\[ \int \frac {\sin ^5(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 b^{5}}{13 f \left (b \sec \left (f x +e \right )\right )^{\frac {13}{2}}}+\frac {4 b^{3}}{9 f \left (b \sec \left (f x +e \right )\right )^{\frac {9}{2}}}-\frac {2 b}{5 f \left (b \sec \left (f x +e \right )\right )^{\frac {5}{2}}} \]
command
integrate(sin(f*x+e)^5/(b*sec(f*x+e))^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {2 \, {\left (45 \, \sqrt {b \cos \left (f x + e\right )} b^{6} \cos \left (f x + e\right )^{6} - 130 \, \sqrt {b \cos \left (f x + e\right )} b^{6} \cos \left (f x + e\right )^{4} + 117 \, \sqrt {b \cos \left (f x + e\right )} b^{6} \cos \left (f x + e\right )^{2}\right )}}{585 \, b^{8} f \mathrm {sgn}\left (\cos \left (f x + e\right )\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________