Let $X_t$ be an Ornstein-Uhlenbeck process solving $dx_t = \theta (\mu-x_t)\,dt + \sigma \,dW_t$. The solution is known and given by: $$ x_t = x_0 e^{-\theta t} + \mu(1-e^{-\theta t}) + \int_0^t \sigma e^{\theta (s-t)} \,dW_s$$

Is there a closed-form formula (both SDE and actual solution) for time integral $\int_0^t X_t\, dt$?

(I know there is a lot of literature on interest theory that analyzes the expectation of this kind of integral, but this is not something I am after)