37.29 Problem number 424

\[ \int \frac {\sin ^7(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx \]

Optimal antiderivative \[ \frac {2 b^{7}}{17 f \left (b \sec \left (f x +e \right )\right )^{\frac {17}{2}}}-\frac {6 b^{5}}{13 f \left (b \sec \left (f x +e \right )\right )^{\frac {13}{2}}}+\frac {2 b^{3}}{3 f \left (b \sec \left (f x +e \right )\right )^{\frac {9}{2}}}-\frac {2 b}{5 f \left (b \sec \left (f x +e \right )\right )^{\frac {5}{2}}} \]

command

integrate(sin(f*x+e)^7/(b*sec(f*x+e))^(3/2),x, algorithm="giac")

Giac 1.9.0-11 via sagemath 9.6 output

\[ \frac {2 \, {\left (195 \, \sqrt {b \cos \left (f x + e\right )} b^{8} \cos \left (f x + e\right )^{8} - 765 \, \sqrt {b \cos \left (f x + e\right )} b^{8} \cos \left (f x + e\right )^{6} + 1105 \, \sqrt {b \cos \left (f x + e\right )} b^{8} \cos \left (f x + e\right )^{4} - 663 \, \sqrt {b \cos \left (f x + e\right )} b^{8} \cos \left (f x + e\right )^{2}\right )}}{3315 \, b^{10} f \mathrm {sgn}\left (\cos \left (f x + e\right )\right )} \]

Giac 1.7.0 via sagemath 9.3 output

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