Evaluation of a tough double integral

Question

This is an integral coming from personal research, and very important to me, but it does not
seem an easy job to do. If a solution is not possible then I'd be glad with a closed form only.

$$\int_{[0,1]^2} \frac{(1-x-y+x y+x \log(x)-x y\log(x)+y \log(y)- x y\log(y)+x y\log(x)\log(y))\log(1+x y)}{x y (1-x) (1-y)\log(x)\log(y)} \ dx \ dy$$

Hopefully this will be seen by Cleo too and maybe we'll find its closed form.

Answer 1

Write the integral as $$ \int_{[0,1]^2}h(x)h(y)\log(1+xy)\,dx\,dy = \sum_{n\geq1}(-1)^{n+1}\frac{a_n^2}{n}, \qquad h(x) = \frac{1-x+x\log x}{x(1-x)\log x}, $$ where the coefficients $a_n$ are given by $$ a_n = \int_0^1 h(x)x^n\,dx = \gamma - H_n + \log n. $$

Using the sums $$ \sum_{n\geq1} \frac{(-1)^{n+1}}{n} = \log2 $$ $$ \sum_{n\geq1} \frac{H_n(-1)^{n+1}}{n} = \tfrac12\big(\zeta(2)-\log^22\big), $$ $$ \sum_{n\geq1}\frac{H_n^2(-1)^{n+1}}{n} = -\tfrac12\zeta(2)\log2+\tfrac13\log^22+\tfrac34\zeta(3), $$ $$ \sum_{n\geq1}\frac{(-1)^{n+1}\log n}{n} = -\gamma\log2+\tfrac12\log^22, $$ $$ \sum\frac{(-1)^{n+1}\log^2n}{n} = -\gamma\log^22+\tfrac13\log^32-2\gamma_1\log2, $$ where $\gamma_1$ is a Stieltjes gamma constant, and noting that the sum $$ \sum_{n\geq 1}(-1)^{n+1}\frac{H_n\log n}{n} = \int_0^1 \frac{du}{u}\big(\lambda_1(u-1)-\lambda_1(-1)\big), \qquad \lambda_s(t) = \frac{\partial \mathrm{Li}_s}{\partial s}(t)$$ has no closed form at all, the integral can be written as $$ -\gamma\zeta(2)-\gamma^2\log2-\tfrac12\zeta(2)\log2+\gamma\log^22+\tfrac23\log^32-2\gamma_1\log2\\+\tfrac34\zeta(3) - 2\sum_{n\geq1}\frac{(-1)^{n+1}H_n\log n}{n}. $$

Answer 2

I can simplify the integrand into;

\begin{equation} \frac{(y\log y + (1-y))(x \log x +(1-x)}{(1-x)(1-y)} \end{equation}

The computation of \begin{equation} \int_{0}^{1} \int_{0}^{1} \frac{(y\log y + (1-y))(x \log x +(1-x))}{(1-x)(1-y)} dx dy \end{equation} is proving to be tougher than expected!