Consider normal random variable $\lambda \sim N(\mu, \sigma^2)$ ($\lambda \in \mathbb{R}$) whose PDF is:
$$ f_\lambda(\lambda) = \frac{1}{\sigma \sqrt{\pi}} \exp \left[ {-\frac{1}{2}\left( \frac{\lambda - \mu}{\sigma} \right)^2} \right] $$
Transformation $g(x) = \frac{1}{a+x}$ is applied and a new random variable $\Lambda$ is generated as:
$$ \Lambda = g(\lambda) \implies \Lambda = \frac{1}{a + \lambda} $$
With $a \in \mathbb{R}$.
The problem
I want to know the expected value of $\Lambda$: $E[\Lambda]$.
My attempt so far was by following the theory. The new PDF is computed according to well known formula:
$$ f_\Lambda(\lambda) = f_\lambda[g^{-1}(\lambda)] \left| \frac{d}{d\lambda}g^{-1}(\lambda) \right| $$
Which means, after a few calculations:
$$ f_\Lambda(\lambda) = \frac{1}{\sigma \sqrt{\pi}} \frac{1}{\lambda^2} \exp \left[ -\frac{1}{2} \left( \frac{1}{\sigma \lambda} - \frac{a}{\sigma} - \frac{\mu}{\sigma} \right)^2 \right] = \frac{\alpha}{\lambda^2} \exp \left[ -\frac{1}{2} \left( \frac{1}{\sigma \lambda} - \beta \right)^2 \right] $$
Having $\alpha = \frac{1}{\sigma \sqrt{\pi}}$ and $\beta = \frac{a}{\sigma} + \frac{\mu}{\sigma}$. Now I can proceed calculating the mean value of $\Lambda$ since I know its PDF by means of: $E[\Lambda] = \int_{-\infty}^{+\infty} \lambda f_\Lambda(\lambda) d\lambda$:
$$ E[\Lambda] = \int_{-\infty}^{+\infty} \frac{\alpha}{\lambda} \exp \left[ -\frac{1}{2} \left( \frac{1}{\sigma \lambda} - \beta \right)^2 \right] d\lambda = \alpha \int_{-\infty}^{+\infty} \frac{1}{\lambda} \exp \left[ -\frac{1}{2} \left( \frac{1}{\sigma \lambda} - \beta \right)^2 \right] d\lambda $$
And here is the problem, this integration is kinda tricky:
$$ \int_{-\infty}^{+\infty} \frac{1}{x} \exp \left[ -\frac{1}{2} \left( \frac{1}{c_1 x} - c_2 \right)^2 \right] dx $$
How to solve this?
Integration attempts
Attempt 1 I have tried with substitutions, but it is horrible because I might get rid of the negative power of the variable in the exponential, but then I get higher order negative powers of the variable multiplying the exponential :(
Attempt 2 I know that the exponential integral function is defined as:
$$ \text{Ei}(x) = -\int_{-x}^{+\infty} \frac{e^{-t}}{t} dt $$
And I have a feeling that the primitive of the integrand might be a function of $\text{Ei}$, but I am not sure.
Attempt 3 Using complex calculus and the Residue Theorem?