The books answer is correct :(
One way to check is to graph both solutions, for example with a graphing calculator, or PC program. If they look nothing like each other then obviously one is wrong!
You had, 1/sqrt(x^2 + 4x + 8)
Or, 1/sqrt{(x+2)^2 + 4}
Or, 1/sqrt(u^2 + a^2), where u = x + 2 and a = 2.
That last expression, 1/sqrt(u^2 + a^2), is a standard integral
and evaluates as sinh^-1(u/a) + c.