I'm basically looking for an appropriate numerical integration method to integrate the following equation:
$\large \int^{\infty}_{-\infty}\frac{1}{\sqrt{2\pi T}}e^{\frac{-x^{2}}{2T}} \frac{1}{\ln(Y + \sigma x + \frac{\sigma^{2}}{2}T) - \lambda K } dx$
where $Y$ is a constant value and $\sigma$, $T$, $K$, and $\lambda$ are given values.
Now, I initially believed that using the Gauss-Hermite Quadrature method would be appropriate to use for this integral since it is from $-\infty$ to $\infty$, however, as pointed out in this previous topic I made, "Two Questions Regarding Gaussian Quadrature," using $\log$ functions are a poor function to have for integration between $-\infty$ to $\infty$. As such, after re-reading all the different possible methods I can use I no longer know what I can do to numerically integrate this equation. If anyone can suggest suggest something I would really appreciate it since I'm at a loss. Thanks in advance.