Expected truncated hitting time of Brownian motion

I'm trying to compute the expected value of a truncated hitting time of a Brownian motion with an one-sided boundary. More specifically, if we let $$B_t$$ be a $$(\mu, \sigma)$$ Brownian motion and $$T= \inf\{t\ge 0: B_t = a\}$$ be a hitting time for some \$a>0, I want to know if there is any closed form formula for the expected truncated hitting time

$$\mathbb{E}[\tau] = \mathbb{E}[T_a \wedge t_0]$$ where $$t_0>0$$ is a constant. I found out that $$T_a$$ follows the inverse Gaussian distribution, but I was not able to obtain a closed form solution for the truncated hitting time.