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