Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 34

Incomplete Elliptic Integral of the First Kind
$34.1 \quad \displaystyle u = F \left({k, \phi}\right) = \int \limits_0^\phi \frac {\mathrm d \phi} {\sqrt{1 - k^2 \sin^2 \phi} } = \int \limits_0^x \frac {\mathrm d v} {\sqrt{\left({1 - v^2}\right) \left({1 - k^2 v^2}\right)} }$

where $\phi = \operatorname{am} u$ is called the amplitude of $u$ and $x = \sin \phi$, ahere here and below $0 < k < 1$.

Complete Elliptic Integral of the First Kind
$34.2 \quad \displaystyle K = F \left({k, \pi / 2}\right) = \int \limits_0^{\pi / 2} \frac {\mathrm d \phi} {\sqrt{1 - k^2 \sin^2 \phi} } = \int \limits_0^1 \frac {\mathrm d v} {\sqrt{\left({1 - v^2}\right) \left({1 - k^2 v^2}\right)} }$

Incomplete Elliptic Integral of the Second Kind
$34.3 \quad \displaystyle E \left({k, \phi}\right) = \int \limits_0^\phi \sqrt{1 - k^2 \sin^2 \phi} \, \mathrm d \phi = \int \limits_0^x \dfrac {\sqrt{1 - k^2 v^2} } {\sqrt{1 - v^2}} \, \mathrm d v$

Complete Elliptic Integral of the Second Kind
$34.4 \quad \displaystyle E = E \left({k, \pi / 2}\right) = \int \limits_0^{\pi / 2} \sqrt{1 - k^2 \sin^2 \phi} \, \mathrm d \phi = \int \limits_0^1 \dfrac {\sqrt{1 - k^2 v^2} } {\sqrt{1 - v^2}} \, \mathrm d v$

Incomplete Elliptic Integral of the Third Kind
$34.5 \quad \displaystyle \Pi \left({k, n, \phi}\right) = \int \limits_0^\phi \frac {\mathrm d \phi} {\left({1 + n \sin^2 \phi}\right) \sqrt{1 - k^2 \sin^2 \phi} } = \int \limits_0^x \frac {\mathrm d v} {\left({1 + n v^2}\right) \sqrt{\left({1 - v^2}\right) \left({1 - k^2 v^2}\right)} }$

Complete Elliptic Integral of the Third Kind
$34.6 \quad \displaystyle \Pi \left({k, n, \pi / 2}\right) = \int \limits_0^{\pi / 2} \frac {\mathrm d \phi} {\left({1 + n \sin^2 \phi}\right) \sqrt{1 - k^2 \sin^2 \phi} } = \int \limits_0^1 \frac {\mathrm d v} {\left({1 + n v^2}\right) \sqrt{\left({1 - v^2}\right) \left({1 - k^2 v^2}\right)} }$