High accurate method to calculate a singular integral related to Hankel transform

In this paper we are interested in the approximation of the integral \[I_0(f,\omega)=\int_0^\infty f(t)\,e^{-t}\,J_0(\omega t)\,dt\] for fairly large $\omega$ values.  This singular integral comes from the Hankel transformation of order $0$, $f(x)$ is a function with which the integral is convergent. 

For fairly large values of $\omega$, the classical quadrature methods are not appropriate, on the other side, these methods are applicable for relatively small values of $\omega$.  Moreover, all quadrature methods are reduced to the evaluation of the function to be integrated into the nodes of the subdivision of the integration interval, hence the obligation to evaluate the exponential function and the Bessel function at rather large nodes of the interval $]0,+\infty[$.

The idea is to have the value of $I_0(f,\omega)$ with great precision for large $\omega$ without having to improve the numerical method of calculation of the integrals, just by studying the behavior of the function $I_0(f,\omega)$ and extrapolating it.

We will use two approaches to extrapolation of $I_0(f,\omega)$.  The first one is the Padé approximant of $I_0(f,\omega)$ and the second one is the rational interpolation.