\[ \int \frac {\sqrt {a-a \sin (e+f x)} (c+c \sin (e+f x))^{3/2}}{x} \, dx \]
Optimal antiderivative \[ c \cosineIntegral \left (f x \right ) \cos \left (e \right ) \sec \left (f x +e \right ) \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {c +c \sin \left (f x +e \right )}+\frac {c \cos \left (2 e \right ) \sec \left (f x +e \right ) \sinIntegral \left (2 f x \right ) \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {c +c \sin \left (f x +e \right )}}{2}-c \sec \left (f x +e \right ) \sinIntegral \left (f x \right ) \sin \left (e \right ) \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {c +c \sin \left (f x +e \right )}+\frac {c \cosineIntegral \left (2 f x \right ) \sec \left (f x +e \right ) \sin \left (2 e \right ) \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {c +c \sin \left (f x +e \right )}}{2} \]
command
integrate((c+c*sin(f*x+e))^(3/2)*(a-a*sin(f*x+e))^(1/2)/x,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {{\left (2 \, c f \cos \left (e\right ) \operatorname {Ci}\left (f x\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + c f \operatorname {Ci}\left (2 \, f x\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (2 \, e\right ) + c f \cos \left (2 \, e\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \operatorname {Si}\left (2 \, f x\right ) - 2 \, c f \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (e\right ) \operatorname {Si}\left (f x\right )\right )} \sqrt {a} \sqrt {c}}{2 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________