 Elliptic integral of the second kind

For real modulus $0\le k\le 1$, the displayed integral is the incomplete elliptic integral of the second kind, with amplitude $\varphi$. The modulus convention matters: some sources instead use the parameter $m=k^2$. The arc length of an ellipse reduces to this integral, because a trigonometric parametrization gives a square root of $1-k^2{\sin }^2\theta$. At $k=0$ it reduces to $\varphi$; at $k=1$ the real integrand is $|\cos \theta |$.