I'm looking for the distribution whose Fourier transform is given by $\varphi(t) = \exp\left(\int_0 ^ 1 \frac{e^{itx} - 1}{x} \right),$ where as usual $\varphi(t) = \int_{- \infty} ^ \infty e^{itx} dF(x)$ and $F$ is a probability distribution function on the real line. The distribution should probably be absolutely continuous and put mass on the positive half-line, so we likely want $f$ satisfying $\varphi(t) = \int_0 ^ \infty e^{itx} f(x) \ dx$, if that helps.
I'm not really sure where to begin with this. I don't know that it has an answer in terms of any well known functions. Mathematica choked when trying to invert it and I haven't been able to find any expression for it in any tables that I've looked in.
For some context, this popped up as the characteristic function of an otherwise unknown random variable of interest.
If possible, I would ask that anyone who posts a known solution to include the source.