Evaluate $\int_0^{\frac{\pi}{2}} \frac{\sin^3{(2x)}}{\ln{\left(\csc{x}\right)}} \mathop{dx}$

You have the correct idea of expanding $\sin{(2x)}$: $$I=8\int_0^{\frac{\pi}{2}} \frac{\sin^3{x}\cos^3{x}}{\ln{\left(\csc{x}\right)}} \; dx$$ Substituting $u=\ln{\left(\csc{x}\right)}$ will be very helpful: $$I=8\int_{\infty}^0 \frac{\sin^3{x}\cos^3{x}}{u} \cdot \frac{du}{-\cot{x}}$$ $$=8\int_{0}^{\infty} \frac{\sin^4{x}\left(1-\sin^2{x}\right)}{u} \; du$$ $$=8\int_{0}^{\infty} \frac{e^{-4u}\left(1-e^{-2u}\right)}{u} \; du$$ $$=8\int_{0}^{\infty} \frac{e^{-4u}-e^{-6u}}{u} \; du$$ Now, this is a simple application of the Frullani integral: $$8 \cdot \ln{\left(\frac{-6}{-4}\right)}=\boxed{8\ln{\left(\frac{3}{2}\right)}}$$ So, it turns out that this integral does have a closed form expression.

Take $$I(a) = \int_0^{\pi /2} \frac{\sin^32x}{\ln \sin x}e^{-a\ln\sin x}\,dx$$ Differentiate w.r.t $a$ and use $\sin 2x = 2\sin x\cos x$ to get: $$I'(a) = -8\int_0^{\pi /2} \sin^{3-a}x\cos^3 x\,dx$$ Put $\sin x = t$ to get: $$I'(a) = -8\int_0^1t^{3-a}(1-t^2)\,dt$$ $$\implies I'(a) = 8\left(\frac1{a-4}-\frac{1}{a-6}\right)$$ Integrating this w.r.t $a$ $$\implies I(a) = 8\ln\left(\frac{a-4}{a-6}\right) + C$$ Now, as $a \rightarrow \infty$, $I \rightarrow 0$ $$\implies C = 0$$ $$\implies -I(0) = -8\ln\frac{2}{3} = \boxed{8\ln \left(\frac{3}{2}\right)}$$