Integrand size = 14, antiderivative size = 23 \[ \int \frac {x^2}{2+x^3+x^6} \, dx=\frac {2 \arctan \left (\frac {1+2 x^3}{\sqrt {7}}\right )}{3 \sqrt {7}} \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1366, 632, 210} \[ \int \frac {x^2}{2+x^3+x^6} \, dx=\frac {2 \arctan \left (\frac {2 x^3+1}{\sqrt {7}}\right )}{3 \sqrt {7}} \]
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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}{2+x+x^2} \, dx,x,x^3\right ) \\ & = -\left (\frac {2}{3} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,1+2 x^3\right )\right ) \\ & = \frac {2 \tan ^{-1}\left (\frac {1+2 x^3}{\sqrt {7}}\right )}{3 \sqrt {7}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{2+x^3+x^6} \, dx=\frac {2 \arctan \left (\frac {1+2 x^3}{\sqrt {7}}\right )}{3 \sqrt {7}} \]
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Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83
method | result | size |
default | \(\frac {2 \arctan \left (\frac {\left (2 x^{3}+1\right ) \sqrt {7}}{7}\right ) \sqrt {7}}{21}\) | \(19\) |
risch | \(\frac {2 \arctan \left (\frac {\left (2 x^{3}+1\right ) \sqrt {7}}{7}\right ) \sqrt {7}}{21}\) | \(19\) |
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Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78 \[ \int \frac {x^2}{2+x^3+x^6} \, dx=\frac {2}{21} \, \sqrt {7} \arctan \left (\frac {1}{7} \, \sqrt {7} {\left (2 \, x^{3} + 1\right )}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {x^2}{2+x^3+x^6} \, dx=\frac {2 \sqrt {7} \operatorname {atan}{\left (\frac {2 \sqrt {7} x^{3}}{7} + \frac {\sqrt {7}}{7} \right )}}{21} \]
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Time = 0.28 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78 \[ \int \frac {x^2}{2+x^3+x^6} \, dx=\frac {2}{21} \, \sqrt {7} \arctan \left (\frac {1}{7} \, \sqrt {7} {\left (2 \, x^{3} + 1\right )}\right ) \]
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Time = 0.38 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78 \[ \int \frac {x^2}{2+x^3+x^6} \, dx=\frac {2}{21} \, \sqrt {7} \arctan \left (\frac {1}{7} \, \sqrt {7} {\left (2 \, x^{3} + 1\right )}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {x^2}{2+x^3+x^6} \, dx=\frac {2\,\sqrt {7}\,\mathrm {atan}\left (\frac {2\,\sqrt {7}\,x^3}{7}+\frac {\sqrt {7}}{7}\right )}{21} \]
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