muse vs cogitate

muse is a noun but cogitate is not a noun.

Word	Noun	Adjective	Verb	Adverb
muse	Yes	No	Yes	No
cogitate	No	No	Yes	No

As verbs, cogitate is a hypernym of muse; that is, cogitate is a word with a broader meaning than muse:

muse: reflect deeply on a subject
cogitate: use or exercise the mind or one's power of reason in order to make inferences, decisions, or arrive at a solution or judgments

Other hypernyms of muse include cerebrate, think.

As verbs, cogitate is a hyponym of muse; that is, cogitate is a word with a more specific, narrower meaning than muse:

muse: reflect deeply on a subject
cogitate: consider carefully and deeply; reflect upon; turn over in one's mind

muse (noun)	cogitate (noun)
the source of an artist's inspiration

muse (verb)	cogitate (verb)
reflect deeply on a subject	use or exercise the mind or one's power of reason in order to make inferences, decisions, or arrive at a solution or judgments
	consider carefully and deeply; reflect upon; turn over in one's mind