New formula for $\ln(e^Ae^B)$ in terms of commutators of $A$ and $B$

We establish the formula
$$ \ln(e^Be^A)=\int_0^t\psi(e^{-\tau ad_A}e^{-\tau ad_B})e^{-\tau ad_A}\,d\tau(A+B), $$
where $\psi(x)=(\ln x)/(x-1)$; here $A$ and $B$ are elements of a. finite-dimensional Lie algebra which satisfy certain conditions. This formula enables us, in particular, to give a simple proof of the Campbell–Hausdorff theorem. We also give a generalization of the formula to the case of an arbitrary number of factors.