\[ \int \frac {\sin ^5(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 b^{5}}{15 f \left (b \sec \left (f x +e \right )\right )^{\frac {15}{2}}}+\frac {4 b^{3}}{11 f \left (b \sec \left (f x +e \right )\right )^{\frac {11}{2}}}-\frac {2 b}{7 f \left (b \sec \left (f x +e \right )\right )^{\frac {7}{2}}} \]
command
integrate(sin(f*x+e)^5/(b*sec(f*x+e))^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {2 \, {\left (77 \, \sqrt {b \cos \left (f x + e\right )} b^{7} \cos \left (f x + e\right )^{7} - 210 \, \sqrt {b \cos \left (f x + e\right )} b^{7} \cos \left (f x + e\right )^{5} + 165 \, \sqrt {b \cos \left (f x + e\right )} b^{7} \cos \left (f x + e\right )^{3}\right )}}{1155 \, b^{10} f \mathrm {sgn}\left (\cos \left (f x + e\right )\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________