Suppose $P(x_1,x_2)$ is the pdf of a bivariate normal distribution, where $x_1\sim\mathcal{N}(\mu_1,\sigma_1^2)$ and $x_2\sim\mathcal{N}(\mu_2,\sigma_2^2)$ and the correlation between $x_1$ and $x_2$ is $\rho$.
I am looking for a closed-form formula or an approximation of the pdf of the RV: $$ Y = \exp{x_1} + \exp{x_2}\,. $$
As far as I know there is probably only an approximation, but I can only find an approximation in the case $\rho=0$, i.e. when $x_1$ and $x_2$ are independent.
Any ideas?
Thanks!