a sz_pat_sincos..nm)rr r )rrrmpatrrr _pat_sincoss    ru piecewisecCst|\}}}}|d}||}|dur0dS||||}}|jrR|jrR|S|jr\|ntj}||}|jsx|jrt} |j|j} } | r| r|dkr|dkrd} d} nP|dkr|dkrd} d} n6|dkr|dkr||k} ||k } n||k} ||k } | r6d| d|dd | |} t||} n2| rh| |d| d|dd} t ||} t | | }| | | }|dkrt ||t |df|dfS||St|t|k} t|t|k} tj}| r|dkr,td|ddD]4}|tj|t|d|t|d||7}qn||dkrBt||}nftd |dt||dt ||dt|d|dtt||dt ||d|}n| r|dkrtd|ddD]4}|tj|t|d|t|d||7}qn||dkrt||}nftd|dt||dt ||dt|d|dtt||dt ||d|}n ||krt t d|tj||}n|| kr|dkr,td|dt||dt ||dt|d|dt t||dt ||d|}nftd |dt||dt ||dt|d|dt t||dt ||d|}|dkrt | ||||t |df|dfS| ||||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] http://en.wikibooks.org/wiki/Calculus/Integration_techniques See Also ======== sympy.integrals.integrals.Integral.doit sympy.integrals.integrals.Integral ZsincosNrTFr r )rZrewritematchis_zerorZZeroZis_odd_ur r rsubsr rabsrangeZ NegativeOner _sin_pow_integrater trigintegrate_cos_pow_integrateZHalf)frZcondsrrrrMzzrZn_Zm_ffuufiZfxresirrrr*s"      "         ,$   ,$   ,$,$ (r*cCs|dkrX|dkrt| Std|t|t||dt|d|t|d|S|dkr|dkrztdt||Std|dt|t||dt|d|dt|d|S|SdS)Nrr r"r!)r rr r)r*rrrrrr)s  $r)cCs|dkrV|dkrt|Std|t|t||dt|d|t|d|S|dkr|dkrxtdt||Std|dt|t||dt|d|dt|d|S|SdS)Nrr r!r")r rr r+r*r4rrrr+(s  $r+N)r )Z sympy.corerrrrrrrZsympy.functionsr r r r Z integralsrrrr%r*r)r+rrrrs$   _-