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