Fel med dubbel substitution?

Ska hitta alla primitiva till:

$$\begin{gathered} \int \frac{1}{x^2-2x+5}dx=\int \frac{{\displaystyle 1}}{{\displaystyle (x-1{)}^2+4}}dx=4\int \frac{{\displaystyle 1}}{{\displaystyle \frac{(x-1{)}^2}{4}+1}}dx \\ t=x-1 \\ dt=dx \\ 4\int \frac{{\displaystyle 1}}{{\displaystyle \frac{(t{)}^2}{4}+1}}dt = 4\int \frac{{\displaystyle 1}}{{\displaystyle (\frac{t}{2}{)}^2+1}}dt \\ z=\frac{t}{2} \\ dz=\frac{1}{2}dt \\ 4*2\int \frac{{\displaystyle 1}}{{\displaystyle z^2+1}}dz =\hspace{0.17em}8 arc\tan \left(\frac{x-1}{2}\right) \end{gathered}$$

Verkar inte stämma, ser inte felet dock.