Optimal. Leaf size=64 \[ -\frac {5}{18} \log \left (x^2-x+1\right )+\frac {x \left (x-x^2\right )}{3 \left (x^3+1\right )}+\log (x)-\frac {4}{9} \log (x+1)-\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
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Rubi [A] time = 0.07, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1829, 1834, 634, 618, 204, 628} \[ \frac {x \left (x-x^2\right )}{3 \left (x^3+1\right )}-\frac {5}{18} \log \left (x^2-x+1\right )+\log (x)-\frac {4}{9} \log (x+1)-\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 1829
Rule 1834
Rubi steps
\begin {align*} \int \frac {1+x^2}{x \left (1+x^3\right )^2} \, dx &=\frac {x \left (x-x^2\right )}{3 \left (1+x^3\right )}-\frac {1}{3} \int \frac {-3-x^2}{x \left (1+x^3\right )} \, dx\\ &=\frac {x \left (x-x^2\right )}{3 \left (1+x^3\right )}-\frac {1}{3} \int \left (-\frac {3}{x}+\frac {4}{3 (1+x)}+\frac {-4+5 x}{3 \left (1-x+x^2\right )}\right ) \, dx\\ &=\frac {x \left (x-x^2\right )}{3 \left (1+x^3\right )}+\log (x)-\frac {4}{9} \log (1+x)-\frac {1}{9} \int \frac {-4+5 x}{1-x+x^2} \, dx\\ &=\frac {x \left (x-x^2\right )}{3 \left (1+x^3\right )}+\log (x)-\frac {4}{9} \log (1+x)+\frac {1}{6} \int \frac {1}{1-x+x^2} \, dx-\frac {5}{18} \int \frac {-1+2 x}{1-x+x^2} \, dx\\ &=\frac {x \left (x-x^2\right )}{3 \left (1+x^3\right )}+\log (x)-\frac {4}{9} \log (1+x)-\frac {5}{18} \log \left (1-x+x^2\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right )\\ &=\frac {x \left (x-x^2\right )}{3 \left (1+x^3\right )}-\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{3 \sqrt {3}}+\log (x)-\frac {4}{9} \log (1+x)-\frac {5}{18} \log \left (1-x+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 65, normalized size = 1.02 \[ \frac {1}{18} \left (-6 \log \left (x^3+1\right )+\log \left (x^2-x+1\right )+\frac {6 \left (x^2+1\right )}{x^3+1}+18 \log (x)-2 \log (x+1)+2 \sqrt {3} \tan ^{-1}\left (\frac {2 x-1}{\sqrt {3}}\right )\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 74, normalized size = 1.16 \[ -\frac {1}{3} \log \left (x^3+1\right )+\frac {1}{18} \log \left (x^2-x+1\right )+\frac {x^2+1}{3 \left (x^3+1\right )}+\log (x)-\frac {1}{9} \log (x+1)-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 x}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.25, size = 73, normalized size = 1.14 \[ \frac {2 \, \sqrt {3} {\left (x^{3} + 1\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + 6 \, x^{2} - 5 \, {\left (x^{3} + 1\right )} \log \left (x^{2} - x + 1\right ) - 8 \, {\left (x^{3} + 1\right )} \log \left (x + 1\right ) + 18 \, {\left (x^{3} + 1\right )} \log \relax (x) + 6}{18 \, {\left (x^{3} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.94, size = 60, normalized size = 0.94 \[ \frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {x^{2} + 1}{3 \, {\left (x^{2} - x + 1\right )} {\left (x + 1\right )}} - \frac {5}{18} \, \log \left (x^{2} - x + 1\right ) - \frac {4}{9} \, \log \left ({\left | x + 1 \right |}\right ) + \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 54, normalized size = 0.84
method | result | size |
risch | \(\frac {\frac {x^{2}}{3}+\frac {1}{3}}{x^{3}+1}-\frac {5 \ln \left (4 x^{2}-4 x +4\right )}{18}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (-1+2 x \right ) \sqrt {3}}{3}\right )}{9}-\frac {4 \ln \left (1+x \right )}{9}+\ln \relax (x )\) | \(54\) |
default | \(\ln \relax (x )-\frac {-1-x}{9 \left (x^{2}-x +1\right )}-\frac {5 \ln \left (x^{2}-x +1\right )}{18}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (-1+2 x \right ) \sqrt {3}}{3}\right )}{9}+\frac {2}{9 \left (1+x \right )}-\frac {4 \ln \left (1+x \right )}{9}\) | \(61\) |
meijerg | \(\frac {x^{2}}{3 x^{3}+3}-\frac {x^{2} \ln \left (1+\left (x^{3}\right )^{\frac {1}{3}}\right )}{9 \left (x^{3}\right )^{\frac {2}{3}}}+\frac {x^{2} \ln \left (1-\left (x^{3}\right )^{\frac {1}{3}}+\left (x^{3}\right )^{\frac {2}{3}}\right )}{18 \left (x^{3}\right )^{\frac {2}{3}}}+\frac {x^{2} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{3}\right )^{\frac {1}{3}}}{2-\left (x^{3}\right )^{\frac {1}{3}}}\right )}{9 \left (x^{3}\right )^{\frac {2}{3}}}-\frac {2 x^{3}}{3 \left (2 x^{3}+2\right )}-\frac {\ln \left (x^{3}+1\right )}{3}+\frac {1}{3}+\ln \relax (x )\) | \(118\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 50, normalized size = 0.78 \[ \frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {x^{2} + 1}{3 \, {\left (x^{3} + 1\right )}} - \frac {5}{18} \, \log \left (x^{2} - x + 1\right ) - \frac {4}{9} \, \log \left (x + 1\right ) + \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 63, normalized size = 0.98 \[ \ln \relax (x)-\frac {4\,\ln \left (x+1\right )}{9}-\ln \left (x-\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {5}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )+\ln \left (x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {5}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )+\frac {\frac {x^2}{3}+\frac {1}{3}}{x^3+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.23, size = 60, normalized size = 0.94 \[ \frac {x^{2} + 1}{3 x^{3} + 3} + \log {\relax (x )} - \frac {4 \log {\left (x + 1 \right )}}{9} - \frac {5 \log {\left (x^{2} - x + 1 \right )}}{18} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3} - \frac {\sqrt {3}}{3} \right )}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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