Nine is an integer multiple of three.
Nine is an integral multiple of three.
Which is more common?
If both are accepted, what's the subtle difference between them?
As a rule if we have a single-word noun, and if there is a single-word adjective with the same meaning that we get from using the noun as the modifier (especially a directly related one), then while using the noun as a modifier for another noun is acceptable, using the adjective is the more popular decision, and tends to read as more graceful English to many.
In this case however:
So while there's no difference in meaning, and you certainly can use integral, there is a good argument in favour of integer here.
I favour integral, but then that's precisely because I'm the sort of person who spends more time on ELU than the mathematics stack-exchange. I'm not the target audience.
Both are accepted. I don't believe there is a difference. As to which is more common, on the arXiv, there are over 7000 hits for "integer multiple" and only around 1500 hits for "integral multiple", so even among mathematicians and physicists, "integer multiple" is more common.