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