Consider one point $E$ and two circle $c_1$ with center $C$ and radius $r_1$ , and circle $c_2$ with center $A$ and radius $r_2$, $E$ is inside two circles, and two circles are internally tangent in $D$. We have $|DE|=a$ and $ r_2>r_1>a>0$, point $A,C,E$ are on the same line, $F$ are any point on circle $c_1$, draw a line through $E,F$ and cross $c_2$ in $G$, then we define a ratio R of distance: $$R=\frac{|EF|}{|EG|}$$ where$|*|$ means the distance between two points.
How to calculate the average of $R$ when point $F$ goes around the whole circle $c_1$? Or in other words, how to calculate the integration below:
$$E(R)=\frac{1}{2\pi}\int_0^{2 \pi } \frac{\sqrt{(r_1-a+r_1 \cos (\theta ))^2+(r_1 \sin (\theta ))^2}}{\sqrt{(r_2-a+r_2 \cos (\theta ))^2+(r_2 \sin (\theta ))^2}} \, d\theta$$
