Calculus Level 5

$\large \int _{ 0 }^{ 1 }{ \dfrac {(\text{Li}_2(x) )^2 }{ { x }^{ 2 } } \, dx }=h\zeta (u)-n\zeta (d)-\dfrac { r }{ e } \zeta (d_1)$

If the equation above holds true for positive integers $$h,u,n,d,r,e$$ and $$d_1$$, where $$r$$ and $$e$$ are coprime, find $$h+u+n+d+r+e+d_1$$.



Notations:

• $$\zeta(\cdot)$$ denotes the Riemann zeta function.

• $${ \text{Li} }_{ n }(a)$$ denotes the polylogarithm function, $${ \text{Li} }_{ n }(a)=\displaystyle\sum _{ k=1 }^{ \infty }{ \dfrac { { a }^{ k } }{ { k }^{ n } } }$$.

×