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