I want to analyze the following integral changing $\sigma$ over the interval $[0,\infty)$

$$\frac{1}{\sigma}\int\_{0}^{1}\frac{(y-1/2)^2}{y(1-y)}\exp\bigg\{-\frac{\big[\log\big(\frac{y}{1-y}\big)\big]^2}{2\sigma^2}\bigg\}dy$$

For large $\sigma$, the adaptive quadrature method has numerical problems (probably due to the singularity of the integrand at $y=1$). Is there a way to evaluate it?

I've tried also to study its equivalent form with the same method, namely

$$\frac{1}{\sigma}\int\_{0}^{\infty}\frac{(t-1)^2}{4t(t+1)^2}\exp\bigg\{-\frac{(\log t)^2}{2\sigma^2}\bigg\}dt$$

where $t=\frac{y}{1-y}$ but I still get numerical problems and find huge differences between the two integrals even for small values of $\sigma$.