Integrand size = 11, antiderivative size = 49 \[ \int \frac {x^3}{1+x^6} \, dx=-\frac {\arctan \left (\frac {1-2 x^2}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{6} \log \left (1+x^2\right )+\frac {1}{12} \log \left (1-x^2+x^4\right ) \]
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Time = 0.03 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {281, 298, 31, 648, 632, 210, 642} \[ \int \frac {x^3}{1+x^6} \, dx=-\frac {\arctan \left (\frac {1-2 x^2}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{6} \log \left (x^2+1\right )+\frac {1}{12} \log \left (x^4-x^2+1\right ) \]
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Rule 31
Rule 210
Rule 281
Rule 298
Rule 632
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x}{1+x^3} \, dx,x,x^2\right ) \\ & = -\left (\frac {1}{6} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,x^2\right )\right )+\frac {1}{6} \text {Subst}\left (\int \frac {1+x}{1-x+x^2} \, dx,x,x^2\right ) \\ & = -\frac {1}{6} \log \left (1+x^2\right )+\frac {1}{12} \text {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,x^2\right )+\frac {1}{4} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,x^2\right ) \\ & = -\frac {1}{6} \log \left (1+x^2\right )+\frac {1}{12} \log \left (1-x^2+x^4\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x^2\right ) \\ & = -\frac {\tan ^{-1}\left (\frac {1-2 x^2}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{6} \log \left (1+x^2\right )+\frac {1}{12} \log \left (1-x^2+x^4\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.51 \[ \int \frac {x^3}{1+x^6} \, dx=\frac {1}{12} \left (-2 \sqrt {3} \arctan \left (\sqrt {3}-2 x\right )-2 \sqrt {3} \arctan \left (\sqrt {3}+2 x\right )-2 \log \left (1+x^2\right )+\log \left (1-\sqrt {3} x+x^2\right )+\log \left (1+\sqrt {3} x+x^2\right )\right ) \]
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Time = 4.45 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.80
method | result | size |
risch | \(-\frac {\ln \left (x^{2}+1\right )}{6}+\frac {\ln \left (x^{4}-x^{2}+1\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (x^{2}-\frac {1}{2}\right ) \sqrt {3}}{3}\right )}{6}\) | \(39\) |
default | \(-\frac {\ln \left (x^{2}+1\right )}{6}+\frac {\ln \left (x^{4}-x^{2}+1\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{2}-1\right ) \sqrt {3}}{3}\right )}{6}\) | \(41\) |
meijerg | \(-\frac {x^{4} \ln \left (1+\left (x^{6}\right )^{\frac {1}{3}}\right )}{6 \left (x^{6}\right )^{\frac {2}{3}}}+\frac {x^{4} \ln \left (1-\left (x^{6}\right )^{\frac {1}{3}}+\left (x^{6}\right )^{\frac {2}{3}}\right )}{12 \left (x^{6}\right )^{\frac {2}{3}}}+\frac {x^{4} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{3}}}{2-\left (x^{6}\right )^{\frac {1}{3}}}\right )}{6 \left (x^{6}\right )^{\frac {2}{3}}}\) | \(80\) |
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Time = 0.27 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.82 \[ \int \frac {x^3}{1+x^6} \, dx=\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} - 1\right )}\right ) + \frac {1}{12} \, \log \left (x^{4} - x^{2} + 1\right ) - \frac {1}{6} \, \log \left (x^{2} + 1\right ) \]
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Time = 0.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.94 \[ \int \frac {x^3}{1+x^6} \, dx=- \frac {\log {\left (x^{2} + 1 \right )}}{6} + \frac {\log {\left (x^{4} - x^{2} + 1 \right )}}{12} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x^{2}}{3} - \frac {\sqrt {3}}{3} \right )}}{6} \]
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Time = 0.30 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.82 \[ \int \frac {x^3}{1+x^6} \, dx=\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} - 1\right )}\right ) + \frac {1}{12} \, \log \left (x^{4} - x^{2} + 1\right ) - \frac {1}{6} \, \log \left (x^{2} + 1\right ) \]
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Time = 0.28 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.82 \[ \int \frac {x^3}{1+x^6} \, dx=\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} - 1\right )}\right ) + \frac {1}{12} \, \log \left (x^{4} - x^{2} + 1\right ) - \frac {1}{6} \, \log \left (x^{2} + 1\right ) \]
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Time = 0.13 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.06 \[ \int \frac {x^3}{1+x^6} \, dx=-\frac {\ln \left (x^2+1\right )}{6}-\ln \left (x^2-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}-\frac {1}{2}\right )\,\left (-\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right )+\ln \left (x^2+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}-\frac {1}{2}\right )\,\left (\frac {1}{12}+\frac {\sqrt {3}\,1{}\mathrm {i}}{12}\right ) \]
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