Various known expressions in terms of hyperbolic functions for the Laplace transforms of random times related to one-dimensional Brownian motion are derived in a unified way by excursion theory and extended to one-dimensional diffusions.
Jim Pitman. Marc Yor. "Laplace transforms related to excursions of a one-dimensional diffusion." Bernoulli 5 (2) 249 - 255, april 1999.