� ��gK+���ddlmZmZmZmZmZmZmZddlm Z m Z m Z m Z m Z ddlmZd�Zed���Zed��Zd d �Zd �Zd �Zd S)�)�cacheit�Dummy�Ne�Integer�Rational�S�Wild)�binomial�sin�cos� Piecewise�Abs�)� integratec�,�t|t��S)N)� isinstancer)�ns �l/home/asafur/pinokio/api/open-webui.git/app/env/lib/python3.11/site-packages/sympy/integrals/trigonometry.py�_integer_instancers�� �a�� !� !�!�c���td�g���}�fd�dD��\}}t|�z��|zt|�z��|zz}||||fS)N�a)�excludec�B��g|]}t|�gtg�����S))r� properties)r r)�.0�s�xs �r� <listcomp>z_pat_sincos.<locals>.<listcomp>s>��� � � �� ��Q�C�->�,?� @� @� @� � � r�nm)r r r )rrr�m�pats` r� _pat_sincosr#sw��� �S�1�#����A� � � � �� � � �D�A�q� �a��c�(�(�A�+��A�a�C���!� � #�C� ��1�a�<�r�u� piecewisec� �t|��\}}}}|�d��}|�|��}|�dS||||}}|jr |jr|S|jr|n tj}||}|js|j�rt} |j|j} } | rK| rI|dkr |dkrd} d} n8|dkr |dkrd} d} n'|dkr|dkr||k} ||k } n ||k} ||k } | r+d| dzz |dz dz z | |zz} t||z��} n+| r)| |zd| dzz |dz dz zz} t||z��} t| | ��}|� | | ��}|dkr%t||z t|d��f|df��S||z St|��t|��k} t|��t|��k} tj}| �r|dkrXtd|dzdz��D]?}|tj|zt#|dz|��zt%|d|zz|��zz }�@�n|dkrt%||��}�nt'd |dz��t|��|dzzzt|��|dz zzt'|dz |dz��t)t|��|dzzt|��|dz zz|��zz}�nr| �r|dkrXtd|dzdz��D]?}|tj|zt#|dz|��zt+|d|zz|��zz }�@�n|dkrt+||��}�n�t'd|dz��t|��|dz zzt|��|dzzzt'|dz |dz��t)t|��|dz zt|��|dzzz|��zz}�ng||kr2ttd|z��tjz|z|��}�n/|| k�r'|dkr�t'd|dz��t|��|dz zzt|��|dzzzt'|dz |dz��tt|��|dz zt|��|dzzz|��zz}n�t'd |dz��t|��|dzzzt|��|dz zzt'|dz |dz��tt|��|dzzt|��|dz zz|��zz}|dkr<t|� |||z��|z t|d��f|df��S|� |||z��|z S) 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 �sincosNrTFr�r%�����)r#�rewrite�match�is_zeror�Zero�is_odd�_ur r r�subsr rr�range� NegativeOner �_sin_pow_integrater� trigintegrate�_cos_pow_integrate�Half)�fr�condsr"rrr!�M�zzr$�n_�m_�ff�uu�fi�fx�res�is rr4r4sM��D�q�>�>�L�C��A�q� � � �(���A� ���� � �A��y��� �Q�4��1��q�A��y��Q�Y�����i� #���Q�V�B� �!��A��x�*�1�8�*� ����1�8�B�� � !�"� !��1�u�u��Q���������Q���1�q�5�5�������a�%�%�A��E�E���U���a�%�[����!�e���a�%�[�� � ��q�!�t�8��A��q�y�)�)�A�q�D�0�B��Q�q�S���B�B�� ��A���Q��T��a�!�e�Q�Y�/�/�B��Q�q�S���B� �r�1� � �� �W�W�Q��^�^�� �K� � ��b�1�f�b��A�h�h�/�"�d��<�<� <��A�v� �" �a�&�&�3�q�6�6�/�B� �a�&�&�3�q�6�6�/�B� �&�C� �hF� �q�5�5��1�a��d�Q�h�'�'� 8� 8���� �q�(�8�A�q�D�!�+<�+<�<�*�1�q��s�7�A�6�6�7�8��� 8��!�V�V�$�Q��*�*�C�C�&�B��A��&�&��Q���!�a�%��8�3�q�6�6�A��E�?�J��A��E�1�q�5�)�)� ��Q���!�a�%���Q���!�a�%��!@�!�D�D�E�E�C�C� �HF� �q�5�5��1�a��d�Q�h�'�'� 8� 8���� �q�(�8�A�q�D�!�+<�+<�<�*�1�q��s�7�A�6�6�7�8��� 8��!�V�V�%�Q��*�*�C�C�&�A�q�1�u�%�%��A����Q���7��A����Q���G��A��E�1�q�5�)�)� ��Q���!�a�%���Q���!�a�%��!@�!�D�D�E�E�C�C� ��6�6��S��1��X�X�a�f�_�q�0�!�4�4�C�C��A�2�g�g��1�u�u�  ��1�q�5�)�)�C��F�F�Q��U�O�;�c�!�f�f�q�1�u�o�M���A��q�1�u�-�-� ��Q���!�a�%��3�q�6�6�A��E�?�!B�A�F�F�G�G��� ��A��E�*�*�S��V�V�a�!�e�_�<�s�1�v�v��A���N���A��q�1�u�-�-� ��Q���!�a�%���Q���!�a�%��!@�!�D�D�E�E�� � ����#�(�(�1�a��c�*�*�Q�.��1�a���9�B��:�F�F�F� �8�8�A�q��s� � �a� �rc��|dkrt|dkrt|�� Std|��t|��zt|��|dz zzt|dz |��t|dz |��zzS|dkr�|dkr t dt|��z |��Std|dz��t|��zt|��|dzzzt|dz|dz��t|dz|��zzS|S)Nrrr)r()r rr r3r4�rrs rr3r3�s���1�u�u� ��6�6���F�F�7�N���Q���#�a�&�&�(�3�q�6�6�A��E�?�:���Q���"�"�%7��A��q�%A�%A�A�B� C� �1�u�u� ��7�7�!��3�q�6�6��1�-�-� -���A��E�"�"�S��V�V�+�c�!�f�f�q�1�u�o�=���Q���A��&�&�);�A��E�1�)E�)E�E�F� G� �rc��|dkrs|dkrt|��Std|��t|��zt|��|dz zzt|dz |��t|dz |��zzS|dkr�|dkr t dt|��z |��Std|dz��t|��zt|��|dzzzt|dz|dz��t|dz|��zzS|S)Nrrr(r))r rr r5r4rDs rr5r5(s���1�u�u� ��6�6��q�6�6�M���A����Q���'�#�a�&�&�1�q�5�/�9���Q���"�"�%7��A��q�%A�%A�A�B� C� �1�u�u� ��7�7� ��3�q�6�6��1�-�-� -���Q��U�#�#�c�!�f�f�,�s�1�v�v��A���>���Q���A��&�&�);�A��E�1�)E�)E�E�F� G� �rN)r%)� sympy.corerrrrrrr �sympy.functionsr r r r r� integralsrrr#r/r4r3r5�rr�<module>rJs���E�E�E�E�E�E�E�E�E�E�E�E�E�E�E�E�E�E�>�>�>�>�>�>�>�>�>�>�>�>�>�>� � � � � � �"�"�"� ��� ��� �U�3�Z�Z��[ �[ �[ �[ �|*�*�*�Z'�'�'�'�'r
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