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There have been a few questions asked on this topic before but my query is slightly different.

Suppose we have a parametric curve

\begin{equation} x(t) = t \\ y(t) = \tanh(t) \end{equation} for some values of $t$ and I am required to re-parametrize it in terms of arc length. To reparametrize in terms of arc length, we have to apply the arc length formula, simplify, express time parameter $t$ in terms of arc length $s$ and substitute this back into the original parametric equations.

Applying the arc length formula yields the integral \begin{equation} \int \sqrt{1+\text{sech}^4(t)} \,\,\mathrm{dt} \end{equation}

For the above example, there is no way to integrate it analytically. Wolfram also does not calculate it analytically. Although we can get a value of the entire curve length through numerical integration, it is not enough to use it for the reparametrization. So my question is, what do we do in such cases? Can the integral still be evaluated by advanced means? I had to abandon some curves and select other curves of similar shapes but every time I ended up with such integrals.

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up vote 0 down vote accepted

Practically one doesn't need the arc-length parametrization of an immersive path $c$, but $\partial_c$, the derivative in respect to its arc-length, defined by $\partial_c:=\dfrac{1}{\|c'\|}\cdot(\dots)'$. Its definition doesn't depend on arc-length parametrization while it gets a simple analytic meaning if it is, in this case we have $\partial_c f=f'$ (where $f$ is another path defined on the same interval as $c$).

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