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Do we have the following $\zeta(4)$ representation included in the mathematical literature?

$$\zeta(4)$$ $$=\frac{32}{45}\int_0^1 x \arctan^2\left(\frac{1}{x}\right) \operatorname{arctanh}^3(x) \textrm{d}x+\frac{16}{45} \pi ^2 \int_0^1 x \arctan^2(x) \operatorname{arctanh}(x) \textrm{d}x$$ $$-\frac{32}{45} \pi \int_0^1 x \arctan^3(x) \operatorname{arctanh}(x)\textrm{d}x+\frac{16}{45} \int_0^1 x \arctan^4(x) \operatorname{arctanh}(x) \textrm{d}x.$$

How about other similar results? Thanks!

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  • $\begingroup$ $$\zeta(4) = \frac{\pi^2}{90}$$, en.wikipedia.org/wiki/… \It also has been proven that $$A_n\zeta(2n) = \pi^{2n}B_n$$ $\endgroup$
    – Masd
    Apr 2 at 23:17
  • $\begingroup$ $\zeta(4)=\dfrac{\pi^4}{90}$ $\endgroup$
    – FDP
    Apr 4 at 11:24
  • $\begingroup$ \begin{align}\zeta(4)=\frac{1}{6}\int_0^1\frac{\ln^3 x}{x-1}dx\end{align} $\endgroup$
    – FDP
    Apr 4 at 11:26
  • $\begingroup$ @FDP haha, okay! :-) $\endgroup$ Apr 4 at 14:04

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