\[ \int \csc ^5(e+f x) (a+b \sin (e+f x))^2 \, dx \]
Optimal antiderivative \[ -\frac {\left (3 a^{2}+4 b^{2}\right ) \arctanh \left (\cos \left (f x +e \right )\right )}{8 f}-\frac {2 a b \cot \left (f x +e \right )}{f}-\frac {2 a b \left (\cot ^{3}\left (f x +e \right )\right )}{3 f}-\frac {\left (3 a^{2}+4 b^{2}\right ) \cot \left (f x +e \right ) \csc \left (f x +e \right )}{8 f}-\frac {a^{2} \cot \left (f x +e \right ) \left (\csc ^{3}\left (f x +e \right )\right )}{4 f} \]
command
integrate(csc(f*x+e)^5*(a+b*sin(f*x+e))^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {3 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 16 \, a b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 24 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 24 \, b^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 144 \, a b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 24 \, {\left (3 \, a^{2} + 4 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}\right ) - \frac {150 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 200 \, b^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 144 \, a b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 24 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 24 \, b^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 16 \, a b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, a^{2}}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4}}}{192 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________