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Can anyone see the equality in the red squere is true? It is taken from the supplementary material for this paper.

I am not sure if it is relevant but here, $\phi(x)$ represents the density function (PDF) of normal distribution and $\Phi(x)$ denotes the cumulative normal distribution function (CDF). i.e. $$\Phi(x) = \int_{-\infty}^x \phi(t)dt$$

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Thanks!

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    It is an integration by parts, of the following type: $$\int_{-\infty}^\infty f'(x)f(x)\, dx= -\int_{-\infty}^\infty f(x)f'(x)\, dx +\left [f(x)^2\right]_{x=-\infty}^{x=+\infty}.$$ – 2017-01-16

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It´s integration by parts:

In $\int_a^b v^{'}udx=[uv]_a^b-\int_a^bvu^{'}dx$ put:

$v^{'}(x)=\phi(x)\Phi(\alpha x)$ and $u(x)=\int_{-\infty}^x\phi(y)\Phi(\alpha y)dy$ and $a=-\infty,b=\infty$ and use the fundamental theorem of calculus.

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    Understood Thank you! – 2017-01-16