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