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