\[ \int \frac {\cot ^3(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx \]
Optimal antiderivative \[ -\frac {\cot ^{2}\left (f x +e \right )}{2 a^{3} f}-\frac {\ln \left (\cos \left (f x +e \right )\right )}{\left (a -b \right )^{3} f}-\frac {\left (a +3 b \right ) \ln \left (\tan \left (f x +e \right )\right )}{a^{4} f}-\frac {b^{2} \left (6 a^{2}-8 a b +3 b^{2}\right ) \ln \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )}{2 a^{4} \left (a -b \right )^{3} f}+\frac {b^{2}}{4 a^{2} \left (a -b \right ) f \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}+\frac {\left (3 a -2 b \right ) b^{2}}{2 a^{3} \left (a -b \right )^{2} f \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )} \]
command
integrate(cot(f*x+e)^3/(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 \[ \text {Exception raised: NotImplementedError} \]_______________________________________________________