What is the meaning of the phrase "bust ones tail"?
Bernie's a little cranky since she's been working, like, 17 hours a day.
And I've got a lot on my plate, too, because I've been busting my tail playing Assassin's Creed. (The Big Bang Theory)

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It means working very hard.
In a more vulgar way, it is also said as "busting my ass."

The image is of someone working so hard that they've broken their backside.

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Do something very diligently.