how to evaluate $$\int_{0}^{1} \int_{0}^{1} \frac{\ln(1 - xy) \cdot \text{Li}_{4}(1 - x)}{x(1 - x)(1 - xy)} \,dy\,dx$$
My attempt
$$ \Omega =\int_{0}^{1} \int_{0}^{1} \frac{\ln(1 - xy) \cdot \text{Li}_{4}(1 - x)}{x(1 - x)(1 - xy)} \,dy\,dx$$
$$ * \Omega = \int_{0}^{1} \frac{\text{Li}_{4}(1 - x)}{x(1 - x)} \,dx \cdot \int_{0}^{1} \frac{\ln(1 - xy)}{1 - xy} \,dy $$
$$ = \int_{0}^{1} \frac{\text{Li}_{4}(1 - x)}{x(1 - x)} \,dx \cdot \left[-\frac{1}{x} \int_{0}^{1} \frac{\ln(1 - xy)}{1 - xy} \,d(1 - xy)\right] $$
$$ = \int_{0}^{1} \frac{\text{Li}_{4}(1 - x)}{x(1 - x)} \,dx \cdot \left[-\frac{1}{2x} \ln^2(1-xy)\bigg|_{0}^{1}\right] $$
$$= - \frac{1}{2} \int_{0}^{1} \frac{\ln^2(1 - x) \cdot \text{Li}_{4}(1 - x)}{x^2(1 - x)} \,dx$$