I'm trying to understand what is happening with the arc length parametrization, but it seems that something is missing.
I considered the curve $a(t)=\left(e^{t}\cdot \sin(t), e^{t}\cdot \cos(t)\right)$, then I computed $$s=\int^{t}_{0}{\sqrt{e^{2u}(\sin^{2}(u)+\cos^{2}(u))}}=e^t-1,$$ so $$t(s)=\ln(s+1).$$ Next I rewrote the curve, so $$b(s)=((s+1)\cdot\sin(\ln(s+1),((s+1)\cdot \cos(\ln(s+1))).$$ Also, I ploted these curves in geogebra.
My question is, why the curves does not have the same length? I know there is a theorem which assume that a re-parametrization of a curve does not change its length.