\[ \int \frac {1}{\sqrt {-\cos (x)+\sec (x)}} \, dx \]
Optimal antiderivative \[ \frac {\arctan \left (\sqrt {\cos }\left (x \right )\right ) \sin \left (x \right )}{\sqrt {\cos \left (x \right )}\, \sqrt {\sin \left (x \right ) \tan \left (x \right )}}-\frac {\arctanh \left (\sqrt {\cos }\left (x \right )\right ) \sin \left (x \right )}{\sqrt {\cos \left (x \right )}\, \sqrt {\sin \left (x \right ) \tan \left (x \right )}} \]
command
integrate(1/(-cos(x)+sec(x))^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{2} \, \arcsin \left (\tan \left (\frac {1}{2} \, x\right )^{2}\right ) - \frac {1}{2} \, \log \left (-\frac {\sqrt {-\tan \left (\frac {1}{2} \, x\right )^{4} + 1} - 1}{\tan \left (\frac {1}{2} \, x\right )^{2}}\right ) \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {1}{\sqrt {-\cos \left (x\right ) + \sec \left (x\right )}}\,{d x} \]________________________________________________________________________________________