Optimal. Leaf size=68 \[ \frac{2 (2-x)}{3 \left (x^2-x+1\right )}-\frac{1}{2 x^2}-2 \log \left (x^2-x+1\right )-\frac{3}{x}+4 \log (x)+\frac{10 \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
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Rubi [A] time = 0.109582, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1646, 1628, 634, 618, 204, 628} \[ \frac{2 (2-x)}{3 \left (x^2-x+1\right )}-\frac{1}{2 x^2}-2 \log \left (x^2-x+1\right )-\frac{3}{x}+4 \log (x)+\frac{10 \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1646
Rule 1628
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1+x+x^2}{x^3 \left (1-x+x^2\right )^2} \, dx &=\frac{2 (2-x)}{3 \left (1-x+x^2\right )}+\frac{1}{3} \int \frac{3+6 x+6 x^2-2 x^3}{x^3 \left (1-x+x^2\right )} \, dx\\ &=\frac{2 (2-x)}{3 \left (1-x+x^2\right )}+\frac{1}{3} \int \left (\frac{3}{x^3}+\frac{9}{x^2}+\frac{12}{x}+\frac{1-12 x}{1-x+x^2}\right ) \, dx\\ &=-\frac{1}{2 x^2}-\frac{3}{x}+\frac{2 (2-x)}{3 \left (1-x+x^2\right )}+4 \log (x)+\frac{1}{3} \int \frac{1-12 x}{1-x+x^2} \, dx\\ &=-\frac{1}{2 x^2}-\frac{3}{x}+\frac{2 (2-x)}{3 \left (1-x+x^2\right )}+4 \log (x)-\frac{5}{3} \int \frac{1}{1-x+x^2} \, dx-2 \int \frac{-1+2 x}{1-x+x^2} \, dx\\ &=-\frac{1}{2 x^2}-\frac{3}{x}+\frac{2 (2-x)}{3 \left (1-x+x^2\right )}+4 \log (x)-2 \log \left (1-x+x^2\right )+\frac{10}{3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 x\right )\\ &=-\frac{1}{2 x^2}-\frac{3}{x}+\frac{2 (2-x)}{3 \left (1-x+x^2\right )}+\frac{10 \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{3 \sqrt{3}}+4 \log (x)-2 \log \left (1-x+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.029881, size = 66, normalized size = 0.97 \[ -\frac{2 (x-2)}{3 \left (x^2-x+1\right )}-\frac{1}{2 x^2}-2 \log \left (x^2-x+1\right )-\frac{3}{x}+4 \log (x)-\frac{10 \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 60, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,{x}^{2}}}-3\,{x}^{-1}+4\,\ln \left ( x \right ) -{\frac{1}{{x}^{2}-x+1} \left ({\frac{2\,x}{3}}-{\frac{4}{3}} \right ) }-2\,\ln \left ({x}^{2}-x+1 \right ) -{\frac{10\,\sqrt{3}}{9}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50708, size = 85, normalized size = 1.25 \begin{align*} -\frac{10}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{22 \, x^{3} - 23 \, x^{2} + 15 \, x + 3}{6 \,{\left (x^{4} - x^{3} + x^{2}\right )}} - 2 \, \log \left (x^{2} - x + 1\right ) + 4 \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74637, size = 250, normalized size = 3.68 \begin{align*} -\frac{66 \, x^{3} + 20 \, \sqrt{3}{\left (x^{4} - x^{3} + x^{2}\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - 69 \, x^{2} + 36 \,{\left (x^{4} - x^{3} + x^{2}\right )} \log \left (x^{2} - x + 1\right ) - 72 \,{\left (x^{4} - x^{3} + x^{2}\right )} \log \left (x\right ) + 45 \, x + 9}{18 \,{\left (x^{4} - x^{3} + x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.187243, size = 71, normalized size = 1.04 \begin{align*} 4 \log{\left (x \right )} - 2 \log{\left (x^{2} - x + 1 \right )} - \frac{10 \sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right )}}{9} - \frac{22 x^{3} - 23 x^{2} + 15 x + 3}{6 x^{4} - 6 x^{3} + 6 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26833, size = 85, normalized size = 1.25 \begin{align*} -\frac{10}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{22 \, x^{3} - 23 \, x^{2} + 15 \, x + 3}{6 \,{\left (x^{2} - x + 1\right )} x^{2}} - 2 \, \log \left (x^{2} - x + 1\right ) + 4 \, \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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