Integrand size = 21, antiderivative size = 25 \[ \int \frac {-x^2+2 x^4}{1+2 x^2} \, dx=-x+\frac {x^3}{3}+\frac {\arctan \left (\sqrt {2} x\right )}{\sqrt {2}} \]
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Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {1607, 470, 327, 209} \[ \int \frac {-x^2+2 x^4}{1+2 x^2} \, dx=\frac {\arctan \left (\sqrt {2} x\right )}{\sqrt {2}}+\frac {x^3}{3}-x \]
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Rule 209
Rule 327
Rule 470
Rule 1607
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^2 \left (-1+2 x^2\right )}{1+2 x^2} \, dx \\ & = \frac {x^3}{3}-2 \int \frac {x^2}{1+2 x^2} \, dx \\ & = -x+\frac {x^3}{3}+\int \frac {1}{1+2 x^2} \, dx \\ & = -x+\frac {x^3}{3}+\frac {\tan ^{-1}\left (\sqrt {2} x\right )}{\sqrt {2}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {-x^2+2 x^4}{1+2 x^2} \, dx=-x+\frac {x^3}{3}+\frac {\arctan \left (\sqrt {2} x\right )}{\sqrt {2}} \]
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Time = 3.49 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84
method | result | size |
default | \(-x +\frac {x^{3}}{3}+\frac {\arctan \left (\sqrt {2}\, x \right ) \sqrt {2}}{2}\) | \(21\) |
risch | \(-x +\frac {x^{3}}{3}+\frac {\arctan \left (\sqrt {2}\, x \right ) \sqrt {2}}{2}\) | \(21\) |
meijerg | \(\frac {\sqrt {2}\, \left (-\frac {2 x \sqrt {2}\, \left (-10 x^{2}+15\right )}{15}+2 \arctan \left (\sqrt {2}\, x \right )\right )}{8}-\frac {\sqrt {2}\, \left (2 \sqrt {2}\, x -2 \arctan \left (\sqrt {2}\, x \right )\right )}{8}\) | \(49\) |
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Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {-x^2+2 x^4}{1+2 x^2} \, dx=\frac {1}{3} \, x^{3} + \frac {1}{2} \, \sqrt {2} \arctan \left (\sqrt {2} x\right ) - x \]
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Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {-x^2+2 x^4}{1+2 x^2} \, dx=\frac {x^{3}}{3} - x + \frac {\sqrt {2} \operatorname {atan}{\left (\sqrt {2} x \right )}}{2} \]
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Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {-x^2+2 x^4}{1+2 x^2} \, dx=\frac {1}{3} \, x^{3} + \frac {1}{2} \, \sqrt {2} \arctan \left (\sqrt {2} x\right ) - x \]
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Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {-x^2+2 x^4}{1+2 x^2} \, dx=\frac {1}{3} \, x^{3} + \frac {1}{2} \, \sqrt {2} \arctan \left (\sqrt {2} x\right ) - x \]
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Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {-x^2+2 x^4}{1+2 x^2} \, dx=\frac {\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x\right )}{2}-x+\frac {x^3}{3} \]
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