\[ \int \cos ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx \]
Optimal antiderivative \[ \frac {a^{2} \left (a +6 b \right ) \arctanh \left (\frac {\sin \left (f x +e \right ) \sqrt {b}}{\sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}\right )}{16 b^{\frac {3}{2}} f}+\frac {\left (a +6 b \right ) \sin \left (f x +e \right ) \left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}}{24 b f}-\frac {\sin \left (f x +e \right ) \left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )^{\frac {5}{2}}}{6 b f}+\frac {a \left (a +6 b \right ) \sin \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{16 b f} \]
command
integrate(cos(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
\[ -\frac {{\left (2 \, {\left (4 \, b \sin \left (f x + e\right )^{2} + \frac {7 \, a b^{4} - 6 \, b^{5}}{b^{4}}\right )} \sin \left (f x + e\right )^{2} + \frac {3 \, {\left (a^{2} b^{3} - 10 \, a b^{4}\right )}}{b^{4}}\right )} \sqrt {b \sin \left (f x + e\right )^{2} + a} \sin \left (f x + e\right ) + \frac {3 \, {\left (a^{3} + 6 \, a^{2} b\right )} \log \left ({\left | -\sqrt {b} \sin \left (f x + e\right ) + \sqrt {b \sin \left (f x + e\right )^{2} + a} \right |}\right )}{b^{\frac {3}{2}}}}{48 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________