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