Assume that $X$, $Y$ are random variables with joint distribution function $H$ and their respective marginals $F$ and $G$.
I'm trying to show that, for $\alpha > 1$: $$ \begin{array} \mathbb{E}|X-Y|^{\alpha} &= \alpha (\alpha -1)\int_{-\infty}^{\infty}\int_{-\infty}^{x} [G(y) - H(x,y)](x-y)^{\alpha - 2}dydx \\&~~~+~ \alpha (\alpha -1)\int_{-\infty}^{\infty}\int_{-\infty}^{y}[F(x)-H(x,y)](y-x)^{\alpha-2}dxdy \end{array}$$
Here we know that
$$ E|X-Y|^{\alpha} = \int\int_{\mathbb{R}^2}|x-y|^{\alpha} \text{d}H(x,y)$$
Any hints/ideas would be appreciated!