Answer
$3 \sqrt 2+3 \ln (\sqrt 2+1)$
Work Step by Step
The length of the curve is given as: $L=\int_{p}^{q}\sqrt{r^2+(\dfrac{dr}{d\theta})^2}d\theta$
Thus, $L=\int_{0}^{\pi/2} \sqrt{(\dfrac{6}{1+\cos \theta})^2+(\dfrac{6 \sin \theta}{(1+\cos \theta)^2})^2} d \theta$
Then, we have $L=(3) \int_{0}^{\pi/2}(\sec^3 \theta/2) d\theta$
or, $L=3 [\tan (\theta/2)\sec (\theta/2)+\ln |\sec (\theta/2)+\tan (\theta/2)]_0^{\pi}$
Thus, $L=3 \sqrt 2+3 \ln (\sqrt 2+1)$
