75.75 Problem number 132

\[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{11/2}} \, dx \]

Optimal antiderivative \[ -\frac {\left (a +a \sec \left (f x +e \right )\right )^{\frac {5}{2}} \tan \left (f x +e \right )}{10 f \left (c -c \sec \left (f x +e \right )\right )^{\frac {11}{2}}}-\frac {\left (a +a \sec \left (f x +e \right )\right )^{\frac {5}{2}} \tan \left (f x +e \right )}{40 c f \left (c -c \sec \left (f x +e \right )\right )^{\frac {9}{2}}}-\frac {\left (a +a \sec \left (f x +e \right )\right )^{\frac {5}{2}} \tan \left (f x +e \right )}{240 c^{2} f \left (c -c \sec \left (f x +e \right )\right )^{\frac {7}{2}}} \]

command

integrate(sec(f*x+e)*(a+a*sec(f*x+e))^(5/2)/(c-c*sec(f*x+e))^(11/2),x, algorithm="giac")

Giac 1.9.0-11 via sagemath 9.6 output

\[ -\frac {{\left (a^{2} - \frac {10 \, {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a\right )}^{2} a^{5} + 5 \, {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a\right )} a^{6} + a^{7}}{a^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10}}\right )} a^{2}}{240 \, \sqrt {-a c} c^{5} f {\left | a \right |} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} \]

Giac 1.7.0 via sagemath 9.3 output

\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________