12 questions linked to/from Is there any integral for the Golden Ratio?
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### Integral $\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \ \mathrm dx$

I need help with this integral: $$I=\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right)\ \mathrm dx.$$ The integrand graph looks like this: $\hspace{1in}$ The ...
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### Elementary solution of $\int_0^1\frac{\ln(1-3x+x^2)\ln x}xdx$
Is there a elementary way to evaluate $$\int_0^1\frac{\ln(1-3x+x^2)\ln x}xdx=\frac85\zeta(3)+\frac{2}{5} \pi ^2 \ln\varphi-2 i \pi\ln^2\varphi$$ where $\varphi$ is the golden ratio? (I posted it ...