\[ \int \frac {\sqrt {a-a \sin (e+f x)} (c+c \sin (e+f x))^{3/2}}{x^3} \, dx \]
Optimal antiderivative \[ -\frac {c \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {c +c \sin \left (f x +e \right )}}{2 x^{2}}-\frac {c \,f^{2} \cosineIntegral \left (f x \right ) \cos \left (e \right ) \sec \left (f x +e \right ) \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {c +c \sin \left (f x +e \right )}}{2}-\frac {c f \cos \left (2 f x +2 e \right ) \sec \left (f x +e \right ) \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {c +c \sin \left (f x +e \right )}}{2 x}-c \,f^{2} \cos \left (2 e \right ) \sec \left (f x +e \right ) \sinIntegral \left (2 f x \right ) \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {c +c \sin \left (f x +e \right )}+\frac {c \,f^{2} \sec \left (f x +e \right ) \sinIntegral \left (f x \right ) \sin \left (e \right ) \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {c +c \sin \left (f x +e \right )}}{2}-c \,f^{2} \cosineIntegral \left (2 f x \right ) \sec \left (f x +e \right ) \sin \left (2 e \right ) \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {c +c \sin \left (f x +e \right )}-\frac {c \sec \left (f x +e \right ) \sin \left (2 f x +2 e \right ) \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {c +c \sin \left (f x +e \right )}}{4 x^{2}}+\frac {c f \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {c +c \sin \left (f x +e \right )}\, \tan \left (f x +e \right )}{2 x} \]
command
integrate((c+c*sin(f*x+e))^(3/2)*(a-a*sin(f*x+e))^(1/2)/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 \[ \text {Exception raised: NotImplementedError} \]_______________________________________________________