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