Optimal. Leaf size=57 \[ \frac {x}{3 \left (1-x^3\right )}+\frac {1}{9} \log \left (x^2+x+1\right )-\frac {2}{9} \log (1-x)+\frac {2 \tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
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Rubi [A] time = 0.02, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {199, 200, 31, 634, 618, 204, 628} \[ \frac {x}{3 \left (1-x^3\right )}+\frac {1}{9} \log \left (x^2+x+1\right )-\frac {2}{9} \log (1-x)+\frac {2 \tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 199
Rule 200
Rule 204
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {1}{\left (-1+x^3\right )^2} \, dx &=\frac {x}{3 \left (1-x^3\right )}-\frac {2}{3} \int \frac {1}{-1+x^3} \, dx\\ &=\frac {x}{3 \left (1-x^3\right )}-\frac {2}{9} \int \frac {1}{-1+x} \, dx-\frac {2}{9} \int \frac {-2-x}{1+x+x^2} \, dx\\ &=\frac {x}{3 \left (1-x^3\right )}-\frac {2}{9} \log (1-x)+\frac {1}{9} \int \frac {1+2 x}{1+x+x^2} \, dx+\frac {1}{3} \int \frac {1}{1+x+x^2} \, dx\\ &=\frac {x}{3 \left (1-x^3\right )}-\frac {2}{9} \log (1-x)+\frac {1}{9} \log \left (1+x+x^2\right )-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right )\\ &=\frac {x}{3 \left (1-x^3\right )}+\frac {2 \tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {2}{9} \log (1-x)+\frac {1}{9} \log \left (1+x+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 49, normalized size = 0.86 \[ \frac {1}{9} \left (-\frac {3 x}{x^3-1}+\log \left (x^2+x+1\right )-2 \log (1-x)+2 \sqrt {3} \tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 56, normalized size = 0.98 \[ -\frac {x}{3 \left (x^3-1\right )}+\frac {1}{9} \log \left (x^2+x+1\right )-\frac {2}{9} \log (x-1)+\frac {2 \tan ^{-1}\left (\frac {2 x}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 58, normalized size = 1.02 \[ \frac {2 \, \sqrt {3} {\left (x^{3} - 1\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + {\left (x^{3} - 1\right )} \log \left (x^{2} + x + 1\right ) - 2 \, {\left (x^{3} - 1\right )} \log \left (x - 1\right ) - 3 \, x}{9 \, {\left (x^{3} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.95, size = 43, normalized size = 0.75 \[ \frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - \frac {x}{3 \, {\left (x^{3} - 1\right )}} + \frac {1}{9} \, \log \left (x^{2} + x + 1\right ) - \frac {2}{9} \, \log \left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 41, normalized size = 0.72
method | result | size |
risch | \(-\frac {x}{3 \left (x^{3}-1\right )}-\frac {2 \ln \left (-1+x \right )}{9}+\frac {\ln \left (x^{2}+x +1\right )}{9}+\frac {2 \sqrt {3}\, \arctan \left (\frac {2 \left (\frac {1}{2}+x \right ) \sqrt {3}}{3}\right )}{9}\) | \(41\) |
default | \(\frac {-1+x}{9 x^{2}+9 x +9}+\frac {\ln \left (x^{2}+x +1\right )}{9}+\frac {2 \arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3}\right ) \sqrt {3}}{9}-\frac {1}{9 \left (-1+x \right )}-\frac {2 \ln \left (-1+x \right )}{9}\) | \(53\) |
meijerg | \(-\frac {\left (-1\right )^{\frac {2}{3}} \left (\frac {3 x \left (-1\right )^{\frac {1}{3}}}{-3 x^{3}+3}-\frac {2 x \left (-1\right )^{\frac {1}{3}} \left (\ln \left (1-\left (x^{3}\right )^{\frac {1}{3}}\right )-\frac {\ln \left (1+\left (x^{3}\right )^{\frac {1}{3}}+\left (x^{3}\right )^{\frac {2}{3}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{3}\right )^{\frac {1}{3}}}{2+\left (x^{3}\right )^{\frac {1}{3}}}\right )\right )}{3 \left (x^{3}\right )^{\frac {1}{3}}}\right )}{3}\) | \(86\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 42, normalized size = 0.74 \[ \frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - \frac {x}{3 \, {\left (x^{3} - 1\right )}} + \frac {1}{9} \, \log \left (x^{2} + x + 1\right ) - \frac {2}{9} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 60, normalized size = 1.05 \[ -\frac {2\,\ln \left (x-1\right )}{9}-\frac {x}{3\,\left (x^3-1\right )}-\ln \left (x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )+\ln \left (2\,x+1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 53, normalized size = 0.93 \[ - \frac {x}{3 x^{3} - 3} - \frac {2 \log {\left (x - 1 \right )}}{9} + \frac {\log {\left (x^{2} + x + 1 \right )}}{9} + \frac {2 \sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3} + \frac {\sqrt {3}}{3} \right )}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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