How can I prove the equality of these two infinite sums: $\sum\limits_{n=1}^{\infty}[ 2\, n\,\pi\coth^{-1}(2n)-\pi]= \frac\pi2-\frac{\pi\,\ln(2)}2$

Question

How can I prove that

$$ \sum_{n=1}^{\infty}[ 2\, n\,\pi\coth^{-1}(2n)-\pi]= \frac{\pi}{2}-\frac{\pi\,\ln(2)}{2}$$

This result comes from using the Infinite product of $\sin(x)$ in the following well-known integral.

$$\int_0^{\frac{\pi}{2}} \ln[\sin(x)] $$

I hope to find a way to directly prove the result , rather than showing it is related to the above integral.

I have spent a great deal of time on this , so any hints or results will be greatly appreciated.

Thank you very much for your help and time.