Recently, I have been discussing **inverses** with a tenth grade class and **integrals** with an eleventh/twelfth grade class, and this has led me to the following wonder:



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> **Wonder.** Is there a "reasonable" way to quantify which of finding an inverse and finding an antiderivative is more "difficult"? Meaning, which of these processes is "easier" or which of these processes is more likely to lead to a "pleasant" inverse or antiderivative?
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For example, one can use composition to create functions that are easy to invert but difficult to find an antiderivative for; e.g., $x \mapsto e^{x^3}$. (More generally, I am thinking about [**Liouville's Theorem**](https://en.wikipedia.org/wiki/Liouville%27s_theorem_(differential_algebra)) on when it's possible to express an antiderivative using elementary functions.) On the other hand, if one simply restricted to the set of polynomials in $\mathbb{R}[x]$, then they would all be straightforward to integrate, but finding a pleasant inverse for those of degree five or higher would be almost always impossible. (In this polynomial context, I am thinking about the [**Abel-Ruffini Theorem**](https://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem).)


Ways to refine the language of this question or pointers to pre-existing explorations of related wonders are all welcome. (Efforts to re-tag will be welcomed, too.)



Quantifying difficulty of integrals versus inverses
Finding integrals is easier, and finding inverses is more difficult, in many ways.


We can consider both integrability and invertibility in two ways:


* *Global integrability*: Is there a bound on $\int\_a^b f(x)\,dx$?
* *Elementary integrability*: Is there an elementary $F$ with $F'=f$?
* *Global invertibility*: Is $f$ a bijection from its domain to its range?
* *Elementary invertibility*: Is there an elementary $g$ with $f(g(x))=g(f(x))=x$ on some domain?


**Comparing the Elementary Concepts**


* It is easy to integrate a cubic $ax^3+bx^2+cx+d$, but awkward to find the elementary inverse. I have never found it worthwhile to memorize the old formulas and doubt many others have.
* A variety of probability distributions have elementary cdfs (which are the integrals of the pdfs), but non-elementary quantiles (which are the inverses of those cdfs). E.g. this occurs for beta distributions $B(a,b)$ with integer $a,b$; for the continuous Bernoulli distribution $\mathcal{CB}(\lambda)$; for the Wigner semicircle distribution; and for Erlang distributions, i.e. gamma distributions with integer $k$. Also, many probability distributions have nice-ish cdfs but not nice-ish quantiles; e.g. for the normal distribution, the first good asymptotic for the cdf is
$$F(x)\sim 1-\exp(-x^2/2))/\sqrt{2\pi x^2}$$ but the first good asymptotic for the quantile function is $$Q(1-p)\sim\sqrt{-\log(4\pi p^2)-\log(-\log(2\pi p^2)/2)}$$
* In the universe of 100 simple elementary expressions of the form $f\_1(x)\star f\_2(x)$, where each $\star$ is one of $+,-,\times,/$ and each $f$ is one of $x,\log (x),e^x,\sqrt{x},x^2$:


	+ 87 have elementary integrals. (The remaining non-elementary integrals can be expressed using Erf, Erfi, and ExpIntegralEi, except for $\int e^x/\log(x)\, dx$, for which Mathematica provides no other expression.)
	+ Only 44 have elementary inverses on appropriate domains. (This includes the four like $x^2\pm\sqrt{x}$, which have elementary inverses so messy that probably few people on this site could find them without assistance.)


**Comparing the Global Concepts**


* The globally integrable functions have nice closure properties: they are a ring. By contrast, the globally invertible functions are not closed under either subtraction or multiplication, and not even closed under monotonic functions like $\log$ or $\tan$ because $\log(\log(x))$ has a smaller domain than $\log$, and $\tan(\tan(x))$ has singularities. (There is one set of invertible functions with nice closure properties: the unbounded monotonic functions on $\mathbb{R}^{\ge 0}$ with $f(0)=0$. That set is closed under addition, multiplication, composition, and functional inverses, but it excludes the very example in the post of $x\to e^{x^3}$, favoring things like $x\to e^{x^3}-1$.)
* The map $f\to f’$ takes any bounded differentiable function to a globally integrable function. The closest analog for globally invertible functions is that the maps $f\to \int f^2$ and $f\to \int e^f$ take almost any function to globally invertible functions (in the first case one has to exclude functions which vanish on an interval), but those maps are not as nice; eg, for reasonable domains, their images may not include the identity.
* For rational functions $p(x)/q(x)$, it is easy to specify which are globally integrable: those where $q$ has no real roots and $\deg(q)\ge \deg(p)+2$. It is awkward to specify which rational functions are globally invertible, e.g. $ax^3+bx^2+cx+d$ is invertible iff (try it without looking first!)



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> $b^2\le 3ac$ and either $a$ or $c$ is non-zero.
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