\[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {3 \csc \left (f x +e \right )}{8 a^{2} c^{2} f \sqrt {a +a \sec \left (f x +e \right )}\, \sqrt {c -c \sec \left (f x +e \right )}}-\frac {\left (\cot ^{2}\left (f x +e \right )\right ) \csc \left (f x +e \right )}{4 a^{2} c^{2} f \sqrt {a +a \sec \left (f x +e \right )}\, \sqrt {c -c \sec \left (f x +e \right )}}-\frac {3 \arctanh \left (\cos \left (f x +e \right )\right ) \tan \left (f x +e \right )}{8 a^{2} c^{2} f \sqrt {a +a \sec \left (f x +e \right )}\, \sqrt {c -c \sec \left (f x +e \right )}} \]
command
integrate(sec(f*x+e)/(a+a*sec(f*x+e))^(5/2)/(c-c*sec(f*x+e))^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\frac {{\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{2} c^{2} - 6 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )} c^{3}}{c^{4}} - \frac {18 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{2} + 28 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )} c + 11 \, c^{2}}{c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4}} + 12 \, \log \left ({\left | c \right |} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}\right ) - 12 \, \log \left ({\left | c \right |}\right ) + 11}{64 \, \sqrt {-a c} a^{2} c f {\left | c \right |} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________