The distribution of the integral of the absolute value of the Brownian bridge was expressed by Cifarelli and independently by Johnson and Killeen in the form of a series. Rice obtained the corresponding probability density by numerical integration. Here we determine the exact tail asymptotics of this distribution, as well as the exact asymptotics of its density function for the large values of the argument.
"Asymptotics of the distribution of the integral of the absolute value of the Brownian bridge for large arguments." Ann. Probab. 28 (1) 132 - 139, January 2000. https://doi.org/10.1214/aop/1019160114