Let $\phi:=\dfrac{\sqrt{5}+1}2$. In the calculation of a certain integral (which has conjecturally a closed form), the following term occured: $$ x=\left(Li_3(-\dfrac1\phi)-Li_3(\phi)\right)\log(\phi)+\left( Li_4(-\dfrac1\phi)+Li_4(\phi)\right).$$ I know that $\Im(x)=\dfrac\pi3\log^3(\phi)$, which seems to indicate that indeed for $\Re(x)$ there should be some closed form too, supposedly in terms of $\phi,\log(\phi), \pi$ (and possibly $\zeta(3)$) $-$ but I can't seem to find it. Any ideas?
2026-03-26 03:12:35.1774494755
Looking for a closed form of a polylog expression involving the golden ratio
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