With the release of Rails 5.1, they included system tests. Which means we can test our JavaScript too in Rails. I see Rails guide explains a sample test creating article
TL;DR: I would go with system tests instead of integration tests in any app I started today. The only advantage of integration tests is speed.
I think system tests have 2 great advantages over integration tests:
I think the only benefit of integration tests is speed. They are much faster indeed (check this experiment I made). For me, the speed difference is not such a big concern because:
I think both local and cloud speed and parallelization are good enough today, and will only get better with time. So I believe system tests is a much safer bet if you are starting a new app today.