Answer
$$\frac{3 \sqrt{3}}{4}$$
Work Step by Step
Since
$$\mathbf{r}=\left(e^{t} \cos t\right) \mathbf{i}+\left(e^{t} \sin t\right) \mathbf{j}+e^{t} \mathbf{k}$$Then
\begin{align*}
\mathbf{v}&=\left(e^{t} \cos t-e^{t} \sin t\right) \mathbf{i}+\left(e^{t} \sin t+e^{t} \cos t\right) \mathbf{j}+e^{t} \mathbf{k}\\
\Rightarrow|v|&=\sqrt{\left(e^{t} \cos t-e^{t} \sin t\right)^{2}+\left(e^{t} \sin t+e^{t} \cos t\right)^{2}+\left(e^{t}\right)^{2}}\\
&=\sqrt{3 e^{2 t}}\\
&=\sqrt{3} e^{t}
\end{align*}
Hence
\begin{align*}
s(t)&=\int_{0}^{t} \sqrt{3} e^{t} d \tau\\
&=\sqrt{3} e^{t}-\sqrt{3}\\
\text{ Length }&=s(0)-s(-\ln 4)\\
&=0-\left(\sqrt{3} e^{-\ln 4}-\sqrt{3}\right)\\
&=\frac{3 \sqrt{3}}{4}\end{align*}
