\[ \int \frac {\csc ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {\left (a -4 b \right ) \arctanh \left (\cos \left (f x +e \right )\right )}{2 a^{3} f}-\frac {\cot \left (f x +e \right ) \csc \left (f x +e \right )}{2 a f \left (a -b +b \left (\sec ^{2}\left (f x +e \right )\right )\right )}-\frac {b \sec \left (f x +e \right )}{a^{2} f \left (a -b +b \left (\sec ^{2}\left (f x +e \right )\right )\right )}-\frac {\left (3 a -4 b \right ) \arctan \left (\frac {\sec \left (f x +e \right ) \sqrt {b}}{\sqrt {a -b}}\right ) \sqrt {b}}{2 a^{3} f \sqrt {a -b}} \]
command
integrate(csc(f*x+e)^3/(a+b*tan(f*x+e)^2)^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {6 \, {\left (a - 4 \, b\right )} \log \left (\frac {{\left | -\cos \left (f x + e\right ) + 1 \right |}}{{\left | \cos \left (f x + e\right ) + 1 \right |}}\right )}{a^{3}} - \frac {12 \, {\left (3 \, a b - 4 \, b^{2}\right )} \arctan \left (-\frac {a \cos \left (f x + e\right ) - b \cos \left (f x + e\right ) - b}{\sqrt {a b - b^{2}} \cos \left (f x + e\right ) + \sqrt {a b - b^{2}}}\right )}{\sqrt {a b - b^{2}} a^{3}} - \frac {3 \, {\left (\cos \left (f x + e\right ) - 1\right )}}{a^{2} {\left (\cos \left (f x + e\right ) + 1\right )}} + \frac {3 \, a^{2} + \frac {4 \, a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {28 \, a b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {16 \, b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {2 \, a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {8 \, a b {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{a^{3} {\left (\frac {a {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {2 \, a {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {4 \, b {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}}{24 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________