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I asked this question on MSE here.

I recently learned that there are many very large numbers that have been defined, such as $\operatorname{TREE}(3)$ and many others that are too big to be written down.

What made me interested is the idea that there is a function that take some finite and small number like $3$ to an absolute beast of a number.

So I wonder if there is some function with an elementary antiderivative that we know its antiderivative is too large to be written down without any shorthand notation like $\sum\limits_{n \in J} c_n f_n(x)$ st that $J$ is a finite set and $f_n $ is some combinations of elementary functions , although we can write down the function itself.

Or in other words: How complicated can an elementary antiderivative get? Is there some example of such functions (examples of functions with elementary antiderivatives that are too big to be written down, although we can write down the functions themselves)? If not, is there a proof why elementary antiderivatives can't get that complicated?

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    $\begingroup$ Something like $\int \sin^n dx$ for large n should result in lots and lots of terms when calculating it in the standard way of repeated partial integrations, but I have no idea how to prove that there is no nicer closed form. $\endgroup$
    – mlk
    Commented Jan 22 at 10:29
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    $\begingroup$ It might be possible to extract an upper bound for the possible length from the Risch algorithm $\endgroup$
    – Daniel Weber
    Commented Jan 22 at 10:39
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    $\begingroup$ I agree with Command Master's comment. See this MO question, and in particular the paper by Daniel Schultz, for some indication of how complicated an elementary antiderivative can get. Offhand, I don't know an upper bound, but my impression is that the most computationally expensive part is some kind of Gröbner basis calculation. So I'd guess that there's a doubly exponential upper bound. Certainly nothing like TREE or even Ackermann. $\endgroup$
    – Timothy Chow
    Commented Jan 22 at 14:01
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    $\begingroup$ I think that at this point it would be best if you stopped the frequent minor edits. If you can't get the question right after 11 edits, then I would stop and reflect about what exactly you're trying to get from it. $\endgroup$
    – Andy Putman
    Commented Jan 23 at 19:11
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    $\begingroup$ @pie: I think it's best to not make so many edits. Each edit pushes the question to the top of the queue again, so it is unfair to other people who are trying to get attention for their questions. $\endgroup$
    – Andy Putman
    Commented Jan 23 at 19:20

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