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`int (3x^2 + x + 4)/(x^4 + 3x^2 + 2) dx` Evaluate the integral

Perform a partial fraction decomposition of `1/(x^4+3x^2+2) = 1/((x^2+1)(x^2+2)),` it is  `1/(x^2+1)-1/(x^2+2).` 



So the integral becomes   `int ((3x^2+x+4)/(x^2+1)-(3x^2+x+4)/(x^2+2)) dx =`


`= int ((3+x/(x^2+1)+1/(x^2+1))-(3+x/(x^2+2)-2/(x^2+2))) dx =`


`= int (x/(x^2+1)-x/(x^2+2)+1/(x^2+1)+2/(x^2+2)) dx.`



For the first two summands perform a substitution `x^2=y,` `xdx=1/2 dy.` The third is from the table and the fourth is almost the same, substitution `y=x/sqrt(2).`


The result is


`1/2 ln(x^2+1) - 1/2 ln(x^2+2) + arctan(x) + sqrt(2)arctan(x/sqrt(2))+C.`

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