Trig Substitution Help

Question

I need to integrate the following using trigonometric substitution. I also know that I need to do the following by completing the square in the denominator, but I can't seem to figure it out.

$$\int \frac {x^2+1}{(x^2-2x+2)^2} dx$$

Answer 1

Complete the square by adding and subtracting $1$ inside the parens to get $$\frac{x^2+1}{(x-1)^2+1}.$$ Then make the substitution $u=x-1$ to get

$$ \int \frac{(u+1)^2+1}{(u^2+1)^2 } \; du.$$ Now trig sub $u=\tan t$.