I have an approximation that I don't know how to prove, I believe it is some simple computations, however, I can't fill in the intermediate steps. $\sum_{k=1}^{n-1}o((\log k)^{-1-\epsilon})\frac{O((k\log\log k)^{1/2})}{k+1}=o(n^{1/2}(\log n)^{-1-\epsilon/2})$.
I know $\log\log k$ grows slower than any positive power of $\log k$, however, I don't know how to simplify so as to compute this summation.