\[ \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 \sqrt {c-c \sec (e+f x)}} \, dx \]
Optimal antiderivative \[ -\frac {\arctan \left (\frac {\sqrt {c}\, \tan \left (f x +e \right ) \sqrt {2}}{2 \sqrt {c -c \sec \left (f x +e \right )}}\right ) \sqrt {2}}{8 a^{3} f \sqrt {c}}+\frac {\tan \left (f x +e \right )}{5 f \left (a +a \sec \left (f x +e \right )\right )^{3} \sqrt {c -c \sec \left (f x +e \right )}}+\frac {\tan \left (f x +e \right )}{6 a f \left (a +a \sec \left (f x +e \right )\right )^{2} \sqrt {c -c \sec \left (f x +e \right )}}+\frac {\tan \left (f x +e \right )}{4 f \left (a^{3}+a^{3} \sec \left (f x +e \right )\right ) \sqrt {c -c \sec \left (f x +e \right )}} \]
command
integrate(sec(f*x+e)/(a+a*sec(f*x+e))^3/(c-c*sec(f*x+e))^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\sqrt {2} {\left (\frac {15 \, \arctan \left (\frac {\sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c}}{\sqrt {c}}\right )}{\sqrt {c}} - \frac {3 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {5}{2}} c^{12} - 5 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {3}{2}} c^{13} + 15 \, \sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c} c^{14}}{c^{15}}\right )}}{120 \, a^{3} f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________