There are multiple ways to achieve this result. First naive option is to use git log and search for a specific commit using grep, but that is not always precise
git log | grep <commit_id>
You are better off to use git branch directly to find all branches containing given COMMIT_ID using
git branch --contains $COMMIT_ID
The next step is finding out current branch which can be done since git 1.8.1 using
git symbolic-ref --short HEAD
And combined together as
git branch $(git symbolic-ref --short HEAD) --contains $COMMIT_ID
But the command above doesn’t return true or false and there is a shorter version that returns exit code 0 if commit is in current branch OR exit code 1 if not
git merge-base --is-ancestor $COMMIT_ID HEAD
Exit code is nice, but as you want string true or false as answer you need to add a bit more and then combined with if from bash you get
if [ 0 -eq $(git merge-base --is-ancestor $COMMIT_ID HEAD) ]; then echo "true"; else echo "false"; fi