Closed Form of $ \int_{0}^{\pi} x^3\ln^8(2\sin{(x)}) dx$

Question

I have need of a closed form expression to this definite integral:

$$ \int\limits_{0}^{\pi} x^3\ln^8(2\sin{(x)}) dx$$

It does exist apparently. Both limits are problematic for the natural logarithm function in the integrand. WolframAlpha PRO gives me a result of 624510 which I take to mean 'Big Number' . By changing limits to a tiny amount left of $\pi$ and a tiny amount right of $0$ , I get a wide range of very large values because from the graph it appears the area is unbounded on the right side. That is as far as my efforts have got me, the past few days.

Answer 1

Putting $x=\frac{\theta}{2}$ the integral $$ I=\int_{0}^{\pi} x^3\ln^8(2\sin{(x)}) \mathrm dx $$ becomes $$ I=\frac{1}{2^{12}}\int_{0}^{\pi/2} \theta^3\ln^8\left(\left|2\sin\left(\tfrac{\theta}{2}\right)\right|^2\right) \mathrm d\theta=-\frac{1}{2^{12}}\operatorname{Ls}^{(3)}_{12}(\pi/2) $$ where $$ \operatorname{Ls}^{(m)}_{n}(\sigma)=-\int_{0}^{\sigma} \theta^m\ln^{n-m-1}\left(\left|2\sin\left(\tfrac{\theta}{2}\right)\right|\right) \mathrm d\theta $$ is the generalized log-sine integral. It involves many advanced tools to be evaluated and I think it's out of your scope.

Answer 2

Not really an answer, but too long for a comment. Agrees nicely with other results. Plouffe's inverter can't find a match.

$Version 

9.0 for Linux x86 (32-bit) (November 20, 2012) 

NIntegrate[x^3*Log[2*Sin[x]]^8,{x,0,Pi}, WorkingPrecision -> 50, 
AccuracyGoal -> 40, PrecisionGoal -> 40] 

624509.97425476323973864321155907856915353495760746