\[ \int \cot (c+b x) \sin (a+b x) \, dx \]
Optimal antiderivative \[ -\frac {\arctanh \left (\cos \left (b x +c \right )\right ) \sin \left (a -c \right )}{b}+\frac {\sin \left (b x +a \right )}{b} \]
command
integrate(cot(b*x+c)*sin(b*x+a),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {2 \, {\left (\frac {{\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, c\right )^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, b x\right ) \tan \left (\frac {1}{2} \, c\right ) - 1 \right |}\right )}{\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + \tan \left (\frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, c\right )} - \frac {{\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right ) - \tan \left (\frac {1}{2} \, c\right )\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, b x\right ) + \tan \left (\frac {1}{2} \, c\right ) \right |}\right )}{\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1} + \frac {\tan \left (\frac {1}{2} \, b x\right ) \tan \left (\frac {1}{2} \, a\right )^{2} - \tan \left (\frac {1}{2} \, b x\right ) - 2 \, \tan \left (\frac {1}{2} \, a\right )}{{\left (\tan \left (\frac {1}{2} \, b x\right )^{2} + 1\right )} {\left (\tan \left (\frac {1}{2} \, a\right )^{2} + 1\right )}}\right )}}{b} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________