Some definite integrals arising from selfdecomposable characteristic functions

In the probability theory, selfdecomposable or class L0distributions play an important role as they are limit distributions of normalized partial sums of sequences of independent, not necessarily identically distributed, random variables. The class L0 is quite large and includes many known classical distributions. For this note, the most important feature of the selfdecomposable variables are their random integral representation with respect to a Lévy process. From those random integral representations we get the equality of logarithms of some characteristic functions. These allow us to get formulas for some definite integrals; some of them were previously unknown, and some are rarely quoted in popular tables of integrals and series.