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Generalized form of this Harmonic Number series $\sum_{n=1}^{\infty} \frac{{H_n}x^{n+1}}{(n+1)^3}$
i've tried to Evaluate $$\int_{0}^{\frac{\pi}{6}}x\ln^2(2\sin(x))dx$$ without using Contour integral
first i changed $2\sin(x)$ into polar form ,and i got $$\int_{0}^{\frac{\pi}{6}}x\ln^2(2\sin(x))dx ...
- 347
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Integral $\int^1_0\frac{\ln{x} \ \mathrm{Li}_2(x)}{1-x}dx$
I would like to know how to evaluate the integral
$$\int^1_0\frac{\ln{x} \ \mathrm{Li}_2(x)}{1-x}dx$$
I tried expanding the integrand as a series but made little progress as I do not know how to ...
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