\[ \int \frac {1}{(a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x)} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (b +c \cos \left (e x +d \right )+a \sin \left (e x +d \right )\right ) \left (a \cos \left (e x +d \right )-c \sin \left (e x +d \right )\right )}{\left (a^{2}-b^{2}+c^{2}\right ) e \left (a +c \cot \left (e x +d \right )+b \csc \left (e x +d \right )\right )^{\frac {3}{2}} \sin \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \sqrt {\frac {\cos \left (d +e x -\arctan \left (c , a\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d}{2}+\frac {e x}{2}-\frac {\arctan \left (c , a\right )}{2}\right ), \sqrt {2}\, \sqrt {\frac {\sqrt {a^{2}+c^{2}}}{b +\sqrt {a^{2}+c^{2}}}}\right ) \left (b +c \cos \left (e x +d \right )+a \sin \left (e x +d \right )\right )^{2}}{\cos \left (\frac {d}{2}+\frac {e x}{2}-\frac {\arctan \left (c , a\right )}{2}\right ) \left (a^{2}-b^{2}+c^{2}\right ) e \left (a +c \cot \left (e x +d \right )+b \csc \left (e x +d \right )\right )^{\frac {3}{2}} \sin \left (e x +d \right )^{\frac {3}{2}} \sqrt {\frac {b +c \cos \left (e x +d \right )+a \sin \left (e x +d \right )}{b +\sqrt {a^{2}+c^{2}}}}} \]
command
integrate(1/(a+c*cot(e*x+d)+b*csc(e*x+d))^(3/2)/sin(e*x+d)^(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 \[ {\rm integral}\left (-\frac {\sqrt {c \cot \left (e x + d\right ) + b \csc \left (e x + d\right ) + a} \sqrt {\sin \left (e x + d\right )}}{a^{2} \cos \left (e x + d\right )^{2} + {\left (c^{2} \cos \left (e x + d\right )^{2} - c^{2}\right )} \cot \left (e x + d\right )^{2} + {\left (b^{2} \cos \left (e x + d\right )^{2} - b^{2}\right )} \csc \left (e x + d\right )^{2} - a^{2} + 2 \, {\left (a c \cos \left (e x + d\right )^{2} - a c\right )} \cot \left (e x + d\right ) + 2 \, {\left (a b \cos \left (e x + d\right )^{2} - a b + {\left (b c \cos \left (e x + d\right )^{2} - b c\right )} \cot \left (e x + d\right )\right )} \csc \left (e x + d\right )}, x\right ) \]