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