\[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {35 \arctan \left (\frac {\sqrt {c}\, \tan \left (f x +e \right ) \sqrt {2}}{2 \sqrt {c -c \sec \left (f x +e \right )}}\right ) \sqrt {2}}{128 a^{2} c^{\frac {5}{2}} f}-\frac {35 \tan \left (f x +e \right )}{48 a^{2} f \left (c -c \sec \left (f x +e \right )\right )^{\frac {5}{2}}}+\frac {\tan \left (f x +e \right )}{3 f \left (a +a \sec \left (f x +e \right )\right )^{2} \left (c -c \sec \left (f x +e \right )\right )^{\frac {5}{2}}}+\frac {7 \tan \left (f x +e \right )}{6 f \left (a^{2}+a^{2} \sec \left (f x +e \right )\right ) \left (c -c \sec \left (f x +e \right )\right )^{\frac {5}{2}}}-\frac {35 \tan \left (f x +e \right )}{64 a^{2} c f \left (c -c \sec \left (f x +e \right )\right )^{\frac {3}{2}}} \]
command
integrate(sec(f*x+e)/(a+a*sec(f*x+e))^2/(c-c*sec(f*x+e))^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\sqrt {2} {\left (105 \, \sqrt {c} \arctan \left (\frac {\sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c}}{\sqrt {c}}\right ) + \frac {8 \, {\left ({\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {3}{2}} c^{2} - 9 \, \sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c} c^{3}\right )}}{c^{3}} - \frac {3 \, {\left (13 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {3}{2}} c + 11 \, \sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c} c^{2}\right )}}{c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4}}\right )}}{384 \, a^{2} c^{3} f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________