\[ \int \csc ^5(e+f x) (a+b \sin (e+f x))^3 \, dx \]
Optimal antiderivative \[ -\frac {3 a \left (a^{2}+4 b^{2}\right ) \arctanh \left (\cos \left (f x +e \right )\right )}{8 f}-\frac {b \left (2 a^{2}+b^{2}\right ) \cot \left (f x +e \right )}{f}-\frac {3 a \left (a^{2}+4 b^{2}\right ) \cot \left (f x +e \right ) \csc \left (f x +e \right )}{8 f}-\frac {3 a^{2} b \cot \left (f x +e \right ) \left (\csc ^{2}\left (f x +e \right )\right )}{4 f}-\frac {a^{2} \cot \left (f x +e \right ) \left (\csc ^{3}\left (f x +e \right )\right ) \left (a +b \sin \left (f x +e \right )\right )}{4 f} \]
command
integrate(csc(f*x+e)^5*(a+b*sin(f*x+e))^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 8 \, a^{2} b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 8 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 24 \, a b^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 72 \, a^{2} b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 32 \, b^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 24 \, {\left (a^{3} + 4 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}\right ) - \frac {50 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 200 \, a b^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 72 \, a^{2} b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 32 \, b^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 8 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 24 \, a b^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 8 \, a^{2} b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a^{3}}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4}}}{64 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________