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Is there a closed form solution for $y\sqrt{y^2 + 1} + \sinh^{-1}(y) = x$?

I would like to invert the arc length of a parabola so I can parameterize it with a constant speed.

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I don't think I've ever seen anybody construct a natural parametrization of the parabola, or even an intrinsic equation. –  J. M. Sep 4 '11 at 5:45
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up vote 0 down vote accepted

Hint: inverse hyperbolic functions can be expressed in terms of logarithms:

$$y+\sqrt{y^2 + 1} + \sinh^{-1}(y) = y+\sqrt{y^2 + 1} + \log(y + \sqrt{y^2+1})= z + log(z) = x$$

where $z = y + \sqrt{y^2+1}$ The last equation can be expressed in a closed form only by using the Lambert W function

Edited: Sorry, I misread the first term (are you sure you got it right?). This turns the equation almost hopeless to find a closed form solution.

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Yes, it's a simplified version of $\int \sqrt{4x^2 + 1}\ dx$. (W|A) –  jnm2 Sep 4 '11 at 2:53
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