Indefinite integral

$\int \frac{x^7+2}{\left(x^2+x+1\right)^2} \, dx \Longrightarrow -2 \log \left(x^2+x+1\right)+\frac{1}{12} x \left(3 x^3-8 x^2+\frac{12}{x^2+x+1}+6 x+24\right)+\frac{2 \tan ^{-1}\left(\frac{2 x+1}{\sqrt{3}}\right)}{\sqrt{3}}$

As that expression at $x=0 \Longrightarrow \frac{\pi }{3 \sqrt{3}}$ , no further correction is needed as an integration constant.

$-\frac{19}{12}-\frac{\pi }{3 \sqrt{3}} \approx -2.18793312141$ . The absolute value was requested as the answer.

$\frac{x^7+2}{\left(x^2+x+1\right)^2} == x^3-2 x^2+\frac{x+2}{\left(x^2+x+1\right)^2}-\frac{2 (2 x+1)}{x^2+x+1}+x+2$ $\int \frac{2 x+1}{x^2+x+1} \, dx, u=x^2+x+1 \Longrightarrow \int \frac{1}{u} \, du \longrightarrow \Longrightarrow \ln{u}$ $\frac{x+2}{\left(x^2+x+1\right)^2} \Longrightarrow \frac{2x+1}{2\left(x^2+x+1\right)^2}+\frac{3}{2\left(x^2+x+1\right)^2}$ $\int \frac{2 x+1}{\left(x^2+x+1\right)^2} \, dx, s=x^2+x+1 \Longrightarrow \int \frac{1}{s^2} \, ds$ $\int \frac{1}{\left(\left(x+\frac{1}{2}\right)^2+\frac{3}{4}\right)^2} \, dx, p=x+\frac12, p=\frac{1}{2} \sqrt{3} (\tan w), \cos ^2(w)=\frac{1}{2} (2 w) \cos +\frac{1}{2} \Longrightarrow \int \frac{16}{9} \left(w \cos ^2\right) \, dw$