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