<s>[INST] Integrate a division of polynomials Hi I have the following integral:
$$\int \frac{2x}{x^2+6x+3}\, dx$$
I made some changes like:
$$\int \dfrac{2x+6-6}{x^2+6x+3}\, dx$$
then I have: 
$$\int \dfrac{2x+6}{x^2+6x+3}\, dx -\int\dfrac{6}{x^2+6x+3}\, dx$$
and thus: $$\ln(x^2+6x+3)-\int\dfrac{6}{x^2+6x+3}\, dx$$
Ok, I have decomposed $$\frac{2x}{x^2+6x+3} $$  in: $$ \frac{3+\sqrt6}{\sqrt6(x+\sqrt 6+3)} + \frac{3-\sqrt6}{\sqrt6 (-x+\sqrt6-3)}$$
How can I integrate this expressions?
 [/INST] $$\frac{2x}{x^2+6x+3}=\frac{A}{x-(-3-\sqrt{6})}+\frac{B}{x-(\sqrt{6}-3)}$$
Find $A$ and $B$ and then:
$$\int \frac{2x}{x^2+6x+3} dx=\int \frac{A}{x-(-3-\sqrt{6})} dx+\int \frac{B}{x-(\sqrt{6}-3)} dx$$
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