Integrand size = 11, antiderivative size = 80 \[ \int \frac {x^4}{1+x^6} \, dx=-\frac {1}{6} \arctan \left (\sqrt {3}-2 x\right )+\frac {\arctan (x)}{3}+\frac {1}{6} \arctan \left (\sqrt {3}+2 x\right )+\frac {\log \left (1-\sqrt {3} x+x^2\right )}{4 \sqrt {3}}-\frac {\log \left (1+\sqrt {3} x+x^2\right )}{4 \sqrt {3}} \]
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Time = 0.20 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {301, 648, 632, 210, 642, 209} \[ \int \frac {x^4}{1+x^6} \, dx=-\frac {1}{6} \arctan \left (\sqrt {3}-2 x\right )+\frac {\arctan (x)}{3}+\frac {1}{6} \arctan \left (2 x+\sqrt {3}\right )+\frac {\log \left (x^2-\sqrt {3} x+1\right )}{4 \sqrt {3}}-\frac {\log \left (x^2+\sqrt {3} x+1\right )}{4 \sqrt {3}} \]
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Rule 209
Rule 210
Rule 301
Rule 632
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \int \frac {-\frac {1}{2}+\frac {\sqrt {3} x}{2}}{1-\sqrt {3} x+x^2} \, dx+\frac {1}{3} \int \frac {-\frac {1}{2}-\frac {\sqrt {3} x}{2}}{1+\sqrt {3} x+x^2} \, dx+\frac {1}{3} \int \frac {1}{1+x^2} \, dx \\ & = \frac {1}{3} \tan ^{-1}(x)+\frac {1}{12} \int \frac {1}{1-\sqrt {3} x+x^2} \, dx+\frac {1}{12} \int \frac {1}{1+\sqrt {3} x+x^2} \, dx+\frac {\int \frac {-\sqrt {3}+2 x}{1-\sqrt {3} x+x^2} \, dx}{4 \sqrt {3}}-\frac {\int \frac {\sqrt {3}+2 x}{1+\sqrt {3} x+x^2} \, dx}{4 \sqrt {3}} \\ & = \frac {1}{3} \tan ^{-1}(x)+\frac {\log \left (1-\sqrt {3} x+x^2\right )}{4 \sqrt {3}}-\frac {\log \left (1+\sqrt {3} x+x^2\right )}{4 \sqrt {3}}-\frac {1}{6} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,-\sqrt {3}+2 x\right )-\frac {1}{6} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {3}+2 x\right ) \\ & = -\frac {1}{6} \tan ^{-1}\left (\sqrt {3}-2 x\right )+\frac {1}{3} \tan ^{-1}(x)+\frac {1}{6} \tan ^{-1}\left (\sqrt {3}+2 x\right )+\frac {\log \left (1-\sqrt {3} x+x^2\right )}{4 \sqrt {3}}-\frac {\log \left (1+\sqrt {3} x+x^2\right )}{4 \sqrt {3}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.91 \[ \int \frac {x^4}{1+x^6} \, dx=\frac {1}{12} \left (-2 \arctan \left (\sqrt {3}-2 x\right )+4 \arctan (x)+2 \arctan \left (\sqrt {3}+2 x\right )+\sqrt {3} \log \left (1-\sqrt {3} x+x^2\right )-\sqrt {3} \log \left (1+\sqrt {3} x+x^2\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.44 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.41
method | result | size |
risch | \(\frac {\arctan \left (x \right )}{3}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-\textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (\textit {\_R}^{3}-\textit {\_R} +x \right )\right )}{6}\) | \(33\) |
default | \(\frac {\arctan \left (x \right )}{3}+\frac {\arctan \left (2 x -\sqrt {3}\right )}{6}+\frac {\arctan \left (2 x +\sqrt {3}\right )}{6}+\frac {\ln \left (1+x^{2}-\sqrt {3}\, x \right ) \sqrt {3}}{12}-\frac {\ln \left (1+x^{2}+\sqrt {3}\, x \right ) \sqrt {3}}{12}\) | \(61\) |
meijerg | \(\frac {x^{5} \sqrt {3}\, \ln \left (1-\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}+\left (x^{6}\right )^{\frac {1}{3}}\right )}{12 \left (x^{6}\right )^{\frac {5}{6}}}+\frac {x^{5} \arctan \left (\frac {\left (x^{6}\right )^{\frac {1}{6}}}{2-\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}\right )}{6 \left (x^{6}\right )^{\frac {5}{6}}}+\frac {x^{5} \arctan \left (\left (x^{6}\right )^{\frac {1}{6}}\right )}{3 \left (x^{6}\right )^{\frac {5}{6}}}-\frac {x^{5} \sqrt {3}\, \ln \left (1+\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}+\left (x^{6}\right )^{\frac {1}{3}}\right )}{12 \left (x^{6}\right )^{\frac {5}{6}}}+\frac {x^{5} \arctan \left (\frac {\left (x^{6}\right )^{\frac {1}{6}}}{2+\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}\right )}{6 \left (x^{6}\right )^{\frac {5}{6}}}\) | \(143\) |
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 173, normalized size of antiderivative = 2.16 \[ \int \frac {x^4}{1+x^6} \, dx=\frac {1}{12} \, \sqrt {2} \sqrt {i \, \sqrt {3} + 1} \log \left ({\left (i \, \sqrt {3} \sqrt {2} - \sqrt {2}\right )} \sqrt {i \, \sqrt {3} + 1} + 4 \, x\right ) - \frac {1}{12} \, \sqrt {2} \sqrt {i \, \sqrt {3} + 1} \log \left ({\left (-i \, \sqrt {3} \sqrt {2} + \sqrt {2}\right )} \sqrt {i \, \sqrt {3} + 1} + 4 \, x\right ) - \frac {1}{12} \, \sqrt {2} \sqrt {-i \, \sqrt {3} + 1} \log \left ({\left (i \, \sqrt {3} \sqrt {2} + \sqrt {2}\right )} \sqrt {-i \, \sqrt {3} + 1} + 4 \, x\right ) + \frac {1}{12} \, \sqrt {2} \sqrt {-i \, \sqrt {3} + 1} \log \left ({\left (-i \, \sqrt {3} \sqrt {2} - \sqrt {2}\right )} \sqrt {-i \, \sqrt {3} + 1} + 4 \, x\right ) + \frac {1}{3} \, \arctan \left (x\right ) \]
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Time = 0.10 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.85 \[ \int \frac {x^4}{1+x^6} \, dx=\frac {\sqrt {3} \log {\left (x^{2} - \sqrt {3} x + 1 \right )}}{12} - \frac {\sqrt {3} \log {\left (x^{2} + \sqrt {3} x + 1 \right )}}{12} + \frac {\operatorname {atan}{\left (x \right )}}{3} + \frac {\operatorname {atan}{\left (2 x - \sqrt {3} \right )}}{6} + \frac {\operatorname {atan}{\left (2 x + \sqrt {3} \right )}}{6} \]
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Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.75 \[ \int \frac {x^4}{1+x^6} \, dx=-\frac {1}{12} \, \sqrt {3} \log \left (x^{2} + \sqrt {3} x + 1\right ) + \frac {1}{12} \, \sqrt {3} \log \left (x^{2} - \sqrt {3} x + 1\right ) + \frac {1}{6} \, \arctan \left (2 \, x + \sqrt {3}\right ) + \frac {1}{6} \, \arctan \left (2 \, x - \sqrt {3}\right ) + \frac {1}{3} \, \arctan \left (x\right ) \]
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Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.75 \[ \int \frac {x^4}{1+x^6} \, dx=-\frac {1}{12} \, \sqrt {3} \log \left (x^{2} + \sqrt {3} x + 1\right ) + \frac {1}{12} \, \sqrt {3} \log \left (x^{2} - \sqrt {3} x + 1\right ) + \frac {1}{6} \, \arctan \left (2 \, x + \sqrt {3}\right ) + \frac {1}{6} \, \arctan \left (2 \, x - \sqrt {3}\right ) + \frac {1}{3} \, \arctan \left (x\right ) \]
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Time = 5.76 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.66 \[ \int \frac {x^4}{1+x^6} \, dx=\frac {\mathrm {atan}\left (x\right )}{3}-\mathrm {atan}\left (\frac {2\,x}{-1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )-\mathrm {atan}\left (\frac {2\,x}{1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right ) \]
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