Combining an identity for harmonic numbers $H_n$, see the penultimate paragraph of Exercise 10 (page 8) of [1], and inspired in an integral, see identity $(22)$ of [2], I wrote using uniform convergence $$\int_0^1\frac{(1-x)\log^2(1-x)}{\sin (\pi x)}dx=2\sum_{n=1}^\infty\frac{H_{n}}{n+1}\int_0^1\frac{(1-x)x^{n+1}}{\sin (\pi x)}dx.\tag{1}$$
I know (using a CAS) some integral occuring in RHS of $(1)$, that are evaluated in terms of particular values of the zeta function $\zeta(s)$. But I think that evaluate for each integer $n\geq 1$ $$\int_0^1\frac{(1-x)x^{n+1}}{\sin (\pi x)}dx\tag{2}$$ is very difficult for me.
Question. Is it possible to evaluate $(2)$ with the purpose to write my identity $(1)$ as a statement for our integral in LHS of $(1)$ expressed as a series involving harmonic numbers and particular values of the Riemann's Zeta function? If these integrals in $(2)$ are known answer this question as a reference request, and I try to search such literature and read it. Thanks in advance.
References:
[1] Jack D’Aurizio, Superior Mathematics from an Elementary point of view, course notes, University of Pisa (2017-2018).
[2] Zurab Silagadze, Sums of Generalized Harmonic Series. For Kids from Five to Fifteen, RESONANCE (September 2015).