\[ \int \frac {(a+a \sin (e+f x))^{3/2}}{x^2} \, dx \]
Optimal antiderivative \[ \frac {3 a f \cos \left (\frac {3 e}{2}+\frac {\pi }{4}\right ) \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sinIntegral \left (\frac {3 f x}{2}\right ) \sqrt {a +a \sin \left (f x +e \right )}}{4}+\frac {3 a f \cosineIntegral \left (\frac {f x}{2}\right ) \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \cos \left (\frac {e}{2}+\frac {\pi }{4}\right ) \sqrt {a +a \sin \left (f x +e \right )}}{4}-\frac {3 a f \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sinIntegral \left (\frac {f x}{2}\right ) \sin \left (\frac {e}{2}+\frac {\pi }{4}\right ) \sqrt {a +a \sin \left (f x +e \right )}}{4}+\frac {3 a f \cosineIntegral \left (\frac {3 f x}{2}\right ) \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sin \left (\frac {3 e}{2}+\frac {\pi }{4}\right ) \sqrt {a +a \sin \left (f x +e \right )}}{4}-\frac {2 a \left (\sin ^{2}\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )\right ) \sqrt {a +a \sin \left (f x +e \right )}}{x} \]
command
integrate((a+a*sin(f*x+e))^(3/2)/x^2,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 \[ \text {Exception raised: NotImplementedError} \]_______________________________________________________