posted by Noel .
I did a long division and got
2x^2/(x^2+4x+8) = 2 - (4x+8)/(x^2 + 4x + 8)
= 2 - 2(2x+4)/(x^2 + 4x + 8)
I noticed that the derivative of x^2 + 4x + 8 is 2x+4 , which I have sitting on top.
=∫ ( 2 - 2(2x+4)/x^2 + 4x + 8) ) dx
= 2x - 4 ln(x^2 + 4x + 8)