We consider a following quantity: \begin{eqnarray} {\mathcal I}^{(p,q)}_r(\xi,t) := \int\limits_\xi^t \frac{[\log(\frac{t}{\eta})]^p}{p!} \cdot \frac{[\log(\frac{\eta}{\xi})]^q}{q!} \cdot \frac{Li_r(\eta)}{\eta} d\eta \end{eqnarray} where $p$,$q$ and $r$ are non-negative integers and $0\le \xi \le t \le 1$ and $L_r(\eta) := \sum\limits_{m=1}^\infty \eta^m/m^r$ is the poly-logarithm.
By using integration by parts we found that the quantities above satisfy following recurrence relations: \begin{eqnarray} {\mathcal I}^{(p,q-p)}_{W-q} &=& 1_{2 p \ge q} \cdot \sum\limits_{j=0}^{q-p-1} \binom{q-p}{j} (-1)^j {\mathcal I}^{(2p-q+j,q-p-j)}_{W-p} + 1_{2p < q}\sum\limits_{j=1}^p \binom{p}{j} (-1)^j {\mathcal I}^{(j,q-p-j)}_{W-q+p} + \\ &&1_{2 p \ge q} \cdot (-1)^{q-p}\left(Li_{W+1}(t) - \sum\limits_{l=1}^{p+1} Li_{W-p+l}(\xi) \cdot \frac{[\log(\frac{t}{\xi})]^{p+1-l}}{(p+1-l)!}\right)-\\ &&1_{2p < q} \cdot (-1)^{q-p}\left(Li_{W+1}(\xi)-\sum\limits_{l=1}^{q-p+1} Li_{W-q+p+l}(t) \frac{\log(\frac{\xi}{t})]^{q-p+1-l}}{(q-p+1-l)!}\right) \end{eqnarray} for $0\le p \le q \le W$. This is a system of $\binom{W+2}{2}$ linear equations for all the unknown quantities $\left\{ {\mathcal I}^{(p,q-p)}_{W-q} \right\}_{0\le p \le q \le W}$ which for a given $W \ge1$ is straightforward to solve on any CAS. For example we have the following: \begin{eqnarray} {\mathcal I}^{(1,2)}_1&=&\frac{1}{2} \text{Li}_3(t) \log ^2\left(\frac{\xi }{t}\right)+2 \text{Li}_4(t) \log \left(\frac{\xi }{t}\right)-\text{Li}_4(\xi ) \log \left(\frac{t}{\xi }\right)+3 \text{Li}_5(t)-3 \text{Li}_5(\xi )\\ {\mathcal I}^{(1,3)}_2&=& -\frac{1}{6} \text{Li}_4(t) \log ^3\left(\frac{\xi }{t}\right)-\text{Li}_5(t) \log ^2\left(\frac{\xi }{t}\right)-3 \text{Li}_6(t) \log \left(\frac{\xi }{t}\right)+\text{Li}_6(\xi ) \log \left(\frac{t}{\xi }\right)-4 \text{Li}_7(t)+4 \text{Li}_7(\xi )\\ {\mathcal I}^{(1,4)}_3 &=& \frac{1}{24} \text{Li}_5(t) \log ^4\left(\frac{\xi }{t}\right)+\frac{1}{3} \text{Li}_6(t) \log ^3\left(\frac{\xi }{t}\right)+\frac{3}{2} \text{Li}_7(t) \log ^2\left(\frac{\xi }{t}\right)+4 \text{Li}_8(t) \log \left(\frac{\xi }{t}\right)-\text{Li}_8(\xi ) \log \left(\frac{t}{\xi }\right)+5 \text{Li}_9(t)-5 \text{Li}_9(\xi ) \end{eqnarray} Now, my question is can we actually find a closed form expression for the quantities in question or otherwise do we always have to resort to CAS to solve the equations in question?
The closed form in question reads: \begin{eqnarray} &&{\mathcal I}^{(p,q)}_r = \frac{1}{p! q!} \sum\limits_{l_1=0}^p \sum\limits_{l_2=0}^q \sum\limits_{l_3=1}^{l_1+l_2+1} \binom{p}{l_1} \binom{q}{l_2} (l_1+l_2)_{(l_3-1)} (-1)^{q-1+l_1+l_2+l_3} \cdot \\ &&\left( Li_{r+l_3}(t) [\log(\xi)]^{q-l_2} [\log(t)]^{p+l_2+1-l_3} - Li_{r+l_3}(\xi) [\log(t)]^{p-l_1} [\log(\xi)]^{q+l_1+1-l_3}\right) \end{eqnarray} and it is obtained by expanding the two terms with logarithms into binomial series then by collecting all powers of logs together and then integrating term by term and by using integration by parts in order to evaluate the integrals.