(1)

\(0<\Re\left(z\right)\lor\Im\left(z\right)\ne0\)
\begin{align*} \int_{\infty}^{z}\frac{\Li_{n}\left(\frac{1}{z}\right)}{z}dz & =\int_{0}^{\frac{1}{z}}\frac{\Li_{n}\left(z\right)}{z^{-1}}d\frac{1}{z}\\ & =-\int_{0}^{\frac{1}{z}}\frac{\Li_{n}\left(z\right)}{z}dz\\ & =-\Li_{n+1}\left(\frac{1}{z}\right)+C \end{align*}

(2)

\begin{align*} \int\Li_{n}\left(z\right)dz & =z\Li_{n}\left(z\right)-\int\left(z\frac{\Li_{n-1}\left(z\right)}{z}\right)dz\\ & =z\Li_{n}\left(z\right)-\int\Li_{n-1}\left(z\right)dz\\ & =\int\sum_{k=0}^{n-1}\left\{ \left(-1\right)^{k}\Li_{n-k}\left(z\right)-\left(-1\right)^{k+1}\Li_{n-1-k}\left(z\right)\right\} dz+\left(-1\right)^{n}\int\Li_{0}\left(z\right)dz\\ & =\sum_{k=0}^{n-1}\left\{ \left(-1\right)^{k}z\Li_{n-k}\left(z\right)\right\} +\left(-1\right)^{n}\left(-z-\Log\left(1-z\right)\right)+C\\ & =\sum_{k=0}^{n-1}\left\{ \left(-1\right)^{n-1-k}z\Li_{k+1}\left(z\right)\right\} -\left(-1\right)^{n}\left(z+\Log\left(1-z\right)\right)+C\cmt{n-1-k\rightarrow k}\\ & =\sum_{k=1}^{n-1}\left\{ \left(-1\right)^{n-1-k}z\Li_{k+1}\left(z\right)\right\} +\left(-1\right)^{n-1}z\Li_{1}\left(z\right)-\left(-1\right)^{n}\left(z+\Log\left(1-z\right)\right)+C\\ & =\sum_{k=0}^{n-2}\left\{ \left(-1\right)^{n-k}z\Li_{k+2}\left(z\right)\right\} +\left(-1\right)^{n}z\Log\left(1-z\right)-\left(-1\right)^{n}\left(z+\Log\left(1-z\right)\right)+C\\ & =\sum_{k=0}^{n-2}\left\{ \left(-1\right)^{n-k}z\Li_{k+2}\left(z\right)\right\} -\left(-1\right)^{n}\left(z+\left(1-z\right)\Log\left(1-z\right)\right)+C\\ & =\sum_{k=0}^{n-2}\left\{ \left(-1\right)^{n-k}z\Li_{k+2}\left(z\right)\right\} -\left(-1\right)^{n}\left(z-\left(1-z\right)\Li_{1}\left(z\right)\right)+C \end{align*}

(3)

\begin{align*} \int\Li_{0}^{n}\left(z\right)dz & =\int\frac{z^{n}}{\left(1-z\right)^{n}}dz\\ & =\frac{1}{n-1}\cdot\frac{z^{n}}{\left(1-z\right)^{n-1}}-\frac{n}{n-1}\int\frac{z^{n-1}}{\left(1-z\right)^{n-1}}dz\\ & =\frac{z}{n-1}\Li_{0}^{n-1}\left(z\right)-\frac{n}{n-1}\int\Li_{0}^{n-1}\left(z\right)dz\\ & =\left(-1\right)^{n}n\sum_{k=2}^{n}\left\{ \frac{\left(-1\right)^{k}}{k}\int\Li_{0}^{k}\left(z\right)dz-\frac{\left(-1\right)^{k-1}}{k-1}\int\Li_{0}^{k-1}\left(z\right)dz\right\} -\left(-1\right)^{n}n\int\Li_{0}\left(z\right)dz\\ & =\left(-1\right)^{n}n\sum_{k=2}^{n}\left\{ \frac{\left(-1\right)^{k}}{k}\left(\frac{z}{k-1}\Li_{0}^{k-1}\left(z\right)\right)\right\} -\left(-1\right)^{n}n\int\frac{z}{1-z}dz\\ & =\left(-1\right)^{n}n\sum_{k=2}^{n}\left\{ \left(-1\right)^{k}\frac{z\Li_{0}^{k-1}\left(z\right)}{k\left(k-1\right)}\right\} -\left(-1\right)^{n}n\int\left(\frac{1}{1-z}-1\right)dz\\ & =\left(-1\right)^{n}n\sum_{k=2}^{n}\left\{ \left(-1\right)^{k}\frac{z\Li_{0}^{k-1}\left(z\right)}{k\left(k-1\right)}\right\} -\left(-1\right)^{n}n\left(-\Log\left(1-z\right)-z\right)+C\\ & =\left(-1\right)^{n}n\sum_{k=1}^{n-1}\left\{ \left(-1\right)^{k+1}\frac{z\Li_{0}^{k}\left(z\right)}{k\left(k+1\right)}\right\} -\left(-1\right)^{n}n\left(\Li_{1}\left(z\right)-z\right)+C \end{align*}

(4)

\begin{align*} \int\Li_{1}^{n}\left(z\right)dz & =z\Li_{1}^{n}\left(z\right)-n\int z\frac{\Li_{1}^{n-1}\left(z\right)}{1-z}dz\\ & =z\Li_{1}^{n}\left(z\right)-n\int\Li_{0}\left(z\right)\Li_{1}^{n-1}\left(z\right)dz\\ & =z\Li_{1}^{n}\left(z\right)-n\left(-\left(n-1\right)!z+\left(n-1\right)!\sum_{k=0}^{n-2}\left\{ \frac{\left(1-z\right)}{\left(k+1\right)!}\Li_{1}^{k+1}\left(z\right)\right\} +\frac{1}{n}\Li_{1}^{n}\left(z\right)\right)+C\\ & =z\Li_{1}^{n}\left(z\right)+n!z-n!\sum_{k=0}^{n-2}\left\{ \frac{\left(1-z\right)}{\left(k+1\right)!}\Li_{1}^{k+1}\left(z\right)\right\} -\Li_{1}^{n}\left(z\right)+C\\ & =n!z-n!\sum_{k=0}^{n-2}\left\{ \frac{\left(1-z\right)}{\left(k+1\right)!}\Li_{1}^{k+1}\left(z\right)\right\} -\left(1-z\right)\Li_{1}^{n}\left(z\right)+C \end{align*}

(5)

\begin{align*} \int\Li_{0}\left(z\right)\Li_{1}^{n}\left(z\right)dz & =\left(-z+\Li_{1}\left(z\right)\right)\Li_{1}^{n}\left(z\right)-n\int\left(-z+\Li_{1}\left(z\right)\right)\frac{\Li_{1}^{n-1}\left(z\right)}{1-z}dz\cmt{\because\int\Li_{0}\left(z\right)dz=-z+\Li_{1}\left(z\right)+C}\\ & =-z\Li_{1}^{n}\left(z\right)+\Li_{1}^{n+1}\left(z\right)+n\int\left(\frac{z}{1-z}\Li_{1}^{n-1}\left(z\right)-\frac{1}{1-z}\Li_{1}^{n}\left(z\right)\right)dz\\ & =-z\Li_{1}^{n}\left(z\right)+\Li_{1}^{n+1}\left(z\right)+n\int\frac{z}{1-z}\Li_{1}^{n-1}\left(z\right)dz-n\int\Li_{1}^{n}\left(z\right)d\Li_{1}\left(z\right)\\ & =-z\Li_{1}^{n}\left(z\right)+\Li_{1}^{n+1}\left(z\right)+n\int\frac{z}{1-z}\Li_{1}^{n-1}\left(z\right)dz-\frac{n}{n+1}\Li_{1}^{n+1}\left(z\right)\\ & =-z\Li_{1}^{n}\left(z\right)+\frac{1}{n+1}\Li_{1}^{n+1}\left(z\right)+n\int\Li_{0}\left(z\right)\Li_{1}^{n-1}\left(z\right)dz\\ & =n!\int\Li_{0}\left(z\right)dz+n!\sum_{k=1}^{n}\left\{ \frac{1}{k!}\int\Li_{0}\left(z\right)\Li_{1}^{k}\left(z\right)dz-\frac{1}{\left(k-1\right)!}\int\Li_{0}\left(z\right)\Li_{1}^{k-1}\left(z\right)dz\right\} \\ & =n!\int\Li_{0}\left(z\right)dz+n!\sum_{k=1}^{n}\left\{ \frac{1}{k!}\left(-z\Li_{1}^{k}\left(z\right)+\frac{1}{k+1}\Li_{1}^{k+1}\left(z\right)\right)\right\} \\ & =n!\left(-z+\Li_{1}\left(z\right)\right)+n!\sum_{k=1}^{n}\left\{ \frac{1}{k!}\left(-z\Li_{1}^{k}\left(z\right)+\frac{1}{k+1}\Li_{1}^{k+1}\left(z\right)\right)\right\} +C\\ & =-n!z+n!\Li_{1}\left(z\right)+n!\sum_{k=1}^{n-1}\left\{ -\frac{z}{\left(k+1\right)!}\Li_{1}^{k+1}\left(z\right)+\frac{1}{\left(k+1\right)!}\Li_{1}^{k+1}\left(z\right)\right\} +n!\left(-z\Li_{1}\left(z\right)\right)+\frac{1}{n+1}\Li_{1}^{n+1}\left(z\right)+C\\ & =-n!z+\left(1-z\right)n!\Li_{1}\left(z\right)+n!\sum_{k=1}^{n-1}\left\{ \frac{\left(1-z\right)}{\left(k+1\right)!}\Li_{1}^{k+1}\left(z\right)\right\} +\frac{1}{n+1}\Li_{1}^{n+1}\left(z\right)+C\\ & =-n!z+n!\sum_{k=0}^{n-1}\left\{ \frac{\left(1-z\right)}{\left(k+1\right)!}\Li_{1}^{k+1}\left(z\right)\right\} +\frac{1}{n+1}\Li_{1}^{n+1}\left(z\right)+C \end{align*}

(6)

\[ \begin{cases} \int\Li_{0}^{n}\left(z\right)dz=\left(-1\right)^{n}n\sum_{k=1}^{n-1}\left\{ \left(-1\right)^{k+1}\frac{z\Li_{0}^{k}\left(z\right)}{k\left(k+1\right)}\right\} -\left(-1\right)^{n}n\left(\Li_{1}\left(z\right)-z\right)+C\\ \int\frac{\Li_{n}\left(z\right)\Li_{n+1}\left(z\right)}{z}=\frac{1}{2}\Li_{n+1}^{2}\left(z\right)+C\\ \int\Li_{0}\left(z\right)dz=-z+\Li_{1}\left(z\right)+C \end{cases} \] を使う。
\begin{align*} \int\Li_{0}^{n}\left(z\right)\Li_{1}\left(z\right)dz & =\left\{ \left(-1\right)^{n}n\sum_{k=1}^{n-1}\left\{ \left(-1\right)^{k+1}\frac{z\Li_{0}^{k}\left(z\right)}{k\left(k+1\right)}\right\} -\left(-1\right)^{n}n\left(\Li_{1}\left(z\right)-z\right)\right\} \Li_{1}\left(z\right)-\int\left\{ \left(-1\right)^{n}n\sum_{k=1}^{n-1}\left(\left(-1\right)^{k+1}\frac{z\Li_{0}^{k}\left(z\right)}{k\left(k+1\right)}\right)-\left(-1\right)^{n}n\left(\Li_{1}\left(z\right)-z\right)\right\} \frac{1}{1-z}dz\\ & =\left(-1\right)^{n}n\sum_{k=1}^{n-1}\left\{ \left(-1\right)^{k+1}\frac{z\Li_{0}^{k}\left(z\right)\Li_{1}\left(z\right)}{k\left(k+1\right)}\right\} -\left(-1\right)^{n}n\left(\Li_{1}^{2}\left(z\right)-z\Li_{1}\left(z\right)\right)-\int\left\{ \left(-1\right)^{n}n\sum_{k=1}^{n-1}\left(\left(-1\right)^{k+1}\frac{\Li_{0}^{k+1}\left(z\right)}{k\left(k+1\right)}\right)-\left(-1\right)^{n}n\left(\frac{\Li_{0}\left(z\right)\Li_{1}\left(z\right)}{z}-\Li_{0}\left(z\right)\right)\right\} dz\\ & =\left(-1\right)^{n}n\sum_{k=1}^{n-1}\left\{ \left(-1\right)^{k+1}\frac{z\Li_{0}^{k}\left(z\right)\Li_{1}\left(z\right)}{k\left(k+1\right)}\right\} -\left(-1\right)^{n}n\left(\Li_{1}^{2}\left(z\right)-z\Li_{1}\left(z\right)\right)-\left\{ \left(-1\right)^{n}n\sum_{k=1}^{n-1}\left\{ \left(-1\right)^{k+1}\frac{1}{k\left(k+1\right)}\left(\left(-1\right)^{k+1}\left(k+1\right)\sum_{j=1}^{k}\left(\left(-1\right)^{j+1}\frac{z\Li_{0}^{j}\left(z\right)}{j\left(j+1\right)}\right)-\left(-1\right)^{k+1}\left(k+1\right)\left(\Li_{1}\left(z\right)-z\right)\right)\right\} -\left(-1\right)^{n}n\left(\frac{1}{2}\Li_{1}^{2}\left(z\right)-\left(-z+\Li_{1}\left(z\right)\right)\right)\right\} +C\\ & =\left(-1\right)^{n}n\sum_{k=1}^{n-1}\left\{ \left(-1\right)^{k+1}\frac{z\Li_{0}^{k}\left(z\right)\Li_{1}\left(z\right)}{k\left(k+1\right)}\right\} -\left(-1\right)^{n}n\left(\Li_{1}^{2}\left(z\right)-z\Li_{1}\left(z\right)\right)-\left(-1\right)^{n}n\sum_{k=1}^{n-1}\left\{ \left(-1\right)^{k+1}\frac{1}{k\left(k+1\right)}\left(\left(-1\right)^{k+1}\left(k+1\right)\sum_{j=1}^{k}\left(\left(-1\right)^{j+1}\frac{z\Li_{0}^{j}\left(z\right)}{j\left(j+1\right)}\right)-\left(-1\right)^{k+1}\left(k+1\right)\left(\Li_{1}\left(z\right)-z\right)\right)\right\} +\left(-1\right)^{n}n\left(\frac{1}{2}\Li_{1}^{2}\left(z\right)+z-\Li_{1}\left(z\right)\right)+C\\ & =\left(-1\right)^{n}n\sum_{k=1}^{n-1}\left\{ \left(-1\right)^{k+1}\frac{z\Li_{0}^{k}\left(z\right)\Li_{1}\left(z\right)}{k\left(k+1\right)}\right\} -\left(-1\right)^{n}n\left(\Li_{1}^{2}\left(z\right)-z\Li_{1}\left(z\right)\right)-\left(-1\right)^{n}n\sum_{k=1}^{n-1}\frac{1}{k}\sum_{j=1}^{k}\left\{ \left(-1\right)^{j+1}\frac{z\Li_{0}^{j}\left(z\right)}{j\left(j+1\right)}\right\} +\left(-1\right)^{n}n\sum_{k=1}^{n-1}\frac{1}{k}\left(\Li_{1}\left(z\right)-z\right)+\left(-1\right)^{n}n\left(\frac{1}{2}\Li_{1}^{2}\left(z\right)+z-\Li_{1}\left(z\right)\right)+C\\ & =\left(-1\right)^{n}n\sum_{k=1}^{n-1}\left\{ \left(-1\right)^{k+1}\frac{z\Li_{0}^{k}\left(z\right)\Li_{1}\left(z\right)}{k\left(k+1\right)}\right\} -\left(-1\right)^{n}n\left(\Li_{1}^{2}\left(z\right)-z\Li_{1}\left(z\right)\right)-\left(-1\right)^{n}n\sum_{j=1}^{n-1}\sum_{k=j}^{n-1}\frac{1}{k}\left\{ \left(-1\right)^{j+1}\frac{z\Li_{0}^{j}\left(z\right)}{j\left(j+1\right)}\right\} +\left(-1\right)^{n}nH_{n-1}\left(\Li_{1}\left(z\right)-z\right)+\left(-1\right)^{n}n\left(\frac{1}{2}\Li_{1}^{2}\left(z\right)+z-\Li_{1}\left(z\right)\right)+C\\ & =\left(-1\right)^{n}n\sum_{k=1}^{n-1}\left\{ \left(-1\right)^{k+1}\frac{z\Li_{0}^{k}\left(z\right)\Li_{1}\left(z\right)}{k\left(k+1\right)}\right\} -\left(-1\right)^{n}n\left(\Li_{1}^{2}\left(z\right)-z\Li_{1}\left(z\right)\right)-\left(-1\right)^{n}n\sum_{j=1}^{n-1}\left(H_{n-1}-H_{j-1}\right)\left(-1\right)^{j+1}\frac{z\Li_{0}^{j}\left(z\right)}{j\left(j+1\right)}+\left(-1\right)^{n}nH_{n-1}\left(\Li_{1}\left(z\right)-z\right)+\left(-1\right)^{n}n\left(\frac{1}{2}\Li_{1}^{2}\left(z\right)+z-\Li_{1}\left(z\right)\right)+C\\ & =\left(-1\right)^{n}nz-\left(-1\right)^{n}nH_{n-1}z+\left(-1\right)^{n}n\sum_{k=1}^{n-1}\left\{ \left(-1\right)^{k+1}\frac{z\Li_{0}^{k}\left(z\right)\Li_{1}\left(z\right)}{k\left(k+1\right)}\right\} -\left(-1\right)^{n}n\sum_{j=1}^{n-1}\left(H_{n-1}-H_{j-1}\right)\left(-1\right)^{j+1}\frac{z\Li_{0}^{j}\left(z\right)}{j\left(j+1\right)}+\left(-1\right)^{n}nH_{n-1}\Li_{1}\left(z\right)-\left(-1\right)^{n}n\Li_{1}\left(z\right)+\left(-1\right)^{n}nz\Li_{1}\left(z\right)-\left(-1\right)^{n}n\Li_{1}^{2}\left(z\right)+\left(-1\right)^{n}n\frac{1}{2}\Li_{1}^{2}\left(z\right)+C\\ & =\left(-1\right)^{n}n\left(1-H_{n-1}\right)z+\left(-1\right)^{n}n\sum_{k=1}^{n-1}\left\{ \left(-1\right)^{k+1}\frac{z\Li_{0}^{k}\left(z\right)\Li_{1}\left(z\right)}{k\left(k+1\right)}\right\} -\left(-1\right)^{n}n\sum_{k=1}^{n-1}\left(H_{n-1}-H_{k-1}\right)\left(-1\right)^{k+1}\frac{z\Li_{0}^{k}\left(z\right)}{k\left(k+1\right)}+\left(-1\right)^{n}n\left(H_{n-1}-1\right)\Li_{1}\left(z\right)+\left(-1\right)^{n}nz\Li_{1}\left(z\right)-\left(-1\right)^{n}\frac{1}{2}n\Li_{1}^{2}\left(z\right)+C\\ & =\left(-1\right)^{n}n\left(1-H_{n-1}\right)z+\left(-1\right)^{n}n\sum_{k=1}^{n-1}\left\{ \left(-1\right)^{k+1}\left(\Li_{1}\left(z\right)-H_{n-1}+H_{k-1}\right)\frac{z\Li_{0}^{k}\left(z\right)}{k\left(k+1\right)}\right\} +\left(-1\right)^{n}n\left(H_{n-1}-1\right)\Li_{1}\left(z\right)+\left(-1\right)^{n}nz\Li_{1}\left(z\right)-\left(-1\right)^{n}\frac{1}{2}n\Li_{1}^{2}\left(z\right)+C \end{align*}

(7)

\[ \begin{cases} \int\Li_{0}^{n}\left(z\right)dz=\left(-1\right)^{n}n\sum_{k=1}^{n-1}\left\{ \left(-1\right)^{k+1}\frac{z\Li_{0}^{k}\left(z\right)}{k\left(k+1\right)}\right\} -\left(-1\right)^{n}n\left(\Li_{1}\left(z\right)-z\right)+C\\ \int\Li_{0}^{n}\left(z\right)\Li_{1}\left(z\right)dz=\left(-1\right)^{n}n\left(1-H_{n-1}\right)z+\left(-1\right)^{n}n\sum_{k=1}^{n-1}\left\{ \left(-1\right)^{k+1}\left(\Li_{1}\left(z\right)-H_{n-1}+H_{k-1}\right)\frac{z\Li_{0}^{k}\left(z\right)}{k\left(k+1\right)}\right\} +\left(-1\right)^{n}n\left(H_{n-1}-1\right)\Li_{1}\left(z\right)+\left(-1\right)^{n}nz\Li_{1}\left(z\right)-\left(-1\right)^{n}\frac{1}{2}n\Li_{1}^{2}\left(z\right)+C\\ \int\frac{\Li_{1}\left(z\right)\Li_{1}\left(z\right)}{z}dz=-2\Li_{3}\left(1-z\right)-2\Li_{2}\left(1-z\right)\Li_{1}\left(z\right)-\Li_{1}\left(1-z\right)\Li_{1}^{2}\left(z\right)+C\\ \int\Li_{1}\left(z\right)dz=z-\left(1-z\right)\Li_{1}\left(z\right)+C\\ \sum_{k=1}^{n-1}\frac{H_{k-1}}{k+1}=\frac{H_{n}^{2}}{2}-\frac{H_{n,2}}{2}+\frac{1}{n}-1 \end{cases} \] を使う。
\begin{align*} \int\Li_{0}^{n}\left(z\right)\Li_{2}\left(z\right)dz & =\left\{ \left(-1\right)^{n}n\sum_{k=1}^{n-1}\left\{ \left(-1\right)^{k+1}\frac{z\Li_{0}^{k}\left(z\right)}{k\left(k+1\right)}\right\} -\left(-1\right)^{n}n\left(\Li_{1}\left(z\right)-z\right)\right\} \Li_{2}\left(z\right)-\int\left\{ \left(-1\right)^{n}n\sum_{k=1}^{n-1}\left(\left(-1\right)^{k+1}\frac{z\Li_{0}^{k}\left(z\right)}{k\left(k+1\right)}\right)-\left(-1\right)^{n}n\left(\Li_{1}\left(z\right)-z\right)\right\} \frac{\Li_{1}\left(z\right)}{z}dz\\ & =\left(-1\right)^{n}n\left\{ \left(\sum_{k=1}^{n-1}\left(\left(-1\right)^{k+1}\frac{z\Li_{0}^{k}\left(z\right)}{k\left(k+1\right)}\right)-\Li_{1}\left(z\right)+z\right)\Li_{2}\left(z\right)-\int\sum_{k=1}^{n-1}\left(\left(-1\right)^{k+1}\frac{z\Li_{0}^{k}\left(z\right)}{k\left(k+1\right)}\right)-\left(\Li_{1}\left(z\right)-z\right)\frac{\Li_{1}\left(z\right)}{z}dz\right\} \\ & =\left(-1\right)^{n}n\left\{ \left(\sum_{k=1}^{n-1}\left(\left(-1\right)^{k+1}\frac{z\Li_{0}^{k}\left(z\right)}{k\left(k+1\right)}\right)-\Li_{1}\left(z\right)+z\right)\Li_{2}\left(z\right)-\int\left(\sum_{k=1}^{n-1}\left(\left(-1\right)^{k+1}\frac{\Li_{0}^{k}\left(z\right)\Li_{1}\left(z\right)}{k\left(k+1\right)}\right)-\frac{\Li_{1}^{2}\left(z\right)}{z}+\Li_{1}\left(z\right)\right)dz\right\} \\ & =\left(-1\right)^{n}n\left\{ \sum_{k=1}^{n-1}\left(\left(-1\right)^{k+1}\frac{z\Li_{0}^{k}\left(z\right)\Li_{2}\left(z\right)}{k\left(k+1\right)}\right)-\Li_{1}\left(z\right)\Li_{2}\left(z\right)+z\Li_{2}\left(z\right)-\left(\sum_{k=1}^{n-1}\left(\left(-1\right)^{k+1}\frac{\left(-1\right)^{k}k}{k\left(k+1\right)}\left(\left(1-H_{k-1}\right)z+\sum_{j=1}^{k-1}\left(\left(-1\right)^{j+1}\left(\Li_{1}\left(z\right)-H_{k-1}+H_{j-1}\right)\frac{z\Li_{0}^{j}\left(z\right)}{j\left(j+1\right)}\right)+\left(H_{k-1}-1\right)\Li_{1}\left(z\right)+z\Li_{1}\left(z\right)-\frac{1}{2}\Li_{1}^{2}\left(z\right)\right)\right)-\left(-2\Li_{3}\left(1-z\right)-2\Li_{2}\left(1-z\right)\Li_{1}\left(z\right)-\Li_{1}\left(1-z\right)\Li_{1}^{2}\left(z\right)\right)+z-\left(1-z\right)\Li_{1}\left(z\right)\right)\right\} +C\\ & =\left(-1\right)^{n}n\left\{ \sum_{k=1}^{n-1}\left(\left(-1\right)^{k+1}\frac{z\Li_{0}^{k}\left(z\right)\Li_{2}\left(z\right)}{k\left(k+1\right)}\right)-\Li_{1}\left(z\right)\Li_{2}\left(z\right)+z\Li_{2}\left(z\right)-\sum_{k=1}^{n-1}\left(\frac{-1}{k+1}\left(\left(1-H_{k-1}\right)z+\sum_{j=1}^{k-1}\left(\left(-1\right)^{j+1}\left(\Li_{1}\left(z\right)-H_{k-1}+H_{j-1}\right)\frac{z\Li_{0}^{j}\left(z\right)}{j\left(j+1\right)}\right)+\left(H_{k-1}+z-1\right)\Li_{1}\left(z\right)-\frac{1}{2}\Li_{1}^{2}\left(z\right)\right)\right)-2\Li_{3}\left(1-z\right)-2\Li_{2}\left(1-z\right)\Li_{1}\left(z\right)-\Li_{1}\left(1-z\right)\Li_{1}^{2}\left(z\right)-z+\left(1-z\right)\Li_{1}\left(z\right)\right\} +C\\ & =\left(-1\right)^{n}n\left\{ \sum_{k=1}^{n-1}\left(\left(-1\right)^{k+1}\frac{z\Li_{0}^{k}\left(z\right)\Li_{2}\left(z\right)}{k\left(k+1\right)}\right)-\Li_{1}\left(z\right)\Li_{2}\left(z\right)+z\Li_{2}\left(z\right)-\sum_{k=1}^{n-1}\left(\frac{-1}{k+1}\left(\left(1-H_{k-1}\right)z+\sum_{j=1}^{k}\left(\left(-1\right)^{j+1}\left(\Li_{1}\left(z\right)-H_{k-1}+H_{j-1}\right)\frac{z\Li_{0}^{j}\left(z\right)}{j\left(j+1\right)}\right)-\left(\left(-1\right)^{k+1}\left(\Li_{1}\left(z\right)-H_{k-1}+H_{k-1}\right)\frac{z\Li_{0}^{k}\left(z\right)}{k\left(k+1\right)}\right)+\left(H_{k-1}+z-1\right)\Li_{1}\left(z\right)-\frac{1}{2}\Li_{1}^{2}\left(z\right)\right)\right)-2\Li_{3}\left(1-z\right)-2\Li_{2}\left(1-z\right)\Li_{1}\left(z\right)-\Li_{1}\left(1-z\right)\Li_{1}^{2}\left(z\right)-z+\left(1-z\right)\Li_{1}\left(z\right)\right\} +C\\ & =\left(-1\right)^{n}n\left\{ \sum_{k=1}^{n-1}\left(\left(-1\right)^{k+1}\frac{z\Li_{0}^{k}\left(z\right)\Li_{2}\left(z\right)}{k\left(k+1\right)}\right)-\Li_{1}\left(z\right)\Li_{2}\left(z\right)+z\Li_{2}\left(z\right)+\sum_{k=1}^{n-1}\frac{1}{k+1}\left(1-H_{k-1}\right)z+\sum_{k=1}^{n-1}\frac{1}{k+1}\sum_{j=1}^{k}\left(\left(-1\right)^{j+1}\left(\Li_{1}\left(z\right)-H_{k-1}+H_{j-1}\right)\frac{z\Li_{0}^{j}\left(z\right)}{j\left(j+1\right)}\right)-\sum_{k=1}^{n-1}\frac{1}{k+1}\left(-1\right)^{k+1}\Li_{1}\left(z\right)\frac{z\Li_{0}^{k}\left(z\right)}{k\left(k+1\right)}+\sum_{k=1}^{n-1}\frac{1}{k+1}\left(H_{k-1}+z-1\right)\Li_{1}\left(z\right)-\sum_{k=1}^{n-1}\frac{1}{k+1}\frac{1}{2}\Li_{1}^{2}\left(z\right)-2\Li_{3}\left(1-z\right)-2\Li_{2}\left(1-z\right)\Li_{1}\left(z\right)-\Li_{1}\left(1-z\right)\Li_{1}^{2}\left(z\right)-z+\left(1-z\right)\Li_{1}\left(z\right)\right\} +C\\ & =\left(-1\right)^{n}n\left\{ \sum_{k=1}^{n-1}\left(\left(-1\right)^{k+1}\frac{z\Li_{0}^{k}\left(z\right)\Li_{2}\left(z\right)}{k\left(k+1\right)}\right)-\Li_{1}\left(z\right)\Li_{2}\left(z\right)+z\Li_{2}\left(z\right)+z\sum_{k=1}^{n-1}\frac{1}{k+1}-z\sum_{k=1}^{n-1}\frac{H_{k-1}}{k+1}+\sum_{j=1}^{n-1}\sum_{k=j}^{n-1}\frac{1}{k+1}\left(\left(-1\right)^{j+1}\left(\Li_{1}\left(z\right)+H_{j-1}-H_{k-1}\right)\frac{z\Li_{0}^{j}\left(z\right)}{j\left(j+1\right)}\right)-\sum_{k=1}^{n-1}\frac{1}{k+1}\left(-1\right)^{k+1}\Li_{1}\left(z\right)\frac{z\Li_{0}^{k}\left(z\right)}{k\left(k+1\right)}+\Li_{1}\left(z\right)\sum_{k=1}^{n-1}\frac{H_{k-1}}{k+1}+\left(z-1\right)\Li_{1}\left(z\right)\sum_{k=1}^{n-1}\frac{1}{k+1}-\frac{1}{2}\Li_{1}^{2}\left(z\right)\sum_{k=1}^{n-1}\frac{1}{k+1}-2\Li_{3}\left(1-z\right)-2\Li_{2}\left(1-z\right)\Li_{1}\left(z\right)-\Li_{1}\left(1-z\right)\Li_{1}^{2}\left(z\right)-z+\left(1-z\right)\Li_{1}\left(z\right)\right\} +C\\ & =\left(-1\right)^{n}n\left\{ \sum_{k=1}^{n-1}\left(\left(-1\right)^{k+1}\frac{z\Li_{0}^{k}\left(z\right)\Li_{2}\left(z\right)}{k\left(k+1\right)}\right)-\Li_{1}\left(z\right)\Li_{2}\left(z\right)+z\Li_{2}\left(z\right)+z\left(H_{n}-1\right)-z\left(\frac{H_{n}^{2}}{2}-\frac{H_{n,2}}{2}+\frac{1}{n}-1\right)+\sum_{j=1}^{n-1}\left(\left(-1\right)^{j+1}\left(\left(H_{n}-H_{j}\right)\left(\Li_{1}\left(z\right)+H_{j-1}\right)-\left(\frac{H_{n}^{2}}{2}-\frac{H_{n,2}}{2}+\frac{1}{n}-1\right)+\left(\frac{H_{j}^{2}}{2}-\frac{H_{j,2}}{2}+\frac{1}{j}-1\right)\right)\frac{z\Li_{0}^{j}\left(z\right)}{j\left(j+1\right)}\right)-\sum_{k=1}^{n-1}\frac{1}{k+1}\left(-1\right)^{k+1}\Li_{1}\left(z\right)\frac{z\Li_{0}^{k}\left(z\right)}{k\left(k+1\right)}+\Li_{1}\left(z\right)\left(\frac{H_{n}^{2}}{2}-\frac{H_{n,2}}{2}+\frac{1}{n}-1\right)+\left(z-1\right)\Li_{1}\left(z\right)\left(H_{n}-1\right)-\frac{1}{2}\Li_{1}^{2}\left(z\right)\left(H_{n}-1\right)-2\Li_{3}\left(1-z\right)-2\Li_{2}\left(1-z\right)\Li_{1}\left(z\right)-\Li_{1}\left(1-z\right)\Li_{1}^{2}\left(z\right)-z+\left(1-z\right)\Li_{1}\left(z\right)\right\} +C\\ & =\left(-1\right)^{n}n\left\{ z\left(H_{n}-1\right)-z\left(\frac{H_{n}^{2}}{2}-\frac{H_{n,2}}{2}+\frac{1}{n}-1\right)-z+\Li_{1}\left(z\right)\left(\frac{H_{n}^{2}}{2}-\frac{H_{n,2}}{2}+\frac{1}{n}-1\right)+\left(z-1\right)\Li_{1}\left(z\right)\left(H_{n}-1\right)+\left(1-z\right)\Li_{1}\left(z\right)-\frac{1}{2}\Li_{1}^{2}\left(z\right)\left(H_{n}-1\right)-\Li_{1}\left(z\right)\Li_{2}\left(z\right)+z\Li_{2}\left(z\right)-2\Li_{3}\left(1-z\right)-2\Li_{2}\left(1-z\right)\Li_{1}\left(z\right)-\Li_{1}\left(1-z\right)\Li_{1}^{2}\left(z\right)+\sum_{k=1}^{n-1}\left(\left(-1\right)^{k+1}\frac{z\Li_{0}^{k}\left(z\right)\Li_{2}\left(z\right)}{k\left(k+1\right)}\right)+\sum_{j=1}^{n-1}\left(-1\right)^{j+1}\left(\left(H_{n}-H_{j}\right)\left(\Li_{1}\left(z\right)+H_{j-1}\right)-\frac{H_{n}^{2}}{2}+\frac{H_{j}^{2}}{2}+\frac{H_{n,2}}{2}-\frac{H_{j,2}}{2}-\frac{1}{n}+\frac{1}{j}\right)\frac{z\Li_{0}^{j}\left(z\right)}{j\left(j+1\right)}-\sum_{k=1}^{n-1}\frac{1}{k+1}\left(-1\right)^{k+1}\frac{z\Li_{0}^{k}\left(z\right)\Li_{1}\left(z\right)}{k\left(k+1\right)}\right\} +C\\ & =\left(-1\right)^{n}n\left\{ z\left(-\frac{H_{n}^{2}}{2}+\frac{H_{n,2}}{2}+H_{n}-\frac{1}{n}-1\right)+\Li_{1}\left(z\right)\left(\frac{H_{n}^{2}}{2}-\frac{H_{n,2}}{2}+\frac{1}{n}-1\right)-\left(1-z\right)\Li_{1}\left(z\right)\left(H_{n}-2\right)-\frac{1}{2}\Li_{1}^{2}\left(z\right)\left(H_{n}-1\right)-\Li_{1}\left(z\right)\Li_{2}\left(z\right)+z\Li_{2}\left(z\right)-2\Li_{3}\left(1-z\right)-2\Li_{2}\left(1-z\right)\Li_{1}\left(z\right)-\Li_{1}\left(1-z\right)\Li_{1}^{2}\left(z\right)+\sum_{k=1}^{n-1}\left(\left(-1\right)^{k+1}\frac{z\Li_{0}^{k}\left(z\right)\Li_{2}\left(z\right)}{k\left(k+1\right)}\right)+\sum_{k=1}^{n-1}\left(-1\right)^{k+1}\left(\left(H_{n}-H_{k}\right)\left(\Li_{1}\left(z\right)+H_{k-1}\right)-\frac{H_{n}^{2}}{2}+\frac{H_{k}^{2}}{2}+\frac{H_{n,2}}{2}-\frac{H_{k,2}}{2}-\frac{1}{n}+\frac{1}{k}\right)\frac{z\Li_{0}^{k}\left(z\right)}{k\left(k+1\right)}-\sum_{k=1}^{n-1}\frac{1}{k+1}\left(-1\right)^{k+1}\frac{z\Li_{0}^{k}\left(z\right)\Li_{1}\left(z\right)}{k\left(k+1\right)}\right\} +C \end{align*}

(8)

\begin{align*} \int z^{\alpha}\Li_{n}\left(z\right)dz & =\frac{z^{\alpha+1}}{\alpha+1}\Li_{n}\left(z\right)-\frac{1}{\alpha+1}\int z^{\alpha+1}\frac{\Li_{n-1}\left(z\right)}{z}dz\\ & =\frac{z^{\alpha+1}}{\alpha+1}\Li_{n}\left(z\right)-\frac{1}{\alpha+1}\int z^{\alpha}\Li_{n-1}\left(z\right)dz\\ & =\int\sum_{k=0}^{n-1}\left\{ \frac{\left(-1\right)^{k}}{\left(\alpha+1\right)^{k}}z^{\alpha}\Li_{n-k}\left(z\right)-\frac{\left(-1\right)^{k+1}}{\left(\alpha+1\right)^{k+1}}z^{\alpha}\Li_{n-1-k}\left(z\right)\right\} dz+\frac{\left(-1\right)^{n}}{\left(\alpha+1\right)^{n}}\int z^{\alpha}\Li_{0}\left(z\right)dz\\ & =\sum_{k=0}^{n-1}\frac{\left(-1\right)^{k}}{\left(\alpha+1\right)^{k}}\frac{z^{\alpha+1}}{\alpha+1}\Li_{n-k}\left(z\right)+\frac{\left(-1\right)^{n}}{\left(\alpha+1\right)^{n}}\int\frac{z^{\alpha+1}}{1-z}dz\\ & =\sum_{k=0}^{n-1}\left(-1\right)^{k}\frac{z^{\alpha+1}}{\left(\alpha+1\right)^{k+1}}\Li_{n-k}\left(z\right)+\frac{\left(-1\right)^{n}}{\left(\alpha+1\right)^{n}}B\left(z;\alpha+2,0\right)+C\\ & =\sum_{k=0}^{n-1}\left(-1\right)^{n-1-k}\frac{z^{\alpha+1}}{\left(\alpha+1\right)^{\left(n-1-k\right)+1}}\Li_{n-\left(n-1-k\right)}\left(z\right)+\frac{\left(-1\right)^{n}}{\left(\alpha+1\right)^{n}}B\left(z;\alpha+2,0\right)+C\\ & =\sum_{k=0}^{n-1}\left(-1\right)^{n-1-k}\frac{z^{\alpha+1}}{\left(\alpha+1\right)^{n-k}}\Li_{k+1}\left(z\right)+\frac{\left(-1\right)^{n}}{\left(\alpha+1\right)^{n}}B\left(z;\alpha+2,0\right)+C \end{align*} また、
\begin{align*} B\left(z;\alpha+2,0\right) & =\int\frac{z^{\alpha+1}}{1-z}dz\\ & =\int z^{\alpha+1}F\left(1;;z\right)dz\\ & =\frac{z^{\alpha+2}}{\alpha+2}F\left(1,\alpha+2;\alpha+3;z\right) \end{align*} となるので、
\[ \int z^{\alpha}\Li_{n}\left(z\right)dz=\sum_{k=0}^{n-1}\left(-1\right)^{n-1-k}\frac{z^{\alpha+1}}{\left(\alpha+1\right)^{n-k}}\Li_{k+1}\left(z\right)+\frac{\left(-1\right)^{n}z^{\alpha+2}}{\left(\alpha+1\right)^{n}\left(\alpha+2\right)}F\left(1,\alpha+2;\alpha+3;z\right)+C \] となる。

(9)

\begin{align*} \int z^{m}\Li_{n}\left(z\right)dz & =\sum_{k=0}^{n-1}\left(-1\right)^{k}\frac{z^{m+1}}{\left(m+1\right)^{k+1}}\Li_{n-k}\left(z\right)+\frac{\left(-1\right)^{n}}{\left(m+1\right)^{n}}B_{z}\left(m+2,0\right)+C\\ & =z^{m+1}\sum_{k=0}^{n-1}\left(-1\right)^{k}\frac{\Li_{n-k}\left(z\right)}{\left(m+1\right)^{k+1}}+\frac{\left(-1\right)^{n}}{\left(m+1\right)^{n}}\left(\Li_{1}\left(z\right)-\sum_{j=1}^{m+1}\frac{z^{j}}{j}\right)+C\\ & =z^{m+1}\sum_{k=0}^{n-1}\left(-1\right)^{n-1-k}\frac{\Li_{k+1}\left(z\right)}{\left(m+1\right)^{n-k}}+\frac{\left(-1\right)^{n}}{\left(m+1\right)^{n}}\left(\Li_{1}\left(z\right)-\sum_{j=1}^{m+1}\frac{z^{j}}{j}\right)+C\cmt{n-1-k\rightarrow k} \end{align*}

(10)

\begin{align*} \int\frac{\Li_{n}\left(z\right)\Li_{n+1}\left(z\right)}{z} & =\Li_{n+1}\left(z\right)\Li_{n+1}\left(z\right)-\int\frac{\Li_{n+1}\left(z\right)\Li_{n}\left(z\right)}{z}dz+C\\ & =\Li_{n+1}^{2}\left(z\right)-\LHS+C\\ & =\frac{1}{2}\Li_{n+1}^{2}\left(z\right)+C\cmt{\frac{C}{2}\rightarrow C} \end{align*}

(11)

\begin{align*} \int\Log^{m}\left(z\right)\frac{\Li_{n}\left(z\right)}{z}dz & =\Log^{m}\left(z\right)\Li_{n+1}\left(z\right)-m\int\Log^{m-1}\left(z\right)\frac{\Li_{n+1}\left(z\right)}{z}dz\\ & =m!\int\sum_{k=0}^{m-1}\left\{ \frac{\left(-1\right)^{k}}{\left(m-k\right)!}\Log^{m-k}\left(z\right)\frac{\Li_{n+k}\left(z\right)}{z}-\frac{\left(-1\right)^{k+1}}{\left(m-1-k\right)!}\Log^{m-1-k}\left(z\right)\frac{\Li_{n+1+k}\left(z\right)}{z}\right\} dz+\left(-1\right)^{m}\left(m!\right)\int\frac{\Li_{n+m}\left(z\right)}{z}dz\\ & =m!\sum_{k=0}^{m-1}\left\{ \frac{\left(-1\right)^{k}}{\left(m-k\right)!}\Log^{m-k}\left(z\right)\Li_{n+1+k}\left(z\right)\right\} +\left(-1\right)^{m}\left(m!\right)\Li_{n+m+1}\left(z\right)+C\\ & =\sum_{k=0}^{m-1}\left\{ \left(-1\right)^{k}P\left(m,k\right)\Log^{m-k}\left(z\right)\Li_{n+1+k}\left(z\right)\right\} +\left(-1\right)^{m}\left(m!\right)\Li_{n+m+1}\left(z\right)+C \end{align*}

(12)

\begin{align*} \int\Log^{m}\left(z\right)\frac{\Li_{n}\left(\frac{1}{z}\right)}{z}dz & =-\int\Log^{m}\left(z\right)\frac{\Li_{n}\left(\frac{1}{z}\right)}{\frac{1}{z}}d\left(\frac{1}{z}\right)\\ & =-\Log^{m}\left(z\right)\Li_{n+1}\left(\frac{1}{z}\right)+m\int\Log^{m-1}\left(z\right)\frac{\Li_{n+1}\left(\frac{1}{z}\right)}{z}dz\\ & =m!\int\sum_{k=0}^{m-1}\left\{ \frac{1}{\left(m-k\right)!}\Log^{m-k}\left(z\right)\frac{\Li_{n+k}\left(\frac{1}{z}\right)}{z}-\frac{1}{\left(m-k-1\right)!}\Log^{m-k-1}\left(z\right)\frac{\Li_{n+k+1}\left(\frac{1}{z}\right)}{z}\right\} dz+m!\int\frac{\Li_{n+m}\left(\frac{1}{z}\right)}{z}dz\\ & =m!\sum_{k=0}^{m-1}\frac{-1}{\left(m-k\right)!}\Log^{m-k}\left(z\right)\Li_{n+k+1}\left(\frac{1}{z}\right)-m!\int\frac{\Li_{n+m}\left(\frac{1}{z}\right)}{\frac{1}{z}}d\left(\frac{1}{z}\right)\\ & =-\sum_{k=0}^{m-1}P\left(m,k\right)\Log^{m-k}\left(z\right)\Li_{n+k+1}\left(\frac{1}{z}\right)-m!\Li_{n+m+1}\left(\frac{1}{z}\right)+C \end{align*}