I would like to know the asymptotic expansion of the sequence of positive numbers given by $$I_{n}:=-\int_{0}^{1}\frac{n^{x-1}}{\Gamma(x-1)}dx,$$ for $n\rightarrow\infty$.

One can easily derive an upper estimation since $$I_{n}\leq\max_{-1<x<0}\frac{1}{|\Gamma(x)|}\int_{0}^{1}n^{x-1}dx\approx0.282\,\frac{n-1}{n\ln n}\leq0.282\,\frac{1}{\ln n}.$$ However, by some numerical experiments, it seems that the sequence $I_{n}$ decreases to zero slightly faster than $1/\ln n$. So far, however, I do not know how to obtain at least the leading term of the asymptotic expansion. Any idea? Many thanks.