<s>[INST] How to prove $\int_0^1\frac{1-x}{(\ln x)(1+x)}\ dx=\ln\left(\frac2{\pi}\right)$? A friend asked me to prove

$$\int_0^1\frac{1-x}{(\ln x)(1+x)}\ dx=\ln\left(\frac2{\pi}\right)$$

and here is my approach:
\begin{align}
I&=\int_0^1\frac{1-x}{\ln x}\frac1{1+x}\ dx\\
&=\int_0^1\left(-\int_0^1x^y\ dy\right)\frac1{1+x}\ dx\\
&=\int_0^1\left(-\int_0^1\frac{x^y}{1+x}\ dx\right)\ dy\\
&=\int_0^1\left((-1)^n\sum_{n=1}^\infty\int_0^1x^{y+n-1}\ dx\right)\ dy\\
&=\int_0^1\left(\sum_{n=1}^\infty\frac{(-1)^n}{y+n}\right)\ dy\\
&=\sum_{n=1}^\infty(-1)^n\int_0^1\frac1{y+n}\ dy\\
&=\sum_{n=1}^\infty(-1)^n\left[\ln(n+1)-\ln(n)\right]\tag{1}
\end{align}

Now how can we finish this alternating sum into $\ln\left(\frac2{\pi}\right)$?
My idea was to use 
$$\operatorname{Li}_a(-1)=(1-2^{-a})\zeta(a)=\sum_{n=1}^\infty\frac{(-1)^n}{n^a}$$
and if we differentiate both sides with respect to $a$ we get
$$2^{1-a}(\zeta^{'}(a)-\ln2\zeta(a))-\zeta^{'}(a)=\sum_{n=1}^\infty\frac{(-1)^{n-1}\ln(n)}{n^a}\tag{2}$$
wolfram says that $\sum_{n=1}^\infty(-1)^n\ln(n)$ is divergent which means we can not take the limit for (2) where $a\mapsto 0$ which means we can not break the sum in (1) into two sums. So any idea how to do (1)?

Other question is, I tried to simplify the sum in (1) as follows
\begin{align}
S&=\sum_{n=1}^\infty(-1)^n\left[\ln(n+1)-\ln(n)\right]\\
&=-\left[\ln(2)-\ln(1)\right]+\left[\ln(3)-\ln(2)\right]-\left[\ln(4)-\ln(3)\right]+\left[\ln(5)-\ln(4)\right]-...\\
&=-2\ln(2)+2\ln(3)-2\ln(4)+...\\
&=2\sum_{n=1}^\infty(-1)^n\ln(n+1)
\end{align}
which is divergent again. what went wrong in my steps?
Thanks.
 [/INST] Note that $$\int_0^1 x^y dy=\frac{x-1}{\ln x}$$using this result our integral becomes $$\begin{aligned} I &=\int_0^1\frac{x-1}{(x+1) \ln x}\\&=\int_0^1\left(\frac{1}{x+1}\int_0^1x^ydy\right)dx \\&=\int_0^1\int_0^1\frac{x^y}{1+x}dy dx\\&=\int_0^1\underbrace{\int_0^1\frac{x^y}{1+x}dx }_{I_1}dy \end{aligned}$$ Since the integral $$I_1=\int_0^1\frac{x^y}{1+x}dx =\frac{1}{2}\left(H_{\frac{
y}{2}}-H_{\frac{y-1}{2}}\right)=\frac{1}{2}\left(\psi^0\left(\frac{2y+1}{2}\right)-\psi^0\left(\frac{y+1}{2}\right)\right)$$ is well know result which I have proved here by polynomial  long division. Integrating $I_1$  we yield $$ \begin{aligned}I &= \int_0^1 I_1 dy \\&=\left[\ln\Gamma\left(\frac{y+2}{2}\right)-\ln\Gamma\left(\frac{y+1}{2}\right)\right]_0^1\\& =\ln\left[
\Gamma\left(\frac{3}{2}\right)\Gamma\left(\frac{1}{2}\right)\right)\cdots(1)\\&=\ln\left(\frac{\sqrt{\pi}}{2}\sqrt{\pi}\right)=\ln\left(\frac{\pi}{2}\right)\end{aligned}$$ we use the half gamma $\displaystyle \Gamma\left(\frac{1}{2}+n\right)=\frac{(2n)!}{4^n n!}\sqrt{\pi}$ argument in $(1)$ or using functional equation of gamma function we can write $(1)$ as $\displaystyle=\ln\left(\frac{1}{2}\Gamma^2\left(\frac{1}{2}\right)\right)=\ln\left(\frac{\pi}{2}\right)$
Alternatively$$\begin{aligned}I_1 & =\int_0^1\frac{x^y}{1+x}dx \\&=\int_0^1x^y \left(\sum_{r=0}^{\infty} (-1)^r x^r\right)dx\\&=\sum_{r=0}^{\infty} \frac{(-1)^r}{y+r+1}=\Phi\left(-1,1,y+1\right)\cdots(3)\end{aligned}$$ where $\Phi(z,s,\alpha )$ is Lerch Transcendent function using the equation 5 and 6 we obtain $$I= \frac{1}{2}\left(\psi^0\left(\frac{2y+1}{2}\right)-\psi^0\left(\frac{y+1}{2}\right)\right)$$ To prove tha relationship  between $ (3)$ and final result we can directly use series formula of digamma function.
The above integral can be  related as

$$\int_0^{\infty} \frac{\operatorname{tanh}x}{e^{2x}} \frac{dx}{x}=\int_0^1\frac{x-1}{(1+x)\ln x} =-\ln\left(\frac{2}{\pi}\right)$$

</s>