                  «       || j                  «      d| j
                  z  «      S )zQ Turn the G function into one of inverse argument
        (i.e. G(1/x) -> G'(x)) c                ó2   — | D �cg c]  }d|z
  ‘Œ	 c}S c c}w rÜ   r’   )Úlr�   s     rg   Útrz_flip_g.<locals>.tr  s   € Ø Ö!˜!��A“Ò!Ð!ùÒ!ó   …rR   )rP   r„   r  r‚   r  rÞ   )ró   r  s     rg   Ú_flip_gr    sB   € ò"ä‘2�a—d‘d“8™R §¡›\©2¨a¯d©d«8±R¸¿¹³\À1ÀQÇZÁZÁ<ÓPÐPrn   c           	     óì  — |dk  rt        t        | «      | «      S t        |j                  «      }t        |j                  «      }t        | |«      \  }} | j                  }|dt        z  d|z
  dz  z  |t        dd«      z  z  z  }|||z  z  }t        |«      D �cg c]
  }|dz   |z  ‘Œ }}|t        | j                  | j                  | j                  t        | j                  «      |z   |«      fS c c}w )a\  
    Let d denote the integrand in the definition of the G function ``g``.
    Consider the function H which is defined in the same way, but with
    integrand d/Gamma(a*s) (contour conventions as usual).

    If ``a`` is rational, the function H can be written as C*G, for a constant C
    and a G-function G.

    This function returns C, G.
    r   r�   rR   rŸ   )Ú_inflate_fox_hr  r   rª   r«   r  rÞ   r   r   r   rP   r‚   r  r„   rº   r  )ró   r�   rª   r«   ÚDr^   rs   Úbss           rg   r  r  "  sà   € ð 	ˆ1‚uÜœg a›j¨1¨"Ó-Ð-Ü	ˆ!�#‰#‹€AÜ	ˆ!�#‰#‹€Aô �a˜Ó�D€A€qØ	�
‰
€AØˆ!ŒB‰$�1�q‘5˜!‘)Ñ	˜Q¤¨¨Q£Ñ/Ñ	/Ñ/€AØˆˆA‰�I€AÜ" 1›XÖ	&˜ˆ1ˆq‰5�!‹)Ð	&€BÐ	&ØŒg�a—d‘d˜AŸH™H a§d¡d¬D°·±«N¸RÑ,?ÀÓCÐCÐCùò 
's   ÂC1zdict[tuple[str, str], Dummy]Ú_dummiesc                óT   — t        | |fi |¤Ž}||j                  v rt        | fi |¤ŽS |S )z¶
    Return a dummy. This will return the same dummy if the same token+name is
    requested more than once, and it is not already in expr.
    This is for being cache-friendly.
    )Ú_dummy_rÈ   r   )ÚnameÚtokenr×   ÚkwargsÚds        rg   Ú_dummyr  ?  s8   € ô 	��eÑ&˜vÑ&€AØˆD×ÑÑÜ�TÑ$˜VÑ$Ð$Ø€Hrn   c                óV   — | |ft         vrt        | fi |¤Žt         | |f<   t         | |f   S )z`
    Return a dummy associated to name and token. Same effect as declaring
    it globally.
    )r  r   )r  r  r  s      rg   r  r  K  s7   € ð
 �%ˆ=œHÑ$Ü"'¨Ñ"7°Ñ"7Œ�$˜�ÑÜ�T˜5�MÑ"Ð"rn   c                ó^   ‡— t        ˆfd„| j                  t        t        «      D «       «       S )zŠ Check if f(x), when expressed using G functions on the positive reals,
        will in fact agree with the G functions almost everywhere c              3  ó:   •K  — | ]  }‰|j                   v –— Œ y ­wrb   )rÈ   ©rd   r×   rz   s     €rg   rh   z_is_analytic.<locals>.<genexpr>X  s   øè ø€ ÒN¨d�1˜×)Ñ)Ô)ÑNùs   ƒ)Úanyrã   r@   r#   )rf   rz   s    `rg   Ú_is_analyticr   U  s$   ø€ ô ÓN°a·g±g¼iÌÓ6MÔNÓNÐNÐNrn   c                óš  ‡‡— |r| j                  d„ t        «      } d}t        | t        «      s| S t	        dt
        ¬«      \  ŠŠ}t        ‰‰k  t        ‰‰«      «      ‰‰k  ft        t        t        ‰«      «      t        k  t        t        ‰«      dt        z  z
  «      t        k  «      t        t        ‰«      t        z
  d«      ft        t        dt        ‰«      z  t        z   «      t        k  t        dt        ‰«      z  t        z
  «      t        k  «      t        t        ‰«      d«      ft        t        dt        ‰«      z  t        z   «      t        k  t        dt        ‰«      z  t        z
  «      t        k  «      t        j                  ft        t        t        ‰«      t        dz  z
  «      t        dz  k  t        t        ‰«      t        dz  z   «      t        dz  k  «      t        t        ‰«      d«      ft        t        t        ‰«      t        dz  z
  «      t        dz  k  t        t        ‰«      t        dz  z   «      t        dz  k  «      t        j                  ft        t        t        ‰dz  dz  dz   «      «      t        k  t        t        t        ‰dz  dz  dz   «      «      t        «      «      t        j                  ft        t        t        ‰dz  dz  dz   «      «      t        k  t        d‰dz  dz  dz   z  d«      «      t        j                  ft        t        t!        ‰«      «      t        k  t        t!        t#        dt        z  t        j$                  z  «      ‰z  «      «      t        k  «      t        t!        t#        t        j$                   t        z  «      ‰z  «      d«      ft        t        t!        ‰«      «      t        dz  k  t        t!        t#        t         t        j$                  z  «      ‰z  «      «      t        dz  k  «      t        t!        t#        t        j$                   t        z  dz  «      ‰z  «      d«      ft        ‰‰k  t        ‰‰k  |«      «      ‰‰k  ft        ‰dz  d«      ‰dz  dkD  z  ‰dz  dkD  ft        d‰z  d«      t'        t        t        ‰«      «      «      t        ‰«      z  dkD  z  t        ‰«      dkD  ft        ‰d«      t'        t        t        ‰«      «      «      t        ‰«      z  dkD  z  t        ‰«      dkD  ft        t        ‰«      «      t        dz  k  t'        t        t        ‰«      «      «      t)        t        ‰dz  «      «      z  dkD  z  ‰dz  dkD  fg} | j*                  | j,                  D �cg c]  }t/        ||«      ‘Œ c}Ž } d	}|�r]d}t1        |«      D �]H  \  }\  }}|j*                  | j*                  k7  rŒ$t1        | j,                  «      D �]  \  }	}
||j,                  d   j2                  v r!|
j5                  |j,                  d   «      }d}n d}|
j5                  |j,                  d   «      }|sŒf|j,                  d
| |j,                  |dz   d
 z   D �cg c]  }|j7                  |«      ‘Œ }}|	g}|D ]Þ  }t1        | j,                  «      D ]Ä  \  }}||v rŒ||k(  r||gz  } Œ0t        |t        «      rE|j,                  d   |k(  r3t        |t        «      r#|j,                  d   |j,                  v r||gz  } Œ…t        |t        «      sŒ~|j,                  d   |k(  sŒ‘t        |t        «      sŒ¢|j,                  d   |j,                  v sŒ¾||gz  } ŒÞ Œà t9        |«      t9        |«      dz   k7  r�Œ¦t1        | j,                  «      D ��cg c]  \  }}||vr|‘Œ c}}|j7                  |«      gz   }t:        r|dvrt=        d|«        | j*                  |Ž } d	} �ŒH �ŒK |r�Œ]ˆˆfd„}| j                  d„ |«      } t:        rt=        d| «       | S c c}w c c}w c c}}w )a®  
    Do naive simplifications on ``cond``.

    Explanation
    ===========

    Note that this routine is completely ad-hoc, simplification rules being
    added as need arises rather than following any logical pattern.

    Examples
    ========

    >>> from sympy.integrals.meijerint import _condsimp as simp
    >>> from sympy import Or, Eq
    >>> from sympy.abc import x, y
    >>> simp(Or(x < y, Eq(x, y)))
    x <= y
    c                ó   — | j                   S rb   ©Úis_Relational©Ú_s    rg   r{   z_condsimp.<locals>.<lambda>o  s
   €  a§o¡o€ rn   Fzp q r)r–   r�   r   rR   éþÿÿÿTN)
r   r�   r¬   é   é   é   é   é   é   é   zused new rule:c                ó²  •— | j                   dk7  s| j                  dk7  r| S | j                  }|j                  t	        ‰«      ‰z  «      }|s&|j                  t        t        ‰«      ‰z  «      «      }|s\t        |t        «      rJ|j                  d   j                  s1|j                  d   t        j                  u r|j                  d   dkD  S | S |‰   dkD  S )Nz==r   rR   )Úrel_opÚrhsÚlhsrë   r"   r)   r'   rê   r*   rk   Úis_polarr   ÚInfinity)ÚrelÚLHSrØ   rª   r«   s      €€rg   Úrel_touchupz_condsimp.<locals>.rel_touchupÂ  s¶   ø€ Ø�:‰:˜Ò §¡¨A¢ØˆJð �g‰gˆØ�I‰I”c˜!“f˜a‘iÓ ˆÙØ—	‘	Ô-¬j¸«m¸QÑ.>Ó?Ó@ˆAÙÜ˜#Ô0Ô1¸#¿(¹(À1¹+×:NÒ:NØŸ™ ™¤q§z¡zÑ1ØŸ™ ™ a™Ð(ØˆJØ�!‘�q‘Ðrn   c                ó   — | j                   S rb   r#  r%  s    rg   r{   z_condsimp.<locals>.<lambda>Ñ  s
   €  !§/¡/€ rn   z_condsimp: )Úreplacer   rê   rX   r   r   rU   r   rT   r#   r"   r   r   Úfalser   Útruer)   r,   r½   r7   r4   rä   rk   Ú	_condsimpÚ	enumeraterÈ   rë   r±   rù   r   Úprint)r‡   Úfirstr    Úrulesr&  ÚchangeÚiruleÚfroÚtors   Úarg1rØ   Únumrz   Ú	otherargsÚ	otherlistÚarg2ÚkÚarg3Úarg_Únewargsr7  rª   r«   s                         @@rg   r<  r<  [  s  ù€ ñ& Ø�|‰|Ñ5Ô7GÓHˆØˆÜ�dœOÔ,ØˆÜ�g¤4Ô(�G€A€qˆ!ô
 
ˆA�‰E”2�a˜“8Ó	˜a 1™fÐ%ô 
ŒS”�Q“‹[œBÑ¤¤C¨£F¨Q¬r©T¡MÓ 2´bÑ 8Ó	9Ü	ŒC�‹F”R‰K˜Ó	ð	ä	ŒS�”3�q“6‘œB‘Ó¤2Ñ%¤s¨1¬S°«V©8´b©=Ó'9¼RÑ'?Ó	@Ü	ŒC�‹F�A‹ð	ä	ŒS�”3�q“6‘œB‘Ó¤"Ñ$¤c¨!¬C°«F©(´R©-Ó&8¼BÑ&>Ó	?Ü	
�‰ð	ä	ŒS”�Q“œ"˜Q™$‘Ó¤2 a¡4Ñ'¬¬S°«V´b¸±d©]Ó);¼rÀ!¹tÑ)CÓ	DÜ	ŒC�‹F�A‹ð	ä	ŒS”�Q“œ"˜Q™$‘Ó¤2 a¡4Ñ'¬¬S°«V´b¸±d©]Ó);¼bÀ¹dÑ)BÓ	CÜ	
�‰ð	ä	ŒS”�Q˜‘T˜!‘V˜a‘Z“Ó!¤BÑ&¬¬3¬s°1°a±4¸±6¸A±:«Ó+?ÄÓ(DÓ	EÜ	
�‰ð	ä	ŒC”�A�q‘D˜‘F˜Q‘J“Ó ¤2Ñ%¤r¨!¨Q°©T°!©V°a©Z©.¸!Ó'<Ó	=Ü	
�‰ð	ä	ŒSÔ$ QÓ'Ó(¬BÑ.ÜÔ"¤9¨R´©U´1·?±?Ñ-BÓ#CÀAÑ#EÓFÓGÌ2ÑMó
Oä	Ô¤	¬1¯?©?Ð*:¼2Ñ*=Ó >¸qÑ @ÓAÀ1Ó	Eð	Gô 
ŒSÔ$ QÓ'Ó(¬B¨q©DÑ0ÜÔ"¤9¬b¨S´·±Ñ-@Ó#AÀ!Ñ#CÓDÓEÌÈAÉÑMó
Oä	Ô¤	¬1¯?©?Ð*:¼2Ñ*=¸aÑ*?Ó @ÀÑ BÓCÀQÓ	Gð	Iô 
ˆA�‰F”C˜˜A™˜q“MÓ	" A¨¡FÐ+Ü	ˆAˆq‰D�!‹˜˜1™˜q™Ñ	! 1 a¡4¨!¡8Ð,Ü	ˆAˆa‰C�‹”sœ3œs 1›v›;Ó'¬¨A«Ñ.°Ñ2Ñ	3´S¸³V¸a±ZÐ@Ü	ˆAˆq‹”SœœS ›V›Ó%¤c¨!£fÑ,¨qÑ0Ñ	1´3°q³6¸A±:Ð>Ü
Œc�!‹f‹+œ˜1™Ñ
¤¤S¬¨Q«£[Ó!1´$´s¸1¸a¹4³y³/Ñ!AÀAÑ!EÑ	FÈÈ1ÉÈqÉÐQð9€Eð< ˆ4�9‰9°D·I±IÖ>¨q”y  EÕ*Ò>Ð?€DØ€FÚ
ØˆÜ )¨%Ó 0ó &	ÑˆE‘9�C˜Ø�x‰x˜4Ÿ9™9Ò$ØÜ$ T§Y¡YÓ/ó #‘��4Ø˜Ÿ™ ™×0Ñ0Ñ0ØŸ
™
 3§8¡8¨A¡;Ó/�AØ‘Cà�CØŸ
™
 3§8¡8¨A¡;Ó/�AÙØØ03·±¸¸#°ÀÇÁÈ#ÐPQÉ'È(ÐASÑ0SÖT¨1˜QŸV™V A�YÐT�	ÐTØ˜C�	Ø%ò "�DÜ#,¨T¯Y©YÓ#7ò "™˜˜4Ø 	™>Ø$Ø 4š<Ø%¨!¨Ñ,˜IÙ!Ü% d¬CÔ0°T·Y±Y¸q±\ÀQÒ5FÜ *¨4´Ô 5¸$¿)¹)ÀA¹,È$Ï)É)Ñ:SØ%¨!¨Ñ,˜IÙ!Ü% d¬CÕ0°T·Y±Y¸q±\ÀQÓ5FÜ *¨4´Õ 5¸$¿)¹)ÀA¹,È$Ï)É)Ò:SØ%¨!¨Ñ,˜IÙ!ñ"ð"ô �y“>¤S¨£^°aÑ%7Ò7ÙÜ1:¸4¿9¹9Ó1E÷ 2¡I Q¨Ø yÑ0ò  ó 2Ø57·W±W¸Q³Z°LñA�åØÐ$FÑFÜÐ.°Ô6Ø �t—y‘y 'Ð*�Ø�ÚòG#ð&	ó õVð �<‰<Ñ1°;Ó?€DÝÜˆm˜TÔ"Ø€Kùò ?ùò  Uùó&2s   Ö`=ÚaÞ7a
c                óX   — t        | t        «      r| S t        | j                  «       «      S )z Re-evaluate the conditions. )rê   Úboolr<  Údoit)r‡   s    rg   Ú
_eval_condrQ  Ö  s"   € ä�$œÔØˆÜ�T—Y‘Y“[Ó!Ð!rn   c                óP   — t        | |«      }|s|j                  t         d„ «      }|S )zû Bring expr nearer to its principal branch by removing superfluous
        factors.
        This function does *not* guarantee to yield the principal branch,
        to avoid introducing opaque principal_branch() objects,
        unless full_pb=True. c                ó   — | S rb   r’   )rz   rú   s     rg   r{   z&_my_principal_branch.<locals>.<lambda>é  s   € ¸€ rn   )r(   r9  )r×   ÚperiodÚfull_pbrm   s       rg   Ú_my_principal_branchrV  á  s)   € ô ˜4 Ó
(€CÙØ�k‰kÔ*©NÓ;ˆØ€Jrn   c           	     ó|  ‡	‡
— t        ||«      \  }Š
t        |j                  |«      \  }Š	|j                  «       }t        ||«      }| t	        ‰	«      |‰
dz   ‰	z  dz
  z  z  z  }ˆ	ˆ
fd„}|t         ||j                  «       ||j                  «       ||j                  «       ||j                  «      ||z  «      fS )z”
    Rewrite the integral fac*po*g dx, from zero to infinity, as
    integral fac*G, where G has argument a*x. Note po=x**s.
    Return fac, G.
    rR   c                óF   •— | D �cg c]  }|d‰z   ‰z  z   dz
  ‘Œ c}S c c}w rÜ   r’   )r  r�   r¥   Úss     €€rg   r  z_rewrite_saxena_1.<locals>.trü  s(   ø€ Ø+,Ö- a��Q˜‘U˜A‘I‘ Ó!Ò-Ð-ùÒ-s   †)
rÙ   rÞ   Ú
get_periodrV  r#   rP   r‚   r  r„   r  )r†   rò   ró   rz   r&  r�   rT  r  r  r¥   rY  s            @@rg   Ú_rewrite_saxena_1r[  í  s®   ù€ ô ˜"˜aÓ �D€A€qÜ˜!Ÿ*™* aÓ(�D€A€qØ�\‰\‹^€FÜ˜Q Ó'€Að 	ŒS�‹V�A˜˜Q™ ™	 A™Ñ&Ñ&Ñ'€Aõ.àŒg‘b˜Ÿ™“h¡ 1§8¡8£©b°·±«h¹¸1¿8¹8»Ø˜‘cóð ð rn   c                ó´
  — | j                   }t        | j                  |«      \  }}t        t	        | j
                  «      t	        | j                  «      t	        | j                  «      t	        | j                  «      g«      \  }}}}	||	kD  r_d„ }
t        t         |
| j
                  «       |
| j                  «       |
| j                  «       |
| j                  «      ||z  «      |«      S | j
                  D �cg c]  }t        |«       dk  ‘Œ c}| j                  D �cg c]  }ddt        |«      z
  k  ‘Œ c}z   }t        |Ž }|| j                  D �cg c]  }t        |«       dk  ‘Œ c}z  }|| j                  D �cg c]  }ddt        |«      z
  k  ‘Œ c}z  }t        |Ž }t        | j                  «       |	dz   |z
  dz  z   |	|z
  kD  }d„ }d„ } |d«        |d||||||	f«        |dt!        | j                  «      t!        | j                  «      f«        |d	t!        | j
                  «      t!        | j                  «      f«        |d
|||f«       g }g }d|k  ||	k  d|k  g}d|k  d|k  t#        |	|dz   «      t%        t        t#        |d«      t#        ||dz   «      «      «      g}d|k  t#        |	|«      g}t'        t)        |dz  «      dz   «      D ]1  }|t+        t-        t/        |«      «      |d|z  z
  t0        z  «      gz  }Œ3 |dkD  t-        t/        |«      «      |t0        z  k  g}t+        |d«      |g}|rg }|||fD ]  }|t        ||z   |z   Ž gz  }Œ ||z  } |d|«       |g}|rg }t        t#        |d«      |dz   |k  ||	k  t-        t/        |«      «      |t0        z  k  g|¢­Ž g}||z  } |d|«       ||g}|rg }t        ||	k  d|k  |dkD  t#        t-        t/        |«      «      |t0        z  «      g|¢­Ž g}|t        ||	dz
  k  t#        |d«      t#        t-        t/        |«      «      d«      g|¢­Ž gz  }||z  } |d|«       g }|t#        ||	«      t#        |d«      t#        t/        |«      d«      t+        |d«      gz  }|s||gz  }g }t3        | j                  | j                  «      D ]  \  }}|||z
  gz  }Œ |t        t5        |Ž «      dk  gz  }t        |Ž }||gz  } |d|g«       t        |dkD  t-        t/        |«      «      |t0        z  k  «      g}|s||gz  }t        |Ž }||gz  } |d|g«       t7        |Ž S c c}w c c}w c c}w c c}w )aV  
    Return a condition under which the mellin transform of g exists.
    Any power of x has already been absorbed into the G function,
    so this is just $\int_0^\infty g\, dx$.

    See [L, section 5.6.1]. (Note that s=1.)

    If ``helper`` is True, only check if the MT exists at infinity, i.e. if
    $\int_1^\infty g\, dx$ exists.
    c                ó2   — | D �cg c]  }d|z
  ‘Œ	 c}S c c}w rÜ   r’   ©r  rz   s     rg   r  z _check_antecedents_1.<locals>.tr  s   € Ø#$Ö%˜a�A˜“EÒ%Ð%ùÒ%r  rR   r�   c                 ó   — t        | Ž  y rb   )Ú_debug)Úmsgs    rg   r\   z#_check_antecedents_1.<locals>.debug"  s	   € Ü�Šrn   c                ó   — t        | |«       y rb   ©Ú_debugf)Ústringr"   s     rg   r]   z$_check_antecedents_1.<locals>.debugf%  s   € Ü�˜Õrn   z$Checking antecedents for 1 function:z*  delta=%s, eta=%s, m=%s, n=%s, p=%s, q=%sz  ap = %s, %sz  bq = %s, %sz"  cond_3=%s, cond_3*=%s, cond_4=%sr   z	  case 1:z	  case 2:z	  case 3:z  extra case:z  second extra case:)r  rÙ   rÞ   r   rù   r„   r‚   rƒ   r…   Ú_check_antecedents_1rP   r  r  r    rT   r¡   rº   r   rW   r   r.   r   r#   r)   r   Úzipr   rU   ) ró   rz   Úhelperr  Úetar&  rØ   rs   rª   r«   r  r¥   r�   ÚtmpÚcond_3Úcond_3_starÚcond_4r\   r]   ÚcondsÚcase1Útmp1Útmp2Útmp3rJ  Úextrar¦   Úcase2Úcase3Ú
case_extrarY  Úcase_extra_2s                                    rg   rf  rf    st  € ð �G‰G€EÜ˜AŸJ™J¨Ó*�F€CˆÜ”C˜Ÿ™“Iœs 1§4¡4›y¬#¨a¯d©d«)´S¸¿¹³YÐ?Ó@�J€A€qˆ!ˆQàˆ1‚uò	&ä#¤G©B¨q¯t©t«H±b¸¿¹³lÙ,.¨q¯t©t«H±b¸¿¹³lÀAÀcÁEó%Kà$%ó'ð 	'ð  !Ÿt™tÖ
$˜!ŒBˆq‹Eˆ6�A‹:Ò
$¸q¿t¹tÖ'D¸!¨¨A´°1³©I«Ò'DÑ
D€CÜ�#ˆY€Fà §¡Ö)˜1ŒR�‹UˆF�Q‹JÒ)Ñ)€CØ 1§8¡8Ö,˜aˆA�”B�q“E‘	‹MÒ,Ñ,€CÜ�s�)€Kä�!—$‘$‹xˆi˜1˜q™5 1™9 a™-Ñ'¨!¨a©%Ñ/€Fòòñ 
Ð
0Ô1Ù
Ð7Ø�3˜˜1˜a Ð#ô%á
ˆ?œT !§$¡$›Z¬¨a¯h©h«Ð8Ô9Ù
ˆ?œT !§$¡$›Z¬¨a¯h©h«Ð8Ô9Ù
Ð/°&¸+ÀvÐ1NÔOà€Eð €EØ�‰F�A˜‘E˜1 ™6Ð"€DØ�‰F�A˜‘FœB˜q ! a¡%›L¬#¬c´"°Q¸³(¼B¸qÀ!ÀaÁ%»LÓ.IÓ*JÐK€DØ�‰F”B�q˜!“HÐ€DÜ”7˜5 ™7Ó# aÑ'Ó(ò FˆØ””CÔ+¨CÓ0Ó1°E¸A¸a¹C±KÄÑ3CÓDÐEÑE‰ðFà�1‰9”cÔ-¨cÓ2Ó3°e¼B±hÑ>Ð
?€CÜ��Q‹Z˜Ð €EÙØˆØ�D˜$Ðò +ˆØ”#˜˜C™ %™Ð)Ð*Ñ*‰ð+à	ˆU�N€EÙ	ˆ+�uÔð ˆH€EÙØˆÜ”�A�q“˜1˜q™5 A™: q¨A¡vÜÔ(¨Ó-Ó.°´r±Ñ9ðCØ<AòCð D€Eà	ˆU�N€EÙ	ˆ+�uÔð �VÐ€EÙØˆÜ��Q‘˜˜Q™ ¨¡	¬2¬cÔ2EÀcÓ2JÓ.KÈUÔSUÉXÓ+Vð Øòð €Eà	Œc�!�q˜1‘u‘*œb ¨›l¬B¬sÔ3FÀsÓ3KÓ/LÈaÓ,PÐYÐSXÒYÐZÑZ€EØ	ˆU�N€EÙ	ˆ+�uÔð €JØ”2�a˜“8œR  q›\¬2Ô.AÀ#Ó.FÈÓ+JÌBÈsÐTUËJÐWÑW€JÙØ�v�hÑˆ
Ø
€AÜ�A—D‘D˜!Ÿ$™$“ò ‰ˆˆ1Ø	ˆa�!‰eˆW‰‰ðà”2”c˜1�g“; ‘?Ð#Ñ#€JÜ�jÐ!€JØ	ˆjˆ\Ñ€EÙ	ˆ/˜J˜<Ô(ä˜ ™	¤3Ô':¸3Ó'?Ó#@À5ÌÁ8Ñ#KÓLÐM€LÙØ˜˜Ñ ˆÜ˜Ð%€LØ	ˆlˆ^Ñ€EÙ	Ð
  < .Ô1ô
 ˆuˆ:Ðùòm %ùÒ'Dùò *ùÚ,s   Ã8UÄUÅUÅ<Uc                óˆ  — ddl m} t        | j                  |«      \  }}d|z  }| j                  D ]  }|t        |dz   «      z  }Œ | j                  D ]  }|t        d|z
  dz
  «      z  }Œ | j                  D ]  }|t        d|z
  dz
  «      z  }Œ | j                  D ]  }|t        |dz   «      z  }Œ  |t        |«      «      S )aƒ  
    Evaluate $\int_0^\infty g\, dx$ using G functions,
    assuming the necessary conditions are fulfilled.

    Examples
    ========

    >>> from sympy.abc import a, b, c, d, x, y
    >>> from sympy import meijerg
    >>> from sympy.integrals.meijerint import _int0oo_1
    >>> _int0oo_1(meijerg([a], [b], [c], [d], x*y), x)
    gamma(-a)*gamma(c + 1)/(y*gamma(-d)*gamma(b + 1))
    r   )Ú	gammasimprR   )
rÒ   ry  rÙ   rÞ   r„   rN   r‚   r  r  r%   )ró   rz   ry  ri  r&  rm   r¥   r�   s           rg   Ú	_int0oo_1rz  r  sÑ   € õ )ä˜AŸJ™J¨Ó*�F€CˆØ
ˆC‰%€Cà�T‰Tò ˆØŒu�Q˜‘U‹|Ñ‰ðà�T‰Tò  ˆØŒu�Q˜‘U˜Q‘YÓÑ‰ð à�X‰Xò  ˆØŒu�Q˜‘U˜Q‘YÓÑ‰ð à�X‰Xò ˆØŒu�Q˜‘U‹|Ñ‰ðá”Z “_Ó%Ð%rn   c                ó  ‡‡‡— ˆˆfd„}t        |‰«      \  }}t        |j                  ‰«      \  }}	t        |j                  ‰«      \  }}
|	dk  dk(  r|	 }	t        |«      }|
dk  dk(  r|
 }
t        |«      }|	j                  r|
j                  sy|	j                  |	j
                  }}|
j                  |
j
                  }}t        ||z  ||z  «      }|||z  z  }|||z  z  }t        ||«      \  }}t        ||«      \  }} ||«      } ||«      }| ||z  z  } t        |j                  ‰«      \  }}t        |j                  ‰«      \  }}|dz   |z  dz
  Š| t        |«      |‰z  z  z  } ˆfd„}t         ||j                  «       ||j                  «       ||j                  «       ||j                  «      |‰z  «      }t        |j                  |j                  |j                  |j                  |‰z  «      }ddlm}  || d¬«      ||fS )	aá  
    Rewrite the integral ``fac*po*g1*g2`` from 0 to oo in terms of G
    functions with argument ``c*x``.

    Explanation
    ===========

    Return C, f1, f2 such that integral C f1 f2 from 0 to infinity equals
    integral fac ``po``, ``g1``, ``g2`` from 0 to infinity.

    Examples
    ========

    >>> from sympy.integrals.meijerint import _rewrite_saxena
    >>> from sympy.abc import s, t, m
    >>> from sympy import meijerg
    >>> g1 = meijerg([], [], [0], [], s*t)
    >>> g2 = meijerg([], [], [m/2], [-m/2], t**2/4)
    >>> r = _rewrite_saxena(1, t**0, g1, g2, t)
    >>> r[0]
    s/(4*sqrt(pi))
    >>> r[1]
    meijerg(((), ()), ((-1/2, 0), ()), s**2*t/4)
    >>> r[2]
    meijerg(((), ()), ((m/2,), (-m/2,)), t/4)
    c                óæ   •— t        | j                  ‰«      \  }}| j                  «       }t        | j                  | j
                  | j                  | j                  t        ||‰«      ‰|z  z  «      S rb   )	rÙ   rÞ   rZ  rP   r‚   r  r„   r  rV  )ró   r�   r¥   ÚperrU  rz   s       €€rg   Úpbz_rewrite_saxena.<locals>.pb«  s^   ø€ Ü˜aŸj™j¨!Ó,‰ˆˆ1Ø�l‰l‹nˆÜ�q—t‘t˜QŸX™X q§t¡t¨Q¯X©XÜ+¨A¨s°GÓ<¸QÀ¹TÑAóCð 	Crn   r   TNrR   c                ó4   •— | D �cg c]  }|‰z   ‘Œ	 c}S c c}w rb   r’   )r  r�   r+   s     €rg   r  z_rewrite_saxena.<locals>.trÐ  s   ø€ Ø!"Ö#˜A��C“Ò#Ð#ùÒ#ó   †©Ú	powdenest©Úpolar)rÙ   rÞ   r  Úis_Rationalrª   r«   r   r  r#   rP   r‚   r  r„   r  rÒ   r‚  )r†   rò   Úg1Úg2rz   rU  r~  r&  rY  Úb1Úb2Úm1Ún1Úm2Ún2ÚtauÚr1Úr2ÚC1ÚC2Úa1r¥   Úa2r  r‚  r+   s       ``                   @rg   Ú_rewrite_saxenar•  �  sä  ú€ õ6Cô ˜"˜aÓ �D€A€qÜ˜2Ÿ;™;¨Ó*�E€A€rÜ˜2Ÿ;™;¨Ó*�E€A€rØ
ˆQ‰�4ÒØˆSˆÜ�R‹[ˆØ
ˆQ‰�4ÒØˆSˆÜ�R‹[ˆØ�>Š> §¢ØØ�T‰T�2—4‘4ˆ€BØ�T‰T�2—4‘4ˆ€BÜ
ˆr�"‰u�b˜‘eÓ
€CØ	ˆr�"‰u‰€BØ	ˆr�"‰u‰€Bä˜˜BÓ�F€BˆÜ˜˜BÓ�F€BˆÙ	ˆB‹€BÙ	ˆB‹€Bàˆ2ˆb‰5�L€CÜ˜2Ÿ;™;¨Ó*�E€BˆÜ˜2Ÿ;™;¨Ó*�E€Bˆð ˆq‰5�!‰)�a‰-€CØ
Œs�1‹v˜˜C™ÑÑ
 €Cô$ä	‘�B—E‘E“™B˜rŸy™y›M©2¨b¯e©e«9±b¸¿¹³mÀRÈÁTÓ	J€BÜ	�—‘˜Ÿ	™	 2§5¡5¨"¯)©)°R¸±TÓ	:€Bå(Ù�S Ô% r¨2Ð-Ð-rn   c                ó¢.  ‡ ‡‡.‡/‡0‡1‡2‡3‡4‡5‡6— t        ‰ j                  |«      \  Š3}t        ‰j                  |«      \  Š/}t        t        ‰ j                  «      t        ‰ j
                  «      t        ‰ j                  «      t        ‰ j                  «      g«      \  }}Š5Š6t        t        ‰j                  «      t        ‰j
                  «      t        ‰j                  «      t        ‰j                  «      g«      \  }}Š0Š2||z   ‰5‰6z   dz  z
  }||z   ‰0‰2z   dz  z
  }	‰ j                  ‰5‰6z
  dz  z   dz   }
‰j                  ‰0‰2z
  dz  z   dz   }‰2‰0z
  ‰6‰5z
  z
  }d‰6‰5z
  z
  |z
  |
z
  }t        ‰2|z
  |z
  z  t        t        ‰/«      «      z   ‰2‰0z
  z  Š1t        ‰6|z
  |z
  z  t        t        ‰3«      «      z   ‰6‰5z
  z  Š4t        d«       t        d‰3||‰5‰6||
f«       t        d‰/||‰0‰2|	|f«       t        d||‰1‰4f«       ˆ ˆfd„} |«       }t        ‰ j                  D ��cg c]'  }‰j                  D ]  }t        d|z   |z   «      dkD  ‘Œ Œ) c}}Ž }t        ‰ j
                  D ��cg c]'  }‰j
                  D ]  }t        d|z   |z   «      dk  ‘Œ Œ) c}}Ž }t        ‰ j
                  D �cg c]2  }‰0‰2z
  t        d|z   dz
  «      z  t        |«      z
  t!        d	d«      kD  ‘Œ4 c}Ž }t        ‰ j                  D �cg c]/  }‰0‰2z
  t        d|z   «      z  t        |«      z
  t!        d	d«      kD  ‘Œ1 c}Ž }t        ‰j
                  D �cg c]2  }‰5‰6z
  t        d|z   dz
  «      z  t        |
«      z
  t!        d	d«      kD  ‘Œ4 c}Ž }t        ‰j                  D �cg c]/  }‰5‰6z
  t        d|z   «      z  t        |
«      z
  t!        d	d«      kD  ‘Œ1 c}Ž }t        |«      dt        |
dz
  ‰2‰0z
  z  ‰6‰5z
  ‰2‰0z
  z  z   |dz
  ‰6‰5z
  z  z   «      z  z   dkD  }t        |«      dt        |
dz
  ‰2‰0z
  z  ‰6‰5z
  ‰2‰0z
  z  z   |dz
  ‰6‰5z
  z  z   «      z  z
  dkD  }t        t        ‰3«      «      |t        z  k  }t#        t        t        ‰3«      «      |t        z  «      }t        t        ‰/«      «      |	t        z  k  }t#        t        t        ‰/«      «      |	t        z  «      }t%        ||	z    t        z  t        j&                  z  «      }t)        |‰/z  ‰3z  «      }t)        |‰3z  ‰/z  «      } |d| z  k(  r]t        t#        |d«      ||	z   dk  t+        t-        |d«      t        ||
z   ‰6z   ‰5z
  «      dk  t        ||
z   ‰2z   ‰0z
  «      dk  «      «      }!néd
„ }"t        t#        |d«      |dz
  |	z   dk  t+        t        t-        |d«       |"|«      «      t        t        ||
z   ‰6z   ‰5z
  «      dk  t#        |d«      «      «      «      }!t        t#        |d«      |	dz
  |z   dk  t+        t        t-        | d«       |"| «      «      t        t        ||
z   ‰2z   ‰0z
  «      dk  t#        | d«      «      «      «      }#t+        |!|#«      }!	 	 ‰2‰0z
  t        ‰/«      d‰2‰0z
  z  z  z  t/        ‰1«      z  ‰6‰5z
  t        ‰3«      d‰6‰5z
  z  z  z  t/        ‰4«      z  z   }$t1        |$dkD  «      dk7  r|$dkD  }%�n¾ˆ/ˆ0ˆ1ˆ2ˆ3ˆ4ˆ5ˆ6fd„}&t3         |&dd«       |&dd«      z  t        t#        t        ‰3«      d«      t#        t        ‰/«      d«      «      f |&t5        t        ‰/«      «      d«       |&t5        t        ‰/«      «      d«      z  t        t#        t        ‰3«      d«      t-        t        ‰/«      d«      «      f |&dt5        t        ‰3«      «      «       |&dt5        t        ‰3«      «      «      z  t        t-        t        ‰3«      d«      t#        t        ‰/«      d«      «      f |&t5        t        ‰/«      «      t5        t        ‰3«      «      «      df«      }'|$dkD  t        t#        |$d«      t-        |'d«      t        |«      dkD  «      t        t#        |$d«      t#        |'d«      t        |«      dkD  «      g}(t+        |(Ž }%|df|df|df|df|df|df|df|df|df|df|df|df|df|!df|%dffD ]  \  })}t        d||)f«       Œ g Š.ˆ.fd„}*‰.t        ||z  |z  |z  dk7  |j8                  du |	j8                  du |||||«      gz  Š. |*d«       ‰.t        t#        ‰5‰6«      t#        |d«      |	j8                  du ‰3j8                  du t        |
«      dk  ||||«	      gz  Š. |*d«       ‰.t        t#        ‰0‰2«      t#        |	d«      |j8                  du ‰/j8                  du t        |«      dk  ||||«	      gz  Š. |*d«       ‰.t        t#        ‰0‰2«      t#        ‰5‰6«      t#        |d«      t#        |	d«      ‰3j8                  du ‰/j8                  du t        |«      dk  t        |
«      dk  t-        ‰3‰/«      |||«      gz  Š. |*d«       ‰.t        t#        ‰0‰2«      t#        ‰5‰6«      t#        |d«      t#        |	d«      ‰3j8                  du ‰/j8                  du t        ||
z   «      dk  t-        ‰/‰3«      |||«      gz  Š. |*d«       ‰.t        ‰0‰2kD  |j8                  du |j8                  du |	dk\  ||||||«
      gz  Š. |*d«       ‰.t        ‰0‰2k  |j8                  du |j8                  du |	dk\  ||||||«
      gz  Š. |*d«       ‰.t        ‰5‰6kD  |j8                  du |	j8                  du |dk\  ||||||«
      gz  Š. |*d«       ‰.t        ‰5‰6k  |j8                  du |	j8                  du |dk\  ||||||«
      gz  Š. |*d«       ‰.t        ‰0‰2kD  t#        ‰5‰6«      t#        |d«      |	dk\  ‰3j8                  du t        |
«      dk  |||||«      gz  Š. |*d«       ‰.t        ‰0‰2k  t#        ‰5‰6«      t#        |d«      |	dk\  ‰3j8                  du t        |
«      dk  |||||«      gz  Š. |*d«       ‰.t        t#        ‰0‰2«      ‰5‰6kD  |dk\  t#        |	d«      ‰/j8                  du t        |«      dk  |||||«      gz  Š. |*d«       ‰.t        t#        ‰0‰2«      ‰5‰6k  |dk\  t#        |	d«      ‰/j8                  du t        |«      dk  |||||«      gz  Š. |*d«       ‰.t        ‰0‰2k  ‰5‰6kD  |dk\  |	dk\  |||||||«      gz  Š. |*d«       ‰.t        ‰0‰2kD  ‰5‰6k  |dk\  |	dk\  |||||||«      gz  Š. |*d«       ‰.t        ‰0‰2kD  ‰5‰6kD  |dk\  |	dk\  |||||||||!«      gz  Š. |*d«       ‰.t        ‰0‰2k  ‰5‰6k  |dk\  |	dk\  |||||||||!«      gz  Š. |*d«       ‰.t        t#        |d«      |j8                  du |j8                  du |j8                  du |||«      gz  Š. |*d «       ‰.t        t#        |d«      |j8                  du |j8                  du |j:                  du |||«      gz  Š. |*d!«       ‰.t        t#        |d«      |j8                  du |	j8                  du |j:                  du |||«      gz  Š. |*d"«       ‰.t        t#        |d«      |j8                  du |	j8                  du |j8                  du |||«      gz  Š. |*d#«       ‰.t        t#        ||z  d«      |j8                  du |	j8                  du |||||«      gz  Š. |*d$«       ‰.t        t#        ||z  d«      |j8                  du |	j8                  du |||||«      gz  Š. |*d%«       t=        ‰ |d¬&«      }+t=        ‰|d¬&«      },‰.t        |,t#        |d«      ‰5|k  |j8                  du ||||«      gz  Š. |*d'«       ‰.t        |,t#        |d«      ‰6|k  |j8                  du ||||«      gz  Š. |*d(«       ‰.t        |+t#        |d«      ‰0|k  |	j8                  du ||||«      gz  Š. |*d)«       ‰.t        |+t#        |d«      ‰2|k  |	j8                  du ||||«      gz  Š. |*d*«       t+        ‰.Ž }-t1        |-«      dk7  r|-S ‰.t        ||z   ‰0kD  t#        |d«      t#        |d«      |j8                  du |j8                  du |	j:                  du t        t        ‰/«      «      ||z   ‰0z
  dz   t        z  k  ||||!|%«      gz  Š. |*d+«       ‰.t        ||z   ‰2kD  t#        |d«      t#        |d«      |j8                  du |j8                  du |	j:                  du t        t        ‰/«      «      ||z   ‰2z
  dz   t        z  k  ||||!|%«      gz  Š. |*d,«       ‰.t        t#        ‰0‰2dz
  «      t#        |d«      t#        |d«      |j8                  du |j8                  du |	dk\  |	t        z  t        t        ‰/«      «      k  ||||!|%«      gz  Š. |*d-«       ‰.t        t#        ‰0‰2dz   «      t#        |d«      t#        |d«      |j8                  du |j8                  du |	dk\  |	t        z  t        t        ‰/«      «      k  ||||!|%«      gz  Š. |*d.«       ‰.t        ‰0‰2dz
  k  t#        |d«      t#        |d«      |j8                  du |j8                  du |	dk\  |	t        z  t        t        ‰/«      «      k  t        t        ‰/«      «      ||z   ‰0z
  dz   t        z  k  ||||!|%«      gz  Š. |*d/«       ‰.t        ‰0‰2dz   kD  t#        |d«      t#        |d«      |j8                  du |j8                  du |	dk\  |	t        z  t        t        ‰/«      «      k  t        t        ‰/«      «      ||z   ‰2z
  dz   t        z  k  ||||!|%«      gz  Š. |*d0«       ‰.t        t#        |d«      t#        |d«      ||z   dkD  |j8                  du |	j8                  du |j:                  du t        t        ‰3«      «      ||z   ‰5z
  dz   t        z  k  ||||!|%«      gz  Š. |*d1«       ‰.t        t#        |d«      t#        |d«      ||z   ‰6kD  |j8                  du |	j8                  du |j:                  du t        t        ‰3«      «      ||z   ‰6z
  dz   t        z  k  ||||!|%«      gz  Š. |*d2«       ‰.t        t#        |d«      t#        |d«      t#        ‰5‰6dz
  «      |j8                  du |	j8                  du |dk\  |t        z  t        t        ‰3«      «      k  t        t        ‰3«      «      |dz   t        z  k  ||||!|%«      gz  Š. |*d3«       ‰.t        t#        |d«      t#        |d«      t#        ‰5‰6dz   «      |j8                  du |	j8                  du |dk\  |t        z  t        t        ‰3«      «      k  t        t        ‰3«      «      |dz   t        z  k  ||||!|%«      gz  Š. |*d4«       ‰.t        t#        |d«      t#        |d«      ‰5‰6dz
  k  |j8                  du |	j8                  du |dk\  |t        z  t        t        ‰3«      «      k  t        t        ‰3«      «      ||z   ‰5z
  dz   t        z  k  ||||!|%«      gz  Š. |*d5«       ‰.t        t#        |d«      t#        |d«      ‰5‰6dz   kD  |j8                  du |	j8                  du |dk\  |t        z  t        t        ‰3«      «      k  t        t        ‰3«      «      ||z   ‰6z
  dz   t        z  k  ||||!|%«      gz  Š. |*d6«       t+        ‰.Ž S c c}}w c c}}w c c}w c c}w c c}w c c}w # t6        $ r d}%Y �ŒÎw xY w)7z> Return a condition under which the integral theorem applies. r�   rR   zChecking antecedents:z1  sigma=%s, s=%s, t=%s, u=%s, v=%s, b*=%s, rho=%sz1  omega=%s, m=%s, n=%s, p=%s, q=%s, c*=%s, mu=%s,z"  phi=%s, eta=%s, psi=%s, theta=%sc                 ó¾   •— ‰‰fD ]U  } t        j                  | j                  | j                  «      D ]&  \  }}||z
  }|j                  sŒ|j
                  sŒ%  y ŒW y)NFT)rþ   rÿ   r‚   r„   Ú
is_integerÚis_positive)ró   re   ÚjÚdiffr†  r‡  s       €€rg   Ú_c1z_check_antecedents.<locals>._c1ú  s^   ø€ Ø�b�ò 	!ˆAÜ!×)Ñ)¨!¯$©$°·±Ó5ò !‘��1Ø˜1‘u�Ø—?“? t×'7Ó'7Ú ñ!ð	!ð
 rn   r   éýÿÿÿc                óL   — | dk7  xr t        t        d| z
  «      «      t        k  S )aã  Returns True if abs(arg(1-z)) < pi, avoiding arg(0).

            Explanation
            ===========

            If ``z`` is 1 then arg is NaN. This raises a
            TypeError on `NaN < pi`. Previously this gave `False` so
            this behavior has been hardcoded here but someone should
            check if this NaN is more serious! This NaN is triggered by
            test_meijerint() in test_meijerint.py:
            `meijerint_definite(exp(x), x, 0, I)`
            rR   )r#   r"   r   )r^   s    rg   Ú_condz!_check_antecedents.<locals>._cond$  s$   € ð ˜‘6Ò2œc¤# a¨!¡e£*›o´Ñ2Ð2rn   Fc                óª   •— | ‰‰z
  z  t        ‰«      d‰‰z
  z  z  z  t        ‰«      z  |‰	‰z
  z  t        ‰«      d‰	‰z
  z  z  z  t        ‰«      z  z   S rÜ   )r#   r8   )
Úc1Úc2Úomegarª   Úpsir«   ÚsigmaÚthetaÚur  s
     €€€€€€€€rg   Ú	lambda_s0z%_check_antecedents.<locals>.lambda_s0S  sc   ø€ Ø˜1˜q™5‘z¤# e£*¨q°!°a±%©yÑ"9Ñ9¼#¸c»(ÑBØ˜!˜a™%‘j¤ U£¨a°°Q±©iÑ!8Ñ8¼¸U»ÑCñDð Drn   rŸ   Tr­   r¬   r(  r)  r*  é   é	   é
   r+  r,  r-  r.  é   z	  c%s: %sc                ó(   •— t        d| ‰d   f«       y )Nz  case %s: %srŸ   rc  )Úcountrn  s    €rg   Úprz_check_antecedents.<locals>.prl  s   ø€ Ü� %¨¨r©Ð!3Õ4rn   r¸   é   é   é   é   é   é   é   )rh  ÚE1ÚE2ÚE3ÚE4é   é   é   é   é   é   é   é   é    é!   é"   é#   )rÙ   rÞ   r   rù   r„   r‚   rƒ   r…   r¡   r   r#   r)   r`  rd  rT   r    r   r   r+   r½   r%   rU   r   r7   rQ  r5   r$   Ú	TypeErrorr™  Úis_negativerf  )7r†  r‡  rz   r&  rY  r¦   rØ   rs   ÚbstarÚcstarÚrhoÚmuÚphiri  rœ  r¡  re   rš  r¢  Úc3Úc4Úc5Úc6Úc7Úc8Úc9Úc10Úc11Úc12Úc13Úz0ÚzosÚzsoÚc14rŸ  Úc14_altÚlambda_cÚc15r¨  Úlambda_srj  r‡   r¯  Ú
mt1_existsÚ
mt2_existsr    rn  r£  rª   r¤  r«   r¥  r¦  r§  r  s7   ``                                            @@@@@@@@@rg   Ú_check_antecedentsrã  Ù  s­  ÿú€ ô ˜bŸk™k¨1Ó-�H€Eˆ1Ü˜bŸk™k¨1Ó-�H€Eˆ1Ü”C˜Ÿ™“J¤ B§E¡E£
¬C°·±«J¼¸B¿E¹E»
ÐCÓD�J€A€qˆ!ˆQÜ”C˜Ÿ™“J¤ B§E¡E£
¬C°·±«J¼¸B¿E¹E»
ÐCÓD�J€A€qˆ!ˆQØ�‰E�Q˜‘U˜A‘IÑ€EØ�‰E�Q˜‘U˜A‘IÑ€EØ
�%‰%�1�q‘5˜!‘)Ñ
˜aÑ
€CØ	�‰�!�a‘%˜‘Ñ	˜QÑ	€BØ
ˆa‰%�1�q‘5‰/€CØ
ˆq�1‰u‰+˜Ñ
˜SÑ
 €CÜˆq�1‰u�q‰y‰>œCÔ 3°EÓ :Ó;Ñ;¸aÀ!¹eÑ
D€CÜ��Q‘˜‘‰^œcÔ"5°eÓ"<Ó=Ñ=ÀÀAÁÑF€Eä
Ð"Ô#ÜÐ?Ø�A�q˜!˜Q  sÐ+ô-äÐ?Ø�A�q˜!˜Q  rÐ*ô,äÐ0°3¸¸SÀ%Ð2HÔIõñ 
‹€BÜ	¨"¯%©%×? Q¸¿¹Ò?°AŒr�!�a‘%˜!‘)‹}˜qÓ Ð?Ð Ó?Ð	@€BÜ	¨b¯e©e×C¨¸R¿U¹UÒC¸Œr�!�a‘%˜!‘)‹}˜uÓ$ÐCÐ$ÓCÐ	D€BÜ	ÈÏÉÖOÀA��A‘”r˜!˜a™% !™)“}Ñ$¤r¨"£vÑ-´¸¸Q³Ó?ÒOÐ	P€BÜ	ÀRÇUÁUÖKÀ��A‘”r˜!˜a™%“yÑ ¤2 b£6Ñ)¬H°R¸«OÓ;ÒKÐ	L€BÜ	È"Ï%É%ÖPÀQ��A‘”r˜!˜a™% !™)“}Ñ$¤r¨#£wÑ.´¸"¸a³Ó@ÒPÐ	Q€BÜ	ÀbÇeÁeÖLÀ��A‘”r˜!˜a™%“yÑ ¤2 c£7Ñ*¬X°b¸!«_Ó<ÒLÐ	M€BÜ
ˆc‹(�Q”r˜3 ™7 Q¨¡UÑ+¨q°1©u°q¸1±u©oÑ=ÀØ
ñBØ�!‰eñAñ ó ñ ñ Øñ€Bä
ˆc‹(�Q”r˜3 ™7 Q¨¡UÑ+¨q°1©u°q¸1±u©oÑ=ÀØ
ñBØ�!‰eñAñ ó ñ ñ Øñ€BäÔ" 5Ó)Ó*¨U´2©XÑ5€CÜ
ŒSÔ$ UÓ+Ó,¨e´B©hÓ
7€CÜÔ" 5Ó)Ó*¨U´2©XÑ5€CÜ
ŒSÔ$ UÓ+Ó,¨e´B©hÓ
7€Cô 
ˆu�u‰}ÐœbÑ ¤§¡Ñ0Ó	1€BÜ
�R˜‘X˜e‘^Ó
$€CÜ
�R˜‘X˜e‘^Ó
$€CØ
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 €r„Fäˆuˆ:ÐùóC @ùÛCùÚOùÚKùÚPùÚLøôv ò Ø‹ðúsB   È,A\
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t        |||	|
||z  «      |z  S )aà  
    Express integral from zero to infinity g1*g2 using a G function,
    assuming the necessary conditions are fulfilled.

    Examples
    ========

    >>> from sympy.integrals.meijerint import _int0oo
    >>> from sympy.abc import s, t, m
    >>> from sympy import meijerg, S
    >>> g1 = meijerg([], [], [-S(1)/2, 0], [], s**2*t/4)
    >>> g2 = meijerg([], [], [m/2], [-m/2], t/4)
    >>> _int0oo(g1, g2, t)
    4*meijerg(((0, 1/2), ()), ((m/2,), (-m/2,)), s**(-2))/s**2
    c                ó.   — | D �cg c]  }| ‘Œ c}S c c}w rb   r’   r^  s     rg   Únegz_int0oo.<locals>.neg  s   € ØŽ�q�’ŠÐùŠs   …
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                  «       ||j                  «       ||j                  «       ||j                  «      |j                  «      fS )z Absorb ``po`` == x**s into g. c                ó:   •— | D �cg c]
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r†   rò   ró   rz   r&  r�   r  r‚  r¥   rY  s
           @@rg   Ú_rewrite_inversionrê  &  s…   ù€ ä˜"˜aÓ �D€A€qÜ˜!Ÿ*™* aÓ(�D€A€qõ$å(Ù�c˜!˜a ™c™(‘l¨$Ô/Ü‘B�q—t‘t“H™b §¡›l©B¨q¯t©t«H±b¸¿¹³lÀAÇJÁJÓOðQð Qrn   c                óú	  ‡ ‡‡‡‡‡— t        d«       ‰ j                  }t        |‰«      \  }}|dk  r t        d«       t        t	        ‰ «      ‰«      S ˆfd„Šˆfd„Št        t        ‰ j                  «      t        ‰ j                  «      t        ‰ j                  «      t        ‰ j                  «      g«      \  }}}}||z   |z
  }	||z
  |z
  }
|	|
z
  dz  }||z
  Š‰dk(  rt
        j                  }n‰dkD  rd}nt
        j                  }d‰z
  dz  t        ‰ j                  Ž z   t        ‰ j                  Ž z
  ‰z  Š‰ j                  }t        d|||||	|
|‰f«       t        d	|‰|f«       ‰ j                  |dz  k\  s|dk\  r||k\  st        d
«       yt!        j"                  ‰ j                  ‰ j                  «      D ]'  \  }}||z
  j$                  sŒ||kD  sŒt        d«        y ||k\  r7t        d«       t'        ‰ j                  D �cg c]  } ‰|dz
  dd|«      ‘Œ c}Ž S ˆ ˆfd„}ˆˆˆfd„}ˆˆˆfd„}ˆˆˆfd„}g }|t'        d|k  d|k  |t(        z  |z
  t(        dz  k\  |dkD   ||t+        t
        j,                  t(        z  |
dz   z  «      z  «      «      gz  }|t'        |dz   |k  |dz   |k  |dkD  |t(        dz  k  |dk(  ||z
  dz   t(        z  |z
  t(        dz  k\   ||t+        t
        j,                  t(        z  ||z
  z  «      z  «       ||t+        t
        j,                   t(        z  ||z
  z  «      z  «      «      gz  }|t'        ||k(  |dk(  |dkD  ‰|z   t(        z  |z
  t(        dz  k\   ||«      «      gz  }|t'        t/        t'        ||dz
  k  d|	k  |	‰dz  k  «      t'        |dz   ||z   k  ||z   ||z   dz  k  «      «      |dkD  |t(        dz  k  |	dz   t(        z  |z
  t(        dz  k\   ||t+        t
        j,                  t(        z  |
z  «      z  «       ||t+        t
        j,                   t(        z  |
z  «      z  «      «      gz  }|t'        d|k  |dkD  |dkD  ||t(        z  z   t(        dz  k  |	|z   t(        z  |z
  t(        dz  k\   ||t+        t
        j,                  t(        z  |
z  «      z  «       ||t+        t
        j,                   t(        z  |
z  «      z  «      «      gz  }||dk(  gz  }t/        |Ž S c c}w )z7 Check antecedents for the laplace inversion integral. z#Checking antecedents for inversion:r   z  Flipping G.c           
     óà  •— t        |‰«      \  }}| |z  } |||z  z  }||z  }g }|t        t        j                  t	        |«      z  t
        z  dz  «      z  }|t        t        j                   t	        |«      z  t
        z  dz  «      z  }	|r|}
n|	}
|t        t        t        |d«      t	        |«      dk  «      t	        | «      dk  «      gz  }|t        t        |d«      t        t        |«      d«      t	        |«      dkD  t	        |
«      dk  «      gz  }|t        t        |d«      t        t        |«      d«      t	        |«      dkD  t	        |
«      dk  t	        | «      dk  «      gz  }t        |Ž S )Nr�   r   rŸ   )rÙ   r+   r   r½   r    r   rT   rU   r   r   r!   )r�   r¥   r¾   r^   ÚplusÚcoeffÚexponentrn  ÚwpÚwmÚwrz   s              €rg   Ústatement_halfz4_check_antecedents_inversion.<locals>.statement_half<  sO  ø€ Ü(¨¨AÓ.‰ˆˆxØ	ˆX‰ˆØ	ˆU�A‰X‰ˆØ	ˆX‰ˆØˆØŒs”1—?‘?¤2 a£5Ñ(¬Ñ+¨AÑ-Ó.Ñ.ˆØŒs”A—O‘OÐ#¤B q£EÑ)¬"Ñ,¨QÑ.Ó/Ñ/ˆÙØ‰AàˆAØ”#”bœ˜A˜q›¤2 a£5¨A¡:Ó.´°1³¸±Ó<Ð=Ñ=ˆØ”#”b˜˜A“h¤¤2 a£5¨!£¬b°«e°a©i¼¸A»À¹ÓCÐDÑDˆØ”#”b˜˜A“h¤¤2 a£5¨!£¬b°«e°a©i¼¸A»À!¹Ü˜“e˜r‘kó#ð $ñ 	$ˆä�5ˆzÐrn   c           
     óD   •— t         ‰| |||d«       ‰| |||d«      «      S )zW Provide a convergence statement for z**a * exp(b*z**c),
             c/f sphinx docs. TF)rT   )r�   r¥   r¾   r^   ró  s       €rg   Ú	statementz/_check_antecedents_inversion.<locals>.statementN  s1   ø€ ô ‘> ! Q¨¨1¨dÓ3Ù! ! Q¨¨1¨eÓ4ó6ð 	6rn   r�   rR   z9  m=%s, n=%s, p=%s, q=%s, tau=%s, nu=%s, rho=%s, sigma=%sz   epsilon=%s, theta=%s, delta=%sz-  Computation not valid for these parameters.Fz  Not a valid G function.z$  Using asymptotic Slater expansion.c                óf   •— t        ‰j                  D �cg c]  } ‰|dz
  dd| «      ‘Œ c}Ž S c c}w r�   )rT   r‚   )r^   r�   ró   rõ  s     €€rg   ÚEz'_check_antecedents_inversion.<locals>.E~  s/   ø€ Ü¸¿¹Ö=°1‘Y˜q 1™u a¨¨AÕ.Ò=Ð>Ð>ùÒ=s   •.c                ó"   •—  ‰‰‰ d‰z  | «      S rÜ   r’   )r^   r¥  rõ  r¦  s    €€€rg   ÚHz'_check_antecedents_inversion.<locals>.H�  s   ø€ Ù˜  ¨¨%©°Ó3Ð3rn   c                ó$   •—  ‰‰‰ d‰z  | d«      S )NrR   Tr’   ©r^   r¥  ró  r¦  s    €€€rg   ÚHpz(_check_antecedents_inversion.<locals>.Hp„  s   ø€ Ù˜e e V¨Q¨u©W°a¸Ó>Ð>rn   c                ó$   •—  ‰‰‰ d‰z  | d«      S )NrR   Fr’   rû  s    €€€rg   ÚHmz(_check_antecedents_inversion.<locals>.Hm‡  s   ø€ Ù˜e e V¨Q¨u©W°a¸Ó?Ð?rn   )r`  rÞ   rÙ   Ú_check_antecedents_inversionr  r   rù   r„   r‚   rƒ   r…   r©   ÚNaNr   r  rd  rþ   rÿ   r˜  rT   r   r+   r½   rU   )ró   rz   r^   r&  rå   rØ   rs   rª   r«   rŽ  r¡   rË  Úepsilonr  r�   r¥   r÷  rù  rü  rþ  rn  r¥  rõ  ró  r¦  s   ``                   @@@@rg   rÿ  rÿ  2  sò  ý€ ä
Ð0Ô1Ø	�
‰
€AÜ˜!˜QÓ�D€A€qØˆ1‚uÜˆÔä+¬G°A«J¸Ó:Ð:ôô$6ô ”C˜Ÿ™“Iœs 1§4¡4›y¬#¨a¯d©d«)´S¸¿¹³YÐ?Ó@�J€A€qˆ!ˆQØ
ˆa‰%�!‰)€CØ	
ˆQ‰�‰€BØ�‰8�Q‰,€CØ�‰E€EØ�‚zÜ—&‘&‰Ø	�ŠØ‰ä—%‘%ˆØ�%‰i˜‰]œS !§$¡$˜ZÑ'¬#¨q¯t©t¨*Ñ4°eÑ;€EØ�G‰G€EÜÐGØ��1�a˜˜b # uÐ-ô/äÐ.°¸%ÀÐ0GÔHð �G‰G�q˜‘sŠN˜q Ašv¨!¨qª&ÜÐ>Ô?Øô
 ×!Ñ! !§$¡$¨¯©Ó-ò ‰ˆˆ1Ø�‰E×Ó ! a£%ÜÐ.Ô/Ùðð 	ˆA‚vÜÐ5Ô6Ü¸¿¹Ö=°1‘Y˜q 1™u a¨¨AÕ.Ò=Ð>Ð>õ?ö4ö?ö@ð €Eà	Œc�!�q‘&˜!˜q™& #¤b¡&¨5¡.´B°q±DÑ"8¸%À!¹)Ù�A”cœ!Ÿ/™/¬"Ñ,¨b°1©fÑ5Ó6Ñ6Ó7ó9ð :ñ :€Eð 
Œc�!�a‘%˜1‘*˜a !™e q™j¨%°!©)°U¼RÀ¹T±\À1ÈÁ6Ø�q‘5˜1‘9œb‘. 5Ñ(¬B¨q©DÑ0Ù�Q”sœ1Ÿ?™?¬2Ñ-¨q°1©uÑ5Ó6Ñ6Ó7Ù�Q”sœAŸO™OÐ+¬BÑ.°°A±Ñ6Ó7Ñ7Ó8ó:ð ;ñ ;€Eð
 
Œc�!�q‘&˜!˜q™& %¨!¡)Ø˜7‘?¤BÑ&¨Ñ.´"°Q±$Ñ6¹¸!»ó>ð ?ñ ?€Eð 
Œc”"”S˜˜a !™e™ Q¨#¡X¨s°e¸A±g©~Ó>Ü˜˜Q™ ! a¡%™¨¨Q©°1°q±5¸!±)Ñ);Ó<ó>à˜!‘)˜U¤R¨¡T™\¨C°!©G´R©<¸%Ñ+?Ä2ÀaÁ4Ñ+GÙ�Q”sœ1Ÿ?™?¬2Ñ-¨bÑ0Ó1Ñ1Ó2Ù�Q”sœAŸO™OÐ+¬BÑ.¨rÑ1Ó2Ñ2Ó3ó	5ð 6ñ 6€Eð 
Œc�!�q‘&˜# ™' 5¨1¡9¨e°c¼"±f©n¼rÀ!¹tÑ.CØ˜‘=¤"Ñ$ uÑ,´°1±Ñ4Ù�Q”sœ1Ÿ?™?¬2Ñ-¨bÑ0Ó1Ñ1Ó2Ù�Q”sœAŸO™OÐ+¬BÑ.¨rÑ1Ó2Ñ2Ó3ó5ð 6ñ 6€Eð
 
ˆa�1‰fˆXÑ€Eô ˆuˆ:Ðùò[ >s   ÈS8c                óÚ   — t        | j                  |«      \  }}t        t        | j                  | j
                  | j                  | j                  |||z  z  «      | «      \  }} ||z  | z  S )zO
    Compute the laplace inversion integral, assuming the formula applies.
    )rÙ   rÞ   r  rP   r‚   r  r„   r  )ró   rz   r¦   r¥   r�   r  s         rg   Ú_int_inversionr  ¬  s\   € ô ˜!Ÿ*™* aÓ(�D€A€qÜœ' !§$¡$¨¯©°!·$±$¸¿¹À!ÀAÀqÁDÁ&ÓIÈAÈ2ÓN�D€A€qØˆQ‰3ˆq‰5€Lrn   c                ó&
  ‡ ‡!— ddl m}mŠ!m}mŠ  t
        si at        t
        «       t        | t        «      r´t        | j                  |«      j                  |«      \  }}t        |«      dkD  ry|d   }|j                  r&|j                  |k7  s|j                  j                   sy||k7  ryddt        | j"                  | j$                  | j&                  | j(                  ||z  «      fgdfS | }| j+                  |t,        «      } t/        | t,        «      }|t
        v �rt
        |   }	|	D �]ò  \  }
}}}| j1                  |
d¬«      }|sŒi }|j3                  «       D ]   \  }}t5        t7        |d¬«      d¬«      ||<   Œ" |}t        |t8        «      s|j+                  |«      }|d	k(  rŒ}t        |t8        t:        f«      st5        |j+                  |«      «      }t=        |«      d	k(  rŒ¼t        |t>        «      s ||«      }g }|D �]  \  }}tA        t5        |j+                  |«      j+                  t,        |«      d¬«      |«      }	 |j+                  |«      j+                  t,        |«      }tE        ||fz   Ž jG                  tH        jJ                  tH        jL                  tH        jN                  «      rŒ¯t        |j"                  |j$                  |j&                  |j(                  t5        |j                  d¬«      «      }|jQ                  ||fz   «       �Œ |s�Œï||fc S  |sytS        d
«       ˆ ˆ!fd„}|} tU        dd| «      }d„ }	  || |||d	d¬«      \  }}} |||||«      }|€gtW        dd«      }|| jX                  vrMt[        | |«      rA	  || j+                  |||z  «      |||dd	¬«      \  }}} |||||«      j+                  |d«      }|�=|jG                  tH        jJ                  tH        j\                  tH        jL                  «      rtS        d«       yt_        j`                  |«      }g }|D ]§  } | j                  |«      \  }}t        |«      dkD  rtc        d«      ‚|d   }tA        |j                  |«      \  }}||dt        |j"                  |j$                  |j&                  |j(                  t5        t7        |d¬«      d¬«      ||z  z  «      fgz  }Œ© tS        d|«       |dfS # tB        $ r Y �Œõw xY w# |$ r d}Y �Œ¡w xY w# |$ r d}Y �ŒFw xY w)aH  
    Try to rewrite f as a sum of single G functions of the form
    C*x**s*G(a*x**b), where b is a rational number and C is independent of x.
    We guarantee that result.argument.as_coeff_mul(x) returns (a, (x**b,))
    or (a, ()).
    Returns a list of tuples (C, s, G) and a condition cond.
    Returns None on failure.
    rR   )Úmellin_transformÚinverse_mellin_transformÚIntegralTransformErrorÚMellinTransformStripErrorNr   T)Úold)Úlift)Úexponents_onlyFz)Trying recursive Mellin transform method.c           
     ó�   •— 	  ‰| |||dd¬«      S # ‰$ r/ ddl m}  ‰ |t        t        | «      «      «      |||dd¬«      cY S w xY w)zÔ Calling simplify() all the time is slow and not helpful, since
            most of the time it only factors things in a way that has to be
            un-done anyway. But sometimes it can remove apparent poles. T)Ú
as_meijergÚneedevalr   )Úsimplify)rÒ   r  rY   r   )ÚFrY  rz   Ústripr  r  r  s        €€rg   Úmy_imtz_rewrite_single.<locals>.my_imt
  s_   ø€ ð
	0Ù+¨A¨q°!°UØ7;ÀdôLð Løà(ò 	0Ý/Ù+Ùœ¤ q£	Ó*Ó+¨Q°°5Ø¨$ô0ò 0ð	0ús   ƒ ‘1AÁArY  zrewrite-singlec           	     ó&  — t        | |d¬«      }|�Wddlm} |\  }}t         ||d¬«      «      }t	        ||ft        | |t        j                  t        j                  f«      df«      S t        | |t        j                  t        j                  f«      S )NT)Úonly_doubler   ©ÚhyperexpandÚnonrepsmall)Úrewrite)	Ú_meijerint_definite_4rÒ   r  Ú_my_unpolarifyr5   rS   r   rÔ   r4  )rf   rz   r    r  rm   r‡   s         rg   Úmy_integratorz&_rewrite_single.<locals>.my_integrator  s�   € Ü! ! Q°DÔ9ˆØˆ=Ý2Ø‰IˆC�Ü ¡¨S¸-Ô!HÓIˆCÜ˜c 4˜[Ü& q¨1¬a¯f©f´a·j±jÐ*AÓBÀDÐIóKð Kä˜˜AœqŸv™v¤q§z¡zÐ2Ó3Ð3rn   )Ú
integratorr  r  r�   )r  r  r  z"Recursive Mellin transform failed.zUnexpected form...z"Recursive Mellin transform worked:)2Ú
transformsr  r  r  r  Ú_lookup_tabler¿   rê   rP   rZ   rÞ   rÓ   rù   rÕ   rÖ   r+   r…  r‚   r  r„   r  r±   r^   r   rë   Úitemsr%   r&   rO  rV   rQ  rº   rÙ   Ú
ValueErrorr
   rl   r   r4  ÚComplexInfinityÚNegativeInfinityr€   r`  r  r  rÈ   r   r   r   rñ   ÚNotImplementedError)"rf   rz   Ú	recursiver  r  rî  rØ   Úf_r¦   r  r�   Útermsr‡   rˆ   r±   Úsubs_rC  rD  rm   r†   ró   r�  r  rY  r  r  r  r&  r�   rk   r¾   r¥   r  r  s"                                   @@rg   Ú_rewrite_singler(  ¼  sÜ  ù€ ÷;ó ;õ ØˆÜœ]Ô+ä�!”WÔÜ˜!Ÿ*™* aÓ(×5Ñ5°aÓ8‰ˆˆqÜˆq‹6�AŠ:ØØˆa‰DˆØ�8Š8Ø�v‰v˜Š{ !§%¡%×"3Ò"3ØØ�!ŠVØØ�A”w˜qŸt™t Q§X¡X¨q¯t©t°Q·X±X¸uÀQ¹wÓGÐHÐIÈ4ÐOÐOà	
€BØ	�‰ˆq”!‹€AÜ�”1‹€AØŒMÒÜ˜!ÑˆØ*+ó #	%Ñ&ˆG�U˜D $Ø—7‘7˜7¨�7Ó-ˆDÚØ�Ø#Ÿz™z›|ò A‘G�C˜Ü!+¬H°R¸dÔ,CØ;?ô"A�E˜#’JðAð �Ü! $¬Ô-ØŸ9™9 T›?�DØ˜5’=ØÜ! $¬¬{Ð(;Ô<Ü% d§i¡i°£oÓ6�DÜ˜dÓ# uÒ,ØÜ! %¬Ô.Ù! $›K�EØ�Ø#ó *‘F�C˜Ü'¬
°3·8±8¸D³>×3FÑ3FÄqÈ!Ó3LØBFô)HØIJóL�Bð!ØŸF™F 4›L×-Ñ-¬a°Ó3˜ô ˜r Q D™yÐ*×.Ñ.¬q¯z©z¼1×;LÑ;LÌa×N`ÑN`ÔaØ Ü §¡ a§h¡h°·±°a·h±hÜ *¨1¯:©:ÀdÔ KóM�Aà—J‘J˜r Q D™yÖ)ð*ó Ø ˜9Ò$ðG#	%ñL ØÜ
Ð6Ô7õ0ð 	€AÜˆsÐ$ aÓ(€Aò4ðÙ& q¨!¨Q¸=Ø05ÀôF‰ˆˆ5�!á�1�a˜˜EÓ"ˆð 	€yô �CÐ)Ó*ˆØ�A—N‘NÑ"¤|°A°qÔ'9ðÙ.¨q¯v©v°a¸¸1¹«~¸qÀ!Ø:GØ8<ÀuôN‘��5˜!ñ ˜1˜a  EÓ*×/Ñ/°°1Ó5�ð 	€y�A—E‘Eœ!Ÿ*™*¤a§e¡e¬Q×->Ñ->Ô?ÜÐ3Ô4ØÜ�=‰=˜Ó€DØ
€CØò 	(ˆØ�~‰~˜aÓ ‰ˆˆ1Üˆq‹6�AŠ:Ü%Ð&:Ó;Ð;Øˆa‰DˆÜ˜aŸj™j¨!Ó,‰ˆˆ1Ø��A”w˜qŸt™t Q§X¡X¨q¯t©t°Q·X±XÜ)¬(Ø#$¨4ô+1ØAEô Gà ! 1¡ñ %ó&ð 'ð (ñ 	(‰ð	(ô Ð/°Ô3Ø�ˆ9Ðøô_ &ò !Ú ð!ûð` "ò Ø‹ðûð *ò Ø“ðús7   É %S'ÍS7 Î	A T Ó'	S4Ó3S4Ó7TÔTÔTÔTc                óZ   — t        | |«      \  }}}t        |||«      }|r|||d   |d   fS y)zÿ
    Try to rewrite ``f`` using a (sum of) single G functions with argument a*x**b.
    Return fac, po, g such that f = fac*po*g, fac is independent of ``x``.
    and po = x**s.
    Here g is a result from _rewrite_single.
    Return None on failure.
    r   rR   N)rô   r(  )rf   rz   r$  r†   rò   ró   s         rg   Ú	_rewrite1r*  J  sC   € ô ˜A˜qÓ!�J€CˆˆQÜ˜˜1˜iÓ(€AÙØ�B˜˜!™˜a ™dÐ"Ð"ð 	rn   c           	     óŽ  ‡— t        | ‰«      \  }}}t        ˆfd„t        |«      D «       «      ryt        |«      }|syt	        t        |ˆfd„ˆfd„ˆfd„g«      «      }t        j                  d|«      D ]N  \  }\  }}t        |‰|«      }	t        |‰|«      }
|	sŒ&|
sŒ)t        |	d   |
d   «      }|dk7  sŒA|||	d	   |
d	   |fc S  y)
a  
    Try to rewrite ``f`` as a product of two G functions of arguments a*x**b.
    Return fac, po, g1, g2 such that f = fac*po*g1*g2, where fac is
    independent of x and po is x**s.
    Here g1 and g2 are results of _rewrite_single.
    Returns None on failure.
    c              3  ó<   •K  — | ]  }t        |‰d «      du –— Œ y­w)FN)r(  r  s     €rg   rh   z_rewrite2.<locals>.<genexpr>a  s   øè ø€ Ò
L°tŒ?˜4  EÓ*¨dÔ2Ñ
Lùs   ƒNc           	     ót   •— t        t        t        | d   ‰«      «      t        t        | d   ‰«      «      «      S ©Nr   rR   )Úmaxrù   rá   ©rª   rz   s    €rg   r{   z_rewrite2.<locals>.<lambda>g  ó.   ø€ ”#”cœ* Q q¡T¨1Ó-Ó.´´J¸qÀ¹tÀQÓ4GÓ0HÓI€ rn   c           	     ót   •— t        t        t        | d   ‰«      «      t        t        | d   ‰«      «      «      S r.  )r/  rù   ræ   r0  s    €rg   r{   z_rewrite2.<locals>.<lambda>h  r1  rn   c           	     ót   •— t        t        t        | d   ‰«      «      t        t        | d   ‰«      «      «      S r.  )r/  rù   rï   r0  s    €rg   r{   z_rewrite2.<locals>.<lambda>i  s3   ø€ ”#”cÔ0°°1±°qÓ9Ó:ÜÔ0°°1±°qÓ9Ó:ó<€ rn   ©FTrR   Fr   )
rô   r  r÷   rû   rº   r   rþ   rÿ   r(  rT   )rf   rz   r†   rò   ró   r  r$  Úfac1Úfac2r†  r‡  r‡   s    `          rg   Ú	_rewrite2r7  X  sÛ   ø€ ô ˜A˜qÓ!�J€CˆˆQÜ
Ó
L¼yÈ»|Ô
LÔLØÜ˜!Ó€AÙØÜŒW�QÛIÛIó	<ð=ó >ó 	?€Aô $-×#4Ñ#4°]ÀAÓ#Fò 3Ñˆ	‘<�D˜$Ü˜T 1 iÓ0ˆÜ˜T 1 iÓ0ˆÚ’"Ü�r˜!‘u˜b ™eÓ$ˆDØ�u‹}Ø˜B  1¡ r¨!¡u¨dÐ2Ò2ñ3rn   c                óv  — t        | «      } g }t        t        | |«      t        j                  hz  t
        ¬«      D ]c  }t        | j                  |||z   «      |«      }|sŒ%|j                  |||z
  «      }t        |t        t        «      r|j                  |«       Œa|c S  | j                  t        «      rnt        d«       t        t!        | «      |«      }|rLt#        |t$        «      s+ddlm}  |t+        |«      |j-                  t.        «      «      S |j1                  |«       |rt3        t5        |«      «      S y)a#  
    Compute an indefinite integral of ``f`` by rewriting it as a G function.

    Examples
    ========

    >>> from sympy.integrals.meijerint import meijerint_indefinite
    >>> from sympy import sin
    >>> from sympy.abc import x
    >>> meijerint_indefinite(sin(x), x)
    -cos(x)
    rÆ   ú*Try rewriting hyperbolics in terms of exp.r   ©ÚcollectN)r   rÌ   rï   r   rÔ   r   Ú_meijerint_indefinite_1r±   rc   rO   rP   r€   rl   r2   r`  Úmeijerint_indefiniter1   rê   rº   Úsympy.simplify.radsimpr;  r   rã   r+   ÚextendÚnextr   )rf   rz   Úresultsr�   rm   Úrvr;  s          rg   r=  r=  u  s  € ô 	�‹
€AØ€GÜÔ*¨1¨aÓ0´A·F±F°8Ñ;ÔAQÔRò ˆÜ% a§f¡f¨Q°°A±Ó&6¸Ó:ˆÙØØ�h‰h�q˜!˜a™%Ó ˆÜ�”UœGÔ$Ø�N‰N˜3ÕàŠJðð 	‡u�uÔÔ ÜÐ;Ô<Ü!Ü'¨Ó*¨Aó/ˆáÜ˜b¤$Ô'Ý:Ùœ|¨BÓ/°·±¼#³Ó?Ð?Ø�N‰N˜2ÔÙÜ”G˜GÓ$Ó%Ð%ð rn   c           	     óx  ‡‡— t        d| d‰«       ddlm}m} t	        | ‰«      }|€y|\  }}}}t        d|«       t
        j                  }	|D �]Þ  \  }
}}t        |j                  ‰«      \  }}t        |‰«      \  }}||z  }||
z  ‰d|z   z  z  |z  }|dz   |z  Št        dd	t
        j                  «      }ˆfd
„}t        d„  ||j                  «      D «       «      rit        t        |j                  «      t        |j                   «      d‰z
  gz   t        |j                  «      ‰ gz   t        |j"                  «      |«       }ngt        t        |j                  «      d‰z
  gz   t        |j                   «      t        |j                  «      t        |j"                  «      ‰ gz   |«      }|j$                  rA| j'                  ‰d«      j)                  t
        j*                  t
        j,                  «      sd}nd} ||j'                  ||‰|z  z  «      |¬«      }|	 |||z  d¬«      z  }	�Œá ˆfd„}t/        |	d¬«      }	|	j0                  r;g }|	j2                  D ]  \  }}t5         ||«      «      }|||fgz  }Œ  t7        |ddiŽ}	nt5         ||	«      «      }	t7        |	t5        |«      ft9        | ‰«      df«      S )z0 Helper that does not attempt any substitution. z,Trying to compute the indefinite integral ofÚwrtr   )r  r‚  Nz could rewrite:rR   r¦   zmeijerint-indefinitec                ó4   •— | D �cg c]  }|‰z   ‘Œ	 c}S c c}w rb   r’   )rª   r�   rË  s     €rg   r  z#_meijerint_indefinite_1.<locals>.trº  s   ø€ Ø%&Ö' �A˜“GÒ'Ð'ùÒ'r€  c              3  óH   K  — | ]  }|j                   xr |d k  dk(  –— Œ y­w)r   TN)r˜  )rd   r¥   s     rg   rh   z*_meijerint_indefinite_1.<locals>.<genexpr>¼  s%   è ø€ ÒC°Qˆq�|‰|Ò0  a¡¨DÑ 0Ó0ÑCùs   ‚ ")ÚplaceTrƒ  c                ó~   •— t        t        | «      d¬«      } t        j                  | j	                  ‰«      d   «      S )aÁ  This multiplies out superfluous powers of x we created, and chops off
        constants:

            >> _clean(x*(exp(x)/x - 1/x) + 3)
            exp(x)

        cancel is used before mul_expand since it is possible for an
        expression to have an additive constant that does not become isolated
        with simple expansion. Such a situation was identified in issue 6369:

        Examples
        ========

        >>> from sympy import sqrt, cancel
        >>> from sympy.abc import x
        >>> a = sqrt(2*x + 1)
        >>> bad = (3*x*a**5 + 2*x - a**5 + 1)/a**2
        >>> bad.expand().as_independent(x)[0]
        0
        >>> cancel(bad).expand().as_independent(x)[0]
        1
        F)ÚdeeprR   )r   rY   r   Ú
_from_argsÚas_coeff_add)rm   rz   s    €rg   Ú_cleanz'_meijerint_indefinite_1.<locals>._cleanÎ  s4   ø€ ô. œ ›¨5Ô1ˆÜ�~‰~˜c×.Ñ.¨qÓ1°!Ñ4Ó5Ð5rn   )ÚevaluaterM  F)r`  rÒ   r  r‚  r*  r   rÔ   rÙ   rÞ   r  r¼   r  r„   rP   rº   r‚   r  r  Úis_extended_nonnegativer±   rl   r   r!  r6   r`   rk   r  r5   rS   )rf   rz   r  r‚  rö   r†   rò   Úglr‡   rm   r  rY  ró   r�   r¥   r&  r¾   Úfac_r¦   r  r    rG  rL  rM  rå   rË  s    `                       @rg   r<  r<  š  s‘  ù€ ä
Ð9¸1¸eÀQÔGß5ä	�1�a‹€BØ	€zààÑ€CˆˆR�Ü
Ð˜bÔ!Ü
�&‰&€CØó %-‰ˆˆ1ˆaÜ˜aŸj™j¨!Ó,‰ˆˆ1Ü˜b !Ó$‰ˆˆ1Ø	ˆQ‰ˆð �Q‰w˜˜Q ™U™Ñ# aÑ'ˆØ�1‰u�a‰iˆô �3Ð.´·±Ó6ˆô	(äÑC¹"¸Q¿T¹T»(ÔCÔCÜÜ�Q—T‘T“
œD §¡›N¨a°©e¨WÑ4´d¸1¿4¹4³jÀSÀDÀ6Ñ6IÌ4ÐPQ×PXÑPXË>Ð[\ó^ð ^‰Aô Ü�Q—T‘T“
˜a ™e˜WÑ$¤d¨1¯8©8£n´d¸1¿4¹4³jÄ$ÀqÇxÁxÃ.ÐUXÐTXÐSYÑBYÐ[\ó^ˆAð ×$Ò$¨Q¯V©V°A°q«\×-=Ñ-=¼a¿e¹eÄQ×EVÑEVÔ-WØ‰EàˆEÙ˜Ÿ™˜q ! A q¡D¡&Ó)°Ô7ˆð 	‰y˜˜a™ tÔ,Ñ,ŠðK%-ôN6ô4 ˜ tÔ
,€CØ
×ÒØˆØ—H‘Hò 	 ‰DˆAˆqÜ™v a›yÓ)ˆAØ˜˜A˜�xÑ‰Gð	 ô ˜Ð1¨5Ñ1‰ä™V C›[Ó)ˆÜ�cœ>¨$Ó/Ð0´8¸A¸q³>À4Ð2HÓIÐIrn   c                óv	  — t        d| |||f«       t        | «      } | j                  t        «      rt	        d«       y| j                  t
        «      rt	        d«       y| |||f\  }}}}t        d«      }| j                  ||«      } |}||k(  rt        j                  dfS g }	|t        j                  u r3|t        j                  ur!t        | j                  || «      || | «      S |t        j                  u �rt	        d«       t        | |«      }
t	        d|
«       t        |
t        d¬	«      t        j                  gz   D ]º  }t	        d
|«       |j                   st	        d«       Œ't#        | j                  |||z   «      |«      }|€t	        d«       ŒTt#        | j                  |||z
  «      |«      }|€t	        d«       Œ�|\  }}|\  }}t%        t'        ||«      «      }|dk(  rt	        d«       Œ±||z   }||fc S  �n|t        j                  u r't        | ||t        j                  «      }|d    |d   fS ||ft        j                  t        j                  fk(  r7t#        | |«      }|�r¤t)        |d   t*        «      r|	j-                  |«       �n~|S |t        j                  u r…t        | |«      D ]v  }||z
  dk\  dk(  sŒt        d|«       t#        | j                  |||z   «      t/        ||z   |z
  «      z  |«      }|sŒOt)        |d   t*        «      r|	j-                  |«       Œt|c S  | j                  |||z   «      } ||z
  }d}|t        j                  urit1        t        j2                  t5        |«      z  «      }t7        |«      }| j                  |||z  «      } | t/        ||z
  «      |z  z  } t        j                  }t	        d||«       t	        d| «       t#        | |«      }|r't)        |d   t*        «      r|	j-                  |«       n|S |j                  t8        «      rt	        d«       t        t;        |«      |||«      }|r[t=        |t>        «      s:ddl m!}  |tE        |d   «      |d   jG                  t0        «      «      f|dd z   }|S |	jI                  |«       |	rtK        tM        |	«      «      S y)aà  
    Integrate ``f`` over the interval [``a``, ``b``], by rewriting it as a product
    of two G functions, or as a single G function.

    Return res, cond, where cond are convergence conditions.

    Examples
    ========

    >>> from sympy.integrals.meijerint import meijerint_definite
    >>> from sympy import exp, oo
    >>> from sympy.abc import x
    >>> meijerint_definite(exp(-x**2), x, -oo, oo)
    (sqrt(pi), True)

    This function is implemented as a succession of functions
    meijerint_definite, _meijerint_definite_2, _meijerint_definite_3,
    _meijerint_definite_4. Each function in the list calls the next one
    (presumably) several times. This means that calling meijerint_definite
    can be very costly.
    z$Integrating %s wrt %s from %s to %s.z+Integrand has DiracDelta terms - giving up.Nz5Integrand has Singularity Function terms - giving up.rz   Tz  Integrating -oo to +oo.z  Sensible splitting points:)rÄ   Úreversez  Trying to split atz  Non-real splitting point.z'  But could not compute first integral.z(  But could not compute second integral.Fz)  But combined condition is always false.r   rR   zTrying x -> x + %szChanged limits tozChanged function tor9  r:  )'rd  r   rl   r?   r`  rQ   r   r±   r   rÔ   r"  r4  Úmeijerint_definiterï   rÌ   r   Úis_extended_realÚ_meijerint_definite_2r<  rT   rc   rP   r€   r@   r+   r½   r"   r#   r2   r1   rê   rº   r>  r;  r   rã   r?  r@  r   )rf   rz   r�   r¥   r%  Úx_Úa_Úb_r  rA  rî   r¾   Úres1Úres2Úcond1Úcond2r‡   rm   ÚsplitrÍ  rB  r;  s                         rg   rS  rS  ô  s_  € ô< Ð2°Q¸¸1¸a°LÔAÜ�‹
€AØ‡u�uŒZÔÜÐ<Ô=Øà‡u�uÔ Ô!ÜÐFÔGØà˜˜1˜a�Z�N€BˆˆB�ô 	ˆc‹
€AØ	�‰ˆq�!‹€AØ	€AàˆA‚vÜ—‘˜ˆ~Ðà€GØŒA×ÑÑ 1¬A¯J©JÑ#6Ü! !§&¡&¨¨Q¨B£-°°Q°B¸¸Ó;Ð;à	
Œa× Ñ Ò	 äÐ*Ô+Ü*¨1¨aÓ0ˆ	ÜÐ-¨yÔ9Ü˜	Ô'7ÀÔFÌ!Ï&É&ÈÑQò 	ˆAÜÐ)¨1Ô-Ø×%Ò%ÜÐ4Ô5ØÜ(¨¯©°°1°q±5Ó)9¸1Ó=ˆDØˆ|ÜÐ@ÔAØÜ(¨¯©°°1°q±5Ó)9¸1Ó=ˆDØˆ|ÜÐAÔBØØ‰KˆD�%Ø‰KˆD�%ÜœS ¨Ó.Ó/ˆDØ�uŠ}ÜÐBÔCØØ˜‘+ˆCØ˜�9Òò)	ð, 
Œa�j‰j‰Ü   A q¬!¯*©*Ó5ˆØ�A‘ˆw˜˜A™ˆÐà
ˆQˆ”A—F‘FœAŸJ™JÐ'Ò	'ä# A qÓ)ˆÚÜ�C˜‘FœGÔ$Ø—‘˜sÖ#à�
ð ”—
‘
‰?Ü/°°1Ó5ò 	'�Ø˜‘I ‘N tÓ+ÜÐ0°%Ô8Ü/°·±°q¸!¸e¹)Ó0DÜ1:¸1¸u¹9Àq¹=Ó1Iñ1JØKLóN�CâÜ  A¡¬Ô0Ø#ŸN™N¨3Õ/à#&šJð	'ð �F‰F�1�a˜!‘eÓˆØ�‰EˆØˆØ”A—J‘JÑÜ”a—o‘o¤c¨!£fÑ,Ó-ˆCÜ�A“ˆAØ—‘�q˜#˜a™%Ó ˆAØ”˜1˜q™5Ó! #Ñ%Ñ%ˆAÜ—
‘
ˆAäÐ" A qÔ)ÜÐ$ aÔ(Ü# A qÓ)ˆÙÜ�C˜‘FœGÔ$Ø—‘˜sÕ#à�
Ø	‡v�vÔ Ô!ÜÐ;Ô<ÜÜ'¨Ó+¨R°°Ró9ˆáÜ˜b¤$Ô'Ý:Ùœl¨2¨a©5Ó1°2°a±5·;±;¼sÓ3CÓDÐFÈÈAÈBÈÑO�Ø�	Ø�N‰N˜2ÔÙÜ”G˜GÓ$Ó%Ð%ð rn   c                óæ  — | dfg}|d   d   }|h}t        |«      }||vr||dfgz  }|j                  |«       t        |«      }||vr||dfgz  }|j                  |«       |j                  t        t
        «      r1t        t        |«      «      }||vr||dfgz  }|j                  |«       |j                  t        t        «      r+ddl	m
}  ||«      }||vr||dfgz  }|j                  |«       |S )	z6 Try to guess sensible rewritings for integrand f(x). zoriginal integrandrŸ   r   r   r   zexpand_trig, expand_mul)Úsincos_to_sumztrig power reduction)r   rŠ   r   rl   r:   r2   r   r7   r8   Úsympy.simplify.fur_  )rf   rz   rm   ÚorigÚsawÚexpandedr_  Úreduceds           rg   Ú_guess_expansionre    s  € àÐ#Ð$Ð
%€Càˆr‰7�1‰:€DØˆ&€CÜ˜$Ó€HØ�sÑØ�˜<Ð(Ð)Ñ)ˆØ�‰�Ôä�d‹|€HØ�sÑØ�˜8Ð$Ð%Ñ%ˆØ�‰�Ôà‡x�xÔ%Ô'9Ô:Üœk¨$Ó/Ó0ˆØ˜3ÑØ�XÐ8Ð9Ð:Ñ:ˆCØ�G‰G�HÔà‡x�x””SÔÝ3Ù Ó%ˆØ˜#ÑØ�WÐ4Ð5Ð6Ñ6ˆCØ�G‰G�GÔà€Jrn   c                óÜ   — t        dd| d¬«      }| j                  ||«      } |}| dk(  rt        j                  dfS t	        | |«      D ]#  \  }}t        d|«       t        ||«      }|sŒ!|c S  y)a€  
    Try to integrate f dx from zero to infinity.

    The body of this function computes various 'simplifications'
    f1, f2, ... of f (e.g. by calling expand_mul(), trigexpand()
    - see _guess_expansion) and calls _meijerint_definite_3 with each of
    these in succession.
    If _meijerint_definite_3 succeeds with any of the simplified functions,
    returns this result.
    rz   zmeijerint-definite2T)Úpositiver   ÚTryingN)r  r±   r   rÔ   re  r`  Ú_meijerint_definite_3)rf   rz   Údummyró   Úexplanationrm   s         rg   rU  rU  Ÿ  sw   € ô  �3Ð-¨q¸4Ô@€EØ	�‰ˆq�%Ó€AØ€AàˆA‚vÜ�v‰v�tˆ|Ðä*¨1¨aÓ0ò ‰ˆˆ;Üˆx˜Ô%Ü# A qÓ)ˆÚØŠJñ	rn   c                óJ  — t        | |«      }|r
|d   dk7  r|S | j                  rzt        d«       | j                  D �cg c]  }t        ||«      ‘Œ }}t	        d„ |D «       «      r9g }t
        j                  }|D ]  \  }}||z  }||gz  }Œ t        |Ž }|dk7  r||fS yyyc c}w )z²
    Try to integrate f dx from zero to infinity.

    This function calls _meijerint_definite_4 to try to compute the
    integral. If this fails, it tries using linearity.
    rR   Fz#Expanding and evaluating all terms.c              3  ó$   K  — | ]  }|d u–— Œ
 y ­wrb   r’   )rd   r    s     rg   rh   z(_meijerint_definite_3.<locals>.<genexpr>Ê  s   è ø€ Ò+ ˆq˜Œ}Ñ+ùs   ‚N)r  Úis_Addr`  rk   rj   r   rÔ   rT   )rf   rz   rm   ró   Úressrn  r    r¾   s           rg   ri  ri  ½  sÁ   € ô    1Ó
%€CÙ
ˆs�1‰v˜ŠØˆ
Ø‡x‚xÜÐ4Ô5Ø56·V±VÖ<°Ô% a¨Õ+Ð<ˆÐ<ÜÑ+ dÔ+Ô+ØˆEÜ—&‘&ˆCØò ‘��1Ø�q‘�Ø˜!˜‘‘ðô �U�ˆAØ�EŠzØ˜A�v�ð ð ,ð ùâ<s   ¾B c                ó*   — t        t        | «      «      S rb   )rQ  r%   )rf   s    rg   r  r  Õ  s   € Ü”j “mÓ$Ð$rn   c                óâ  — ddl m} t        d| «       |sÊt        | |d¬«      }|�º|\  }}}}t        d|||«       t        j
                  }	|D ]U  \  }
}} |
dk(  rŒt        ||
z  |||z  z  | |«      \  }
} |	|
t        | |«      z  z  }	t        |t        | |«      «      }|dk(  sŒU n t        |«      }|dk(  rt        d«       nt        d	|	«       t         ||	«      «      |fS t        | |«      }|��d
D ]ü  }|\  }}}}}t        d||||«       t        j
                  }	|D ]ˆ  \  }}}|D ]{  \  }}}t        ||z  |z  ||||z   z  z  ||||«      }|€t        d«          y|\  }
}}t        d|
||«       t        |t        |||«      «      }|dk(  r n|	|
t        |||«      z  z  }	Œ} Œˆ n t        |«      }|dk(  rt        d|«       ŒÔt        d|	f«       |r|	|fc S t         ||	«      «      |fc S  yy)a�  
    Try to integrate f dx from zero to infinity.

    Explanation
    ===========

    This function tries to apply the integration theorems found in literature,
    i.e. it tries to rewrite f as either one or a product of two G-functions.

    The parameter ``only_double`` is used internally in the recursive algorithm
    to disable trying to rewrite f as a single G-function.
    r   r  ÚIntegratingF)r$  Nú#Could rewrite as single G function:úBut cond is always False.z&Result before branch substitutions is:r4  z!Could rewrite as two G functions:zNon-rational exponents.zSaxena subst for yielded:z&But cond is always False (full_pb=%s).z)Result before branch substitutions is: %s)rÒ   r  r`  r*  r   rÔ   r[  rz  rT   rf  r  r7  r•  rã  rç  rd  )rf   rz   r  r  rö   r†   rò   ró   r‡   rm   r  rY  rU  r†  r‡  r‘  Ús1Úf1r’  Ús2Úf2r    Úf1_Úf2_s                           rg   r  r  Ù  sX  € õ +ä
ˆ=˜!ÔáÜ�q˜! uÔ-ˆØˆ>Ø!ÑˆC��Q˜ÜÐ8¸#¸rÀ1ÔEÜ—&‘&ˆCØò ‘��1�aØ˜’6ØÜ(¨¨Q©°°1°a±4±¸¸AÓ>‘��1Ø�qœ 1 a›Ñ(Ñ(�Ü˜4Ô!5°a¸Ó!;Ó<�Ø˜5“=Ùðô " $Ó'ˆDØ�uŠ}ÜÐ2Õ3äÐ?ÀÔEÜ%¡k°#Ó&6Ó7¸Ð=Ð=ô 
�1�a‹€BØ	�~Ø$ò 	>ˆGØ$&Ñ!ˆC��R˜˜TÜÐ6¸¸RÀÀRÔHÜ—&‘&ˆCØ ò ‘
��B˜Ø"$ò ‘J�B˜˜BÜ'¨¨B©¨r©	°2°a¸"¸r¹'±l±?Ø(*¨B°°7ó<�Aà�yÜÐ8Ô9ÛØ"#‘K�A�s˜CÜÐ6¸¸3ÀÔDÜ˜tÔ%7¸¸SÀ!Ó%DÓE�DØ˜u’}ÙØ˜1œW S¨#¨qÓ1Ñ1Ñ1‘Cðð Ùðô  " $Ó'ˆDØ�uŠ}ÜÐ@À'ÕJäÐCÀcÀWÔMÙØ ˜9Ò$Ü%¡k°#Ó&6Ó7¸Ð=Ò=ñ7	>ð rn   c           	     óè  — | }|}t        dd¬«      }| j                  ||«      } t        d| «       t        | |«      st        d«       yt        j
                  }| j                  rt        | j                  «      }nt        | t        «      r| g}nd}|�r\g }g }|�rE|j                  «       }	t        |	t        «      rmt        |	«      }
|
j                  r||
j                  z  }ŒJ	 t        |	j                  d   |«      \  }}|dk(  r|j                  «       nÆ|j                  |	«       n´|	j                   r—t        |	«      }
|
j                  r||
j                  z  }ŒÃ||	j"                  j$                  vrF	 t        |	j                  |«      \  }}|dk(  r'|j                  t'        |	j"                  «      z  «       |j                  |	«       n|j                  |	«       |r�ŒEt)        |Ž }t+        |Ž } || j$                  vrnt        d	| |«       t-        t/        |«      d«      }|d
k(  rt        d«       y| t1        ||z   «      z  }t        d||«       t3        |j                  ||«      |f«      S t5        | |«      }|��U|\  }}}}t        d|||«       t        j
                  }|D ]P  \  }}} t7        ||z  |||z  z  | |«      \  }} ||t9        | ||«      z  z  }t;        |t=        | |«      «      }|d
k(  sŒP n t?        |«      }|d
k(  rt        d«       yt        d|«       ddl m!} t?         ||«      «      }|jE                  tF        «      s|tG        |«      z  }|j                  |||z   «      }t        |tH        «      s|j                  |||z   «      }ddl%m&} t3        |j                  ||«      |f ||j                  ||«      ||d«      df«      S y# t        $ r d}Y �Œàw xY w# t        $ r d}Y �Œdw xY w)aê  
    Compute the inverse laplace transform
    $\int_{c+i\infty}^{c-i\infty} f(x) e^{tx}\, dx$,
    for real c larger than the real part of all singularities of ``f``.

    Note that ``t`` is always assumed real and positive.

    Return None if the integral does not exist or could not be evaluated.

    Examples
    ========

    >>> from sympy.abc import x, t
    >>> from sympy.integrals.meijerint import meijerint_inversion
    >>> meijerint_inversion(1/x, x, t)
    Heaviside(t)
    r¦   Trƒ  zLaplace-invertingzBut expression is not analytic.Nr   rR   z.Expression consists of constant and exp shift:Fz3but shift is nonreal, cannot be a Laplace transformz1Result is a delta function, possibly conditional:rs  rt  z"Result before branch substitution:r  )ÚInverseLaplaceTransform)'r   r±   r`  r   r   rÔ   Úis_Mulrº   rk   rê   r+   Úpopr   rÙ   rÎ   r€   rÕ   rÖ   rÈ   r-   r   r   r   r!   r?   r5   r*  rê  r  rT   rÿ  r  rÒ   r  rl   r@   rO  r  r|  )rf   rz   r¦   r%  Út_Úshiftrk   rM  Úexponentialsr"   rI  r�   r¥   r‡   rm   rö   r†   rò   ró   r  rY  r  r|  s                          rg   Úmeijerint_inversionr‚  !  s»  € ð$ 
€BØ	
€BÜˆc˜Ô€AØ	�‰ˆr�1‹€AÜ
Ð Ô"Ü˜˜1ÔÜÐ0Ô1Øô �F‰F€Eà‡x‚xÜ�A—F‘F‹|‰Ü	�A”sÔ	Øˆs‰àˆâØˆØˆÚØ—(‘(“*ˆCÜ˜#œsÔ#Ü˜c“{�Ø—;’;Ø˜DŸI™IÑ%�DØðÜ)¨#¯(©(°1©+°qÓ9‘D�A�qð ˜’6Ø ×'Ñ'¨Õ*à—N‘N 3Õ'Ø—’Ü˜c“{�Ø—;’;Ø˜DŸI™IÑ%�DØØ˜CŸH™H×1Ñ1Ñ1ðÜ-¨c¯g©g°qÓ9™˜˜1ð ˜A’vØ$×+Ñ+¨A¬c°#·(±(«m©OÔ<Ø—‘˜sÕ#à—‘˜sÔ#ó; ô< �\Ð"ˆÜ�ˆMˆà�—‘ÑÜÐ?ÀÀEÔJÜ”"�U“)˜QÓˆØ�5Š=ÜÐHÔIØØ”
˜1˜u™9Ó%Ñ%ˆÜÐBÀCÈÔNä˜#Ÿ(™( 1 b›/¨4Ð0Ó1Ð1ä	�1�a‹€BØ	�~ØÑˆˆR��DÜÐ4°c¸2¸qÔAÜ�f‰fˆØò 	‰GˆAˆq�!Ü% c¨!¡e¨R°°1±©W°a¸Ó;‰DˆAˆqØ�1”^ A q¨!Ó,Ñ,Ñ,ˆCÜ�tÔ9¸!¸QÓ?Ó@ˆDØ�u‹}Ùð	ô ˜dÓ#ˆØ�5Š=ÜÐ.Õ/äÐ7¸Ô=Ý2Ü ¡¨SÓ!1Ó2ˆCØ—7‘7œ9Ô%Ø”y “|Ñ#�Ø—(‘(˜1˜a %™iÓ(ˆCÜ˜d¤DÔ)Ø—y‘y  A¨¡IÓ.�Ý;Ü˜cŸh™h q¨"›o¨tÐ4Ù5°b·g±g¸aÀ³nÀaÈÈTÓRÐTXÐYó[ð [ð/ øôI +ò Ø“Aðûô /ò Ø›ðús$   Ã"O Å3O" ÏOÏOÏ"O1Ï0O1)rf   r   rz   r   rÇ   ztuple[type[Basic], ...]r¹   )F)²rÏ   Ú
__future__r   rþ   Úsympyr   Ú
sympy.corer   r   Úsympy.core.addr   Úsympy.core.basicr   Úsympy.core.cacher	   Úsympy.core.containersr
   Úsympy.core.exprtoolsr   Úsympy.core.functionr   r   r   r   r   Úsympy.core.mulr   Úsympy.core.intfuncr   Úsympy.core.numbersr   r   Úsympy.core.relationalr   r   r   Úsympy.core.sortingr   r   Úsympy.core.symbolr   r   r   r   Úsympy.core.sympifyr   Ú(sympy.functions.combinatorial.factorialsr   Ú$sympy.functions.elementary.complexesr    r!   r"   r#   r$   r%   r&   r'   r(   r)   r*   Ú&sympy.functions.elementary.exponentialr+   r,   r-   Ú#sympy.functions.elementary.integersr.   Ú%sympy.functions.elementary.hyperbolicr/   r0   r1   r2   Ú(sympy.functions.elementary.miscellaneousr4   Ú$sympy.functions.elementary.piecewiser5   r6   Ú(sympy.functions.elementary.trigonometricr7   r8   r9   r:   Úsympy.functions.special.besselr;   r<   r=   r>   Ú'sympy.functions.special.delta_functionsr?   r@   Ú*sympy.functions.special.elliptic_integralsrA   rB   Ú'sympy.functions.special.error_functionsrC   rD   rE   rF   rG   rH   rI   rJ   rK   rL   rM   Ú'sympy.functions.special.gamma_functionsrN   Úsympy.functions.special.hyperrO   rP   Ú-sympy.functions.special.singularity_functionsrQ   Ú	integralsrS   Úsympy.logic.boolalgrT   rU   rV   rW   rX   Úsympy.polysrY   rZ   Úsympy.utilities.iterablesr[   Úsympy.utilities.miscr\   r`  r]   rd  r^   rc   r¿   Úsympy.utilities.timeutilsrÀ   Útimeitr   r   rÎ   rÙ   rá   ræ   rï   rô   r÷   rû   r  r  r  r  Ú__annotations__r  r  r   r<  rQ  rV  r[  rf  rz  r•  rã  rç  rê  rÿ  r  r  r(  r*  r7  r=  r<  rS  re  rU  ri  r  r  r‚  r’   rn   rg   ú<module>rª     s'  ðòõ8 #Û å ß Ý Ý "Ý $Ý 'Ý -÷8õ 8å Ý #ß +ß :Ñ :ß 8ß :Ó :Ý &Ý >÷÷ ÷ ñ ÷ GÑ FÝ 7÷9ó 9å 9ß J÷ó ç MÓ Mß Iß M÷6÷ 6÷ 6ñ 6å 9ß 8Ý MÝ ß JÕ Jß &Ý 9Ý 0Ý 2ñ 
ˆ#ƒJ€òòJPõb /Ù	�)Ó	€ó	Nô	˜*ô 	òHòDòBIò
!òH$òNò.Iò>0ò QòDð4 +-€Ð
&Ó ,ò	ò#òOóyòv"ó	òó*mò`&ó<F.òRjò`	2ò:	Qòwòtð €ð 	ØòIó ó 	ðIóX#ò3ò:"&òJWJðt ñG&ó ðG&òTò@ò<ò0%ð òD>ó ðD>óNn[rn   