∫1/(1+x^3)

 1+x^3 =(1+x)(1+x^2-x)

  now expression becomes

   dx/(x+1)(x^2+1-x)

  applying partial fraction

   1/3logx+1 -1/6log(x^2-x+1) +[taninverse{2x-1/(3^1/2)}]/(3^1/2)