Arc length of polar curves

Let $R$ be the $\blue{\text{circular segment}}$ which lies inside the circle $\purple{r=4\cos\theta}$ and to the left of the line $\red{r=\sec\theta}$ as pictured below.  **What is the perimeter of region $R$?** *Enter the exact answer as an expression involving radicals and $\pi$.* [[☃ expression 1]]