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### Evaluating $\int_0^1 \frac{\log x \log \left(1-x^4 \right)}{1+x^2}dx$

I am trying to prove that \int_{0}^{1}\frac{\log\left(x\right) \log\left(\,{1 - x^{4}}\,\right)}{1 + x^{2}} \,\mathrm{d}x = \frac{\pi^{3}}{16} - 3\mathrm{G}\log\left(2\right) \tag{1} \...
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### Integral $\int_0^1 \ln\left(\frac{1-x}{1+x}\right)\ln\left(\frac{1-x^2}{1+x^2}\right)\frac{dx}{x}$

Greetings I saw here (among the last integrals) that: $$\int_0^1 \ln\left(\frac{1-x}{1+x}\right)\ln\left(\frac{1-x^2}{1+x^2}\right)\frac{dx}{x}=\pi C$$ Where $C$ is Catalan's constant. Did this ...
### Find : $\int_0^{\pi/4}x\ln(\sin x)\mathrm dx$
I'm try to find this integral $$\int_0^{\pi/4}x\ln(\sin x)\mathrm dx$$ My try use : $\ln(\sin x)=-\ln2-\sum\limits_{n=1}^{\infty}\frac{\cos (2nx)}{n}$ But I don't know how to complete summation ......