The law of the hitting time of a 1-dimensional Brownian motion $W$ is well known, but I can't find any information on the density of the hitting time of $|W|$.

I define $T=\inf \{t>0,|W|(t)= 1\}$. One can find some useful informations about this random variables like its expectation 1, variance $\frac{2}{3}$ or Laplace transform $\mathbb{E}[e^{-xT}]=\frac{1}{\cosh(\sqrt{2x})}$ (see Revuz and Yor).

How can I know if $T$ has a density ? Can I inverse the Laplace transform easily (numerically could be enough) ? Any additional information on $T$ would be appreciated.