Landen's Transformation

If $x\sin\alpha =\sin(2\beta -\alpha)$, then

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

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

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Ascending Landen Transformation'' and ``Landen's Transformation.'' §16.14 and 17.5 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 573-574 and 597-598, 1972.