SimplifyPolyLog[y] performs several simplifications assuming that the variables occuring in the Log and PolyLog functions are between 0 and 1.
The simplifications will in general not be valid if the arguments are complex or outside the range between 0 and 1.
SimplifyPolyLog[PolyLog[2, 1/x]]\zeta (2)+\text{Li}_2(1-x)-\frac{1}{2} \log ^2(x)+\log (1-x) \log (x)+i \pi \log (x)
SimplifyPolyLog[PolyLog[2, x]]\zeta (2)-\text{Li}_2(1-x)-\log (1-x) \log (x)
SimplifyPolyLog[PolyLog[2, 1 - x^2]]-\zeta (2)+2 \;\text{Li}_2(1-x)-2 \;\text{Li}_2(-x)-2 \log (x) \log (x+1)
SimplifyPolyLog[PolyLog[2, x^2]]2 \zeta (2)-2 \;\text{Li}_2(1-x)+2 \;\text{Li}_2(-x)-2 \log (1-x) \log (x)
SimplifyPolyLog[PolyLog[2, -x/(1 - x)]]-\zeta (2)+\text{Li}_2(1-x)-\frac{1}{2} \log ^2(1-x)+\log (x) \log (1-x)
SimplifyPolyLog[PolyLog[2, x/(x - 1)]]-\zeta (2)+\text{Li}_2(1-x)-\frac{1}{2} \log ^2(1-x)+\log (x) \log (1-x)
SimplifyPolyLog[Nielsen[1, 2, -x/(1 - x)]]S_{12}(x)-\frac{1}{6} \log ^3(1-x)
SimplifyPolyLog[PolyLog[3, -1/x]]\text{Li}_3(-x)+\zeta (2) \log (x)+\frac{\log ^3(x)}{6}
SimplifyPolyLog[PolyLog[3, 1 - x]]\text{Li}_3(1-x)
SimplifyPolyLog[PolyLog[3, x^2]]4 \;\text{Li}_3(-x)-4 \;\text{Li}_2(1-x) \log (x)-4 S_{12}(1-x)+4 \zeta (2) \log (x)-2 \log (1-x) \log ^2(x)+4 \zeta (3)
SimplifyPolyLog[PolyLog[3, -x/(1 - x)]]-\text{Li}_3(1-x)+\text{Li}_2(1-x) \log (x)+S_{12}(1-x)+\zeta (2) \log (1-x)-\zeta (2) \log (x)+\frac{1}{6} \log ^3(1-x)-\frac{1}{2} \log (x) \log ^2(1-x)+\frac{1}{2} \log ^2(x) \log (1-x)
SimplifyPolyLog[PolyLog[3, 1 - 1/x]]\text{Li}_2(1-x) \log (x)-\text{Li}_2(1-x) \log (1-x)+S_{12}(1-x)+S_{12}(x)+\frac{\log ^3(x)}{6}-\frac{1}{2} \log ^2(1-x) \log (x)-\zeta (3)
SimplifyPolyLog[PolyLog[4, -x/(1 - x)]]-\text{Li}_4(x)+\frac{1}{2} \;\text{Li}_2(1-x) \log ^2(1-x)-\text{Li}_2(1-x) \log (x) \log (1-x)-S_{13}(x)+S_{22}(x)-S_{12}(1-x) \log (1-x)-S_{12}(x) \log (1-x)-\frac{1}{2} \zeta (2) \log ^2(1-x)+\zeta (2) \log (x) \log (1-x)+\zeta (3) \log (1-x)-\frac{1}{24} \log ^4(1-x)+\frac{1}{2} \log (x) \log ^3(1-x)-\frac{1}{2} \log ^2(x) \log ^2(1-x)
SimplifyPolyLog[Log[a + b/c]]\log \left(\frac{a c+b}{c}\right)
SimplifyPolyLog[Log[1/x]]-\log (x)
SimplifyPolyLog[ArcTanh[x]]\frac{1}{2} \log \left(-\frac{x+1}{1-x}\right)
SimplifyPolyLog[ArcSinh[x]]\log \left(\sqrt{x^2+1}+x\right)
SimplifyPolyLog[ArcCosh[x]]\log \left(\sqrt{x^2-1}+x\right)