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The phrase "are busy doing" is correct and usable in written English.
It can be used to indicate that someone is currently engaged in an activity or task.
Example: "They are busy doing their homework right now, so please do not disturb them."
Alternatives: "are occupied with" or "are engaged in".
Dictionary
are busy doing
adjective
Crowded with business or activities; having a great deal going on.
Exact(43)
Or the people there are busy doing something else.
They don't get a lot of attention, nor do they seek it, but they are busy doing the work this country needs doing.
"Customers like the idea of the designers, who are busy doing their shows, taking time to go to department stores," Ms. Keating said.
Finally, grab those opportunities because once you are busy doing - albeit stressing - you will build momentum and may well surprise yourself.
I didn't have to touch anything, because my hands are busy doing something in the kitchen, and I can just get it to play music while I wait.
"Most parents are busy, doing laundry, running around like me, and we're trusting the schools to reflect public policy," she said.
Similar(16)
If your friends are busy, do fun things with your brothers and sisters instead.
They were busy doing what needs doing.
Tom Waits has been busy doing the dishes.
We were busy doing other work".
I have to be busy doing something.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com