Closed form for $\sum_{i=1}^{n}\binom{n}{i}\frac{x^{i}}{i}$

Question

Asked a similar question earlier, turned out the maths I did behind it was dumb. Now, in a much simpler form, how can one find the closed form of the below? $$\sum_{i=1}^{n}\binom{n}{i}\frac{u^{i}}{i}$$ This is a rearrangement and reduction of $$\sum^{i=1}_{k-1}\binom{k-1}{i}(1+p)^{k-1-i}\frac{(2x-1-p)^{i}}{2^{k-1}i}$$ with the constant coefficients excluded. All I'm getting is the hypergeometric function, which isn't really an answer, given it's just shorthand for a summation. Can this otherwise be closed?