$$X \sim \ln\mathcal{N}(\mu_X,\,\sigma_X)$$ $$Y \sim \mathcal{N}(0,\,1)$$ $$Z = X + Y$$
I want to find the probability density functions and cumulative distribution functions of $Z$.
As the below is not integrable, I assume I can only find an approximation.
$$\int_{-\infty }^{\infty } \frac{e^{-\frac{\left(\log (x)-\mu _X\right){}^2}{2 \sigma _X^2}}}{\sqrt{2 \pi } \sigma _X x} \frac{e^{-\frac{\left(z-x\right){}^2}{2}}}{\sqrt{2 \pi }} \, dx$$
What approaches are there to finding an approximation?
I've tried using Mathematica to integrate the first few terms of the Taylor series about $x=0$, but it returns $0$.
What else can I try?