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