Optimal. Leaf size=61 \[ \frac {2 (x+1)}{3 \left (x^2-x+1\right )}-\frac {3}{2} \log \left (x^2-x+1\right )-\frac {1}{x}+3 \log (x)-\frac {7 \tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
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Rubi [A] time = 0.13, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1646, 1628, 634, 618, 204, 628} \[ \frac {2 (x+1)}{3 \left (x^2-x+1\right )}-\frac {3}{2} \log \left (x^2-x+1\right )-\frac {1}{x}+3 \log (x)-\frac {7 \tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 1628
Rule 1646
Rubi steps
\begin {align*} \int \frac {1+x+x^2}{x^2 \left (1-x+x^2\right )^2} \, dx &=\frac {2 (1+x)}{3 \left (1-x+x^2\right )}+\frac {1}{3} \int \frac {3+6 x+2 x^2}{x^2 \left (1-x+x^2\right )} \, dx\\ &=\frac {2 (1+x)}{3 \left (1-x+x^2\right )}+\frac {1}{3} \int \left (\frac {3}{x^2}+\frac {9}{x}+\frac {8-9 x}{1-x+x^2}\right ) \, dx\\ &=-\frac {1}{x}+\frac {2 (1+x)}{3 \left (1-x+x^2\right )}+3 \log (x)+\frac {1}{3} \int \frac {8-9 x}{1-x+x^2} \, dx\\ &=-\frac {1}{x}+\frac {2 (1+x)}{3 \left (1-x+x^2\right )}+3 \log (x)+\frac {7}{6} \int \frac {1}{1-x+x^2} \, dx-\frac {3}{2} \int \frac {-1+2 x}{1-x+x^2} \, dx\\ &=-\frac {1}{x}+\frac {2 (1+x)}{3 \left (1-x+x^2\right )}+3 \log (x)-\frac {3}{2} \log \left (1-x+x^2\right )-\frac {7}{3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right )\\ &=-\frac {1}{x}+\frac {2 (1+x)}{3 \left (1-x+x^2\right )}-\frac {7 \tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{3 \sqrt {3}}+3 \log (x)-\frac {3}{2} \log \left (1-x+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 61, normalized size = 1.00 \[ \frac {2 (x+1)}{3 \left (x^2-x+1\right )}-\frac {3}{2} \log \left (x^2-x+1\right )-\frac {1}{x}+3 \log (x)+\frac {7 \tan ^{-1}\left (\frac {2 x-1}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.20, size = 85, normalized size = 1.39 \[ \frac {14 \, \sqrt {3} {\left (x^{3} - x^{2} + x\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - 6 \, x^{2} - 27 \, {\left (x^{3} - x^{2} + x\right )} \log \left (x^{2} - x + 1\right ) + 54 \, {\left (x^{3} - x^{2} + x\right )} \log \relax (x) + 30 \, x - 18}{18 \, {\left (x^{3} - x^{2} + x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 55, normalized size = 0.90 \[ \frac {7}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {x^{2} - 5 \, x + 3}{3 \, {\left (x^{3} - x^{2} + x\right )}} - \frac {3}{2} \, \log \left (x^{2} - x + 1\right ) + 3 \, \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 55, normalized size = 0.90 \[ \frac {7 \sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{9}+3 \ln \relax (x )-\frac {3 \ln \left (x^{2}-x +1\right )}{2}-\frac {1}{x}-\frac {-\frac {2 x}{3}-\frac {2}{3}}{x^{2}-x +1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 54, normalized size = 0.89 \[ \frac {7}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {x^{2} - 5 \, x + 3}{3 \, {\left (x^{3} - x^{2} + x\right )}} - \frac {3}{2} \, \log \left (x^{2} - x + 1\right ) + 3 \, \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.13, size = 68, normalized size = 1.11 \[ 3\,\ln \relax (x)-\frac {\frac {x^2}{3}-\frac {5\,x}{3}+1}{x^3-x^2+x}-\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {3}{2}+\frac {\sqrt {3}\,7{}\mathrm {i}}{18}\right )+\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {3}{2}+\frac {\sqrt {3}\,7{}\mathrm {i}}{18}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.20, size = 65, normalized size = 1.07 \[ \frac {- x^{2} + 5 x - 3}{3 x^{3} - 3 x^{2} + 3 x} + 3 \log {\relax (x )} - \frac {3 \log {\left (x^{2} - x + 1 \right )}}{2} + \frac {7 \sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3} - \frac {\sqrt {3}}{3} \right )}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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