The following integral is due to Ramamnujan (or so I was told). I have not been able to find a reference for it, but I imagine this would have been how it was derived.
\[\begin{align*} \int_0^1\frac{\ln\left(\frac{1+\sqrt{4x+1}}{2}\right)}{x}\,\mathrm dx &= \int_1^\phi\frac{(2u-1)\ln(u)}{u^2-u}\,\mathrm du,\qquad x=u^2-u\\ &= \int_1^\phi\left(\frac{\ln(u)}{u}+\frac{\ln(u)}{1-u}\right)\,\mathrm du\\ &= \left[\frac{\ln^2(u)}{2}-\text{Li}_2(1-u)\right]\Bigg\vert_1^\phi\\ &= \frac{\ln^2(\phi)}{2}-\text{Li}_2(\bar{\phi})\\ &= \frac{\pi^2}{15}.\end{align*}\]Several special values (as seen on page 6 of Zagier’s write-up) can be obtained from the functional equations mentioned on pages 8 and 9.