\[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {10 a^{3} \arctan \left (\frac {\sqrt {c}\, \tan \left (f x +e \right ) \sqrt {2}}{2 \sqrt {c -c \sec \left (f x +e \right )}}\right ) \sqrt {2}}{c^{\frac {3}{2}} f}-\frac {a \left (a +a \sec \left (f x +e \right )\right )^{2} \tan \left (f x +e \right )}{f \left (c -c \sec \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {10 a^{3} \tan \left (f x +e \right )}{c f \sqrt {c -c \sec \left (f x +e \right )}}-\frac {5 \left (a^{3}+a^{3} \sec \left (f x +e \right )\right ) \tan \left (f x +e \right )}{3 c f \sqrt {c -c \sec \left (f x +e \right )}} \]
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
\[ -\frac {2 \, a^{3} {\left (\frac {15 \, \sqrt {2} \arctan \left (\frac {\sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c}}{\sqrt {c}}\right )}{c^{\frac {3}{2}}} + \frac {2 \, \sqrt {2} {\left (6 \, c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 7 \, c\right )}}{{\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {3}{2}} c} + \frac {3 \, \sqrt {2} \sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c}}{c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}}\right )}}{3 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________