Optimal. Leaf size=46 \[ -\frac {x^2}{2}-\frac {1}{2} \log \left (x^2+x+1\right )-\log (1-x)-\frac {\tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.08, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {1593, 1887, 1875, 31, 634, 618, 204, 628} \[ -\frac {x^2}{2}-\frac {1}{2} \log \left (x^2+x+1\right )-\log (1-x)-\frac {\tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 618
Rule 628
Rule 634
Rule 1593
Rule 1875
Rule 1887
Rubi steps
\begin {align*} \int \frac {2 x^2+x^4}{1-x^3} \, dx &=\int \frac {x^2 \left (2+x^2\right )}{1-x^3} \, dx\\ &=\int \left (-x+\frac {x (1+2 x)}{1-x^3}\right ) \, dx\\ &=-\frac {x^2}{2}+\int \frac {x (1+2 x)}{1-x^3} \, dx\\ &=-\frac {x^2}{2}+\frac {1}{3} \int \frac {-3-3 x}{1+x+x^2} \, dx+\int \frac {1}{1-x} \, dx\\ &=-\frac {x^2}{2}-\log (1-x)-\frac {1}{2} \int \frac {1}{1+x+x^2} \, dx-\frac {1}{2} \int \frac {1+2 x}{1+x+x^2} \, dx\\ &=-\frac {x^2}{2}-\log (1-x)-\frac {1}{2} \log \left (1+x+x^2\right )+\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right )\\ &=-\frac {x^2}{2}-\frac {\tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{\sqrt {3}}-\log (1-x)-\frac {1}{2} \log \left (1+x+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 54, normalized size = 1.17 \[ \frac {1}{6} \left (-4 \log \left (1-x^3\right )-3 x^2+\log \left (x^2+x+1\right )-2 \log (1-x)-2 \sqrt {3} \tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 37, normalized size = 0.80 \[ -\frac {1}{2} \, x^{2} - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - \frac {1}{2} \, \log \left (x^{2} + x + 1\right ) - \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 38, normalized size = 0.83 \[ -\frac {1}{2} \, x^{2} - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - \frac {1}{2} \, \log \left (x^{2} + x + 1\right ) - \log \left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 38, normalized size = 0.83 \[ -\frac {x^{2}}{2}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{3}-\ln \left (x -1\right )-\frac {\ln \left (x^{2}+x +1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.90, size = 37, normalized size = 0.80 \[ -\frac {1}{2} \, x^{2} - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - \frac {1}{2} \, \log \left (x^{2} + x + 1\right ) - \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 51, normalized size = 1.11 \[ -\ln \left (x-1\right )+\ln \left (x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )-\ln \left (x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )-\frac {x^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.31, size = 46, normalized size = 1.00 \[ - \frac {x^{2}}{2} - \log {\left (x - 1 \right )} - \frac {\log {\left (x^{2} + x + 1 \right )}}{2} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3} + \frac {\sqrt {3}}{3} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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