Conditional Distribution of a Brownian Motion with terminal point in an interval.

Question

Let $\mathrm{d}X_t=\mu\mathrm{d}t+\sigma \mathrm{d}B_t$ where $B_t$ is a standard Brownian motion. We know that $X_0=c$ and $X_T\in (a,b)$ where $a,b$ are two known real number and $T$ is a fixed time. I want to find the conditional distribution of $X_t|X_0=c,X_T \in(a,b)$. Indeed what I am interested is to find the conditional mean and variance. For $t>T$, it is pretty simple, since it is just a truncated normal distribution $X_T|X_T\in(a,b)$ plus an independent increment $X_t-X_T$. However, for $t\in(0,T)$, I cannot figure it out. I guess it could be obtain use some similar techniques as obtaining Brownian bridge with $X_T=x$, however it turns out to be out of my ability.