\[ \int \csc ^3(e+f x) \left (a+b \tan ^2(e+f x)\right ) \, dx \]
Optimal antiderivative \[ -\frac {\left (a +2 b \right ) \arctanh \left (\cos \left (f x +e \right )\right )}{2 f}-\frac {a \cot \left (f x +e \right ) \csc \left (f x +e \right )}{2 f}+\frac {b \sec \left (f x +e \right )}{f} \]
command
integrate(csc(f*x+e)^3*(a+b*tan(f*x+e)^2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2 \, {\left (a + 2 \, b\right )} \log \left (\frac {{\left | -\cos \left (f x + e\right ) + 1 \right |}}{{\left | \cos \left (f x + e\right ) + 1 \right |}}\right ) - \frac {a {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {a + \frac {14 \, b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {a {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {2 \, b {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}}{\frac {\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1} + \frac {{\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}}}{8 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________