\[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^{5/2} \sqrt {c-c \sec (e+f x)}} \, dx \]
Optimal antiderivative \[ \frac {\tan \left (f x +e \right )}{4 f \left (a +a \sec \left (f x +e \right )\right )^{\frac {5}{2}} \sqrt {c -c \sec \left (f x +e \right )}}+\frac {\tan \left (f x +e \right )}{4 a f \left (a +a \sec \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {c -c \sec \left (f x +e \right )}}-\frac {\arctanh \left (\cos \left (f x +e \right )\right ) \tan \left (f x +e \right )}{4 a^{2} f \sqrt {a +a \sec \left (f x +e \right )}\, \sqrt {c -c \sec \left (f x +e \right )}} \]
command
integrate(sec(f*x+e)/(a+a*sec(f*x+e))^(5/2)/(c-c*sec(f*x+e))^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {c^{2} {\left (\frac {2 \, \log \left ({\left | c \right |} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}\right )}{c} - \frac {2 \, \log \left ({\left | c \right |}\right )}{c} + \frac {{\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{2} c^{3} - 2 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )} c^{4}}{c^{6}}\right )}}{16 \, \sqrt {-a c} a^{2} f {\left | c \right |} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________