\[ \int \frac {x^2 \sqrt {a-a \sin (e+f x)}}{(c+c \sin (e+f x))^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 a x}{c \,f^{2} \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {c +c \sin \left (f x +e \right )}}+\frac {2 a \arctanh \left (\sin \left (f x +e \right )\right ) \cos \left (f x +e \right )}{c \,f^{3} \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {c +c \sin \left (f x +e \right )}}+\frac {2 a \cos \left (f x +e \right ) \ln \left (\cos \left (f x +e \right )\right )}{c \,f^{3} \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {c +c \sin \left (f x +e \right )}}-\frac {a \,x^{2} \sec \left (f x +e \right )}{c f \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {c +c \sin \left (f x +e \right )}}+\frac {2 a x \sin \left (f x +e \right )}{c \,f^{2} \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {c +c \sin \left (f x +e \right )}}+\frac {a \,x^{2} \tan \left (f x +e \right )}{c f \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {c +c \sin \left (f x +e \right )}} \]
command
integrate(x^2*(a-a*sin(f*x+e))^(1/2)/(c+c*sin(f*x+e))^(3/2),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 \[ \int \frac {\sqrt {-a \sin \left (f x + e\right ) + a} x^{2}}{{\left (c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}\,{d x} \]_______________________________________________________