Answer
$$\pi \sqrt{4\pi ^2+1}+\frac{1}{2}\ln \left(2\pi +\sqrt{4\pi ^2+1}\right)$$
Work Step by Step
\begin{aligned}
s &=\int_{a}^{b} \sqrt{x^{\prime}(t)^{2}+y^{\prime}(t)^{2}} d t \\
&=\int_{0}^{2 \pi} \sqrt{ (\cos \left(t\right)-t\sin \left(t\right))^2+(\sin \left(t\right)+t\cos \left(t\right))^2} d t \\
&= \int_{0}^{2 \pi} \sqrt{ t^2\cos ^2\left(t\right)+t^2\sin ^2\left(t\right)+1} d t\\
&= \int_{0}^{2 \pi} \sqrt{ t^2 +1} d t\\
&= \frac{1}{2} t \sqrt{1+t^{2}}+\frac{1}{2} \ln (t+\sqrt{1+t^{2}})\bigg|_{0}^{2\pi}\\
&= \pi \sqrt{4\pi ^2+1}+\frac{1}{2}\ln \left(2\pi +\sqrt{4\pi ^2+1}\right)
\end{aligned}
