The marginal of a multivariate Gaussian can be computed in closed form, i.e.,

$p(x) = \int_y \mathcal{N}((x,y);\mu,\Sigma)$

is simple. But what I need is

$L(x) = \int_y \mathcal{N}((x\mid y); \mu(y),\Sigma_{\mid y})$,

i.e., the integral over the variable that is being conditioned on. I know that's not a distribution in $x$, but can the integral be computed in closed form? Or approximated efficiently without sampling?