\[ \int \frac {\cot ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx \]
Optimal antiderivative \[ -\frac {b^{5}}{4 a^{3} \left (a +b \right )^{3} f \left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right )^{2}}+\frac {b^{4} \left (5 a +2 b \right )}{2 a^{3} \left (a +b \right )^{4} f \left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right )}+\frac {\left (2 a +5 b \right ) \left (\csc ^{2}\left (f x +e \right )\right )}{2 \left (a +b \right )^{4} f}-\frac {\csc ^{4}\left (f x +e \right )}{4 \left (a +b \right )^{3} f}+\frac {b^{3} \left (10 a^{2}+5 a b +b^{2}\right ) \ln \left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right )}{2 a^{3} \left (a +b \right )^{5} f}+\frac {\left (a^{2}+5 a b +10 b^{2}\right ) \ln \left (\sin \left (f x +e \right )\right )}{\left (a +b \right )^{5} f} \]
command
integrate(cot(f*x+e)^5/(a+b*sec(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 \[ \text {Exception raised: NotImplementedError} \]_______________________________________________________