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The phrase "total sum of" is correct and usable in written English.
You can use it to refer to the collective combined sum of particular items or amounts. For example, "The total sum of the donations was $5,000."
Exact(59)
This quantity is known as the total sum of squares.
It is the total sum of all our carbon emissions that will determine the impacts.
Mamma Mia and Cats currently make up the total sum of musical theatre in China.
At the start of the 2012-13 financial year, the total sum of the WICB's retained earnings was $128,090.
That's the total sum of Dnipro's shirt numbers, thanks in particular to Boyko (71), Fedetsky (44) and Matheus (99).
But the total sum of £9m has been described as "woefully inadequate" by Unite's assistant general secretary Tony Burke.
The document lists more than 1,000 investors who are late paying what they owe — a total sum of more than $65 million.
Total sum of squares.
Total sum of days in OPAT regimen was 11,698.
The PSS questionnaire had a total sum of 56.
The total sum of square is computed from Eq. 10.
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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