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)}...

Here I have a question I just received, and still trying to find a proper starting point $$\int_0^1 \frac{(1-x+x\log(x))\operatorname{Li_3(x)}}{(x-1) x \log(x)} \ dx$$ What starting point would you propose?

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Thanks to @David H's comment I got that $$\int_0^1 \frac{(1-x+x\log(x))\operatorname{Li_3(x)}}{(x-1) x \log(x)} \ dx=\frac{5}{4}\zeta(4)-\gamma \zeta(3)+\zeta'(3)$$ that is proving to be numerically correct.