\[ \int \frac {\sqrt {a-a \cos (e+f x)}}{\sqrt {-\cos (e+f x)}} \, dx \]
Optimal antiderivative \[ -\frac {2 \arcsin \left (\frac {\sin \left (f x +e \right ) \sqrt {a}}{\sqrt {a -a \cos \left (f x +e \right )}}\right ) \sqrt {a}}{f} \]
command
integrate((a-a*cos(f*x+e))^(1/2)/(-cos(f*x+e))^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {4 \, \sqrt {a} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2 \, {\left (2 \, \sqrt {2} - \sqrt {-\tan \left (\frac {1}{4} \, f x + \frac {1}{4} \, e\right )^{4} + 6 \, \tan \left (\frac {1}{4} \, f x + \frac {1}{4} \, e\right )^{2} - 1}\right )}}{\tan \left (\frac {1}{4} \, f x + \frac {1}{4} \, e\right )^{2} - 3}\right )}\right ) \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________