Relationship between integral of x*log(sin(x)), and Apéry's constant, pi, and logarithm

Theorem

The following formula holds: $$\displaystyle\int_0^{\frac{\pi}{2}} x \log(\sin(x)) dx = \dfrac{7}{16}\zeta(3) - \dfrac{\pi^2}{8} \log(2),$$ where $\pi$ denotes pi, $\log$ denotes the logarithm, $\sin$ denotes sine, and $\zeta(3)$ denotes Apéry's constant.