log log integral inverse sech

In this entry we will prove the following result



\begin{align*} \int_0^1 \ln\left(\ln\left(\frac{1+\sqrt{1-x^2}}{x} \right) \right)\,dx&=-\gamma-2\ln(2)-2\ln\left( \frac{ \Gamma\left(\frac34 \right)}{\Gamma\left(\frac14 \right)}\right) \end{align*}


Click here to read it.


Some previous established results were used, i.e.:

Lerch´s formula


\begin{align*} \ln\left( \Gamma(x)\right)=\zeta^\prime(0,x)-\zeta^\prime(0) \end{align*}

Click here for the proof.

Relation between Bernoulli´s polynomials and Hurwitz zeta function


\begin{align*} \zeta(-n,x)=-\frac{\operatorname{B}_{n+1}(x)}{n+1} \end{align*}

click here for the proof (equations 8.22 and 8.30)

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