in any case vs at any rate

in any case and at any rate both are adverbs.

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

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

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

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

in any case (adverb)	at any rate (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
making an additional point; anyway	if nothing else (`leastwise' is informal and `leastways' is colloquial)