I am trying to solve an integral of the form
\begin{align*} \int\limits_{0<t_0<t_1<\cdots<t_r<1} \frac{\log^{m_0}(1-t_0)\left\{\prod\limits_{j=1}^r\log^{m_j}\left(\frac{1-t_{j-1}}{1-t_{j}}\right)\right\}\log^{m_{r+1}}(1-t_r)}{t_0t_1\cdots t_r}dt_0dt_1\cdots dt_r, \end{align*} where $m_j\in \{1,2,3,\cdots\},\ (j=0,1,2,\cdots,r+1).$