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#### DreamWeaver

##### Well-known member

- Sep 16, 2013

- 337

Assuming the following classic result - due to Vardi - holds...

[tex]\int_{\pi/4}^{\pi/2}\log\log(\tan x)\,dx=\frac{\pi}{2}\log\left[\sqrt{2\pi}\frac{\Gamma(3/4)}{\Gamma(1/4)}\right][/tex]

Prove that:

[tex]\int_{\pi/4}^{\pi/2}\log^2[\log(\tan x)]\,dx=[/tex]

[tex]\beta''(1)+\frac{\pi^3}{24}-\frac{\pi\gamma^2}{4}-\pi\gamma\log\left[\sqrt{2\pi}\frac{\Gamma(3/4)}{\Gamma(1/4)}\right][/tex]

and...

[tex]\int_{\pi/4}^{\pi/2}\log^3[\log(\tan x)]\,dx=[/tex]

[tex]\beta'''(1)-3\gamma\,\beta''(1)+\frac{\pi\gamma^3}{2}-\frac{\pi}{2}\zeta(3) + \left(\frac{\pi^3}{4}+\frac{3\pi\gamma^2}{2}\right)\log\left[\sqrt{2\pi}\frac{\Gamma(3/4)}{\Gamma(1/4)}\right][/tex]

Where \(\displaystyle \beta(x)\,\) is the Dirichlet Beta function, defined by:

\(\displaystyle \beta(x)=\sum_{k=0}^{\infty}\frac{(-1)^k}{(2k+1)^x}\)