Closed-form of integral $\int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $

I'm looking for a closed form of this definite iterated integral.

$$I = \int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy. $$

From Vladimir Reshetnikov we already know it, that the numerical value of it is

$$I\approx4.49076009892257799033708885767243640685411695804791115741588093621176851...$$

There are similar integrals having closed forms:

$$ \begin{align} J_1 = & \int_0^1 \int_0^1 {\frac {\arcsin \left( \sqrt {1-s}\sqrt {y} \right) }{\sqrt {1-y} \sqrt {sy-y+1}}}\,ds\,dy = 2\pi -2\pi \ln 2. \\ J_2 = & \int_0^1 \int_0^1 {\frac {\arcsin \left( \sqrt {1-s}\sqrt {y} \right) }{\sqrt {1-s} \sqrt {y}\sqrt {sy-y+1}}}\,ds\,dy = -\frac{7}{4}\zeta\left( 3 \right)+\frac{1}{2}{\pi }^{2}\ln 2. \end{align}$$

Solution 1:

I think Math-fun's second approach based on changing the order of integration is a good strategy. Appropriate use of substitutions and trig identities along the way clean up a lot of the resulting "mess":

$$\begin{align} \mathcal{I} &=\int_{0}^{1}\mathrm{d}y\int_{0}^{1}\mathrm{d}s\,\frac{\arcsin{\left(\sqrt{1-s}\sqrt{y}\right)}}{\left(sy-y+1\right)\sqrt{1-y}}\\ &=\int_{0}^{1}\mathrm{d}y\int_{0}^{1}\mathrm{d}t\,\frac{\arcsin{\left(\sqrt{ty}\right)}}{\left(1-ty\right)\sqrt{1-y}};~~~\small{\left[1-s=t\right]}\\ &=\int_{0}^{1}\mathrm{d}y\int_{0}^{y}\mathrm{d}u\,\frac{\arcsin{\left(\sqrt{u}\right)}}{\left(1-u\right)y\sqrt{1-y}};~~~\small{\left[yt=u\right]}\\ &=\int_{0}^{1}\mathrm{d}u\int_{u}^{1}\mathrm{d}y\,\frac{\arcsin{\left(\sqrt{u}\right)}}{\left(1-u\right)y\sqrt{1-y}}\\ &=\int_{0}^{1}\mathrm{d}u\,\frac{\arcsin{\left(\sqrt{u}\right)}}{1-u}\int_{0}^{\sqrt{1-u}}\frac{2\,\mathrm{d}x}{1-x^2};~~~\small{\left[\sqrt{1-y}=x\right]}\\ &=\int_{0}^{1}\mathrm{d}u\,\frac{2\arcsin{\left(\sqrt{u}\right)}}{1-u}\cdot\operatorname{arctanh}{\left(\sqrt{1-u}\right)}\\ &=\int_{0}^{1}\frac{2\arcsin{\left(\sqrt{1-v}\right)}\operatorname{arctanh}{\left(\sqrt{v}\right)}}{v}\,\mathrm{d}v;~~~\small{\left[1-u=v\right]}\\ &=\int_{0}^{1}\frac{4\arcsin{\left(\sqrt{1-w^2}\right)}\operatorname{arctanh}{\left(w\right)}}{w}\,\mathrm{d}w;~~~\small{\left[\sqrt{v}=w\right]}\\ &=4\int_{0}^{1}\frac{\arccos{\left(w\right)}\operatorname{arctanh}{\left(w\right)}}{w}\,\mathrm{d}w\\ &=4\int_{0}^{1}\frac{\operatorname{Li}{\left(w\right)}-\operatorname{Li}{\left(-w\right)}}{2\sqrt{1-w^2}}\,\mathrm{d}w\\ &=4\,{_4F_3}{\left(\frac12,\frac12,1,1;\frac32,\frac32,\frac32;1\right)}.\\ \end{align}$$

And so we see that the above integral is intimately connected to this fun problem, which has generated so much discussion and so many spin-off questions that it wouldn't make sense for me to try to rehash everything here. And given the participation of this question's author in said discussions, I can't help but wonder if he suspected this integral's closed form value all along. =)

Solution 2:

This is not an answer, but I would like to share what I did on this. Let $s=1-w^2$ and $y=z^2$ to have $$I=4\int_0^1\int_0^1\frac{wz\arcsin wz}{\sqrt{1-z^2}(1-w^2z^2)}dwdz$$ Now using $wz=u$ and $z=v$ we obtain $$I=4\int_0^1\int_0^v\frac{u\arcsin u}{v\sqrt{1-v^2}(1-u^2)}dudv$$ Approach $1$: write both $\arcsin u$ and $(1-u^2)^{-1}$ in terms of Taylor's expansions and integrate. This results in an ugly double sum.

Approach $2$: change the order of integration to obtain $$\begin{align}I&=4\int_0^1\int_{v=u}^1\frac{u\arcsin u}{v\sqrt{1-v^2}(1-u^2)}dvdu\\ &=4\int_0^1\frac{u\arcsin u}{(1-u^2)}\ln{\frac{1+\sqrt{1-u^2}}{u}} du\\ &=16\int_0^{\pi/4}x \tan{2x}\ln{\cot{x}} du \end{align}$$ where for the last equality I used $u=\sin x$ and made some trigonometric manipulations.

I tried different Taylor's expansions for this second approach but did not manage to "see" a neatly summable result. Out of what I tried for this second what I liked most was an expansion of $\ln \cot x$ around $\frac{\pi}{4}$, but no success so far.

I thought to do some integrations by part and that might work ... maybe somebody else would do it faster though!