Let $l\ge 0$ be an integer. Using integration by parts and the definition of the polylogarithm we have found the following identity: \begin{equation} \int \frac{[\log(\xi)]^l}{(1+\xi)^2} \log(1+\xi) d \xi = \sum\limits_{p=0}^l l_{(p)} \cdot [\log(\xi)]^{l-p} \cdot \phi_p(\xi) \cdot (-1)^{p+1} \end{equation} where the functions $\phi_p$ are given as follows: \begin{eqnarray} \phi_0(\xi) &:=& \frac{\log (\xi +1)+1}{\xi +1} \\ \phi_1(\xi) &:=& -\text{Li}_2(-\xi )-\frac{1}{2} \log ^2(\xi +1)-\log (\xi +1)+\log (\xi ) \\ \phi_2(\xi) &:=& -\text{Li}_3(-\xi )+\text{Li}_3(\xi +1)+\text{Li}_2(-\xi ) (\log (\xi +1)+1)+\frac{\log ^2(\xi )}{2}+\frac{1}{2} \log ^2(\xi +1) \log (\xi )+\frac{1}{2} i \pi \log ^2(\xi +1)-\frac{1}{6} \pi ^2 \log (\xi +1) \\ \phi_3(\xi) &:=& -2 \text{Li}_4(-\xi )-\text{Li}_4\left(\frac{\xi }{\xi +1}\right)+\text{Li}_4(\xi +1)+\text{Li}_3(-\xi ) (\log (\xi +1)+1)+ \frac{1}{72} \left( -72 \zeta (3) \log (\xi +1)+ 72 \zeta (3) \log (\xi )- 3 \log ^4(\xi +1)+ 12 \log(\xi ) \log ^3(\xi +1)+ \\ 12 i \pi \log ^3(\xi +1)+ 12 \log ^3(\xi )- 6 \pi ^2 \log ^2(\xi +1)+ 72 i \pi \zeta (3)+\pi ^4 \right) \end{eqnarray} Here $l_{(p)}:=l(l-1)\cdot \dots \cdot (l-p+1)$ is the lower Pochhammer symbol.

Unfortunately we got stuck at $l=3$ because we lack knowledge about a reflection formula for the polylogarithm of order four and therefore we cannot simplify the terms involving $\mbox{Li}_4$ in the last equation above. Now, the question is therefore twofold. Firstly, what is the generic formula for $\phi_p(\xi)$ when $p\ge 4$ and secondly where can I find the functional identities (reflection formulae) for polylogarithms of order greater or equal to five).


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