chase after vs trail

chase after is not a noun while trail is a noun.

Word	Noun	Adjective	Verb	Adverb
chase after	No	No	Yes	No
trail	Yes	No	Yes	No

As verbs, chase after and trail are synonyms defined as:

Other synonyms of chase after include chase, dog, give chase, go after, tag, tail, track.

chase after (noun)	trail (noun)
	a path or track roughly blazed through wild or hilly country
	evidence pointing to a possible solution
	a track or mark left by something that has passed

chase after (verb)	trail (verb)
go after with the intent to catch	drag loosely along a surface; allow to sweep the ground
pursue someone sexually or romantically	hang down so as to drag along the ground
	go after with the intent to catch
	move, proceed, or walk draggingly or slowly
	to lag or linger behind