 Solution

Near zero, the absolute value of the integrand is comparable towhich is integrable for $z>0$. At infinity, the factor $e^{-t}$ dominates every power of $t$ and $\log t$, so the integral also converges there.

Direct use of Laplace's method is inconvenient because the phasedoes not have the form $z\varphi (t)$ with a fixed phase: its stationary point $t=z$ moves to infinity as $z\to \infty$. Rescaling is needed to fix the maximum.