Answer
The length of the curve is:
$L\approx 2.4221$
Work Step by Step
The length of a curve with polar equation of one loop of the curve
$$
r=\cos(2\theta), \quad\quad -\frac{\pi}{4} \leq \theta \leq \frac{\pi}{4}
$$
is given by the following:
$$
\begin{split}
L=& \int_{-\pi / 4}^{\pi / 4} \sqrt{r^{2}+\left(\frac{d r}{d \theta}\right)^{2}} d \theta\\
&=\int_{-\pi / 4}^{\pi / 4} \sqrt{\cos ^{2} 2 \theta+(-2 \sin 2 \theta)^{2}} d \theta\\
&=\int_{-\pi / 4}^{\pi / 4} \sqrt{\cos ^{2} 2 \theta+4 \sin ^{2} 2 \theta} d \theta\\
&= \int_{-\pi / 4}^{\pi / 4} \sqrt{1+3 \sin ^{2} 2 \theta} d \theta \\
&\approx 2.4221
\end{split}
$$
