\[ \int \frac {\sqrt {a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {a \tan \left (f x +e \right )}{f \left (c -c \sec \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {a +a \sec \left (f x +e \right )}}+\frac {a \ln \left (1-\cos \left (f x +e \right )\right ) \tan \left (f x +e \right )}{c f \sqrt {a +a \sec \left (f x +e \right )}\, \sqrt {c -c \sec \left (f x +e \right )}} \]
command
integrate((a+a*sec(f*x+e))^(1/2)/(c-c*sec(f*x+e))^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\sqrt {2} {\left (\frac {2 \, \sqrt {2} \sqrt {-a c} a \log \left (2 \, {\left | a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} \right |}\right )}{c^{2} {\left | a \right |}} - \frac {2 \, \sqrt {2} \sqrt {-a c} a \log \left ({\left | -a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a \right |}\right )}{c^{2} {\left | a \right |}} - \frac {\sqrt {2} {\left (2 \, {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a\right )} \sqrt {-a c} a + \sqrt {-a c} a^{2}\right )}}{a c^{2} {\left | a \right |} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}}\right )}}{4 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________