Expected sojourn time of Brownian motion

Question

I am trying to determine the expected sojourn time of Brownian motion below a pre-specified barrier. Let $B=(B_t)_{t\geq 0}$ be a standard Brownian motion starting from $a\in \mathbb{R}$, and let $u\geq a$ be the barrier. Fixing $t>0$ and defining $X_t=\int_0^t \mathbb{1}_{[0,u]}(B_s)ds$, my aim is to calculate $\mathbb{E}(X_t)=\int_{\mathbb{R}}x f_{X_t}(x)dx$, where $f_{X_t}$ is the corresponding density function. According to Borodin and Salminen (Handbook of Brownian Motion - Facts and Formulae, p. 160), it is possible to apply the Feynman-Kac formula to obtain the following representation of $f_{X_t}$: $$f_{X_t}(x)=\frac{1}{\pi \sqrt{x(t-x)}}\exp {\left(-\frac{(u-a)^2}{2x}\right)}\mathbb{1}_{[0,t]}(x)$$ I am therefore left to calculate the following: $$\mathbb{E}(X_t)=\int_0^t \frac{x}{\pi \sqrt{x(t-x)}}\exp {\left(-\frac{(u-a)^2}{2x}\right)}dx.$$ Having reached this stage, maybe I am missing an obvious trick, but I am unsure how to move forward and whether or not it is possible to obtain a simplified expression for this integral (possibly expressing it in terms of the erf or erfc functions)? Any ideas or references to the literature would be greatly appreciated.