This post evaluates the integral,

$$ I = int_0 ^ 1 frac { ln ^ 2 (x) , ln ^ 3 (1 + x)} xdx $$

just like

$$ I

= – frac { pi ^ 6} {252} -18 zeta ( bar {5}, 1) +3 zeta ^ 2 (3) tag1 $$

where,

$$ zeta ( bar {5}, 1) = frac {1} {24} int ^ 1_0 frac { ln ^ 4 {x} ln (1 + x)} {1 + x} { rm d} x $$

More succinctly,

$$ I = -12 , S_ {3,3} (- 1) tag2 $$

with *Nielsen polilogaritmo widespread.* $ S_ {n, p} (z) $.

**Question:** How do we show that? $ zeta ( bar {5}, 1) $ is *Only a generalized polilogaritmo of Nielsen disguised?* Or, in general,

$$ begin {aligned} S_ {n, p} (- 1)

& = C_1 int_0 ^ 1 frac {( ln x) ^ {n-1} big ( ln (1 + x) big) ^ p} {x} dx \

& = C_2 int_0 ^ 1 frac {( ln x) ^ {n} big ( ln (1 + x) big) ^ {p-1}} {1 + x} dx end {aligned} $$

where,

$$ C_1 = frac {(- 1) ^ {n + p-1}} {(n-1)! , P!}, Qquad C_2 = frac {(- 1) ^ {n + p}} {n! , (p-1)!} $$

Thus,

$$ zeta ( bar {5}, 1) = S_ {4,2} (- 1) $$