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The phrase "long been shown" is correct and usable in written English.
It is typically used to indicate that something has been proven or demonstrated for a significant period of time. You can use it when discussing a commonly accepted fact or when providing evidence for a statement. Example: It has long been shown that regular exercise can improve overall health and well-being.
Exact(54)
Cannabinoids have long been shown to have a range of potential therapeutic effects, including antiemetic actions, analgesia, and anxiolysis.
It has long been shown that insiders realize significant positive abnormal returns on their transactions, in percentage terms.
"The SAT has long been shown to correlate poorly with academic success," he said.
Advertising rates for the awards show, which has long been shown on CBS, have risen commensurately.
The notion of the "rogue shark" with a taste for human flesh that needs to be hunted down and eliminated has long been shown to be a myth.
In one case, it had long been shown conclusively in public that the United States abducted an innocent man and sent him to Syria, where he was tortured.
Similar(6)
Seemingly unperturbed by the whiff of scandal, celebrities have long been showing their love for the cartoon.
Mysterious charges — for an assortment of memberships and services — have long been showing up on wire-line telephone bills.
Riding the Scenic Railway - although it had long been showing its years and looked more than a little forlorn - was a wonderfully genteel experience.
On the G.O.P. side, the polls had long been showing Trump, Cruz, and Rubio filling the top three places, which is what happened.
This 50-year-old technology has long been showing its age – despite the levels of complexity there are major password breaches reported weekly – but it has proved hard to replace.
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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