I'm looking for the distribution of
$X = \int_0^T e^{-W_t} dt \int_0^T e^{W_t}dt$
and
$Y = \frac{\int_0^T e^{-W_t} dt}{ \int_0^T e^{W_t}dt}$
(where $W_t$ is a standard brownian motion)
On most of the papers i find on the internet they only tackle the integrals by separte using Bougerol's identity. But not exactly the symmetric problem i'm studying with the two integrals.
thanks.