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