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 !
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).