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EDIT: I asked this question on MO here.

I recently learned that there are many very large numbers that have been defined, such as $TREE(3)$ and many others that we cannot write down.

What made me interested is the idea that there is a function that take some finite and small number 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, but we can write down the function itself.

What I mean by writing down the antiderivative: Is to write it without any shorthand notation like $\sum_n f(x_n)c_n$ Like how we can't layout the digits of $TREE(3)$ if we wanted to But the antiderivative is finite.

What I mean to be too large is the humanity can't write it down without any shorthand notation because the antiderivative has a lot of terms (say for example $10^{10000}$ term) and composition of functions.

What I mean by writing down the function is: It is possible to write down its distributive form without any shorthand notation like $(1+x)^4$ i will count this as shorthand and its distributive form is $1+4x+6x^2+4x^3+x^4$ so functions like $x^{TREE(3)},\ {TREE(3)} $ doesn't count.

pie
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    $\int \mathrm{TREE}(3) dx$? – Randall Jan 19 '24 at 15:58
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    @Randall Isn't that just $x \operatorname{TREE}(3)$? –  Jan 19 '24 at 16:01
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    Right, but $\mathrm{TREE}(3)$ can't be written down, so.... – Randall Jan 19 '24 at 16:02
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    Or you can argue that the integrand shouldn't be allowed to be the unwritable $\mathrm{TREE}(3)$ either, but then how about $\int x^{\mathrm{TREE}(3)-1} \ dx$. Maybe $\mathrm{TREE}(3)-1$ can be written down, but then its integral cannot. Where do you draw the line between writable and not? – Randall Jan 19 '24 at 16:06
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    This amusing TREE(3) discussion makes me realize that this question should really be about the complexity of the antidifferentiation operator. – Lee Mosher Jan 19 '24 at 16:25
  • @Randall I edited the question , I hope it is clear now , since english is not my langue I have a hard time wrtinig the question if you can correct my choice of words like "distributive " that would be great – pie Jan 19 '24 at 16:26
  • A good starting point would be Liouville's theorem Based on the form that functions with elementary antiderivatives should have, it doesn't look like their antiderivatives can get that much more complicated that the original function. Though it can also depend on what is considered complicated, because some antiderivatives may require computing roots of high-degree polynomials. – Tob Ernack Jan 20 '24 at 17:56
  • Doest it count https://math.stackexchange.com/questions/4842694/closed-form-of-integral-int-0-infty-frace-x1x-cdotsdx-with-cont ? – Miss and Mister cassoulet char Jan 21 '24 at 18:39
  • @DesmosTutu the dominator is infinite so no – pie Jan 21 '24 at 20:51
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    Now posted to MO, https://mathoverflow.net/questions/462641/how-much-complicated-can-an-elementary-antiderivative-get – Gerry Myerson Jan 22 '24 at 10:46
  • @GerryMyerson I forgot to update this question – pie Jan 22 '24 at 10:47

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