
    \hK+                         S SK JrJrJrJrJrJrJr  S SKJ	r	J
r
JrJrJr  SSKJr  S r\S 5       r\" S5      rSS jrS	 rS
 rg)    )cacheitDummyNeIntegerRationalSWild)binomialsincos	PiecewiseAbs   )	integratec                 "    [        U [        5      $ )N)
isinstancer   )ns    T/var/www/auris/envauris/lib/python3.13/site-packages/sympy/integrals/trigonometry.py_integer_instancer      s    a!!    c           	          [        SU /S9nS Vs/ s H  n[        X /[        /S9PM     snu  p4[        X-  5      U-  [        X-  5      U-  -  nXQX44$ s  snf )Na)excludenm)r   
properties)r	   r   r   r   )xr   sr   mpats         r   _pat_sincosr       si    S1#AA C->,?@DA
ac(A+AC!
#C1<s   Auc                 j
   [        U5      u  p4pVU R                  S5      n U R                  U5      nUc  gXu   Xv   peUR                  (       a  UR                  (       a  U$ UR                  (       a  UO[        R
                  nXt   nUR                  (       d  UR                  (       Ga  [        n	UR                  UR                  pU
(       aH  U(       aA  US:  a  US:  a  SnSn
O0US:  a  US:  a  Sn
SnOUS:  a  US:  a
  XV:  n
XV:  + nO	XV:  n
XV:  + nU
(       a%  SU	S-  -
  US-
  S-  -  * X-  -  n[        XA-  5      nO*U(       a#  X-  SU	S-  -
  US-
  S-  -  -  n[        XA-  5      n[        WU	5      nUR                  U	W5      nUS:X  a  [        X-  [        US5      4US45      $ X-  $ [        U5      [        U5      :  n
[        U5      [        U5      :  n[        R
                  nU
(       a  US:  aT  [        SUS-  S-   5       H<  nU[        R                   U-  [#        US-  U5      -  [%        USU-  -   U5      -  -  nM>     GOUS:X  a  [%        XQ5      nGO['        S	US-   5      [        U5      US-   -  -  [        U5      US-
  -  -  ['        US-
  US-   5      [)        [        U5      US-   -  [        U5      US-
  -  -  U5      -  -   nGOU(       a  US:  aT  [        SUS-  S-   5       H<  nU[        R                   U-  [#        US-  U5      -  [+        USU-  -   U5      -  -  nM>     GOUS:X  a  [+        Xa5      nGO['        SUS-   5      [        U5      US-
  -  -  [        U5      US-   -  -  ['        US-
  US-   5      [)        [        U5      US-
  -  [        U5      US-   -  -  U5      -  -   nGO'Xe:X  a-  [        [        SU-  5      [        R,                  -  U-  U5      nOXe* :X  a  US:  au  ['        SUS-   5      [        U5      US-
  -  -  [        U5      US-   -  -  ['        US-
  US-   5      [        [        U5      US-
  -  [        U5      US-   -  -  U5      -  -   nOt['        S	US-   5      [        U5      US-   -  -  [        U5      US-
  -  -  ['        US-
  US-   5      [        [        U5      US-   -  [        U5      US-
  -  -  U5      -  -   nUS:X  a/  [        UR                  XU-  5      U-  [        US5      4US45      $ UR                  XU-  5      U-  $ )
a\  
Integrate f = Mul(trig) over x.

Examples
========

>>> from sympy import sin, cos, tan, sec
>>> from sympy.integrals.trigonometry import trigintegrate
>>> from sympy.abc import x

>>> trigintegrate(sin(x)*cos(x), x)
sin(x)**2/2

>>> trigintegrate(sin(x)**2, x)
x/2 - sin(x)*cos(x)/2

>>> trigintegrate(tan(x)*sec(x), x)
1/cos(x)

>>> trigintegrate(sin(x)*tan(x), x)
-log(sin(x) - 1)/2 + log(sin(x) + 1)/2 - sin(x)

References
==========

.. [1] https://en.wikibooks.org/wiki/Calculus/Integration_techniques

See Also
========

sympy.integrals.integrals.Integral.doit
sympy.integrals.integrals.Integral
sincosNr   TFr      	piecewise)r    rewritematchis_zeror   Zerois_odd_ur   r   r   subsr   r   r   rangeNegativeOner
   _sin_pow_integrater   trigintegrate_cos_pow_integrateHalf)fr   condsr   r   r   r   Mzzr!   n_m_ffuufifxresis                     r   r1   r1      s#   D q>LCA			(A	Ay4qyyQYYiiQVVB	Axx1888188B " 1uQQ1q5 a%AEU[ e[ q!t8Aqy))AD0BQSB QTa!eQY//BQSBr1WWQ^KbfbAh/"d<<v" a&3q6/B
a&3q6/B
&&C	 q51adQh'q(8AqD!+<<*1qs7A67 8 ( !V$Q*C& BA&Q!a%83q6AE?JAE1q5) Q!a%Q!a%!@!DEEC 
 q5 1adQh'q(8AqD!+<<*1qs7A67 8 ( !V %Q*C& Aq1u%AQ7AQGAE1q5) Q!a%Q!a%!@!DEEC
 6S1Xaff_q0!4C2g1u
  1q5)CFQUO;c!fq1uoMAq1u- Q!a%3q6AE?!BAFGG  AE*SVa!e_<s1vANAq1u- Q!a%Q!a%!@!DEE #((1c*Q.1a9B:FF88Asar   c                    U S:  a\  U S:X  a  [        U5      * $ [        SU 5      [        U5      -  [        U5      U S-
  -  -  [        U S-
  U 5      [        U S-
  U5      -  -   $ U S:  an  U S:X  a  [	        S[        U5      -  U5      $ [        SU S-   5      [        U5      -  [        U5      U S-   -  -  [        U S-   U S-   5      [        U S-   U5      -  -   $ U$ )Nr   r   r&   r$   )r   r   r   r0   r1   r   r   s     r   r0   r0      s    1u6F7N Q#a&(3q6AE?:Q"%7Aq%AAB 	C 	1u7 !3q61-- AE"SV+c!fq1uo=QA&);AE1)EEF 	G r   c                    U S:  a[  U S:X  a  [        U5      $ [        SU 5      [        U5      -  [        U5      U S-
  -  -  [        U S-
  U 5      [        U S-
  U5      -  -   $ U S:  an  U S:X  a  [	        S[        U5      -  U5      $ [        SU S-   5      [        U5      -  [        U5      U S-   -  -  [        U S-   U S-   5      [        U S-   U5      -  -   $ U$ )Nr   r   r$   r&   )r   r   r   r2   r1   rA   s     r   r2   r2   (  s    1u6q6M AQ'#a&1q5/9Q"%7Aq%AAB 	C 	1u7 3q61-- QU#c!f,s1vA>QA&);AE1)EEF 	G
 r   N)r%   )
sympy.corer   r   r   r   r   r   r	   sympy.functionsr
   r   r   r   r   	integralsr   r   r    r,   r1   r0   r2    r   r   <module>rG      sM    E E E > >  " 	 	 3Z[ |*Z'r   