\[ \int \frac {\cot ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx \]
Optimal antiderivative \[ \frac {\left (a +3 b \right ) \left (\cot ^{2}\left (f x +e \right )\right )}{2 a^{4} f}-\frac {\cot ^{4}\left (f x +e \right )}{4 a^{3} f}+\frac {\ln \left (\cos \left (f x +e \right )\right )}{\left (a -b \right )^{3} f}+\frac {\left (a^{2}+3 a b +6 b^{2}\right ) \ln \left (\tan \left (f x +e \right )\right )}{a^{5} f}+\frac {b^{3} \left (10 a^{2}-15 a b +6 b^{2}\right ) \ln \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )}{2 a^{5} \left (a -b \right )^{3} f}-\frac {b^{3}}{4 a^{3} \left (a -b \right ) f \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}-\frac {\left (4 a -3 b \right ) b^{3}}{2 a^{4} \left (a -b \right )^{2} f \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )} \]
command
integrate(cot(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 \[ \text {Exception raised: NotImplementedError} \]_______________________________________________________