Adding Unique Days In A Range

We have a client that has to total the number of days that people received medical care (for insurance purposes). BUT...they can't count duplicate days. Some of the dates show a range of days, others only one day. They've asked for my help. (We're actually a computer services firm, but I'm pretty good with Excel...until now!)

In column C I have placed the correct totals for the date ranges, according to the payment policies. But I can't figure out how to write the formula that accommodates all the variables (mostly since all the unique dates aren't entered--only ranges).

Each line is a different medical charge, so line 1 might be 22 days of hospital stay. Line 2 is for a different charge, but covers the same dates, so it can't count. Lines 3, 4, and 5 all occur within the previously calculated range, so they don't count. The date range in line 6 is six days but 4 of them have been counted already, only 2 days fall outside an already calculated range.

It would be easy enough to do this manually for a small list, but this patient has 1367 entries in 4 months of care! And there are others that are much larger.

Fortunately, all the dates are in calendar order. Also, fortunately, they don't need line-by-line totals, just the grand total at the end of all the unique days in the range. So if I can figure out how many unique days are in the columns ....