Kac has considered the integral of the absolute value of a Brownian motion process and determined the Laplace-Stieltjes transform of the distribution function of this integral. In this paper explicit expressions are given for the distribution and the moments of this integral.
"On the Distribution of the Integral of the Absolute Value of the Brownian Motion." Ann. Appl. Probab. 3 (1) 186 - 197, February, 1993. https://doi.org/10.1214/aoap/1177005514