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