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.
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.
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.