\[ \int \frac {\csc ^3(e+f x)}{a+b \tan ^2(e+f x)} \, dx \]
Optimal antiderivative \[ -\frac {\left (a -2 b \right ) \arctanh \left (\cos \left (f x +e \right )\right )}{2 a^{2} f}-\frac {\cot \left (f x +e \right ) \csc \left (f x +e \right )}{2 a f}-\frac {\arctan \left (\frac {\sec \left (f x +e \right ) \sqrt {b}}{\sqrt {a -b}}\right ) \sqrt {a -b}\, \sqrt {b}}{a^{2} f} \]
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
\[ \frac {\frac {2 \, {\left (a - 2 \, b\right )} \log \left (\frac {{\left | -\cos \left (f x + e\right ) + 1 \right |}}{{\left | \cos \left (f x + e\right ) + 1 \right |}}\right )}{a^{2}} - \frac {8 \, \sqrt {a b - b^{2}} \arctan \left (-\frac {a \cos \left (f x + e\right ) - b \cos \left (f x + e\right ) - b}{\sqrt {a b - b^{2}} \cos \left (f x + e\right ) + \sqrt {a b - b^{2}}}\right )}{a^{2}} + \frac {{\left (a - \frac {2 \, a {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {4 \, b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1}\right )} {\left (\cos \left (f x + e\right ) + 1\right )}}{a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}} - \frac {\cos \left (f x + e\right ) - 1}{a {\left (\cos \left (f x + e\right ) + 1\right )}}}{8 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________