\[ \int \csc ^4(e+f x) (a+b \sin (e+f x)) \, dx \]
Optimal antiderivative \[ -\frac {b \arctanh \left (\cos \left (f x +e \right )\right )}{2 f}-\frac {a \cot \left (f x +e \right )}{f}-\frac {a \left (\cot ^{3}\left (f x +e \right )\right )}{3 f}-\frac {b \cot \left (f x +e \right ) \csc \left (f x +e \right )}{2 f} \]
command
integrate(csc(f*x+e)^4*(a+b*sin(f*x+e)),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 3 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 12 \, b \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}\right ) + 9 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \frac {22 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 9 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3}}}{24 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________