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