Integrand size = 16, antiderivative size = 23 \[ \int \frac {x^2}{1-x^3+x^6} \, dx=-\frac {2 \arctan \left (\frac {1-2 x^3}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
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Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1366, 632, 210} \[ \int \frac {x^2}{1-x^3+x^6} \, dx=-\frac {2 \arctan \left (\frac {1-2 x^3}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
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Rule 210
Rule 632
Rule 1366
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,x^3\right ) \\ & = -\left (\frac {2}{3} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x^3\right )\right ) \\ & = -\frac {2 \tan ^{-1}\left (\frac {1-2 x^3}{\sqrt {3}}\right )}{3 \sqrt {3}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{1-x^3+x^6} \, dx=\frac {2 \arctan \left (\frac {-1+2 x^3}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
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Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83
method | result | size |
default | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (2 x^{3}-1\right ) \sqrt {3}}{3}\right )}{9}\) | \(19\) |
risch | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (2 x^{3}-1\right ) \sqrt {3}}{3}\right )}{9}\) | \(19\) |
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Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78 \[ \int \frac {x^2}{1-x^3+x^6} \, dx=\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{3} - 1\right )}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {x^2}{1-x^3+x^6} \, dx=\frac {2 \sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x^{3}}{3} - \frac {\sqrt {3}}{3} \right )}}{9} \]
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Time = 0.28 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78 \[ \int \frac {x^2}{1-x^3+x^6} \, dx=\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{3} - 1\right )}\right ) \]
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Time = 0.32 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78 \[ \int \frac {x^2}{1-x^3+x^6} \, dx=\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{3} - 1\right )}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {x^2}{1-x^3+x^6} \, dx=-\frac {2\,\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}}{3}-\frac {2\,\sqrt {3}\,x^3}{3}\right )}{9} \]
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