Is the inverse for the arc length of a parabola (say, $f(x)=\dfrac{x^2}{2}$) not discovered, or not possible to express given elementary functions and product log ($W(x)$)? If the latter is so, is there a proof?
I ask because Wolfram Alpha offers not an expression, even using the product log, of the inverse arc length $L$ of a parabola of the simplest form: the inverse of $$ L=\frac{x\sqrt{1+x^2}}2+\frac{\ln(x+\sqrt{1+x^2})}2.$$
I've also seen other questions of a similar nature, but none address why there is no inverse; is it a lack of an ingenious way to find it or a lack of possibility (or a lack of an ingenious proof of the latter)