∫x/(x³+1) dx
=1/3*∫(x+1)/(x²-x+1)-1/3*∫1/(x+1)
=1/6*∫(2x-1)/(x²-x+1)+1/2*∫1/(x²-x+1)
=1/6*ln(x²-x+1)+1/2*∫1/[(x-1/2)²+3/4]
=1/6*ln(x²-x+1)-1/3*ln(x+1)+1/√3*arctan[(2x-1)/√3]
∫x/(x³+1) dx
=1/3*∫(x+1)/(x²-x+1)-1/3*∫1/(x+1)
=1/6*∫(2x-1)/(x²-x+1)+1/2*∫1/(x²-x+1)
=1/6*ln(x²-x+1)+1/2*∫1/[(x-1/2)²+3/4]
=1/6*ln(x²-x+1)-1/3*ln(x+1)+1/√3*arctan[(2x-1)/√3]