For clarity, archiving services will only work for pages you need to sign in for if the service has a smurf account for that website being archived running on their bot (archive often does), and the sign in requirement is not more complex that being required to sign in with any account.
If sign in is required because of a requirement like "must be following" or a whitelisted set of users (like with youtube private videos), the archive bot won't work other than to archive that the URL did lead to a sign in page (which is useful, but incomplete).
Your best bet at that point is saving the .html from "save page as (complete)" while signed in, and then also the content such as video separately, though be aware these options will show your login in the archive so they're best kept for non-public release unfortunately. Less complete but more releaseable is screenshots of the page.