\[ \int \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx \]
Optimal antiderivative \[ -\frac {\arctanh \left (\frac {\sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {a}}\right ) \sqrt {a}}{f}+\frac {\arctanh \left (\frac {\sqrt {a +b \left (\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {a -b}}\right ) \sqrt {a -b}}{f} \]
command
integrate(cot(f*x+e)*(a+b*tan(f*x+e)^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {{\left (\frac {2 \, a \arctan \left (-\frac {\sqrt {a - b} \sin \left (f x + e\right )^{2} - \sqrt {a \sin \left (f x + e\right )^{4} - b \sin \left (f x + e\right )^{4} - 2 \, a \sin \left (f x + e\right )^{2} + b \sin \left (f x + e\right )^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + \sqrt {a - b} \log \left ({\left | -2 \, {\left (\sqrt {a - b} \sin \left (f x + e\right )^{2} - \sqrt {a \sin \left (f x + e\right )^{4} - b \sin \left (f x + e\right )^{4} - 2 \, a \sin \left (f x + e\right )^{2} + b \sin \left (f x + e\right )^{2} + a}\right )} {\left (a - b\right )} + {\left (2 \, a - b\right )} \sqrt {a - b} \right |}\right )\right )} \mathrm {sgn}\left (\sin \left (f x + e\right )^{2} - 1\right )}{2 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________