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

I=∫∞0(∫101√1−y2√1+x2y2dy)3dx.

The inner integral can be represented using the hypergeometric function 2F1 or the complete elliptic integral of the 1st kind K with an imaginary argument:

∫101√1−y2√1+x2y2dy=π22F1(12,12;1;−x2)=K(x√−1).

My conjecture is the integral I has a closed-form representation:I?=3Γ(14)81280π2=7.09022700484626946098980237…,

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?

**Answer**

Using

K(ik)=1√1+k2K(√k2k2+1)

and a substitution t2=k21+k2, rewrite the integral as

∫∞0K(ik)3dk=∫10K(t)3dt.

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

∫10K(k)3dk=35K(1/√2)4=3Γ(14)81280π2,

using K(1/√2)=14π−1/2Γ(14)2.

**Attribution***Source : Link , Question Author : Vladimir Reshetnikov , Answer Author : Kirill*