\[ \int \csc (e+f x) (a+b \sin (e+f x))^3 \, dx \]
Optimal antiderivative \[ \frac {b \left (6 a^{2}+b^{2}\right ) x}{2}-\frac {a^{3} \arctanh \left (\cos \left (f x +e \right )\right )}{f}-\frac {5 a \,b^{2} \cos \left (f x +e \right )}{2 f}-\frac {b^{2} \cos \left (f x +e \right ) \left (a +b \sin \left (f x +e \right )\right )}{2 f} \]
command
integrate(csc(f*x+e)*(a+b*sin(f*x+e))^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}\right ) + {\left (6 \, a^{2} b + b^{3}\right )} {\left (f x + e\right )} + \frac {2 \, {\left (b^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 6 \, a b^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - b^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 6 \, a b^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}^{2}}}{2 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________