PlusDistribution[1/(1 - x)]
denotes a distribution (in the sense of the “+” prescription).
[1/(1 - x)] PlusDistribution
[Log[1 - x]/(1 - x)] PlusDistribution
[PlusDistribution[1/(1 - x)], {x, 0, 1}] Integrate2
[PlusDistribution[Log[1 - x]/(1 - x)], {x, 0, 1}] Integrate2
[PlusDistribution[Log[1 - x]^2/(1 - x)], {x, 0, 1}] Integrate2
[Log[x (1 - x)]/(1 - x)] PlusDistribution