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Look at what I found! $$\int_{-\pi/4}^{\pi/4}x\csc x\ \mathrm dx=\sum_{n\geq1}\pi n\log\bigg(\frac{4n+1}{4n-1}\bigg)\prod_{k\geq1\\k\neq n}\frac{k^2}{k^2-n^2}$$ I want to know if it is valid, and if there is a nice closed form to go with my series.

Here's my proof.

Recall the famous product representation $$\sin x=x\prod_{k\geq1}\bigg(1-\frac{x^2}{\pi^2k^2}\bigg)$$ Hence we have that $$I=\int_{-\pi/4}^{\pi/4}x\csc x\ \mathrm dx=\int_{-\pi/4}^{\pi/4}\prod_{k\geq1}\frac{\pi^2k^2}{\pi^2k^2-x^2}\mathrm dx$$ We may now preform a fraction decomposition, and start with $$\prod_{k\geq1}\frac{\pi^2k^2}{\pi^2k^2-x^2}=\sum_{n\geq1}\frac{\pi^2n^2}{\pi^2n^2-x^2}b_n$$ Multiplying both sides by $\prod_{k\geq1}\frac{\pi^2k^2-x^2}{\pi^2k^2}$, $$1=\sum_{n\geq1}\frac{\pi^2n^2}{\pi^2n^2-x^2}b_n\prod_{k\geq1}\frac{\pi^2k^2-x^2}{\pi^2k^2}$$ $$1=\sum_{n\geq1}b_n\prod_{k\geq1\\k\neq n}\frac{\pi^2k^2-x^2}{\pi^2k^2}$$ Which in turn implies that, for any $m\in\Bbb N$, $$1=b_m\prod_{k\geq1\\k\neq m}\frac{\pi^2k^2-\pi^2m^2}{\pi^2k^2}$$ $$b_m=\prod_{k\geq1\\k\neq m}\frac{k^2}{k^2-m^2}$$ Hence we can begin the process of integration: $$I=\int_{-\pi/4}^{\pi/4}\sum_{n\geq1}\frac{\pi^2n^2}{\pi^2n^2-x^2}b_n\mathrm dx$$ $$I=\sum_{n\geq1}\pi^2n^2b_n\int_{-\pi/4}^{\pi/4}\frac{\mathrm dx}{\pi^2n^2-x^2}$$ This final integral evaluates to $$\int_{-\pi/4}^{\pi/4}\frac{\mathrm dx}{\pi^2n^2-x^2}=\frac1{\pi n}\log\bigg(\frac{4n+1}{4n-1}\bigg)$$ So finally we have $$I=\sum_{n\geq1}\pi n\log\bigg(\frac{4n+1}{4n-1}\bigg)\prod_{k\geq1\\k\neq n}\frac{k^2}{k^2-n^2}$$ Does this work? Is there a closed form value for $I$? Is there another way to prove this?

Update:

Integrating form $0$ to $\pi/2$ instead of $-\pi/4$ to $\pi/4$ gives, according to Wolfram Alpha, $$\sum_{n\geq1}\frac{\pi n}2\log\bigg(\frac{2n+1}{2n-1}\bigg)\prod_{k\geq1\\k\neq n}\frac{k^2}{k^2-n^2}=2G$$ Where $G$ is Catalan's constant. Isn't that cool?! That's crazy cool!

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    $\begingroup$ Yes it is 'crazy-cool' ... & there are lirerally thousands of crazy-cool results like that! I can't get enough of 'em. $\endgroup$ – AmbretteOrrisey Dec 4 '18 at 22:00

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