integration - Any ideas to the integral $int_0^frac{pi}{2}xln(sin x)~dx$ ?

I've corrected typing error in the integral.
I apologize for my mistake.

Reedited question:

Can anybody solve integral: $\int_0^\frac{\pi}{2}x\ln(\sin x)~dx$ ?
I'm just trying to guess some simple formula for $\zeta(3)$. My "strategy" is simple: Find some conjectures and check them numerically.

Value of similar integral is well known:
$\int_0^\pi x\ln(\sin x)~dx=-\ln(2)\dfrac{\pi^2}{2}$ .
Is there any idea for antiderivative of $x\ln(\sin x)$ ? It's to difficult to solve it for me.
Any ideas?