\[ \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx \]
Optimal antiderivative \[ -\frac {3 \left (a^{2}-12 a b +16 b^{2}\right ) \arctanh \left (\cos \left (f x +e \right )\right )}{8 a^{5} f}-\frac {\left (5 a -8 b \right ) \cot \left (f x +e \right ) \csc \left (f x +e \right )}{8 a^{2} f \left (a -b +b \left (\sec ^{2}\left (f x +e \right )\right )\right )^{2}}-\frac {\left (\cot ^{3}\left (f x +e \right )\right ) \csc \left (f x +e \right )}{4 a f \left (a -b +b \left (\sec ^{2}\left (f x +e \right )\right )\right )^{2}}-\frac {\left (7 a -12 b \right ) b \sec \left (f x +e \right )}{8 a^{3} f \left (a -b +b \left (\sec ^{2}\left (f x +e \right )\right )\right )^{2}}-\frac {3 \left (a -2 b \right ) b \sec \left (f x +e \right )}{2 a^{4} f \left (a -b +b \left (\sec ^{2}\left (f x +e \right )\right )\right )}-\frac {3 \left (5 a^{2}-20 a b +16 b^{2}\right ) \arctan \left (\frac {\sec \left (f x +e \right ) \sqrt {b}}{\sqrt {a -b}}\right ) \sqrt {b}}{8 a^{5} f \sqrt {a -b}} \]
command
integrate(csc(f*x+e)^5/(a+b*tan(f*x+e)^2)^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Exception raised: NotImplementedError} \]_______________________________________________________