How can I find the missing integers in a unique and sequential list (one per line) in a unix terminal?

Using awk you can do this:

awk '{for(i=p+1; i<$1; i++) print i} {p=$1}' file

2
6
7

Using seq and grep:

seq $(head -n1 file) $(tail -n1 file) | grep -vwFf file -

seq creates the full sequence, grep removes the lines that exists in the file from it.