
    g?                     :   d Z ddlmZmZ ddlmZmZmZmZm	Z	m
Z
mZmZmZmZmZmZmZmZmZmZ ddlmZ ddlmZ 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%  ed          \  Z&Z'Z(Z)Z*d Z+d Z,d Z-d Z.d Z/d Z0d Z1d Z2d Z3d Z4d Z5d Z6d Z7d Z8d Z9dS )z?Most of these tests come from the examples in Bronstein's book.    )DifferentialExtension
derivation)prde_normal_denomprde_special_denomprde_linear_constraintsconstant_system	prde_spdeprde_no_cancel_b_largeprde_no_cancel_b_smalllimited_integrate_reducelimited_integrate
is_deriv_kis_log_deriv_k_t_radicalparametric_log_deriv_heu!is_log_deriv_k_t_radical_in_fieldparam_poly_rischDEparam_rischDEprde_cancel_liouvillian)
PolyMatrix)Rational)S)symbols)QQ)Poly)xtnzt:4 kc                     t          dt          dt                    t          dt          dz  z   t                    gi          } t          dt                    }t          t          t                    }t          t          t                    t          dt          dz  z   t                    ft          dt                    t          t          t          t          dz  z  z   t                    fg}t	          ||||           t          t          t          d          t          dt          d          t          dt          d          ft          t          t          z  t          d          t          t          dz  dz   t          d          ft          dt          d          t          t          dz  dz   t          d          fgt          dt          d          fk    sJ t          t          t                    t          t          dz  dt          z  z   dz   t                    ft          t          t          z  t                    t          t          dz  dt          z  z   dz   t                    ft          t          t          dz  z  t                    t          t          dz  dt          z  z   dz   t                    fg}t          dt          dt                    t          t          t                    gi          } t	          t          t          t                    t          dt                    ||           t          t          dz   t                    t          dt          z   t          z  t          z   t                    t          dt          d          ft          t          t                    t          dt                    ft          t          t          z  t                    t          dt          d          ft          t          t          dz  z  t                    t          dt          d          fgt          t          dz   t                    fk    sJ d S )	ND      	extensionZZ(x)domainZZ[x])r   r   r   r   r   )DEfafdGs       [/var/www/html/ai-engine/env/lib/python3.11/site-packages/sympy/integrals/tests/test_prde.pytest_prde_normal_denomr.      sL   	#Q

DQT1<M<M/N)O	P	P	PB	aB	aB
q!**d1q!t8Q''	(41::tA!Q$J7J7J*KLARQ++	a7	#	#	#d1a&@&@&@$q!C C C & $QqS!G < < <	adQh'	*	*	* ,.21a.H.H.H!Q$(Ag....0126q!G2L2L2L	NN N N N
 q!**d1a4!A#:>1--	.ac1QTAaCZ!^Q1!#'!Q$??DA!a4K4K"L	NA	#Q

DAJJ/G)H	I	I	IBT!QZZaQ;;	a!eQ$QzA~q1141W3M3M3MNRVWXZ[R\R\Q

R1Q3llDAg$>$>$>?$qAvq//Q'"""B$Q%&*1q5!nn	66 6 6 6 6 6    c            
         t          t          dz   t                    } t          t          dz  t                    }t          dt                    }t          t          t                    t          dt                    ft          t          dz  t                    t          dt                    ft          t          dz  t                    t          dt                    fg}t          dt          dt                    t          t          t                    gi          }t	          | ||||          t          t          dz   t                    t          t          dz  t                    t          t          t                    t          dt                    ft          t          dz  t                    t          dt                    ft          t          dz  t                    t          dt                    fgt          dt                    fk    sJ t          t          t                    t          dt                    ft          dt                    t          t          t                    fg}t	          t          dt                    t          t          dz  t                    t          dt                    ||          t          dt                    t          t          dz  dz
  t                    t          t          dz  t                    t          dt                    ft          dt                    t          dt                    fgt          t          t                    fk    sJ t          dt          dt                    t          dt          z  t
          z  t
                    gi          }|                                 t          t          t                    t          t          dz  t                    ft          dt          z  t                    t          t          t                    fg}t	          t          dt          z  t          z  dz   t                    t          t          dz  dt          dz  z  t          z  z   t                    t          t          dz  dz   t                    ||          t          dt          z  t          z  dz   t                    t          dt          d	
          t          t          t                    t          t          dz  t                    ft          dt          z  t                    t          t          t                    fgt          dt                    fk    sJ t          dt          dt                    t          t          dz  dz   dz  t          z  t                    gi          }t          t          t          z   t                    t          t          t          z  t                    ft          dt          z  t                    t          t          dz  t                    fg}t	          t          dt          z  t          z  dz   t                    t          t          dz  dt          dz  z  t          z  z   t                    t          t          dz  t                    ||          t          dt          z  t          z  dz   t                    t          dt          d	
          t          t          t          z   t                    t          t          t          z  t                    ft          dt          z  t          t                    t          t          dz  t          t                    fgt          dt                    fk    sJ t	          t          t          dz   t                    t          t          dz  t                    t          t          dz  t                    ||          t          t          dz   t                    t          dt          d	
          t          t          t          z   t                    t          t          t          z  t                    ft          dt          z  t          t                    t          t          dz  t          t                    fgt          dt                    fk    sJ d S )Nr    r!      r   r"      r   r(   r%   )r   r   r   r   r   t0decrement_level)ababdr,   r)   s        r-   test_prde_special_denomr9   )   s   QUAA	adAB	aB
q!**d1ajj	!DAqMM41::#>adAPTUVXYPZPZ@[\A	#Q

DAJJ/G)H	I	I	IBaRB//	a!eQadA$q!**d1ajj)A	adAQ

#d1a4mmT!QZZ%@)BCG1::	OO O O O q!**d1ajj	!DAJJQ

#;<Ad1ajj$q!tQ--aQKK	aT!Q$(A&&$q!tQ--a)D	aT!QZZ )"#'1::	// / / / 
#Q

DAb"<M<M/N)O	P	P	PB
q!**d1a4mm	$tAaC||T!QZZ&@AAd1Q3q519a00$q!ta1fQh2J2JDQRTUQUXYQY[\L]L]_`bdee	ac!eai		T!Qw77741::tAqDRS}}:U	ac1tAqzz":$%)!QZZ	11 1 1 1 
#Q

D!Q$(AaQR<S<S/T)U	V	V	VB
q1ua..$qsA,,	'$qsA,,QT1)FGAd1Q3q519a00$q!ta1fQh2J2JDQRTUQUWXMM[\^`aa	ac!eai		T!Qw7774Aq>>4PQRSPSUV<<:X	ac1a$q!tQ**+:-.21ajj	:: : : : d1q5!nnd1a4mmT!Q$]]ArRR	a!eQa7333tAE1~~tAaCQR||6TW[\]^_\_abdeWfWfQT1aW 6 AJJ	(( ( ( ( ( (r/   c                     t          dt          dt                    gi          } t          dt          dz  z  dt          z  z   dz   t                    t          t          dz  dz
  t                    ft          dt                    t          t          dz
  t                    ft          dt                    t          t          dz   t                    fg}t          t          dt                    t          dt                    ||           t          dt          z  t          d          t          dt          d          t          dt          d          ft	          g d	g d
gt                    fk    sJ t          t
          t
                    t          dt
                    ft          t
          dz  t
                    t          dt
                    ft          t
          dz  t
                    t          dt
                    fg}t          dt          dt                    t          t
          t
                    gi          } t          t          t
          dz   t
                    t          t
          dz  t
                    ||           t          t
          t
          d          t          t
          dz  t
          d          t          t
          dz  t
          d          ft	          ddg t
                    fk    sJ t          dt          z  t
                    t          t
          t
                    ft          t           t
                    t          t
          t
                    fg}t          dt          dt                    t          dt          z  t
                    gi          } t          t          dt
                    t          dt
                    ||           t          dt
          d          t          dt
          d          ft	          dt          z  t           ggt
                    fk    sJ d S )Nr   r    r"   r!   r1   r   r   r%   )r    r    r'   )r3   r    r    QQ[x])r   r   r   r   Matrixr   r)   r,   s     r-   test_prde_linear_constraintsr>   F   s3   	#Q

|)<	=	=	=B
qAv!|a
#
#T!Q$(A%6%6	7$q!**d1q5RSnn9U	aT!a%^^$	&A"41::tAqzz1bAA
qsAd
#
#
#T!Qt%<%<%<d1aPT>U>U>U	VJJJ			*A..	00 0 0 0 q!**d1ajj	!DAqMM41::#>adAPTUVXYPZPZ@[\A	#Q

DAJJ/G)H	I	I	IB"4Aq>>41a==!RHH
q!D
!
!
!41a#=#=#=tAqD!TX?Y?Y?Y	Z1aQ	!! ! ! ! qsA,,Q

	#dA2qkk41::%>?A	#Q

D1aLL/I)J	K	K	KB"41::tAqzz1bAA
q!G
$
$
$d1a&@&@&@	A6AaCRSQS9+WXCYCYZ[ [ [ [ [ [r/   c                  ~   t          t          dz    t          dz
  z  t          dz   t          dz
  z  dgt           dz
  t          dz   t          dz
  gdt          dz   z  t          dz
  z  ddggt                    } t          t          dz   t          dz
  z  gt          dz   gdggt                    }t          dt	          dt                    gi          }t          j        t                    t                   }t          | ||          t          g dg dg d	g d
g|          t          g d|          fk    sJ d S )Nr1   r    r!   r   r   r"   )r    r   r   )r   r    r   )r   r   r   )r   r   r    ring)r   r    r   r   )r<   r   r   r   r   r   
frac_fieldr   )Aur)   Rs       r-   test_constant_systemrF   X   sX   1q51q5!AEAE?A6aQA&AEAE"Aq)+,-	/ 	/A 	!a%!a%!AE7QC0!44A	#Q

|)<	=	=	=B
aA1a$$	 #$
% 
% 
% '-\\\&B&B&B	DD D D D D Dr/   c            
         t          t          t                    t          t           t          z  t                    g} t          dt          dt                    t          dt          z  t                    gi          }t	          t          t          t                    t          dt          z  t                    | t
          |          t          t          t                    t          dt          d          t          dt          z  t          d          t          t           t          d          gt          t          dz   t          d          t          dt          d          gt
          dz
  fk    sJ d S )	Nr   r    r"   r'   r   r$   r%   r!   )r   r   r   r   r	   r   )r   r)   s     r-   test_prde_spderH   f   s   	aT1"Q$]]#A	#Q

D1aLL/I)J	K	K	KBT!QZZbdA1b99	aT!Qw///	ac1W	%	%	%tQB''B'B'BC	q!teQw	'	'	'a7)C)C)CDa!e	MM M M M M Mr/   c                     t          dt          dt                    gi          } t          t          dt                    t          t          dz  t                    t          dt                    gd|           t          t          dz  dt          z  z
  dz   t                    t          dt                    gt	          g dg dgt                    fk    sJ t          t          dt                    t          t          dz  t                    t          dt                    gd|           t          t          dz  dt          dz  z  z
  dt          z  z   dz
  t                    t          dt                    gt	          g dg dgt                    fk    sJ t          t          t          t                    t          t          dz  t                    t          dt                    gd|           t          t          t          d	
          t          dt          d	
          gt	          g dg dg dgt                    fk    sJ t          dt          dt                    t          t
          dz  dz   t
                    gi          } t          t
          dz  t
                    t          t          t
          dz  z  t
                    t          t
          dz  t
                    t          t          t
          dz  z  t
                    t          dt          z   t
                    g}t          j        t                    t
                   }t          t          t          t
          z  t
                    |d|           t          t
          dz  dz  t          dz  t
          dz  z  z
  t          dz  dz  t
          dz  z  z   t          dd          t          dz  dz  z
  t
          z  z   t          dz  z   t
                    t          t          dz  t
          dz  z  t          dz  dz  t
          dz  z  z
  t          dd          t          dz  dz  z   t
          z  z   t          dz  z
  t
                    t          t
          t
                    t          dt
                    t          dt
                    gt	          g dddt          dd          dddddddg
g dg dg dg dg dg dg dg dg
|          fk    sJ t          dt          dt                    t          dt
          dz  z   t
                    gi          } t          dt          dz  z  t
          d          }d t          d          D             }t          ||d|           \  }}|                                }t          j        t                    t
                   }t          |          dk    sJ |d         t	          t          dd          dddddgdz  |          k    sJ t	          |g          |d         dd d d f         z  d         t          t          dz  dz  t
          d 
          k    sJ t	          |g          |d         d dd d f         z  d         t          t          t          j        z
  t
          d 
          k    sJ d S )!Nr   r    r"   r!   )r    r   r'   r   )r   r    r   r'   r1      ZZr%   r   )r    r'   r   r   r3            ir'   )
r    r   r'   r   r   r   r   r   r   r   )
r   r   r   r   r   r   r   r   r   r   )
r   r   r   r    r   r   r   r   r   r   )
r   r   r   r   r    r   r   r   r   r   )
r    r   r   r   r   r'   r   r   r   r   )
r   r    r   r   r   r   r'   r   r   r   )
r   r   r    r   r   r   r   r'   r   r   )
r   r   r   r    r   r   r   r   r'   r   )
r   r   r   r   r    r   r   r   r   r'   r@   Tfieldc                     g | ]>}t          d           D ],}t          t          |z  t          |z  z  t          d          -?S )r1   TrO   )ranger   r   r   ).0ijs      r-   
<listcomp>z'test_prde_no_cancel.<locals>.<listcomp>   sF    LLLA588LLaad1a4i$	'	'	'LLLLr/   QQ(x))r   r   r   r
   r<   r   r   rB   r   r   rR   	nullspacelenr   Half)r)   r,   rE   bqhrC   Vs           r-   test_prde_no_cancelr_   p   s   	#Q

|)<	=	=	=B!$q!**tAqD!}}d1ajj.I1bQQ
q!taczA~q
!
!41::	.8E8GHI1K 1K 	LL L L L "$q!**tAqD!}}d1ajj.I1bQQ
q!ta1f}qs"Q&
*
*DAJJ	7ANAPQR:T :T 	UU U U U "$q!**tAqD!}}d1ajj.I1bQQ
q!D
!
!
!41T#:#:#:	;V___DSOODSOOEUVW>Y >Y 	ZZ Z Z Z 
#Q

DA1<M<M/N)O	P	P	PB 
adAQq!tVQadAQq!tVQaRSeUVXA
aA!$qsA,,1b99
q!tAv"QT	!AqDGAqDL0HS"4E4E1R4OQR3RRUVWYUYY[\
]
]QqSAX1Qq!t#xAAa'?&BBQqSH!LLdSTVWjjQ

DAJJ
 !')Z)Z)Z*+QQAAPQTUXY)Z)Z)Z)Z)Z)Z)Z)Z)Z)Z)Z)Z)Z)Z)Z)Z)Z)Z)Z)Z)Z)Z)Z)Z)Z	)\ cd	"e 	"e 	"e	ff f f f 
#Q

DQT1<M<M/N)O	P	P	PBR1Wat$$$ALLqLLLA!!Q2..DAq	A
aAq66Q;;;;Q468B??Aq!Q:1<1EEEEEEEA3KK!QRRU#Q'41Q'+J+J+JJJJJA3KK!RaRU#Q'4AF
Ag+N+N+NNNNNNNr/   c                     t          dt          dt                    t          dt          z  t                    gi          } t          dt          d          }t          t          dz
  t          z  t          d          }t          t          dz
  t          d          }t          t          d	z   t          z   t          d          }t	          t          d
t          dz
  z  t                    t          t           dz   t                    t          dt                    gd|           \  }}|                                }|||||||||||||||||gk    sJ |                                dk    sJ t          |g          |d         d dd d f         z  t          t          dt          d          gg          k    sJ t          dt          dt                    t          t           t                    gi          } t	          t          dt          d          t          dt          d          gd|           t          dt          d          t          t          t          d          gt          g dg| j                  fk    sJ d S )Nr   r    r"   r   TrO   r$   r%   r!   r'      rW   r;   r   )r'   r   r    )r   r   r   r   r   rX   rankr<   )r)   p0p1p2p3r]   rC   r^   s           r-   test_prde_cancel_liouvillianrg      sH    
#Q

D1aLL/I)J	K	K	KB	a$			B	q1uai7	+	+	+B	a!eQw	'	'	'B	q!teai7	+	+	+B"4AE
A#6#6qb1fa$qRS**8UWXZ\]]DAq	ARRRRRRRRPPPPP6688r>>>>A3KK!SbS!!!V$$q!G2L2L2L1M0N)O)OOOOO
 
#Q

D!QKK/H)I	J	J	JB"41W#=#=#=QRY@Z@Z@Z?[]^`bcc1a%%%tAq'A'A'ABFJJJ<Y[Y]D^D^_` ` ` ` ` `r/   c                     t          dt          dt                    gi          } t          t          dz  t          z
  t          d          }t          dt          d          }t          t          t          d          t          t          dz  t          d          g}t          |||d|           \  }}|                                t          g d| j                  gk    sJ |d	         |d         z   t          t          t          d
          k    sJ t          t          dz  t          z
  t          d          }t          t          dz  dt          z  z
  dz   t          d          }t          dt          d          t          t          t          d          t          t          dz  t          d          g}t          |||d|           \  }}|                                t          g d| j                  gk    sJ t          d| j                   |d	         z  |d         z   |d         z   }|t          ||           z  ||z  z   t          t          dz  dt          z  z
  dz   t          d
          k    sJ d S )Nr   r    r"   r!   TrO   r1   )r   r    r    r    r   r   r%   r3   )r1   r    ri   r    r    )r   r   r   r   rX   r<   r   r   )r)   r6   r[   r\   r]   rC   ps          r-   test_param_poly_rischDErk      s4   	#Q

|)<	=	=	=BQTAXq%%%AQA	a$			adAT!:!:!:;AaAq"--DAq;;==VLLL"$7788888 Q4!A$;$q!D1111111 	QTAXq%%%AQTAaCZ!^Qd+++A	a$			a$!7!7!7	adAT	"	"	"	$AaAq"--DAq;;==V$8$8$8"$??@@@@@	aqtad"QqT)AZ21$QTAaCZ!^Qt(L(L(LLLLLLLr/   c                  l   t          dt          dt                    gi          } t          dt          d          t          t          t          d          }}||f||f||fg}t          | t          t          dz  t          d          ||           \  }}t	          |          dk    sJ d |D             }|                                }t	          |          dk    sJ |d	         t          g d
| j                  k    sJ |d	          |d         z   d	|d         z  z   }|                    t                    |t          dz  z  z
  ddt          z  z
  k    sJ t          dt          dt                    t          t          t                    gi          } t          t          t          z   t          d          t          dt          d          ft          d	t          d          t          dt          d          fg}t          t          t           dz
  t          d          t          t          t          z   t          d          ||           \  }}t	          |          dk    sJ d |D             }|                                }t	          |          dk    sJ |d	         t          g d| j                  k    sJ d	|d	         z  d	|d         z  z   d|d         z  z   d	|d         z  z   d	|d         z  z   }|                    t                    |t          t          z   z  z
  d	k    sJ d S )Nr   r    r"   TrO   r!   r1   c                 x    g | ]7}|d                                           |d                                          z  8S r   r    as_exprrS   his     r-   rV   z&test_param_rischDE.<locals>.<listcomp>   5    555RAA	(555r/   r   )r'   r    r   r'   r    r   r$   r%   r   r3   c                 x    g | ]7}|d                                           |d                                          z  8S rn   ro   rq   s     r-   rV   z&test_param_rischDE.<locals>.<listcomp>   rs   r/   )r   r   r   r   r    r   r   rL   )	r   r   r   r   rY   rX   r<   r   diff)	r)   rd   pxr,   r]   rC   rj   r^   ys	            r-   test_param_rischDErx      s   	#Q

|)<	=	=	=B!Qd###T!Qd%;%;%;B
bB8b"X&A"d1a4$777B??DAqq66Q;;;;551555A	Aq66Q;;;;Q46...555555	
1!q1vA66!99qAvQqS(((( 
#Q

DAJJ/G)H	I	I	IB
q1ua
(
(
($q!D*A*A*A	BT!QW[E\E\E\^bcdfgpt^u^u^uDvwAqb1fat444d1q5!46P6P6PRSUWXXDAqq66Q;;;;551555A	Aq66Q;;;;Q46///666666	!A$1Q4!AaD& 1QqT6)AadF2A66!99q!a%y A%%%%%%r/   c                     t          dt          dt                    t          dt          z  t                    gi          } t	          t          t          t                    t          t          dz  t                    t          t          t                    t          t          t                    fg|           t          t          t                    t          dt          z  t                    t          t          t                    dt          t          t                    t          dt          d          ft          t           t          z  t                    t          dt          d          fgfk    sJ d S )Nr   r    r"   r!   r'   r(   r%   )r   r   r   r   r   r)   s    r-   test_limited_integrate_reducer{      s   	#Q

D1aLL/I)J	K	K	KB#DAJJQT1aAJJA @ 	aT"Q$]]DAJJDAJJQRY@Z@Z@Z3[
r!tQ--a7333	45	77 7 7 7 7 7r/   c                     t          dt          dt                    gi          } t          t          t                    t          t          dz   t                    fg}t          t          dt          z   dt          dz  z  z   dt          dz  z  z
   t                    t          dt          z
  t          dz  z
  t          dz  z   t                    ||           t          t          dz  t          z
  dz   t                    t          t          dz
  t          d          fdgfk    sJ t          dt                    t          t          t                    fg}t          t          dt          dz  z  t                    t          dt                    ||           t          dt          dz  z  d	z  t                    t          dt          d          fd
gfk    sJ d S )Nr   r    r"   r3   r!   r1   r   r%   	   r   )r   r   r   r   r=   s     r-   test_limited_integrater~      s   	#Q

|)<	=	=	=B
q!**d1q5!nn	%&ATAEAadFNQq!tV$;"<a@@QA1	a  !R) )
q!tax!|Q

a!eQt!<!<!<	=sCD D D D q!**d1ajj	!"AT!AqD&!__d1ajj!R@@
qAvax

T!Qt444	5s;< < < < < <r/   c                  \   t          t          dt                    gd gd gd          } t          t          dt          z  t                    t          dt                    |           J t          t          dt                    t          dt          z  t                    t          dt          z  t
                    gg dd dt          z  t          gd          } t          t          t          t
          dz  z   t
                    t          dt
                    |           t          dft          dfgt          t          z  ddfk    sJ t          t          dt                    t          t          t                    t          dt          z  t                    gg dd t          t          gd          } t          t          t          t          dz  z   dz   t                    t          dt                    |           t          dft          dfgt          t          dz  z  ddfk    sJ d S )Nr    r   extsextargsr"   r!   )Nexplogr   r1   )r   r   r   r   t1t2r4   r   rz   s    r-   test_is_log_deriv_k_t_radicalr      s   	Q

|dV6* * 
 
 
B#D1aLL$q!**bAAIII	Q

D2rNNDQRSTQTVXMM/Z$$$$!Q*A *A 
B 
B 
BB#DRT2$6$6QRHHq'Aq6	BqD!Q'( ( ( ( 
Q

DRLL$qQRsTU,,/W$$$$1*? *? 
@ 
@ 
@B#DQqS1a$8$8$q!**bIIq'Aq6	Ab!eGQ*+ + + + + +r/   c            	         t          t          dt                    t          dt          z  t                    t          dt          dz   z  t                    gg dd t          t          dz   gd          } t          t          dt          dz  z  dt          z  z   t                    t          dt                    |           t          dft          dfgt          t          z   dfk    sJ t          t          dt                    t          dt          z  t                    t          t          t                    gg dd t          t          gd          } t          t          t          dz  t          dz  z  t                    t          dt                    |           t          dft          dfgdt          z  dt          z  z   dfk    sJ t          t          dt                    t          dt          z  t                    gd dgd t          dz  gd          } t          t          t          t                    t          dt                    |           t          t          j        fgt          dz  dfk    sJ t          t          dt                    t          ddt          z   z  t                    gd dgd t          dz  dt          z  z   dz   gd          } t          t          dt          z   t                    t          dt                    |           t          t          j        fgt          dz  dfk    sJ t          t          dt                    t          d	t          z  t                    gd dgd dt          z  gd          } t          t          dt                    t          t          t                    |           t          dfgt          dfk    sJ d S )
Nr    )Nr   r   r   r"   r!   )Nr   r   r1   r   r'   )
r   r   r   r   r   r   r   rZ   r4   r   rz   s    r-   test_is_deriv_kr     sa   	Q

D1bMM4PQSTWXSXPY[]K^K^/_$$$$1q51A*C *C 
D 
D 
DBd1QT6AaC<,,d1bkk2>>q'B7	R"Wa() ) ) ) 
Q

D1bMM4PRTV<</X$$$$1*? *? 
@ 
@ 
@Bd1a4A:r**DBKK<<a&2q'	AbD1Q3J*+ + + + 
Q

D1bMM/Ju4A,*8 *8 
9 
9 
9Bd1bkk42;;33qv,Aq!" " " " 
Q

DAEB<O<O/Pu4A!a*@*B *B 
C 
C 
CBd1q5"ootAr{{B77qv,Aq!" " " "
 
Q

DAqMM/Ju41+*7 *7 
8 
8 
8Bd1ajj$q!**b111vh15EEEEEEEr/   c            	         t          dt          dt                    t          dt          z  t                    gi          } t	          t          dt          z  dz   t                    t          dt          z  t          z  t                    |           dt          t          dz  z  fk    sJ t	          t          ddt          z  z   t                    t          dt          z  t          z  t                    |           dt          dz  t          dz  z  fk    sJ t          dt          dt                    t          t           t          dz  z  t                    gi          } t	          t          ddt          z  z    t                    t          dt          dz  z  dt          dz  z  t          z  z   t                    |           dt          t          dz  z   fk    sJ t	          t          dt                    t          t          dz  t                    |           dt          fk    sJ t	          t          dt                    t          dt          dz  z  t                    |           ddt          z  fk    sJ d S )Nr   r    r"   r3   r!   r1   r'   )r   r   r   r   r   rz   s    r-   &test_is_log_deriv_k_t_radical_in_fieldr   +  s    
#Q

D1aLL/I)J	K	K	KB,T!A#'1-=-=tAaCE1~~rRR	
AadF   ,T!ac'1-=-=tAaCE1~~rRR	
AqDAI    
#Q

D!AqD!<L<L/M)N	O	O	OB,TA!G*a-@-@1a4!AqD&(	A$ $	
A1H    -T"a[[$q!tQ--LL	
A   ,T!QZZa1fa"MM	
AaC     r/   c            
         t          dt          dt                    t          dt          z  t                    gi          } t	          t          dt          dz  z  t          z   dz
  t                    t          dt          z  t          dz  z  t                    t          dt                    t          t          t          dz  z  t                    |           ddt          t          dz  z  fk    sJ d S )Nr   r    r"   r3   r!   rJ   r'   )r   r   r   r   r   rz   s    r-   test_parametric_log_derivr   >  s    	#Q

D1aLL/I)J	K	K	KB#D1a4!a$;$;T!A#ad(A=N=NQKKa1fa"& &	
AqAv     r/   N):__doc__sympy.integrals.rischr   r   sympy.integrals.prder   r   r   r   r	   r
   r   r   r   r   r   r   r   r   r   r   sympy.polys.polymatrixr   r<   sympy.core.numbersr   sympy.core.singletonr   sympy.core.symbolr   !sympy.polys.domains.rationalfieldr   sympy.polys.polytoolsr   	sympy.abcr   r   r   r4   r   r   t3kr.   r9   r>   rF   rH   r_   rg   rk   rx   r{   r~   r   r   r   r    r/   r-   <module>r      s~   E E C C C C C C C C                                    8 7 7 7 7 7 ' ' ' ' ' ' " " " " " " % % % % % % 0 0 0 0 0 0 & & & & & &          GG$$ BB6 6 6&( ( (:[ [ [$D D DM M M-O -O -O`` ` `.M M M0& & &67 7 7< < <+ + +"F F F:  &    r/   