Integral of an infinite product

In this article our goal it to prove the following integral


\begin{align*} \int_0^\infty e^{-x}\left(\prod_{n=1}^\infty\left(1-e^{-24\!\;n\!\;x}\right)\right)dx=\frac{\pi ^2}{6 \sqrt{3}} \end{align*}


To this end we will rely on two results: Euler´s pentagonal theorem which states that


\begin{align*} \prod_{n=1}^\infty\left(1-q^{n}\right)=\sum_{n=-\infty}^{\infty}(-1)^nq^{\large \frac{3n^2-n}2} \end{align*}


we have proved this before (see here). The second tool that we will use regards the evaluation of alternating infinite series using the residues theorem to evaluate the following infinite series


\begin{align*} \sum_{n=-\infty}^{\infty}\frac{(-1)^n}{(6n-1)^2}=\frac{\pi ^2}{6 \sqrt{3}} \end{align*}


Click here for the proof.

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