The following is a [re-post from MSE](https://math.stackexchange.com/questions/1354791/integral-formula-for-int-0-inftye-3-pi-x2-sinh-pi-x-sinh-3-pi) because I did not get any answer even after offering a bounty.


Towards the end of G. N. Watson's (one of the joint authors of famous book "A Course of Modern Analysis") paper "The Final Problem: An Account of the Mock Theta Functions" the following formula of Ramanujan is mentioned:
\begin{align}&\int\_{0}^{\infty}e^{-3\pi x^{2}}\frac{\sinh \pi x}{\sinh 3\pi x}\,dx\notag\\
&\,\,\,\,\,\,\,\,= \frac{1}{e^{2\pi/3}\sqrt{3}}\sum\_{n = 0}^{\infty}\frac{e^{-2n(n + 1)\pi}}{(1 + e^{-\pi})^{2}(1 + e^{-3\pi})^{2}\dots(1 + e^{-(2n + 1)\pi})^{2}}\tag{1}
\end{align} where the term corresponding to $n = 0$ in the sum on the right is $1$.


Is there way to establish this exotic integral formula? Or a reference to any existing proof of $(1)$ would be of great help.



Integral formula for $\int_{0}^{\infty}e^{-3\pi x^{2}}((\sinh \pi x)/(\sinh 3\pi x))\,dx$ by Ramanujan
This relation seems to be part of the story of Ramanujan's mock theta function. The identity you discuss is a specialization of transformation properties of a mock theta function evaluated at $\tau =i$. See this paper of [Zwegers](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.164.7621&rep=rep1&type=pdf) for proofs of the general transformation identity (and he gives other references to earlier work of Watson): in particular, look at section 3 and Lemma 3.2 of Zwegers's paper. Other surveys that you may find of interest are [Ono](http://www.ams.org/notices/201011/rtx101101410p.pdf) and [Duke](http://www.math.ucla.edu/~wdduke/preprints/mocktheta.pdf). 


