Define g(x) = (1+x) ln(1+x) - x. One can check that g is strictly monotonically increasing for x>=0 by checking its derivative is ln(1+x). So g is invertible and its inverse is also strictly monotically increasing.

Is there an explicit closed form for its inverse?

With a page of calculations I can prove that (1/2) x/ln(1+sqrt(x)) <= g^{-1}(x) <= 2 x/ln(1+sqrt(x)) for all x>=0. Is this obvious? Can estimates like this be found in the literature?