This is related to this question and my partial answer to it. I've found that the proof of the 2nd identity reduces to showing that $$i\int_0^{\pi}\left[\mathrm{Li}_2\left(-1-e^{ix}\right)-\mathrm{Li}_2\left(-1-e^{-ix}\right)\right]dx=\frac{7}{3}\zeta(3) \tag{1} $$ Here $\mathrm{Li}_2(z)$ is the dilogarithm function defined by analytic continuation of the series $\sum\limits_{n=1}^{\infty}\frac{z^n}{n^2}$ to the cut plane $\mathbb{C}\backslash[1,\infty)$ and $\zeta(3)=\sum\limits_{n=1}^{\infty}\frac{1}{n^3}$ is the Apéry's constant.
I am looking for a proof of $(1)$ or an alternative proof of the 2nd identity here. Thank you in advance.