Complexity of union-find with path-compression, without rank

Question

Wikipedia says union by rank without path compression gives an amortized time complexity of $O(\log n)$, and that both union by rank and path compression gives an amortized time complexity of $O(\alpha(n))$ (where $\alpha$ is the inverse of the Ackerman function). However, it does not mention the running time of path compression without union rank, which is what I usually implement myself.

What's the amortized time complexity of union-find with the path-compression optimization, but without the union by rank optimization?