Answer
$$ \displaystyle\int \dfrac{-x+2}{{x}^{2}+x+2}\, \mathrm d x = -{{\sqrt{7}\,\ln \left(x^2+x+2\right)-10\,\arctan \left({{2\,x+1 }\over{\sqrt{7}}}\right)}\over{2\,\sqrt{7}}} $$
Explanation
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