\[ \int \frac {1}{(a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {d \left (4 c^{3}-21 c^{2} d +62 c \,d^{2}+147 d^{3}\right ) \cos \left (f x +e \right )}{30 a^{3} \left (c -d \right )^{4} \left (c +d \right ) f \sqrt {c +d \sin \left (f x +e \right )}}-\frac {\cos \left (f x +e \right )}{5 \left (c -d \right ) f \left (a +a \sin \left (f x +e \right )\right )^{3} \sqrt {c +d \sin \left (f x +e \right )}}-\frac {2 \left (c -4 d \right ) \cos \left (f x +e \right )}{15 a \left (c -d \right )^{2} f \left (a +a \sin \left (f x +e \right )\right )^{2} \sqrt {c +d \sin \left (f x +e \right )}}-\frac {\left (4 c^{2}-21 c d +65 d^{2}\right ) \cos \left (f x +e \right )}{30 \left (c -d \right )^{3} f \left (a^{3}+a^{3} \sin \left (f x +e \right )\right ) \sqrt {c +d \sin \left (f x +e \right )}}+\frac {\left (4 c^{3}-21 c^{2} d +62 c \,d^{2}+147 d^{3}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}}\, \EllipticE \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ), \sqrt {2}\, \sqrt {\frac {d}{c +d}}\right ) \sqrt {c +d \sin \left (f x +e \right )}}{30 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) a^{3} \left (c -d \right )^{4} \left (c +d \right ) f \sqrt {\frac {c +d \sin \left (f x +e \right )}{c +d}}}-\frac {\left (4 c^{2}-21 c d +65 d^{2}\right ) \sqrt {\frac {1}{2}+\frac {\sin \left (f x +e \right )}{2}}\, \EllipticF \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ), \sqrt {2}\, \sqrt {\frac {d}{c +d}}\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c +d}}}{30 \sin \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) a^{3} \left (c -d \right )^{3} f \sqrt {c +d \sin \left (f x +e \right )}} \]
command
integrate(1/(a+a*sin(f*x+e))^3/(c+d*sin(f*x+e))^(3/2),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 \[ {\rm integral}\left (\frac {\sqrt {d \sin \left (f x + e\right ) + c}}{4 \, a^{3} c^{2} + 8 \, a^{3} c d + 4 \, a^{3} d^{2} + {\left (2 \, a^{3} c d + 3 \, a^{3} d^{2}\right )} \cos \left (f x + e\right )^{4} - {\left (3 \, a^{3} c^{2} + 10 \, a^{3} c d + 7 \, a^{3} d^{2}\right )} \cos \left (f x + e\right )^{2} + {\left (a^{3} d^{2} \cos \left (f x + e\right )^{4} + 4 \, a^{3} c^{2} + 8 \, a^{3} c d + 4 \, a^{3} d^{2} - {\left (a^{3} c^{2} + 6 \, a^{3} c d + 5 \, a^{3} d^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}, x\right ) \]