I have encountered the following integral in my research which does not give-in to my attempts: $$ \int_\mathbb{R} x \left( \frac{1}{\sigma_1} \phi\left(\frac{x}{\sigma_1}\right) \Phi\left(\frac{x-\mu}{\sigma_2}\right) + \frac{1}{\sigma_2} \phi\left(\frac{x-\mu}{\sigma_2}\right) \Phi\left(\frac{x}{\sigma_1}\right) \right) dx $$ where $\phi(x)$ denote probability density function and $\Phi(x)$ a cumulative density function of the standard normal distribution. $\sigma_1$ and $\sigma_2$ are positive, and $\mu$ is real.
I would appreciate hints on how to evaluate it. Thank you!