Conjecture about integral $\int_0^1 K\left(\sqrt{\vphantom1x}\right)\,K\left(\sqrt{1-x}\right)\,x^ndx$ I'm interested in the following integral:
$$\mathcal J(n)=\int_0^1 K\left(\sqrt{\vphantom1x}\right)\,K\left(\sqrt{1-x}\right)\,x^ndx,\tag1$$
where $K(z)$ is the complete elliptic integral of the 1ˢᵗ kind:
$$K(z)={_2F_1}\left(\begin{array}c\tfrac12,\tfrac12\\1\end{array}\middle|\ z^2\right)\cdot\frac\pi2.\tag2$$
Using numerical integration and inverse symbolic calculation techniques, I found a conjectural formula for this integral, that only though seems to work for $n\in\mathbb N_0$:
$$\mathcal J(n)\stackrel?=\frac{\pi^3}{8\cdot4^n}\left(\frac{(2n-1)!!}{n!}\right)^2{_4F_3}\left(\begin{array}c\tfrac12,\tfrac12,-n,-n\\1,\tfrac12-n,\tfrac12-n\end{array}\middle|\ 1\right).\tag3$$
This formula produces the following sequence of results for $n=0,1,2,...$
$$\frac{\pi^3}8,\frac{\pi^3}{16},\frac{11\,\pi^3}{256},\frac{17\,\pi^3}{512},\frac{1787 \,\pi^3}{65536},\frac{3047\,\pi^3}{131072},\frac{42631\,\pi^3}{2097152},\frac{75937\,\pi^3}{4194304},...\tag4$$



*

*Can we prove the conjecture $(3)$ is correct?

*Is it possible to find a generalization that works for non-integer values of $n$ as well?


Following my answer in this question, we have
$$
\begin{align*}&\; \int^1_0x^nK\left(\sqrt{\vphantom1x}\right)K\left(\sqrt{1-x}\right)dx\\
&=\int^1_0\left(\frac\pi2\sum^{\infty}_{m=0}\frac{(2m)!^2}{2^{4m}(m!)^4}x^{m+n}\right)K\left(\sqrt{1-x}\right)dx\\
&=\frac\pi2\sum^{\infty}_{m=0}\frac{(2m)!^2}{2^{4m}(m!)^4}\int^1_0x^{m+n}K\left(\sqrt{1-x}\right)dx\\
&=\frac\pi2\sum^{\infty}_{m=0}\frac{(2m)!^2}{2^{4m}(m!)^4}\frac{2^{4m+4n+1}((m+n)!)^4}{(2m+2n+1)!^2}\\
&=2^{4n}\pi\sum^{\infty}_{m=0}\frac{(2m)!^2}{(m!)^4}\frac{((m+n)!)^4}{(2m+2n+1)!^2}\\
&=\frac{2^{4n}\pi(n!)^4}{(2n+1)!^2}{}_4F_3\left(\frac12,\frac12,n+1,n+1;1,n+\frac32,n+\frac32\middle|\,1\right).
\end{align*}
$$
This form checks numerically for non-integer $n$, and I think it should be related to OP's conjectured form via some $
_4F_3$ transformation.
