at any rate vs in any case

at any rate and in any case both are adverbs.

Word	Noun	Adjective	Verb	Adverb
at any rate	No	No	No	Yes
in any case	No	No	No	Yes

As adverbs, at any rate and in any case are synonyms defined as:

at any rate and in any case: used to indicate that a statement explains or supports a previous statement

Other synonyms of at any rate include anyhow, anyway, anyways, in any event.

at any rate (adverb)	in any case (adverb)
used to indicate that a statement explains or supports a previous statement	used to indicate that a statement explains or supports a previous statement
if nothing else (`leastwise' is informal and `leastways' is colloquial)	making an additional point; anyway