I'm working with a hierarchical statistical model, whereby the output of a log-normal distribution affects the argument of an exponential distribution. I need to marginalize, obtaining the following integral of interest:
$$ I(y;\mu,\sigma,\lambda) \equiv \frac{\lambda}{\sigma\sqrt{2\pi}}\int_{0}^\infty \frac{1}{x^2} \exp\left(-\frac{(\ln x - \mu)^2}{2\sigma^2}\right)\exp\left(-\frac{\lambda}{x}y\right)dx$$ where $\mu, \sigma,\lambda>0$.
I've tried changing variables: $x \rightarrow \exp\theta$, but this doesn't seem to help, as the integral contains both terms like $\ln x$ and powers of $x$.
Q: Can this integration be made in analytic form (series are acceptable, but closed form preferred)?