\[ \int \frac {1}{\sqrt {-1+\cos ^2(x)}} \, dx \]
Optimal antiderivative \[ -\frac {\arctanh \left (\cos \left (x \right )\right ) \sin \left (x \right )}{\sqrt {-\left (\sin ^{2}\left (x \right )\right )}} \]
command
integrate(1/(-1+cos(x)^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {i \, \log \left (-i \, \tan \left (\frac {1}{2} \, x\right )\right )}{\mathrm {sgn}\left (-\tan \left (\frac {1}{2} \, x\right )^{3} - \tan \left (\frac {1}{2} \, x\right )\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {1}{\sqrt {\cos \left (x\right )^{2} - 1}}\,{d x} \]________________________________________________________________________________________