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I have just known about the elliptic functions and I saw three nice examples as following :

$$\int_{0}^{\frac{\pi}{2}}\frac{\tan x}{4\ln^{2}\tan x+ \pi^{2}}{\rm d}x\cong\frac{1}{4}$$ $$\int_{0}^{1}\frac{{\rm d}x}{\sqrt{1- x^{3}}}\cong\frac{1}{\sqrt[4]{3}}\int_{0}^{\sqrt{4\sqrt{3}- 6}}\frac{{\rm d}x}{\sqrt{1- x^{2}}\sqrt{1- \frac{\sqrt{3}+ 2}{4}\cdot x^{2}}}$$ $$\int_{0}^{\frac{\pi}{3}}\int_{0}^{\frac{\pi}{3}}\frac{{\rm d}y{\rm d}x}{\sin^{2}x+ \cos^{2}y}\cong\frac{4}{\pi}$$ I'm looking forward to similar problems, any comments and solutions are welcome and appreciated

Thanks a real lot !

Harry Peter
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    Maybe you can search similar problems. – River Li Feb 04 '21 at 12:40
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    Searching for this kind of approximations is entertaining, but hardly useful. I would also suggest providing the desired accuracy to which they should work. 1-3 digits is not very interesting, so something like 10 digits or more? – Yuriy S Feb 12 '21 at 12:18
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    If you're interested in approximations related to elliptic integrals, you'll want to play with their relation to the AGM iteration. – A rural reader Feb 13 '21 at 02:34

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Substitution $$x=\arctan y$$ presents the first integral in the form of $$\int\limits_0^\infty \dfrac{y\,\text d y}{(4\ln^2y+\pi^2)(1+y^2)} = \dfrac14$$ (see WA integration).

Yuri Negometyanov
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