\[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {7 \arctan \left (\frac {\sqrt {c}\, \tan \left (f x +e \right ) \sqrt {2}}{2 \sqrt {c -c \sec \left (f x +e \right )}}\right ) \sqrt {2}}{32 a^{3} c^{\frac {3}{2}} f}-\frac {7 \tan \left (f x +e \right )}{16 a^{3} f \left (c -c \sec \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {\tan \left (f x +e \right )}{5 f \left (a +a \sec \left (f x +e \right )\right )^{3} \left (c -c \sec \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {7 \tan \left (f x +e \right )}{30 a f \left (a +a \sec \left (f x +e \right )\right )^{2} \left (c -c \sec \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {7 \tan \left (f x +e \right )}{12 f \left (a^{3}+a^{3} \sec \left (f x +e \right )\right ) \left (c -c \sec \left (f x +e \right )\right )^{\frac {3}{2}}} \]
command
integrate(sec(f*x+e)/(a+a*sec(f*x+e))^3/(c-c*sec(f*x+e))^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\sqrt {2} {\left (105 \, \sqrt {c} \arctan \left (\frac {\sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c}}{\sqrt {c}}\right ) - \frac {15 \, \sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c}}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}} - \frac {2 \, {\left (3 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {5}{2}} c^{8} - 10 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {3}{2}} c^{9} + 45 \, \sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c} c^{10}\right )}}{c^{10}}\right )}}{480 \, a^{3} c^{2} f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________