How to find Sum of # of days w/o counting overlapping days twice

Fun challenge, use this formula.

Row\Column	A	B	C	D
1	Start date	End date	Duration	Duration (no overlap)
2	6-Jan	10-Jan	5	19
3	13-Jan	20-Jan	8
4	12-Jan	22-Jan	11
5	18-Jan	25-Jan	8

Formula in D1 (array formula)

Formula:

=SUM(1*(MMULT(((MIN(A2:A5)-ROW(A2)+ROW(OFFSET(A2,0,0,MAX(B2:B5)-MIN(A2:A5)+1,1)))>=TRANSPOSE(A2:A5))*((MIN(A2:A5)-ROW(A2)+ROW(OFFSET(A2,0,0,MAX(B2:B5)-MIN(A2:A5)+1,1)))<=TRANSPOSE(B2:B5)),1*(ROW(A2:A5)>0))<>0))

This is an array formula, please copy in the formula, then hit CTRL SHIFT ENTER to finalise the formula rather than just enter.