(* This is formula 4.262.1 *)

Integrate[(Log[x])^3/(1+x),{x,0,1}]

(* This is the value in the table *)

N[-7*Pi^4/120]

(* Numerical value of the answer *)

NIntegrate[(Log[x])^3/(1+x),{x,0,1}]

(* Everything matches *)