Integration $\int_{-1}^1 \sqrt{\frac{r^2-x^2}{1-x^2}}dx$ I am interested in the following integral: (r is a constant)
$$\int_{-1}^1 \sqrt{\frac{r^2-x^2}{1-x^2}}dx$$
Initially I thought of a trigonometric substitution, or a substitution like $z^2=r^2-x^2$, but to no avail. Is it possible to find an analytical solution? 

The antiderivative involves elliptic integrals (which are not the nicest I know). From there, the integral is given by 

$$\int_{-1}^1 \sqrt{\frac{r^2-x^2}{1-x^2}}dx=2 r E\left(\frac{1}{r^2}\right)$$

provided that $\Re(r)\geq 1\lor \Re(r)\leq -1\lor r\notin \mathbb{R}$ 
