I believe I have a challenge for the integration community.
Show directly: $$2\pi\int_{0}^{\infty}\frac{\eta(4ix)^6\eta(20ix)^6}{\eta(2ix)^2\eta(8ix)^2\eta(10ix)^2\eta(40ix)^2}\,\mathrm{d}x=\frac{1}{\sqrt{\varphi}}K\left(\frac{i}{\varphi}\right),$$ where $\varphi:=\frac{\sqrt{5}+1}{2}$ is the golden ratio.
As a reminder, here $\eta(\tau):=q^{1/24}\prod_{n=1}^{\infty}\left(1-q^n\right)$ with $q:=e^{2\pi i\tau}$ is the Dedekind eta function with $\Im(\tau)>0$ and $K(k):=\int_{0}^{\frac{\pi}{2}}\frac{\mathrm{d}\theta}{\sqrt{1-k^2\sin^2\theta}}$ is the complete elliptic integral of the first kind.
What I mean by "show directly" is a method that does not utilise the fact that the integrand is the modular form associated with the elliptic curves in the isogeny class with Cremona label $80b$. In particular, it must not use the fact that these elliptic curves are of rank $0$ and thus have a general integral formula for their associated $L$-function at $s=1$ given by some rational factor multiplied by the integral of the associated Néron differential over $E\left(\mathbb{R}\right)$ (which can, in general, be expressed in terms of $K$ and the roots of the cubic of the elliptic curve in simplified form, say).
I am after a direct approach that utilises, for example, substitutions that transform the integrand from being in terms of $\eta$ into being in terms of say $K,K',k,k'$ and evaluating the resulting integral(s) through standard integration techniques.