The following identity arose while I was working on a recent MO question:

$-\sum_{n=1}^{\infty}\frac{1}{n}\frac{(-x)^n}{1-x^n}=\sum_{n=1}^{\infty}\frac{1}{n}\frac{x^n}{1-x^{2n}}.$

I have no doubt that the identity is true, but I am not able to prove it. Can anyone help?

It is easy to prove by Taylor expansion that the left-hand-side of the identity can equivalently be written as $\sum_{n=1}^{\infty}\ln(1+x^n)$, which is the logarithm of the q-Pochhammer symbol $(-x,x)_{\infty}$, so an alternative way to pose my question is to ask for a proof of the series expansion

$\ln(-x,x)_{\infty}=\sum_{n=1}^{\infty}\frac{1}{n}\frac{x^{n}}{1-x^{2n}}.$