So i thought about applying the Watson lemma to determine the asymptotic behavior of the integral $$ I(x)=\int_{0}^{\infty} \frac{e^{-x(t-\ln(t))}}{(1+t^2)} dt, $$ as $x \rightarrow \infty$. I think it should be possible to substitute the term $t-\ln(t)$ so that we can apply the lemma. Has anyone an idea on how this could be worked out?