"Simplest" symbolic equation with "most complex" symbolic answer

Question

The below was inspired by this question, in which the OP was surprised that the "simple" equation $x^3 + x = 1$ could have such a "complex" solution.

---

In rough form, the question here is a bit more general:

• What is an example of the "simplest" symbolic equation whose solution is "most complex"?

That rough statement is of course inadequate to constrain or compare candidate answers. Thus, to be specific, here are the conditions:

• The equation involves a single variable, say $x$, and any number of constants.
• The "complexity" of the problem equation and of the symbolic solution is the number of bytes in its expression (for instance in a MathJax representation).
• The global criterion to be optimized is the ratio of the number of bytes in the solution to the number of bytes in the problem equation.
• Both the problem and solution must be represented in their simplest canonical form. For instance, one must use $\sin x$ rather than its infinite series representation.
• All standard special functions and symbols (factorial !, power ^, etc.) are allowed, where the complexity is in the number of bytes in their canonical MathJax representation.
• Both the problem and the solution must be symbolic, not numerical. (Otherwise simple problem equations such as $x^2 = 2$, which have an "infinitely complex" numerical solution, would trivially optimize the criterion.)
• There must be a true, single closed-form solution. (Thus one cannot include, for example, $x^{9!} = 2$ and get a large list of solutions.)
• The solution is not a simple numerical calculation, for instance $x = 9 \uparrow \uparrow \uparrow$ (using Knuth's uparrow notation) or $x = (9!)^{9!}$, etc. As illustrated by the linked question above, it is rather the algebraic complexity of a solution.

One can get lost in nit picking (e.g., deciding which is to be preferred, \sqrt{x} versus x^{1/2}, or a/b versus \frac{a}{b}, and such). I suspect there are some dramatic examples where such minutia are irrelevant.

Answer 1

The unique real root $x$ of $x^5-5x+12=0$ is given by $$-cx=\root5\of{(a+c)^2(b-c)}+\root5\of{(-a+c)(b-c)^2}+\root5\of{(a+c)(b+c)^2}-\root5\of{(-a+c)^2(b+c)}$$ where $c=\root4\of5$, $a=\sqrt{2\phi^{-1}}$, $b=\sqrt{2\phi}$, and $\phi=(1+\sqrt5)/2$, according to Wikipedia.

Answer 2

Problem: express $\sin(2\pi/17)$ in algebraic terms.

Solution: Let $\epsilon=\sqrt{17+\sqrt{17}}$.

Let $\epsilon'=\sqrt{17-\sqrt{17}}$.

Let $\delta=\sqrt{17}-1$.

Let $\alpha=\sqrt{34+6\sqrt{17}+\sqrt2\delta\epsilon'-8\sqrt2\epsilon}$.

Then $\sin(2\pi/17)={1\over16}\sqrt2\sqrt{4\epsilon'^2-2\sqrt2\delta\epsilon'+8\sqrt2\epsilon-(\sqrt2\delta+2\epsilon')\alpha}$

Note: it's even worse for $\sin(2\pi/257)$, and far worse for $\sin(2\pi/65537)$. For the first, see https://math.stackexchange.com/q/517218 – for the second, you're on your own.