\[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {3 \csc \left (f x +e \right )}{8 a^{2} c f \sqrt {a +a \sec \left (f x +e \right )}\, \sqrt {c -c \sec \left (f x +e \right )}}+\frac {\tan \left (f x +e \right )}{4 f \left (a +a \sec \left (f x +e \right )\right )^{\frac {5}{2}} \left (c -c \sec \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {3 \arctanh \left (\cos \left (f x +e \right )\right ) \tan \left (f x +e \right )}{8 a^{2} c f \sqrt {a +a \sec \left (f x +e \right )}\, \sqrt {c -c \sec \left (f x +e \right )}} \]
command
integrate(sec(f*x+e)/(a+a*sec(f*x+e))^(5/2)/(c-c*sec(f*x+e))^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {2 \, {\left (3 \, c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}}{c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}} - \frac {{\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{2} c^{2} - 4 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )} c^{3}}{c^{4}} - 6 \, \log \left ({\left | c \right |} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}\right ) + 6 \, \log \left ({\left | c \right |}\right ) - 4}{32 \, \sqrt {-a c} a^{2} f {\left | c \right |} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________