Answer
$$ \displaystyle\int \dfrac{1}{{x}^{3}-1}\, \mathrm d x = -{{\sqrt{3}\,\ln \left(x^2+x+1\right)+6\,\arctan \left({{2\,x+1 }\over{\sqrt{3}}}\right)-2\,\sqrt{3}\,\ln \left(x-1\right)}\over{2 \,3\sqrt{3}}} $$
Explanation
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