Hi all, I would like to share some curious/strange identities with you. Maybe in the future will be skilled enough to prove them?

This thread is not for proving identities but to simply state them. If you know any other crazy identities or want to comment don't hesitate to post.

1. \(\displaystyle \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)\cosh(10n+5)\pi}=\frac{1}{2} \sin^{-1} \left[\left\{ \left( 3-2\sqrt2\right)\left( 2+\sqrt5\right) \left( \sqrt{10}-3\right)\left( -\sqrt2+\sqrt[4]5\right)^2 \right\}^2 \right]\)

2. \(\displaystyle \sum_{n=0}^\infty (-1)^n \frac{(2n+1)^2}{\sinh (2n+1)\pi }=\frac{\sqrt{2}-1}{8}\frac{\pi^{3/2}}{\Gamma^6 \left(\frac{3}{4}\right)}\)

3. \(\displaystyle \prod_{k=1}^\infty \left(1+e^{-\frac{k\pi}{3}} \right)=\frac{e^{\frac{\pi}{72}}}{\sqrt[8]{2}}\sqrt[24]{\frac{\sqrt{2}+\sqrt[4]{3}}{(\sqrt{2}-\sqrt[4]{3})^5}}\)

4. \(\displaystyle \int_0^1 K(k)^3 \; dk = \frac{3}{1280\pi^2}\Gamma^8\left( \frac{1}{4}\right)\) where \(K(k)\) is the complete elliptic integral of the first kind.

5. \(\displaystyle \sum_{k=0}^\infty \frac{2k+1}{\left\{ 25+\frac{(2k+1)^4}{100}\right\}(1+e^{(2k+1)\pi}) }=\frac{4689}{11890}-\frac{\pi}{8}\text{coth}^2 \left(\frac{5\pi}{2} \right)\)

6.\(\displaystyle \int_0^\infty \frac{1}{\prod\limits_{n=0}^\infty (1+e^{-10 n \pi}x^2)}dx=\frac{\pi^{\frac{3}{4}}\Gamma \left(\frac{3}{4}\right)}{2e^{\frac{5\pi}{8}}}\sqrt{5}\sqrt[8]{2}\left( 1+\sqrt[4]{5}\right)\sqrt{\frac{1+\sqrt{5}}{2}}\)