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The phrase "have long shown" is grammatically correct and usable in written English.
You can use this phrase when discussing a topic that has been observed or studied over a long period of time. For example, "Studies have long shown that children learn best through hands-on activities."
Exact(56)
In medical terms, studies have long shown more unequal societies have higher mortality rates than more equal ones.
Both players are feisty, provocative, volatile characters, as their records for their clubs, and their national teams, have long shown.
Black voters have long shown a willingness to support white candidates far more often than whites have voted for blacks.
But studies have long shown that people's spending is largely a function of what they view as their "permanent income".
These have long shown up in Google search results, but are not part of the new Search Plus Your World.
Among his data-driven findings: American consumers have long shown an "exceptional willingness" to buy, for instance, technology products before their utility is clear.
While women in Hollywood have long shown interest in closing the wage gap, the issue has become more personal in the recent months.
The House members' resistance was not surprising, because lawmakers have long shown an unwillingness to put any oversight responsibility in the hands of those outside the institution.
The findings of the two main employment surveys done by the Bureau of Labor Statistics have long shown a large discrepancy.
Studies have long shown that taking part in a group activity can improve confidence, self esteem and ability to relate to other people in a social setting.
Similar(1)
Get enough sleep Studies have long showed a link between inadequate sleep and cognitive troubles.
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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