Proof of $\int_0^\infty\frac{\left (1- e^{\pi\sqrt3x}\cos(\pi x )\right )e^{-2\pi x/\sqrt3}}{x(1+x^3)(1+x^3/2^3)(1+x^3/3^3)\dots}~dx=0.$

Question

I have the following integral in my notebook: $$\int_0^\infty\frac{\left (1- e^{\pi\sqrt3x}\cos(\pi x )\right )e^{-2\pi x/\sqrt3}}{x\prod_{j=1}^\infty (1+ x^3/j^3)}\ \mathsf dx=0.$$ Though after going through all my bookmarks, I can't find where I got it from, and I certainly do not know where to begin evaluating this integral. WolframAlpha offers no useful simplification of the integrand. Any help would be appreciated.

Edit: ArXiv link to the original paper found! Here: https://arxiv.org/abs/1712.07456.

Answer 1

This integral is proved here https://arxiv.org/abs/1712.07456 .