How many integers are in the solution of the inequality $|x+4|<9$ ?

Step 1 ：First, consider two cases: (1) $x+4\geq 0$ (or $x\geq -4$), and (2) $x+4<0$ (or $x<-4$).

Step 2 ：For case (1), the inequality becomes $x+4<9$, which simplifies to $x<5$.

Step 3 ：For case (2), the inequality becomes $-(x+4)<9$, which simplifies to $x+4>-9$, and further to $x>-13$.

Step 4 ：Combining both cases, the solution is $-13

Step 5 ：The integers in this solution are -1 through -12 (12 integers), 1 through 4 (4 integers), and 0 (1 integer).

Step 6 ：Add the integers from both cases and the 0: $12+4+1=\boxed{17}$.