Answer
$r(s)=(\dfrac{s}{\sqrt 3}+1) \lt 1, \sin (\ln (\dfrac{s}{\sqrt 3}+1)), \cos (\ln (\dfrac{s}{\sqrt 3}+1)) \gt$
Work Step by Step
Here, $r(t)=\lt e^t , e^t \sin t, e^t \cos t \gt $
and $|r'(t)| = e^t\lt 1, \sin t, \cos t\gt$;
$|r'(t)|=e^t\lt 1, \cos t+\sin t, \cos t-\sin t\gt $
Further, $|r'(t)|^2=e^{2t}\lt 1, (\cos t+\sin t)^2, (\cos t-\sin t)^2 \gt=3e^{2t} $
$\implies |r'(t)|=\sqrt 3e^{t} $
We need to solve for $t$
$e^t=\dfrac{s}{\sqrt 3}+1$ and $t=\ln (\dfrac{s}{\sqrt 3}+1)$
Hence, we have
$r(s)=(\dfrac{s}{\sqrt 3}+1) \lt 1, \sin (\ln (\dfrac{s}{\sqrt 3}+1)), \cos (\ln (\dfrac{s}{\sqrt 3}+1)) \gt$
