Depends how gnarly the functions you're integrating are. The greatest goods in the top-level post sound time-independent (except for when they start & stop); if so one just integrates exp(-r**t) over different t intervals, which is extremely fast & straightforward.
The first greatest good for the greatest number for the greatest number will start "first" (by whatever measurement is applied) but ends before the second greatest good ends and doesn't last as long (in total) as the third greatest good.
The second greatest good for the greatest number will start end "last" (by whatever measurement is applied), but does not last as long as the third greatest good (in total)and doesn't start as soon as the first greatest good.
The third greatest good for the greatest number lasts the longest (in total), but ends before the second greatest good ends and starts after the first greatest good starts.
What within utilitarianism allows for selecting between these three greatest good for the greatest number?