\[ \int \frac {\sec (e+f x) \sqrt {c-c \sec (e+f x)}}{(a+a \sec (e+f x))^3} \, dx \]
Optimal antiderivative \[ \frac {2 c \tan \left (f x +e \right )}{5 f \left (a +a \sec \left (f x +e \right )\right )^{3} \sqrt {c -c \sec \left (f x +e \right )}} \]
command
integrate(sec(f*x+e)*(c-c*sec(f*x+e))^(1/2)/(a+a*sec(f*x+e))^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\sqrt {2} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {5}{2}} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{20 \, a^{3} c^{2} f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________