∫(2x+4)/(x^2+2x+6) dx
= ∫(2x+2)/(x^2+2x+6) dx + 2∫dx/(x^2+2x+6)
=ln|x^2+2x+6| + 2∫dx/(x^2+2x+6)
consider
x^2+2x+6 = (x+1)^2 +5
let
x+1 = √5.tany
dx =√5(secy)^2 dy
∫dx/(x^2+2x+6)
=(√5/5)∫ dy
=(√5/5)y
=(√5/5)arctan[(x+1)/√5]
∫(2x+4)/(x^2+2x+6) dx
=ln|x^2+2x+6| + 2∫dx/(x^2+2x+6)
=ln|x^2+2x+6| + (2√5/5)arctan[(x+1)/√5] + C