TITLE: How to calculate $\int\frac{1}{x + 1 + \sqrt{x^2 + 4x + 5}}\ dx$?
QUESTION [8 upvotes]: How to calculate $$\int\frac{1}{x + 1 + \sqrt{x^2 + 4x + 5}}dx?$$ I really don't know how to attack this integral. I tried $u=x^2 + 4x + 5$ but failed miserably. Help please.

REPLY [2 votes]: Another approach : (The shortest one)
Using Euler substitution by setting $t-x=\sqrt{x^2+4x+5}$, we will obtain 
$x=\dfrac{t^2-5}{2t+4}$ and $dx=\dfrac{t^2+4t+5}{2(t+2)^2}\ dt$, then the integral turns out to be
$$
-\int\dfrac{t^2+4t+5}{2(t+2)(t+3)}\ dt=\int\left[\frac1{t+3}-\frac1{2(t+2)}-\frac12\right]\ dt.
$$
The last part uses partial fraction decomposition and the rest should be easy to be solved.