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