<s>[INST] Integration of this function I have done the integration of functions by completing the squares. this question is also done in the similar fashion, but I am wondering. I cannot solve the following integral
$$\int \dfrac{4x + 7}{(x^2 - 2x + 3)^2}\,dx$$
 [/INST] In the second integral substitute $x-1=t$ to get $\int\frac{11}{(t^2+2)^2}dt$
$=\frac{11}{2}\int\frac{2+t^2-t^2}{(t^2+2)^2}dt=\frac{11}{2}\int\frac{dt}{t^2+2}-\frac{11}{4}\int\frac{t.2t}{(t^2+2)^2}dt$
$=\frac{11}{4}\int\frac{dt}{(t/2)^2+1}-\frac{11}{4}\int\frac{t}{(t^2+2)^2}d(t^2+2)$
$=\frac{11}{2}\int\frac{d(t/2)}{(t/2)^2+1}-\frac{11}{4}\int t d\left(-\frac{1}{t^2+2}\right)$
The fist integral is $tan^{-1}$ in the second apply integrating by parts.
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