# Closed form for ∫∞0(∫101√1−y2√1+x2y2dy)3dx.\int_0^\infty\left(\int_0^1\frac1{\sqrt{1-y^2}\sqrt{1+x^2\,y^2}}\mathrm dy\right)^3\mathrm dx.

I need to find a closed form for these nested definite integrals:

The inner integral can be represented using the hypergeometric function $_2F_1$ or the complete elliptic integral of the 1st kind $K$ with an imaginary argument:

My conjecture is the integral $I$ has a closed-form representation:
but I was neither able to find a proof of it, nor disprove the equality using numerical integration. Could you please help me with resolving this question?

Using

and a substitution $t^2 = \frac{k^2}{1+k^2}$, rewrite the integral as

There is a paper “Moments of elliptic integrals and critical L-values”
by Rogers, Wan and Zucker (http://arxiv.org/abs/1303.2259; also one of
the authors’ earlier papers: http://arxiv.org/abs/1101.1132), and the
authors, by relating this integral to an L-series of a modular form (their theorems 1 and 2),
show that

using $K(1/\sqrt{2}) = \frac14 \pi^{-1/2}\Gamma(\frac14)^2$.