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