75.41 Problem number 83

\[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{\sqrt {c-c \sec (e+f x)}} \, dx \]

Optimal antiderivative \[ -\frac {8 a^{3} \arctan \left (\frac {\sqrt {c}\, \tan \left (f x +e \right ) \sqrt {2}}{2 \sqrt {c -c \sec \left (f x +e \right )}}\right ) \sqrt {2}}{f \sqrt {c}}+\frac {8 a^{3} \tan \left (f x +e \right )}{f \sqrt {c -c \sec \left (f x +e \right )}}+\frac {2 a \left (a +a \sec \left (f x +e \right )\right )^{2} \tan \left (f x +e \right )}{5 f \sqrt {c -c \sec \left (f x +e \right )}}+\frac {4 \left (a^{3}+a^{3} \sec \left (f x +e \right )\right ) \tan \left (f x +e \right )}{3 f \sqrt {c -c \sec \left (f x +e \right )}} \]

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

\[ \frac {8 \, a^{3} {\left (\frac {15 \, \sqrt {2} \arctan \left (\frac {\sqrt {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c}}{\sqrt {c}}\right )}{\sqrt {c}} + \frac {\sqrt {2} {\left (15 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{2} - 5 \, {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )} c + 3 \, c^{2}\right )}}{{\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c\right )}^{\frac {5}{2}}}\right )}}{15 \, f} \]

Giac 1.7.0 via sagemath 9.3 output

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