I need help verifying that the following equality holds:
$$Li_2(-2-2\sqrt2)+Li_2(3-2\sqrt2)+Li_2(\frac{1}{\sqrt2})-Li_2(-\frac{1}{\sqrt2})-Li_2(2-\sqrt2)-Li_2(-1-\sqrt2)-2Li_2(-3+2\sqrt2)$$
$$=$$
$$\frac{\pi^2}{6}-\log(1+\sqrt2)\log(2+\sqrt2)-\log(3+2\sqrt2)\log(4-2\sqrt2)$$
using identities for the dilogarithm. I've tried every formula that I know of but haven't managed to make any progress.