\[ \int \frac {(c-c \sec (e+f x))^{7/2}}{(a+a \sec (e+f x))^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {c^{4} \ln \left (\cos \left (f x +e \right )\right ) \tan \left (f x +e \right )}{a^{2} f \sqrt {a +a \sec \left (f x +e \right )}\, \sqrt {c -c \sec \left (f x +e \right )}}+\frac {2 c^{4} \ln \left (1+\sec \left (f x +e \right )\right ) \tan \left (f x +e \right )}{a^{2} f \sqrt {a +a \sec \left (f x +e \right )}\, \sqrt {c -c \sec \left (f x +e \right )}}-\frac {4 c^{4} \tan \left (f x +e \right )}{a^{2} f \left (1+\sec \left (f x +e \right )\right )^{2} \sqrt {a +a \sec \left (f x +e \right )}\, \sqrt {c -c \sec \left (f x +e \right )}}+\frac {4 c^{4} \tan \left (f x +e \right )}{a^{2} f \left (1+\sec \left (f x +e \right )\right ) \sqrt {a +a \sec \left (f x +e \right )}\, \sqrt {c -c \sec \left (f x +e \right )}} \]
command
integrate((c-c*sec(f*x+e))^(7/2)/(a+a*sec(f*x+e))^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {{\left ({\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{2} \sqrt {-a c} a^{2} {\left | c \right |} + 2 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )} \sqrt {-a c} a^{2} c {\left | c \right |}\right )} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{a^{5} f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________