Perhaps it´s for emphasis
, and because it have an adverb with negative meaning (albeit normally that happens with the subject and the auxiliary verb). The fact is that it the two forms are used, you won´t see a native from an English-speaking country blush because he used the "I never will give up", or then Tony Blair and Obama should shame on them!
Seems to me, that is for emphasis (hence the widely usage in lyrics), although it is used less than the other form.https://dictionary.cambridge.org/pt/gra ... /inversion
Here some examples of usage of both ways:http://www.wcgs-sutton.co.uk/Culture-and-EthosWe believe that we will do what it takes to achieve excellence and we will not give up until we are satisfied that we have given our all.https://techcrunch.com/2015/07/16/microsoft-will-never-give-up-on-mobile
/http://www.telegraph.co.uk/news/2017/10 ... las-vegas/Why Americans will never give up our guns – even after a massacre like Las Vegas.
(Inside the item, the headline itself.)http://www.telegraph.co.uk/news/politic ... ttack.html
of-speech-David-Cameron-says-after-Charlie-Hebdo-attackhttp://www.telegraph.co.uk/finance/pers ... -work.html
Employees-think-they-will-never-give-up-workhttp://www.telegraph.co.uk/rugby-union/ ... r-england/
In a New York Times item: The Clintons never will give up
trying to get Hillary elected as the first woman president–that means in 2012,http://snltranscripts.jt.org/10/10iobama.phtml..."In closing," the "president" [Obama] says, "let me reassure you that however long it takes, this nation's current troubles will pass because you Americans never have and never will give up. I say 'you Americans,' because even though I always thought I was born here, uh, lately I begin to have my doubts."https://www.azlyrics.com/lyrics/alisonk ... iveup.html
...No I never will give up, no no no!
In an academical text? I would favor the first, unless it was a dramatic statement that deserved the unconventional way (and also from a quotation, as I pointed from presidents, etc). Outside of that, I think it would be seen as normal, just less used.