\[ \int \cot (e+f x) \left (a+b \tan ^2(e+f x)\right )^2 \, dx \]
Optimal antiderivative \[ \frac {\left (a -b \right )^{2} \ln \left (\cos \left (f x +e \right )\right )}{f}+\frac {a^{2} \ln \left (\tan \left (f x +e \right )\right )}{f}+\frac {b^{2} \left (\tan ^{2}\left (f x +e \right )\right )}{2 f} \]
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
\[ \frac {a^{2} \log \left (\sin \left (f x + e\right )^{2}\right ) - {\left (2 \, a b - b^{2}\right )} \log \left ({\left | \sin \left (f x + e\right )^{2} - 1 \right |}\right ) + \frac {2 \, a b \sin \left (f x + e\right )^{2} - b^{2} \sin \left (f x + e\right )^{2} - 2 \, a b}{\sin \left (f x + e\right )^{2} - 1}}{2 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________