
    \h?                     2   S r SSKJrJr  SSKJrJrJrJrJ	r	J
r
JrJrJrJrJrJrJrJrJrJr  SSKJr  SSKJr  SSKJr  SSKJr  SSKJr  SS	K J!r!  SS
K"J#r#J$r$J%r%  \" S5      u  r&r'r(r)r*S r+S r,S r-S r.S r/S r0S r1S r2S r3S r4S r5S r6S r7S r8S r9g)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                     [        S[        S[        5      [        S[        S-  -   [        5      /0S9n [        S[        5      n[        [        [        5      n[        [        [        5      [        S[        S-  -   [        5      4[        S[        5      [        [        [        [        S-  -  -   [        5      4/n[	        XX05      [        [        [        SS9[        S[        SS9[        S[        SS94[        [        [        -  [        SS9[        [        S-  S-   [        SS94[        S[        SS9[        [        S-  S-   [        SS94/[        S[        SS94:X  d   e[        [        [        5      [        [        S-  S[        -  -   S-   [        5      4[        [        [        -  [        5      [        [        S-  S[        -  -   S-   [        5      4[        [        [        S-  -  [        5      [        [        S-  S[        -  -   S-   [        5      4/n[        S[        S[        5      [        [        [        5      /0S9n [	        [        [        [        5      [        S[        5      X05      [        [        S-   [        5      [        S[        -   [        -  [        -   [        5      [        S[        SS94[        [        [        5      [        S[        5      4[        [        [        -  [        5      [        S[        SS94[        [        [        S-  -  [        5      [        S[        SS94/[        [        S-   [        5      4:X  d   eg )	ND      	extensionZZ(x)domainZZ[x])r   r   r   r   r   )DEfafdGs       W/var/www/auris/envauris/lib/python3.13/site-packages/sympy/integrals/tests/test_prde.pytest_prde_normal_denomr.      s   	#Q
DQT1<M/N)O	PB	aB	aB
q!*d1q!t8Q'	(41:tA!Q$J7J*KLARQ+	a7	#d1a&@$q!C & $QqS!G <	adQh'	* ,.21a.H!Q$(Ag..0126q!G2L	NN N N
 q!*d1a4!A#:>1-	.ac1QTAaCZ!^Q1!#'!Q$?DA!a4K"L	NA	#Q
DAJ/G)H	IBT!QZaQ;	a!eQ$QzA~q141W3MNRVWXZ[R\Q
R1Q3lDAg$>?$qAvq/Q'"B$Q%&*1q5!n	66 6 6    c            
         [        [        S-   [        5      n [        [        S-  [        5      n[        S[        5      n[        [        [        5      [        S[        5      4[        [        S-  [        5      [        S[        5      4[        [        S-  [        5      [        S[        5      4/n[        S[        S[        5      [        [        [        5      /0S9n[	        XX#U5      [        [        S-   [        5      [        [        S-  [        5      [        [        [        5      [        S[        5      4[        [        S-  [        5      [        S[        5      4[        [        S-  [        5      [        S[        5      4/[        S[        5      4:X  d   e[        [        [        5      [        S[        5      4[        S[        5      [        [        [        5      4/n[	        [        S[        5      [        [        S-  [        5      [        S[        5      X45      [        S[        5      [        [        S-  S-
  [        5      [        [        S-  [        5      [        S[        5      4[        S[        5      [        S[        5      4/[        [        [        5      4:X  d   e[        S[        S[        5      [        S[        -  [
        -  [
        5      /0S9nUR                  5         [        [        [        5      [        [        S-  [        5      4[        S[        -  [        5      [        [        [        5      4/n[	        [        S[        -  [        -  S-   [        5      [        [        S-  S[        S-  -  [        -  -   [        5      [        [        S-  S-   [        5      X45      [        S[        -  [        -  S-   [        5      [        S[        S	S
9[        [        [        5      [        [        S-  [        5      4[        S[        -  [        5      [        [        [        5      4/[        S[        5      4:X  d   e[        S[        S[        5      [        [        S-  S-   S-  [        -  [        5      /0S9n[        [        [        -   [        5      [        [        [        -  [        5      4[        S[        -  [        5      [        [        S-  [        5      4/n[	        [        S[        -  [        -  S-   [        5      [        [        S-  S[        S-  -  [        -  -   [        5      [        [        S-  [        5      X45      [        S[        -  [        -  S-   [        5      [        S[        S	S
9[        [        [        -   [        5      [        [        [        -  [        5      4[        S[        -  [        [        5      [        [        S-  [        [        5      4/[        S[        5      4:X  d   e[	        [        [        S-   [        5      [        [        S-  [        5      [        [        S-  [        5      X45      [        [        S-   [        5      [        S[        S	S
9[        [        [        -   [        5      [        [        [        -  [        5      4[        S[        -  [        [        5      [        [        S-  [        [        5      4/[        S[        5      4:X  d   eg )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!*d1aj	!DAqM41:#>adAPTUVXYPZ@[\A	#Q
DAJ/G)H	IBaRB/	a!eQadA$q!*d1aj)A	adAQ
#d1a4mT!QZ%@)BCG1:	OO O O q!*d1aj	!DAJQ
#;<Ad1aj$q!tQ-aQK	aT!Q$(A&$q!tQ-a)D	aT!QZ )"#'1:	// / / 
#Q
DAb"<M/N)O	PB
q!*d1a4m	$tAaC|T!QZ&@AAd1Q3q519a0$q!ta1fQh2JDQRTUQUXYQY[\L]_`e	ac!eai	T!Qw741:tAqDRS}:U	ac1tAqz":$%)!QZ	11 1 1 
#Q
D!Q$(AaQR<S/T)U	VB
q1ua.$qsA,	'$qsA,QT1)FGAd1Q3q519a0$q!ta1fQh2JDQRTUQUWXM[\a	ac!eai	T!Qw74Aq>4PQRSPSUV<:X	ac1a$q!tQ*+:-.21aj	:: : : d1q5!nd1a4mT!Q$]AR	a!eQa73tAE1~tAaCQR|6TW[\]^_\_abdeWfQT1aW 6 AJ	(( ( (r/   c            	      n   [        S[        S[        5      /0S9n [        S[        S-  -  S[        -  -   S-   [        5      [        [        S-  S-
  [        5      4[        S[        5      [        [        S-
  [        5      4[        S[        5      [        [        S-   [        5      4/n[        [        S[        5      [        S[        5      X5      [        S[        -  [        SS9[        S[        SS9[        S[        SS94[	        / S	Q/ S
Q/[        5      4:X  d   e[        [
        [
        5      [        S[
        5      4[        [
        S-  [
        5      [        S[
        5      4[        [
        S-  [
        5      [        S[
        5      4/n[        S[        S[        5      [        [
        [
        5      /0S9n [        [        [
        S-   [
        5      [        [
        S-  [
        5      X5      [        [
        [
        SS9[        [
        S-  [
        SS9[        [
        S-  [
        SS94[	        SS/ [
        5      4:X  d   e[        S[        -  [
        5      [        [
        [
        5      4[        [        * [
        5      [        [
        [
        5      4/n[        S[        S[        5      [        S[        -  [
        5      /0S9n [        [        S[
        5      [        S[
        5      X5      [        S[
        SS9[        S[
        SS94[	        S[        -  [        * //[
        5      4:X  d   eg )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   s|   	#Q
|)<	=B
qAv!|a
#T!Q$(A%6	7$q!*d1q5RSn9U	aT!a%^$	&A"41:tAqz1A
qsAd
#T!Qt%<d1aPT>U	VJ	*A.	00 0 0 q!*d1aj	!DAqM41:#>adAPTUVXYPZ@[\A	#Q
DAJ/G)H	IB"4Aq>41a=!H
q!D
!41a#=tAqD!TX?Y	Z1aQ	!! ! ! qsA,Q
	#dA2qk41:%>?A	#Q
D1aL/I)J	KB"41:tAqz1A
q!G
$d1a&@	A6AaCRSQS9+WXCYZ[ [ [r/   c                     [        [        S-   * [        S-
  -  [        S-   [        S-
  -  S/[        * S-
  [        S-   [        S-
  /S[        S-   -  [        S-
  -  SS//[        5      n [        [        S-   [        S-
  -  /[        S-   /S//[        5      n[        S[	        S[        5      /0S9n[
        R                  " [        5      [           n[        XU5      [        / SQ/ SQ/ S	Q/ S
Q/US9[        / SQUS94:X  d   eg )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   s   1q51q5!AEAE?A6aQA&AEAE"Aq)+,-	/A 	!a%!a%!AE7QC0!4A	#Q
|)<	=B
aA1$	 #$
% '-\&B	DD D Dr/   c                  2   [        [        [        5      [        [        * [        -  [        5      /n [        S[        S[        5      [        S[        -  [        5      /0S9n[	        [        [        [        5      [        S[        -  [        5      U [
        U5      [        [        [        5      [        S[        SS9[        S[        -  [        SS9[        [        * [        SS9/[        [        S-  * [        SS9[        S[        SS9/[
        S-
  4:X  d   eg )	Nr   r    r"   r'   r   r$   r%   r!   )r   r   r   r   r	   r   )r   r)   s     r-   test_prde_spderH   f   s    	aT1"Q$]#A	#Q
D1aL/I)J	KBT!QZbdA1b9	aT!Qw/	ac1W	%tQB''BC	q!teQw	'a7)CDa!e	MM M Mr/   c                  <   [        S[        S[        5      /0S9n [        [        S[        5      [        [        S-  [        5      [        S[        5      /SU 5      [        [        S-  S[        -  -
  S-   [        5      [        S[        5      /[	        / SQ/ SQ/[        5      4:X  d   e[        [        S[        5      [        [        S-  [        5      [        S[        5      /SU 5      [        [        S-  S[        S-  -  -
  S[        -  -   S-
  [        5      [        S[        5      /[	        / SQ/ SQ/[        5      4:X  d   e[        [        [        [        5      [        [        S-  [        5      [        S[        5      /SU 5      [        [        [        S	S
9[        S[        S	S
9/[	        / SQ/ SQ/ SQ/[        5      4:X  d   e[        S[        S[        5      [        [
        S-  S-   [
        5      /0S9n [        [
        S-  [
        5      [        [        [
        S-  -  [
        5      [        [
        S-  [
        5      [        [        [
        S-  -  [
        5      [        S[        -   [
        5      /n[        R                  " [        5      [
           n[        [        [        [
        -  [
        5      USU 5      [        [
        S-  S-  [        S-  [
        S-  -  -
  [        S-  S-  [
        S-  -  -   [        SS5      [        S-  S-  -
  [
        -  -   [        S-  -   [
        5      [        [        S-  [
        S-  -  [        S-  S-  [
        S-  -  -
  [        SS5      [        S-  S-  -   [
        -  -   [        S-  -
  [
        5      [        [
        [
        5      [        S[
        5      [        S[
        5      /[	        / SQSS[        SS5      SSSSSSS/
/ SQ/ SQ/ SQ/ SQ/ SQ/ SQ/ SQ/ SQ/
US94:X  d   e[        S[        S[        5      [        S[
        S-  -   [
        5      /0S9n [        S[        S-  -  [
        SS9n[        S5       VVs/ s H6  n[        S5        H#  n[        [        U-  [
        U-  -  [
        SS9PM%     M8     nnn[        X6SU 5      u  pxUR                  5       n	[        R                  " [        5      [
           n[        U	5      S:X  d   eU	S   [	        [        SS5      SSSSS/S-  US9:X  d   e[	        U/5      U	S   SS 2S S 24   -  S   [        [        S-  S-  [
        SS
9:X  d   e[	        U/5      U	S   S S2S S 24   -  S   [        [        [        R                  -
  [
        SS
9:X  d   eg s  snnf ) 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fieldQQ(x))r   r   r   r
   r<   r   r   rB   r   r   range	nullspacelenr   Half)
r)   r,   rE   bijqhrC   Vs
             r-   test_prde_no_cancelr\   p   s   	#Q
|)<	=B!$q!*tAqD!}d1aj.I1bQ
q!taczA~q
!41:	.8E8GHI1K 	LL L L "$q!*tAqD!}d1aj.I1bQ
q!ta1f}qs"Q&
*DAJ	7ANAPQR:T 	UU U U "$q!*tAqD!}d1aj.I1bQ
q!D
!41T#:	;V_DSDSEUVW>Y 	ZZ Z Z 
#Q
DA1<M/N)O	PB 
adAQq!tVQadAQq!tVQaRSeUVXA
aA!$qsA,1b9
q!tAv"QT	!AqDGAqDL0HS"4E1R4OQR3RRUVWYUYY[\
]QqSAX1Qq!t#xAAa'?&BBQqSH!LdSTVWjQ
DAJ
 !')Z*+QQAAPQTUXY)Z)Z)Z)Z)Z)Z)Z)Z)Z	)\ cd	"e	ff f f 
#Q
DQT1<M/N)O	PBR1Wat$A16qLA58aad1a4i$	'8	'AL!!2.DA	A
aAq6Q;;Q468B?Aq!Q:1<1EEEEA3K!QRU#Q'41Q'+JJJJA3K!RaRU#Q'4AFF
Ag+NNNN 	Ms   =Vc                     [        S[        S[        5      [        S[        -  [        5      /0S9n [        S[        SS9n[        [        S-
  [        -  [        SS9n[        [        S-
  [        SS9n[        [        S	-  * [        -   [        SS9n[	        [        S
[        S-
  -  [        5      [        [        * S-   [        5      [        S[        5      /SU 5      u  pVUR                  5       nXQXXXXXXAXU/:X  d   eUR                  5       S:X  d   e[        U/5      US   S S2S S 24   -  [        [        S[        SS9//5      :X  d   e[        S[        S[        5      [        [        * [        5      /0S9n [	        [        S[        SS9[        S[        SS9/SU 5      [        S[        SS9[        [        [        SS9/[        / SQ/U R                  5      4:X  d   eg )Nr   r    r"   r   TrO   r$   r%   r!   r'      rQ   r;   r   )r'   r   r    )r   r   r   r   r   rS   rankr<   )r)   p0p1p2p3rZ   rC   r[   s           r-   test_prde_cancel_liouvillianrd      s    
#Q
D1aL/I)J	KB	a$	B	q1uai7	+B	a!eQw	'B	q!teai7	+B"4AE
A#6qb1fa$qRS*8UWXZ\]DA	ARRRRRRRRPPPP668r>>A3K!SbS!V$$q!G2L1M0N)OOOO
 
#Q
D!QK/H)I	JB"41W#=QRY@Z?[]^`bc1a%tAq'ABFJ<Y[Y]Y]D^_` ` `r/   c                     [        S[        S[        5      /0S9n [        [        S-  [        -
  [        SS9n[        S[        SS9n[        [        [        SS9[        [        S-  [        SS9/n[        XUSU 5      u  pEUR	                  5       [        / SQU R                  5      /:X  d   eUS	   US   -   [        [        [        S
S9:X  d   e[        [        S-  [        -
  [        SS9n[        [        S-  S[        -  -
  S-   [        SS9n[        S[        SS9[        [        [        SS9[        [        S-  [        SS9/n[        XUSU 5      u  pEUR	                  5       [        / SQU R                  5      /:X  d   e[        SU R                  5      * US	   -  US   -   US   -   nU[        X`5      -  X&-  -   [        [        S-  S[        -  -
  S-   [        S
S9:X  d   eg )Nr   r    r"   r!   TrO   r1   )r   r    r    r    r   r   r%   r3   )r1   r    rf   r    r    )r   r   r   r   rS   r<   r   r   )r)   r6   rV   rY   rZ   rC   ps          r-   test_param_poly_rischDErh      s   	#Q
|)<	=BQTAXq%AQA	a$	adAT!:;AaAq"-DA;;=VL"$$78888 Q4!A$;$q!D1111 	QTAXq%AQTAaCZ!^Qd+A	a$	a$!7	adAT	"	$AaAq"-DA;;=V$8"$$?@@@@	aqtad"QqT)AZ$QTAaCZ!^Qt(LLLLr/   c            	         [        S[        S[        5      /0S9n [        S[        SS9[        [        [        SS9p!X4X4X!4/n[        U* [        [        S-  [        SS9X05      u  pE[	        U5      S:X  d   eU Vs/ s H)  ofS   R                  5       US   R                  5       -  PM+     nnUR                  5       n[	        U5      S:X  d   eUS   [        / S	QU R                  5      :X  d   eUS   * US   -   SUS   -  -   n	U	R                  [        5      U	[        S-  -  -
  SS[        -  -
  :X  d   e[        S[        S[        5      [        [        [        5      /0S9n [        [        [        -   [        S
S9[        S[        SS94[        S[        SS9[        S[        SS94/n[        [        [        * S-
  [        SS9[        [        [        -   [        SS9X05      u  pE[	        U5      S:X  d   eU Vs/ s H)  ofS   R                  5       US   R                  5       -  PM+     nnUR                  5       n[	        U5      S:X  d   eUS   [        / SQU R                  5      :X  d   eSUS   -  SUS   -  -   SUS   -  -   SUS   -  -   SUS   -  -   n	U	R                  [        5      U	[        [        -   -  -
  S:X  d   eg s  snf s  snf )Nr   r    r"   TrO   r!   r1   r   )r'   r    r   r'   r    r   r$   r%   r   r3   )r   r   r   r   r    r   r   rL   )
r   r   r   r   rT   as_exprrS   r<   r   diff)
r)   ra   pxr,   rZ   rC   hirg   r[   ys
             r-   test_param_rischDEro      s   	#Q
|)<	=B!Qd#T!Qd%;
B8bX&A"d1a4$7?DAq6Q;;3451RAA	(1A5	Aq6Q;;Q46.5555	
1!q1vA66!9qAvQqS((( 
#Q
DAJ/G)H	IB
q1ua
($q!D*A	BT!QW[E\^bcdfgpt^uDvwAqb1fat4d1q5!46PRSXDAq6Q;;3451RAA	(1A5	Aq6Q;;Q46/6666	!A$1Q4!AaD& 1QqT6)AadF2A66!9q!a%y A%%%% 	6 	6s   =0K
=0Kc                  <   [        S[        S[        5      [        S[        -  [        5      /0S9n [	        [        [        [        5      [        [        S-  [        5      [        [        [        5      [        [        [        5      4/U 5      [        [        [        5      [        S[        -  [        5      [        [        [        5      S[        [        [        5      [        S[        SS94[        [        * [        -  [        5      [        S[        SS94/4:X  d   eg )Nr   r    r"   r!   r'   r(   r%   )r   r   r   r   r   r)   s    r-   test_limited_integrate_reducerr      s    	#Q
D1aL/I)J	KB#DAJQT1aAJA @	aT"Q$]DAJDAJQRY@Z3[
r!tQ-a73	45	77 7 7r/   c                     [        S[        S[        5      /0S9n [        [        [        5      [        [        S-   [        5      4/n[        [        S[        -   S[        S-  -  -   S[        S-  -  -
  * [        5      [        S[        -
  [        S-  -
  [        S-  -   [        5      X5      [        [        S-  [        -
  S-   [        5      [        [        S-
  [        SS94S/4:X  d   e[        S[        5      [        [        [        5      4/n[        [        S[        S-  -  [        5      [        S[        5      X5      [        S[        S-  -  S	-  [        5      [        S[        SS94S
/4:X  d   eg )Nr   r    r"   r3   r!   r1   r   r%   	   r   )r   r   r   r   r=   s     r-   test_limited_integrateru      s>   	#Q
|)<	=B
q!*d1q5!n	%&ATAEAadFNQq!tV$;"<a@QA1	a !)
q!tax!|Q
a!eQt!<	=sCD D D q!*d1aj	!"AT!AqD&!_d1aj!@
qAvax
T!Qt4	5s;< < <r/   c                  n   [        [        S[        5      /S /S /S.S9n [        [        S[        -  [        5      [        S[        5      U 5      b   e[        [        S[        5      [        S[        -  [        5      [        S[        -  [
        5      // SQS S[        -  [        /S.S9n [        [        [        [
        S-  -   [
        5      [        S[
        5      U 5      [        S4[        S4/[        [        -  SS4:X  d   e[        [        S[        5      [        [        [        5      [        S[        -  [        5      // SQS [        [        /S.S9n [        [        [        [        S-  -   S-   [        5      [        S[        5      U 5      [        S4[        S4/[        [        S-  -  SS4:X  d   eg )Nr    r   extsextargsr"   r!   )Nexplogr   r1   )r   r   r   r   t1t2r4   r   rq   s    r-   test_is_log_deriv_k_t_radicalr~      sp   	Q
|dV6* 
B#D1aL$q!*bAIII	Q
D2rNDQRSTQTVXM/Z$$!Q*A 
BB#DRT2$6QRHq'Aq6	BqD!Q'( ( ( 
Q
DRL$qQRsTU,/W$$1*? 
@B#DQqS1a$8$q!*bIq'Aq6	Ab!eGQ*+ + +r/   c            	      >   [        [        S[        5      [        S[        -  [        5      [        S[        S-   -  [        5      // SQS [        [        S-   /S.S9n [        [        S[        S-  -  S[        -  -   [        5      [        S[        5      U 5      [        S4[        S4/[        [        -   S4:X  d   e[        [        S[        5      [        S[        -  [        5      [        [        [        5      // SQS [        [        /S.S9n [        [        [        S-  [        S-  -  [        5      [        S[        5      U 5      [        S4[        S4/S[        -  S[        -  -   S4:X  d   e[        [        S[        5      [        S[        -  [        5      /S S/S [        S-  /S.S9n [        [        [        [        5      [        S[        5      U 5      [        [        R                  4/[        S-  S4:X  d   e[        [        S[        5      [        SS[        -   -  [        5      /S S/S [        S-  S[        -  -   S-   /S.S9n [        [        S[        -   [        5      [        S[        5      U 5      [        [        R                  4/[        S-  S4:X  d   e[        [        S[        5      [        S	[        -  [        5      /S S/S S[        -  /S.S9n [        [        S[        5      [        [        [        5      U 5      [        S4/[        S4:X  d   eg )
Nr    )Nr{   r{   rw   r"   r!   )Nr{   rz   r1   r{   r'   )
r   r   r   r|   r}   r   r   rU   r4   r   rq   s    r-   test_is_deriv_kr     s   	Q
D1bM4PQSTWXSXPY[]K^/_$$1q51A*C 
DBd1QT6AaC<,d1bk2>q'B7	R"Wa() ) ) 
Q
D1bM4PRTV</X$$1*? 
@Bd1a4A:r*DBK<a&2q'	AbD1Q3J*+ + + 
Q
D1bM/Ju4A,*8 
9Bd1bk42;3qvv,Aq!" " " 
Q
DAEB<O/Pu4A!a*@*B 
CBd1q5"otAr{B7qvv,Aq!" " "
 
Q
DAqM/Ju41+*7 
8Bd1aj$q!*b11vh15EEEEr/   c            	         [        S[        S[        5      [        S[        -  [        5      /0S9n [	        [        S[        -  S-   [        5      [        S[        -  [        -  [        5      U 5      S[        [        S-  -  4:X  d   e[	        [        SS[        -  -   [        5      [        S[        -  [        -  [        5      U 5      S[        S-  [        S-  -  4:X  d   e[        S[        S[        5      [        [        * [        S-  -  [        5      /0S9n [	        [        SS[        -  -   * [        5      [        S[        S-  -  S[        S-  -  [        -  -   [        5      U 5      S[        [        S-  -   4:X  d   e[	        [        S[        5      [        [        S-  [        5      U 5      S[        4:X  d   e[	        [        S[        5      [        S[        S-  -  [        5      U 5      SS[        -  4:X  d   eg )Nr   r    r"   r3   r!   r1   r'   )r   r   r   r   r   rq   s    r-   &test_is_log_deriv_k_t_radical_in_fieldr   +  s    
#Q
D1aL/I)J	KB,T!A#'1-=tAaCE1~rR	
AadF  ,T!ac'1-=tAaCE1~rR	
AqDAI   
#Q
D!AqD!<L/M)N	OB,TA!G*a-@1a4!AqD&(	A$	
A1H   -T"a[$q!tQ-L	
A  ,T!QZa1fa"M	
AaC  r/   c            
         [        S[        S[        5      [        S[        -  [        5      /0S9n [	        [        S[        S-  -  [        -   S-
  [        5      [        S[        -  [        S-  -  [        5      [        S[        5      [        [        [        S-  -  [        5      U 5      SS[        [        S-  -  4:X  d   eg )Nr   r    r"   r3   r!   rJ   r'   )r   r   r   r   r   rq   s    r-   test_parametric_log_derivr   >  s    	#Q
D1aL/I)J	KB#D1a4!a$;T!A#ad(A=NQK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\   rd   rh   ro   rr   ru   r~   r   r   r    r/   r-   <module>r      s    E C     8 ' " % 0 &  G$ BB6&(:[$DM-O``.M0&67<+"F:&r/   