\[ \int \csc ^2(c+b x) \sin (a+b x) \, dx \]
Optimal antiderivative \[ -\frac {\arctanh \left (\cos \left (b x +c \right )\right ) \cos \left (a -c \right )}{b}-\frac {\csc \left (b x +c \right ) \sin \left (a -c \right )}{b} \]
command
integrate(csc(b*x+c)^2*sin(b*x+a),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\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 )^{2} + 4 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, c\right )^{2} + 1\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, b x + \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 + \frac {1}{2} \, c\right ) \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, b x + \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} \, b x + \frac {1}{2} \, c\right ) \tan \left (\frac {1}{2} \, a\right ) - \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) \tan \left (\frac {1}{2} \, c\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 + \frac {1}{2} \, c\right ) \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) \tan \left (\frac {1}{2} \, a\right )^{2} + 4 \, \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) + \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right ) + \tan \left (\frac {1}{2} \, a\right ) - \tan \left (\frac {1}{2} \, c\right )}{{\left (\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\right )} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, c\right )}}{b} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________