Optimal. Leaf size=64 \[ \frac {x \left (x-x^2\right )}{3 \left (1+x^3\right )}-\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{3 \sqrt {3}}+\log (x)-\frac {4}{9} \log (1+x)-\frac {5}{18} \log \left (1-x+x^2\right ) \]
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Rubi [A]
time = 0.05, 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 = {1843, 1848,
648, 632, 210, 642} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {1-2 x}{\sqrt {3}}\right )}{3 \sqrt {3}}-\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) \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 1843
Rule 1848
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} \text {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 = 0.02, size = 65, normalized size = 1.02 \begin {gather*} \frac {1}{18} \left (\frac {6 \left (1+x^2\right )}{1+x^3}+2 \sqrt {3} \tan ^{-1}\left (\frac {-1+2 x}{\sqrt {3}}\right )+18 \log (x)-2 \log (1+x)+\log \left (1-x+x^2\right )-6 \log \left (1+x^3\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 61, normalized size = 0.95
method | result | size |
risch | \(\frac {\frac {x^{2}}{3}+\frac {1}{3}}{x^{3}+1}-\frac {5 \ln \left (4 x^{2}-4 x +4\right )}{18}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{9}-\frac {4 \ln \left (1+x \right )}{9}+\ln \left (x \right )\) | \(54\) |
default | \(\ln \left (x \right )+\frac {2}{9 \left (1+x \right )}-\frac {4 \ln \left (1+x \right )}{9}-\frac {-1-x}{9 \left (x^{2}-x +1\right )}-\frac {5 \ln \left (x^{2}-x +1\right )}{18}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{9}\) | \(61\) |
meijerg | \(\frac {x^{2}}{3 x^{3}+3}-\frac {x^{2} \ln \left (1+\left (x^{3}\right )^{\frac {1}{3}}\right )}{9 \left (x^{3}\right )^{\frac {2}{3}}}+\frac {x^{2} \ln \left (1-\left (x^{3}\right )^{\frac {1}{3}}+\left (x^{3}\right )^{\frac {2}{3}}\right )}{18 \left (x^{3}\right )^{\frac {2}{3}}}+\frac {x^{2} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{3}\right )^{\frac {1}{3}}}{2-\left (x^{3}\right )^{\frac {1}{3}}}\right )}{9 \left (x^{3}\right )^{\frac {2}{3}}}+\frac {1}{3}+\ln \left (x \right )-\frac {2 x^{3}}{3 \left (2 x^{3}+2\right )}-\frac {\ln \left (x^{3}+1\right )}{3}\) | \(118\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.94, size = 50, normalized size = 0.78 \begin {gather*} \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 \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.49, size = 73, normalized size = 1.14 \begin {gather*} \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 \left (x\right ) + 6}{18 \, {\left (x^{3} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 60, normalized size = 0.94 \begin {gather*} \frac {x^{2} + 1}{3 x^{3} + 3} + \log {\left (x \right )} - \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.49, size = 60, normalized size = 0.94 \begin {gather*} \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 ) \end {gather*}
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 \begin {gather*} \ln \left (x\right )-\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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