Two definite integrals These integrals are closely related since $\frac{\pi^2}{8}=\sum_{n=0}^{\infty} \frac{1}{(2n+1)^2}$ and $G=\sum_{n\mathop=0}^{\infty} \frac{(-1)^{n}}{(2n+1)^2}$. I'm not able to prove them though.

Show that
  $$\int_0^1 \frac{1}{\sqrt{x^2-1}}\log\left(\frac{\sqrt{x+1}+\sqrt{x-1}}{\sqrt{x+1}-\sqrt{x-1}}\right)\,\mathrm{d}x =\frac{\pi^2}{8}$$
  $$\int_1^{\infty} \frac{1}{x\sqrt{x^2-1}}\log\left(\frac{\sqrt{x+1}+\sqrt{x-1}}{\sqrt{x+1}-\sqrt{x-1}}\right)\,\mathrm{d}x =2G$$
where $G$ is Catalan's constant.


The first integral is easier than it looks.  Write
$$\begin{align}I &= \int_0^1 \frac{dx}{\sqrt{x^2-1}} \log{\left (\frac{\sqrt{x+1}+\sqrt{x-1}}{\sqrt{x+1}-\sqrt{x-1}} \right )} \\ &= -i \int_0^1 \frac{dx}{\sqrt{1-x^2}} \log{\left (x+i \sqrt{1-x^2}\right )}\\ &= \int_0^1 \frac{dx}{\sqrt{1-x^2}} \arccos{x}\\ &= -\frac12 \left [\arccos^2{x} \right ]_0^1\\ &= \frac{\pi^2}{8}\end{align}$$
The second integral is also straightforward, using the sub $x=\cosh{t}$:
$$\begin{align}J &= \int_1^{\infty} \frac{dx}{x \sqrt{x^2-1}} \log{\left (\frac{\sqrt{x+1}+\sqrt{x-1}}{\sqrt{x+1}-\sqrt{x-1}} \right )} \\ &= \int_0^{\infty} dt \frac{t}{\cosh{t}}\\ &=2 \sum_{k=0}^{\infty} (-1)^k \int_0^{\infty} dt \, t \, e^{-(2 k+1) t}\\ &= 2 \sum_{k=0}^{\infty} \frac{(-1)^k}{(2 k+1)^2} \\ &= 2 G\end{align}$$
