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