\[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {5 \arctan \left (\frac {\sqrt {c}\, \tan \left (f x +e \right ) \sqrt {2}}{2 \sqrt {c -c \sec \left (f x +e \right )}}\right ) \sqrt {2}}{16 a^{2} c^{\frac {3}{2}} f}-\frac {5 \tan \left (f x +e \right )}{8 a^{2} f \left (c -c \sec \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {\tan \left (f x +e \right )}{3 f \left (a +a \sec \left (f x +e \right )\right )^{2} \left (c -c \sec \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {5 \tan \left (f x +e \right )}{6 f \left (a^{2}+a^{2} \sec \left (f x +e \right )\right ) \left (c -c \sec \left (f x +e \right )\right )^{\frac {3}{2}}} \]
command
integrate(sec(f*x+e)/(a+a*sec(f*x+e))^2/(c-c*sec(f*x+e))^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\sqrt {2} {\left (15 \, \sqrt {c} \arctan \left (\frac {\sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c}}{\sqrt {c}}\right ) - \frac {3 \, \sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c}}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}} + \frac {2 \, {\left ({\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {3}{2}} c^{2} - 6 \, \sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c} c^{3}\right )}}{c^{3}}\right )}}{48 \, a^{2} c^{2} f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________