\[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {63 \arctan \left (\frac {\sqrt {c}\, \tan \left (f x +e \right ) \sqrt {2}}{2 \sqrt {c -c \sec \left (f x +e \right )}}\right ) \sqrt {2}}{256 a^{3} c^{\frac {5}{2}} f}-\frac {21 \tan \left (f x +e \right )}{32 a^{3} f \left (c -c \sec \left (f x +e \right )\right )^{\frac {5}{2}}}+\frac {\tan \left (f x +e \right )}{5 f \left (a +a \sec \left (f x +e \right )\right )^{3} \left (c -c \sec \left (f x +e \right )\right )^{\frac {5}{2}}}+\frac {3 \tan \left (f x +e \right )}{10 a f \left (a +a \sec \left (f x +e \right )\right )^{2} \left (c -c \sec \left (f x +e \right )\right )^{\frac {5}{2}}}+\frac {21 \tan \left (f x +e \right )}{20 f \left (a^{3}+a^{3} \sec \left (f x +e \right )\right ) \left (c -c \sec \left (f x +e \right )\right )^{\frac {5}{2}}}-\frac {63 \tan \left (f x +e \right )}{128 a^{3} c f \left (c -c \sec \left (f x +e \right )\right )^{\frac {3}{2}}} \]
command
integrate(sec(f*x+e)/(a+a*sec(f*x+e))^3/(c-c*sec(f*x+e))^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\sqrt {2} {\left (315 \, \sqrt {c} \arctan \left (\frac {\sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c}}{\sqrt {c}}\right ) - \frac {5 \, {\left (17 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c} c^{2}\right )}}{c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4}} - \frac {8 \, {\left ({\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {5}{2}} c^{8} - 5 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {3}{2}} c^{9} + 30 \, \sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c} c^{10}\right )}}{c^{10}}\right )}}{1280 \, a^{3} c^{3} f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________