\begin{align*}
D_{1,(F_X, F_Y)}(G_X)  &=\frac{1}{(\int_{-\infty}^{\infty} (F_Y (t) - F_X (t))^2 dt)^2}\cdot\\&\Big[ \left(\int_{-\infty}^{\infty} (F_Y (t) - F_X (t))^2 dt\right) \left(\int_{-\infty}^\infty 2 (F_Y (t) - F_X (t))_+ G_X dt\right)\\& - \left(\int_{-\infty}^{\infty} (F_Y (t) - F_X (t))_+^2 dt\right)\left(\int_{-\infty}^\infty 2 (F_Y (t) - F_X (t)) G_X dt\right)\Big].
\end{align*}
