\[ \int \frac {\sin ^3(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx \]
Optimal antiderivative \[ \frac {\left (a -b \right ) \arctan \left (\frac {\cos \left (f x +e \right ) \sqrt {b}}{\sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}}\right )}{2 b^{\frac {3}{2}} f}-\frac {\cos \left (f x +e \right ) \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}}{2 b f} \]
command
integrate(sin(f*x+e)^3/(a+b*sin(f*x+e)^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \cos \left (f x + e\right )}{2 \, b f} + \frac {{\left (a - b\right )} \log \left ({\left | \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} + \frac {\sqrt {-b f^{2}} \cos \left (f x + e\right )}{f} \right |}\right )}{2 \, \sqrt {-b} b {\left | f \right |}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________