I need to evaluate the following integral with a high precision:
$$
I=\int\_{0}^{\infty}\left[%
{\pi^{2} \over 6} - {\rm Li}\_2\left({\rm e}^{-x}\right)
-{\rm Li}\_2\left({\rm e}^{-1/x}\right)\right]\,{{\rm d}x \over x},
$$
where ${\rm Li}\_{2}$ denotes the
[dilogarithm](http://en.wikipedia.org/wiki/Dilogarithm)
$\displaystyle{%
\left(~\mbox{note that}\ {\rm Li}\_{2}\left(1\right) = {\pi^{2} \over 6}~\right)}$.

Unfortunately, a numerical integration in my CAS is only able to produce $3$ stable digits $I \approx 3.77\ldots$ that I do not even sure to be provably correct. 

So, if only I were so lucky that a closed form existed for this integral, then, hopefully, it could be used to easily evaluate $I$ with a much higher precision. Could you suggest how to find a closed form ( if one exists )?