\[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{\sqrt {c-c \sec (e+f x)}} \, dx \]
Optimal antiderivative \[ -\frac {4 a^{2} \arctan \left (\frac {\sqrt {c}\, \tan \left (f x +e \right ) \sqrt {2}}{2 \sqrt {c -c \sec \left (f x +e \right )}}\right ) \sqrt {2}}{f \sqrt {c}}+\frac {16 a^{2} \tan \left (f x +e \right )}{3 f \sqrt {c -c \sec \left (f x +e \right )}}-\frac {2 a^{2} \sqrt {c -c \sec \left (f x +e \right )}\, \tan \left (f x +e \right )}{3 c f} \]
command
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^2/(c-c*sec(f*x+e))^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {4 \, a^{2} {\left (\frac {3 \, \sqrt {2} \arctan \left (\frac {\sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c}}{\sqrt {c}}\right )}{\sqrt {c}} + \frac {\sqrt {2} {\left (3 \, c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 4 \, c\right )}}{{\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {3}{2}}}\right )}}{3 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________