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I will answer a simplified version, so leave the generalization as an exercise. Let $Z$ be a standard normal random variable so $X=e^Z$ is standard lognormal. Since $X>0$ we have $Y=\frac1{1+X}$ is in the unit interval. Let $\phi, \Phi$ be the density and cdf (cumulative distribution ) functions of the standard normal, then we find  \...