What is the integral of the square of a general polylogarithm with gaussian argument?

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I'm attempting an integration of such a function. Mathematica won't help me and I've tried several series decompositions which haven't yielded much yet.

So I'm after: $\int_{-\infty}^{\infty}\mathrm{Li}_n^2(-\alpha e^{-\beta x^2})\,\mathrm{d}x$

If the general Polylogarithm can't be done I could suffice with $Li_1$, but not ideal.

Edit: If this is not the appropriate place for such a question let me know and I will remove it, googling of Polylogarithms brought me here (!).