Let J=-3/2∫dx/(x2+x+1)=
-3/2∫dx/(x2+x+¼+¾)=
-3/2∫dx/((x+½)2+¾).
Let x+½=(√3/2)tanθ, (x+½)2=¾tan2θ, (x+½)2+¾=¾tan2θ+¾=¾sec2θ;
dx=(√3/2)sec2θdθ; θ=tan-1{(2x+1)/√3}
J=-¾√3∫sec2θdθ/[¾sec2θ]=-√3∫dθ=-√3θ+C, where C is integration constant.
J=-√3tan-1{(2x+1)/√3}+C.