ugh, it took me an embarrassing amount of time to spot my error:
$$2\int_0^1 \int_0^x xx' \ln\left(\frac{x'+x}{x-x'}\right)dx'dx=2\int_0^1 \int_0^1 x^3 u \ln\left(\frac{1+u}{1-u}\right)\,du\,dx=2\int_0^1 x^3 \,dx \int_0^1 u \ln\left(\frac{1+u}{1-u}\right)\,du$$