Say I have i.i.d data $X\_1,\ldots,X\_m\sim F$ and $Y\_1,\ldots,X\_n\sim F$, where $F$ is the distribution function.

How can I calculate the following integral? 
$$
W=\int\_{-\infty}^\infty \int\_{-\infty}^s F(t)-F(t)F(s) \, d F(t) \, dF(s)
$$

**Edit** See solution below. As a side note for those interested, this was used to derive the asymptotic variance of the Mann-Whitney statistic using empirical process theory.