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I'm trying to simplify this integral

$$ \int_{-\infty}^{-a}e^{-\frac {x^2}{2}} Q(x+a) dx $$

where $ Q(x) = \frac {1}{\sqrt{2\pi}} \int_{x} ^{\infty}\exp(-\frac{v^2}{2})dv $, and $ a>0$

so that the simplified expression will contain only Q-function without integration on it
(I suspect that it can somehow be written as an expression involving $Q^2(\cdot)$)

I would like some help with this.

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    Well, where are you stuck? To simplify (if it's indeed possible) you just need to substitute the integral definition of $Q$ inside your expression, then you try to simplify the resulting double integral and see if you can integrate w.r.t. $x$ – 2017-02-02
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    @YuriyS I'm stuck with simplifying the resulting double integral... – 2017-02-02
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    I see. It might not be possible to simplify this further, maybe I'll try it later or check with Mathematica. Good luck – 2017-02-02
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    @ dorsh605, I suggest you write the double integral you obtained in the body of the question. The more of your own steps you show the easier it would be for other users to help – 2017-02-02
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    try integration by parts... – 2017-02-02
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    @tired I also tried integration by parts but with no luck – 2017-02-02

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