If I could find an $f(t)$ in the $xy$ plane
where $\displaystyle\int\frac{\sqrt{f'_x(t)^2+f'_y(t)^2}}{f_x(t)^2+f_y(t)^2}\, dt$ is known (basically $\displaystyle\frac{\text{arclength}}{\text{distance}^2}$ integrated)
I have tried to use both quadratic and cubic bezier curves, but the results got waay too messy for my realtime application. And I gave up when I found a full explanation of how to find the arc length of bezier curves (I found that it's quite difficult or impossible).
My idea now is that I could design my curve with some other kinds of parameters, so that my calculation will be possible. Does such kind of curve exist??
Circular arcs. And you can string these together smoothly to get free-form “spline” curves. Look up biarc curves to find out more. https://en.wikipedia.org/wiki/Biarc
Another option is a so-called Pythagorean hodograph curve. These are specifically designed to ensure that the square root term in your integrand simplifies nicely. Rida Farouki wrote an entire book about these creatures: https://faculty.engineering.ucdavis.edu/farouki/wp-content/uploads/sites/41/2013/02/Introduction-to-PH-curves.pdf