Wish to evaluate $\int_{0}^{1}\frac{x^4-3x^2+2x}{\sin(\pi x)} dx$ We want to evaluate this integral,$\newcommand{\cosec}{\operatorname{cosec}}$
$$\int_{0}^{1}\frac{x^4-3x^2+2x}{\sin(\pi x)}\mathrm dx\tag1$$
$$\int_{0}^{1}[x^4\cosec(\pi x)-3x^2\cosec(\pi x)+2x\cosec(\pi x)]\mathrm dx$$
Apply integration by parts, first part:
$$\int x^4\cosec(\pi x)\mathrm dx=-\frac{x^4}{\pi}\ln[\cot(\pi x/2)]-\frac{4}{\pi}\int x^3\ln[\cot(\pi x/2)]\mathrm dx$$ again we apply integration by parts to this remaining integral,
$$\int x^3\ln[\cot(\pi x/2)]\mathrm dx$$
Integrate this part it is very difficult $\int \ln[\cot(\pi x/2)]\mathrm dx$
I am stuck at this point. I guess there must be another method to deal with this integral $(1)$.
I am looking forward to see how you would go about to evaluates $(1)$

Too long for comments.
@user10354138 provided a very nice solution for the problem.
Since similar questions would probably happen, I tabulated the expressions of
$$I_n=\int_0^1 x^{n-2}\,\frac{x^2-x}{\sin(\pi x)}\,dx$$
$$\left(
\begin{array}{cc}
n & I_n \\
 2 & -\frac{7 \zeta (3)}{\pi ^3} \\
 3 & -\frac{7 \zeta (3)}{2 \pi ^3} \\
 4 & -\frac{21 \zeta (3)}{2 \pi ^3}+\frac{93 \zeta (5)}{\pi ^5} \\
 5 & -\frac{14 \zeta (3)}{\pi ^3}+\frac{279 \zeta (5)}{2 \pi ^5} \\
 6 & -\frac{35 \zeta (3)}{2 \pi ^3}+\frac{465 \zeta (5)}{\pi ^5}-\frac{5715 \zeta
   (7)}{2 \pi ^7} \\
 7 & -\frac{21 \zeta (3)}{\pi ^3}+\frac{930 \zeta (5)}{\pi ^5}-\frac{28575 \zeta
   (7)}{4 \pi ^7} \\
 8 & -\frac{49 \zeta (3)}{2 \pi ^3}+\frac{3255 \zeta (5)}{2 \pi ^5}-\frac{120015
   \zeta (7)}{4 \pi ^7}+\frac{160965 \zeta (9)}{\pi ^9} \\
 9 & -\frac{28 \zeta (3)}{\pi ^3}+\frac{2604 \zeta (5)}{\pi ^5}-\frac{80010 \zeta
   (7)}{\pi ^7}+\frac{1126755 \zeta (9)}{2 \pi ^9} \\
 10 & -\frac{63 \zeta (3)}{2 \pi ^3}+\frac{3906 \zeta (5)}{\pi ^5}-\frac{360045
   \zeta (7)}{2 \pi ^7}+\frac{2897370 \zeta (9)}{\pi ^9}-\frac{29016225 \zeta
   (11)}{2 \pi ^{11}}
\end{array}
\right)$$
and we see how linear combinations of the $I_n$'s can lead to nice simplifications by cancellations.
