74.44 Problem number 98

\[ \int \frac {(a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{3/2}} \, dx \]

Optimal antiderivative \[ -\frac {2 a^{2} \tan \left (f x +e \right )}{f \left (c -c \sec \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {a +a \sec \left (f x +e \right )}}+\frac {a^{2} \ln \left (1-\cos \left (f x +e \right )\right ) \tan \left (f x +e \right )}{c f \sqrt {a +a \sec \left (f x +e \right )}\, \sqrt {c -c \sec \left (f x +e \right )}} \]

command

integrate((a+a*sec(f*x+e))^(3/2)/(c-c*sec(f*x+e))^(3/2),x, algorithm="giac")

Giac 1.9.0-11 via sagemath 9.6 output

\[ -\frac {\frac {\sqrt {-a c} a^{2} \log \left ({\left | a \right |} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}\right )}{c^{2} {\left | a \right |}} - \frac {\sqrt {-a c} a^{2} \log \left ({\left | -a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a \right |}\right )}{c^{2} {\left | a \right |}} - \frac {{\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a\right )} \sqrt {-a c} a}{c^{2} {\left | a \right |} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}}}{f} \]

Giac 1.7.0 via sagemath 9.3 output

\[ \text {Timed out} \]________________________________________________________________________________________