so i thought about applying the watson lemma to 
$\int_{0}^{\infty} \frac{e^{-x(t-ln(t))}}{(1+t^2)} dt$, $x \rightarrow \infty$.
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 work?