My take is that notation allows to squeeze more into our limited "thinking memory". It's easy to remember "integral of that plus integral of this equals something", but it's nearly impossible to memorize the expanded definition that uses only basic terms. I guess, good notation is like a box: it lets us move a bunch of things at once.
That makes sense: perhaps "abstraction" is the word? An n-ary operator with a well-understood conceptual effect, and a reasonably small number of arguments linked to it (enough to fit into a person's point-in-time memory as you mention).
(addendum/edit: and hopefully those abstractions are clearly implemented and fixable in practice.. hello, software libraries)
