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