Elliptic integrals are integrals of the form

with R a rational function and y a polynomial of degree 3 or 4. This is the algebraic form of an elliptic integral. There are also trig forms (rational functions of sin and cos and a square root of a quadratic polynomial in sin and cos) and hyperbolic trig forms.

Elliptic integrals are reduced to their Legendre normal form in terms of elementary functions and the Elliptic functions EllipticF, EllipticE, and EllipticPi (or their complete versions).

${\∫}_0^{\frac{1}{2}}\frac{\sqrt{1\+x^4}}{1-x^4}\ⅆx$

$-\frac{\sqrt{2}\ln \left(\sqrt{17}-2\sqrt{2}\right)}{8}+\frac{\sqrt{2}\ln \left(\sqrt{17}+2\sqrt{2}\right)}{8}-\frac{\sqrt{2}\arctan \left(\frac{\sqrt{17}\sqrt{2}}{4}\right)}{4}+\frac{\mathrm{\pi }\sqrt{2}}{8}$

$\mathrm{assume}\left(0<k\&comma;k<1\right)$

${\&int;}_0^k\frac{x^2}{\sqrt{\left(1-x^2\right)\left(1-k^2{x}^2\right)}}\&DifferentialD;x$

$\frac{\mathrm{EllipticF}\left(\mathrm{k~}\&comma;\mathrm{k~}\right)}{{\mathrm{k~}}^2}-\frac{\mathrm{EllipticE}\left(\mathrm{k~}\&comma;\mathrm{k~}\right)}{{\mathrm{k~}}^2}$

$\mathrm{ans}≔{\&int;}_0^{\frac{1}{4}}\frac{1}{\left(x^4\&plus;2\right)\sqrt{4-5x^2\&plus;x^4}}\&DifferentialD;x$

$\mathrm{ans}≔\frac{\left(\sum _{\mathrm{\_\&alpha;}=\mathrm{RootOf}\left({\mathrm{\_Z}}^4+2\right)}\mathrm{EllipticPi}\left(\frac{1}{4}\&comma;-\frac{{\mathrm{\_\&alpha;}}^2}{2}\&comma;\frac{1}{2}\right)\right)}{16}$

$\mathrm{evalf}\left(\mathrm{ans}\right)$

$0.06331207100+0.\mathrm{I}$

$\mathrm{evalf}\left(\mathrm{ans}\&comma;20\right)$

$0.063312071018173992738+0.\mathrm{I}$

${\&int;}_0^{\frac{\mathrm{Pi}}{2}}\sqrt{1\&plus;2\sin \left(x\right)}\&DifferentialD;x$

$-\mathrm{EllipticK}\left(\frac{\sqrt{3}}{2}\right)+\mathrm{EllipticF}\left(\frac{\sqrt{2}\sqrt{3}}{3}\&comma;\frac{\sqrt{3}}{2}\right)+4\mathrm{EllipticE}\left(\frac{\sqrt{3}}{2}\right)-\mathrm{EllipticPi}\left(\frac{\sqrt{2}\sqrt{3}}{3}\&comma;\frac{3}{4}\&comma;\frac{\sqrt{3}}{2}\right)$

$\mathrm{Itrig}≔\&int;\frac{1}{\sqrt{1\&plus;2\cos \left(x\right)}}\&DifferentialD;x$

$\mathrm{Itrig}≔\frac{2\sqrt{3}\mathrm{InverseJacobiAM}\left(\frac{x}{2}\&comma;\frac{2\sqrt{3}}{3}\right)}{3}$

$\mathrm{simplify}\left(\mathrm{combine}\left(\frac{\partial }{\partial x}\mathrm{Itrig}-\frac{1}{\sqrt{1\&plus;2\cos \left(x\right)}}\&comma;\mathrm{trig}\right)\right)$

$0$

Labahn, G., and Mutrie, M. "Reduction of Elliptic Integrals to Legendre Normal Form." University of Waterloo Tech Report 97-21, Department of Computer Science, 1997.