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