$\int_{0}^{1}x^{x^{\alpha}} dx$ for negative $\alpha$

Question

I'm trying to come up with a formula for $$\int_{0}^{1}x^{x^{\alpha}} dx,$$ basically the Sophomore's Dream.

$$\sum_{k=0}^{\infty}\frac{(-1)^k}{(\alpha k+1)^{k+1}}$$

However it only works for positive $\alpha$, the sum converges to $1$ for $\alpha$ approaching $-\infty$ while the integral approaches $0$.

This might have something to do with the poles at $\alpha=\frac{-1}{k}$?