\[ \int \frac {\cot (e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx \]
Optimal antiderivative \[ \frac {\ln \left (\cos \left (f x +e \right )\right )}{\left (a -b \right )^{2} f}+\frac {\ln \left (\tan \left (f x +e \right )\right )}{a^{2} f}+\frac {\left (2 a -b \right ) b \ln \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )}{2 a^{2} \left (a -b \right )^{2} f}-\frac {b}{2 a \left (a -b \right ) f \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )} \]
command
integrate(cot(f*x+e)/(a+b*tan(f*x+e)^2)^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {{\left (2 \, a b - b^{2}\right )} \log \left ({\left | -a \sin \left (f x + e\right )^{2} + b \sin \left (f x + e\right )^{2} + a \right |}\right )}{a^{4} - 2 \, a^{3} b + a^{2} b^{2}} - \frac {2 \, a b \sin \left (f x + e\right )^{2} - b^{2} \sin \left (f x + e\right )^{2} - 2 \, a b}{{\left (a^{3} - a^{2} b\right )} {\left (a \sin \left (f x + e\right )^{2} - b \sin \left (f x + e\right )^{2} - a\right )}} + \frac {\log \left (\sin \left (f x + e\right )^{2}\right )}{a^{2}}}{2 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________