Check login before serving cached page

Is there a way to first check if someone is logged in before serving a cached page? Yes, I do want to serve cached pages to logged in users because I have a few pages that have a lot of videos that take a while to load. There’s nothing user specific on these pages so it’s fine. What I don’t want is to serve these cached pages with private videos to anybody who isn’t logged in.

I’ve tried many cacheing plugins ( w3 super cache, quick cache, db cache, etc…), most of have an option “don’t cache pages for logged in users” which I don’t want.