\[ \int \frac {(c+d \sin (e+f x))^{3/2}}{(a+a \sin (e+f x))^3} \, dx \]
Optimal antiderivative \[ -\frac {\left (c -d \right ) \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}}{5 f \left (a +a \sin \left (f x +e \right )\right )^{3}}-\frac {2 \left (c +2 d \right ) \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}}{15 a f \left (a +a \sin \left (f x +e \right )\right )^{2}}-\frac {\left (4 c^{2}+5 c d -3 d^{2}\right ) \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}}{30 \left (c -d \right ) f \left (a^{3}+a^{3} \sin \left (f x +e \right )\right )}+\frac {\left (4 c^{2}+5 c d -3 d^{2}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}}\, \EllipticE \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ), \sqrt {2}\, \sqrt {\frac {d}{c +d}}\right ) \sqrt {c +d \sin \left (f x +e \right )}}{30 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) a^{3} \left (c -d \right ) f \sqrt {\frac {c +d \sin \left (f x +e \right )}{c +d}}}-\frac {\left (c +d \right ) \left (4 c +5 d \right ) \sqrt {\frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}}\, \EllipticF \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ), \sqrt {2}\, \sqrt {\frac {d}{c +d}}\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c +d}}}{30 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) a^{3} f \sqrt {c +d \sin \left (f x +e \right )}} \]
command
integrate((c+d*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^3,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ {\rm integral}\left (-\frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}{3 \, a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3} + {\left (a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3}\right )} \sin \left (f x + e\right )}, x\right ) \]