Chapter 8: Techniques of Integration - Section 8.6 - Integral Tables and Computer Algebra Systems - Exercises 8.6 - Page 481: 37

$\sinh^{-1}(\dfrac{x+1}{2})+C$

Use the formula: $\int \frac{dx}{\sqrt{a^2+x^2}}=\sinh^{-1}\frac{x}{a}+C$ Let $I=\int\dfrac{1}{\sqrt{x^2+2x+5}}dx \\ =\int\dfrac{1}{\sqrt{(x+1)^2+4}} \space dx $ Suppose $u=x+1 \implies u=dx$ Therefore, $I=\int\dfrac{1}{\sqrt{u^2+4}}du=\sinh^{-1}\dfrac{u}{2}+C\\=\sinh^{-1}(\dfrac{x+1}{2})+C$