Gauss's Transformation

If

$$(1+x\sin^2\alpha)\sin\beta = (1+x)\sin\alpha,$$

then

$$(1+x)\int_0^\alpha {d\phi\over \sqrt{1-x^2\sin^2\phi}}=\int_0^\beta {d\phi \over\sqrt{1-{4x\over(1+x)^2}\sin^2\phi}}.$$

See also Elliptic Integral of the First Kind, Landen's Transformation