CHALLENGING INTEGRAL INTEGRALBOT

In this post we will prove the following integral


\begin{align*} \int_0^\infty \frac{\cos(x) \ln(x)}{\cosh(x)-\sin(x)}\,dx &=-\ln^2(2)-\frac{\pi^2}{16}-\frac{\pi}{4}\ln\left( \frac{\Gamma^4\left(\frac14\right)}{4\pi^3}\right) \end{align*}


Click here to see the proof.


We will need the following results previously established


\begin{align*} \beta^\prime(1)= \sum_{k=1}^{\infty} \frac{(-1)^{k+1} \ln (2 k+1)}{2 k+1}=\frac{\pi}{4}\left(\gamma+2 \ln 2+3 \ln \pi-4 \ln \Gamma\left(\frac{1}{4}\right)\right)\\ \end{align*}


proved here, and


\begin{align*} &\eta^\prime(1)=\sum_{n=1}^{\infty} \frac{(-1)^{n} \ln (n)}{n}=\gamma \ln (2)-\frac{\ln ^{2}(2)}{2} \end{align*}

proved here

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