\[ \int \frac {(a+a \sin (e+f x))^{3/2}}{x^3} \, dx \]
Optimal antiderivative \[ \frac {9 a \,f^{2} \cosineIntegral \left (\frac {3 f x}{2}\right ) \cos \left (\frac {3 e}{2}+\frac {\pi }{4}\right ) \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a +a \sin \left (f x +e \right )}}{16}-\frac {3 a \,f^{2} \cos \left (\frac {e}{2}+\frac {\pi }{4}\right ) \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sinIntegral \left (\frac {f x}{2}\right ) \sqrt {a +a \sin \left (f x +e \right )}}{16}-\frac {9 a \,f^{2} \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sinIntegral \left (\frac {3 f x}{2}\right ) \sin \left (\frac {3 e}{2}+\frac {\pi }{4}\right ) \sqrt {a +a \sin \left (f x +e \right )}}{16}-\frac {3 a \,f^{2} \cosineIntegral \left (\frac {f x}{2}\right ) \csc \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sin \left (\frac {e}{2}+\frac {\pi }{4}\right ) \sqrt {a +a \sin \left (f x +e \right )}}{16}-\frac {3 a f \cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \sqrt {a +a \sin \left (f x +e \right )}}{2 x}-\frac {a \left (\sin ^{2}\left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )\right ) \sqrt {a +a \sin \left (f x +e \right )}}{x^{2}} \]
command
integrate((a+a*sin(f*x+e))^(3/2)/x^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 \[ \text {Exception raised: NotImplementedError} \]_______________________________________________________