Say we are given an oscillatory integral of the form

$\Psi(x)=\int_{-\infty}^\infty e^{i\psi(x,t)} a(t)dt$.

where $a(t)$ is a sufficiently nice function. When, for instance, $|\psi(x,t)_t| \gg x^\alpha$ uniformly in $t$ for some $\alpha>0$, one can prove that $\Psi(x)$ is rapidly decreasing in $x\to \infty$.

My question here is to determine if we can find a constant $c>0$ and $C>0$ such that $\Psi(x)\ll e^{-cx^C}$.

For example, if we put

$f(x)=\int_{-\infty}^\infty \frac{\cos xt dt}{\sqrt{1+t^2}}$

then it is known through the asymptotic expansion of the modified Bessel function of the second kind that

$f(x) \ll x^{-1/2}e^{-x}.$

Other example will be the following integral:

$K_{ix}(1)=\int_{-\infty}^\infty e^{-\cosh t}\cos xt dt$

which is $\ll x^{-1/2}e^{-\frac{\pi x}{2}}$.

However, if I were to use the integration by parts, as far as I know, I can only prove that these functions are rapidly decaying in $x$.

To summarize: I would like to learn if there is any general method which can go beyond polynomial-decay of the given oscillatory integral.

Thank you.