A hard integral Looking for a solution for an integral:
$$I(k)=\int_0^{\infty } \frac{e^{-\frac{(\log (u)-k)^2}{2 s^2}}}{\sqrt{2 \pi } s \left(1+u\right)} \, du .$$
So far I tried substitutions and by parts to no avail.

If $k$ is an integer multiple of $s^2$, then it appears you can use the result $I(0)=\frac12$ to obtain $I(k)$ as a sum of a finite number of terms.
