What's a good Numerical Integration Method for this Equation?

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.

• For $x$ sufficiently negative won't you be taking the log of a negative number? – Rahul Sep 1 '16 at 1:46
• Yup, that's basically the problem I'm having - I need to find some way to convert this equation so that there are no negative values or something, or maybe find some numerical integration scheme which can work around it, but the problem is I don't know what to do, so I thought I'd ask here for some suggestions or possible ways to work around this issue – ThePlowKing Sep 1 '16 at 1:48
• That's definitely possible, but the problem is is that I actually have to solve for $Y$, so I'm not sure how I can use it to limit the domain of $x$. I omitted it because I didn't think it was relevant, but the full equation is $\large \int^{\infty}_{-\infty}\frac{1}{\sqrt{2\pi T}}e^{\frac{-x^{2}}{2T}} \frac{1}{\ln(Y + \sigma x + \frac{\sigma^{2}}{2}T) - \lambda K } dx = \frac{1}{S}$ where I basically have to solve the whole equation for $Y$ and where $T$, $\sigma$, $\lambda$ and $K$ are all known. Is it possible to use these constants to restrict the domain of $x$? – ThePlowKing Sep 1 '16 at 2:08