$\begin{align}  & \int{\frac{dx}{{{x}^{2}}-2x+5}}=\int{\frac{dx}{{{x}^{2}}-2x+1+5-1}} \\ & \int{\frac{dx}{{{x}^{2}}-2x+5}}=\int{\frac{dx}{{{(x-1)}^{2}}+4}}=\int{\frac{dx}{{{(x-1)}^{2}}+{{2}^{2}}}} \\ & \int{\frac{dx}{{{x}^{2}}-2x+5}}=\frac{1}{2}\arctan \frac{x-1}{2}+C \\\end{align}$