Inspired by this post:

I was amazed to see that (at least according to wolframalpha) the inverse of such a nice and simple function as $f(x)=x^3+x$ is: $$ f^{-1}(x) = \sqrt[3]{\frac{2}{3( \sqrt{81x^2+12}- 9x)}} - \sqrt[3]{\frac{\sqrt{81x^2+12}- 9x}{ 18}} $$

Now there may be a way to simplify that that I'm not seeing...

But regardless, I was wondering if there are any other seemingly simple functions with crazy, ugly inverses.

Is there any way to know ahead of time whether a function will have a nice inverse?

I know that not all functions even have inverses (over the reals). But is there any rhyme or reason as to why such a simple function would have such a crazy complex inverse? And is there a more general criteria for know whether other functions will be similar in this respect?

EDIT: To make this a little easier to answer, I've been told I need a better definition than ugly. Let's go with non-analytic, just because I'm interested. But if anyone has a better idea, please let me know.

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    $\begingroup$ Most polynomials have awful inverses. $\endgroup$ – symplectomorphic Mar 10 '16 at 7:00
  • $\begingroup$ And yes, of course there's a rhyme and reason: it is not at all obvious how to untangle the mapping $x\mapsto x^3+x$. Compare this to the function $x\mapsto x^3+1$, which cubes the input and then adds 1. Obviously to go in the reverse you first subtract 1 and then take the cube root, so the inverse is $x\mapsto\sqrt[3]{x-1}$. $\endgroup$ – symplectomorphic Mar 10 '16 at 7:03
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    $\begingroup$ And starting from degree 5 many of them are so ugly that you can't even write them analytically. $\endgroup$ – pointer Mar 10 '16 at 7:03
  • $\begingroup$ Why is that? I mean anything of the form $f(x)=x^n+a$, $n \in \mathbb{Z}$, $a,x \in \mathbb{R}$ has a nice inverse. Why do adding the extra terms of of order $x^{n-1}$ to $x$ muck with things? $\endgroup$ – D. W. Mar 10 '16 at 7:03
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    $\begingroup$ That's largely a silly question. To get a precise answer you'd need a precise definition of "ugly." The vast majority of "arbitrary" functions don't even have simple analytic representations. The short answer is: the inverse is "nice" if you can easily solve the appropriate equation. But most equations are very hard to solve. $\endgroup$ – symplectomorphic Mar 10 '16 at 7:17

If you treat the function $f(x)=x+x^3$ as a series expansion and perform a series reversion then $$ f^{-1}(x) = x - x^3 + 3x^5 - 12 x^7 + 55x^9 - 273 x^{11} + \cdots $$ alternatively $$ f^{-1}(x) = \sum_{n=0}^\infty \binom{3n}{n}\frac{(-1)^n x^n}{2n+1} $$ with the coefficients identified thanks to OEIS entry A001764. The fact that the inverse series expansion has all integers with a simple formula is quite nice in my opinion, but if you look in the OEIS entry there are many combinatoric interpretations of this series. Viewed in the world of generating functions this is a 'nice' inverse. The function can be written in two other ways $$ f^{-1}(x) = \frac{2}{\sqrt{3}}\sinh\left(\frac{1}{3}\mathrm{arcsinh}\left(\frac{3\sqrt{3}x}{2}\right)\right) $$ and $$ f^{-1}(x) = x \;_2F_1\left( \frac{1}{3},\frac{2}{3}; \frac{3}{2};\frac{3^3 x^2}{2^2}\right) $$ In the language of hyperbolic trig functions the $\sinh$ and $\mathrm{arcsinh}$ almost cancel out if it wasn't for the factor of $1/3$, that's at least interesting, but not pretty. In the language of hypergeometric functions, this inverse is quite beautiful and mysterious. polynomial inverses often generate nice hypergeometric representations, and for higher orders that is one of the few languages that is capable of writing the inverses, (see my answer to this post for an example, note the $\frac{5^5}{4^4}$ pattern continues).

The reason it looks so ugly in it's surd form is probably because the function is not as special when written in that language. (I don't think many things look pretty with roots really).

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    $\begingroup$ This is a very thoughtful and thorough answer to an interesting question, and I'm disappointed that it hasn't gotten more attention. (+1) $\endgroup$ – Frpzzd Apr 4 '18 at 23:23

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