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