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