I would like to compute the expectation of the following expectation
$\mathbb{E}[\int_a^\infty e^{-rt}\min(x_t,c)\,dt]\,$
where a, r, c are constants, $dx_t = \mu x_t dt + \sigma x_t dW_t$ is a geometrical Brownian motion with $(\mu < r)$ and $\min(x_t,c)$ denotes the minimum of $x_t$ and c. Any help would be much appreciated!