Find the entire arc length of the curve $r=2acos^3(θ/3)$

Question

Question: Find the entire arc length of the curve $r=2a \cos^3(\frac{\theta}{3})$

My attempt: Given, $r = 2a \cos^3(\frac{\theta}{3})$

Using chain rule while differentiating with respect to $\theta$,

$ \displaystyle \frac{dr}{dθ} = \frac{d}{d\theta} (2a \cos^3 \frac{\theta}{3})$

        = 2a * d / dθ [cos^3(θ/3)

        = 2a * 3 cos^2(θ/3) * d/dθ [cos(θ/3)]

        = 6a cos^3(θ/3) * [-sin(θ/3)] * d/dθ [θ/3]

        = 6a cos^3(θ/3) * [-sin(θ/3)] * 1/3

        = -2a*cos^2(θ/3)*sin(θ/3)

Now, As Tom K mentioned in his comment,

$ \cos(3π + \theta) = - \cos \theta$

$(\cos (3 π + \theta), \sin(3 π + \theta)) = - (\cos \theta, \sin\theta)$

This means values of x and y repeat every $3 \pi$.

Now, for the arc length $L$, integrate $ \displaystyle \sqrt {r^2 + \bigg(\frac{dr}{d\theta}\bigg)^2} \ d\theta \ $ with $ \ \theta \in (0,3\pi)$.

(I have slove this far.Now,what should i d0.Seeking help from seniors)