In my research on quantum groups I have the following conjecture: \begin{equation} \int_0^\infty\frac{\eta (2 i x)^8}{\eta (i x)^2 \eta (4 i x)^2}\,dx\,{\stackrel?=}\,\frac{K(\tfrac{1}{\sqrt{2}})}{\pi}\tag{1} \end{equation} where \begin{equation} \eta(ix)=e^{-\frac{\pi x}{12}}\prod_{n=1}^\infty (1-e^{-2n\pi x}) \end{equation} is the Dedekind eta function and \begin{equation} K(\tfrac{1}{\sqrt{2}})=\frac{\Gamma^2(\tfrac14)}{4\sqrt{\pi}} \end{equation} is the Elliptic integral singular value.
This value has been guessed using Inverse symbolic calculator and then checked numerically to a high precision, but I do not have a proof.
Question: Is (1) true?