Integrand size = 16, antiderivative size = 19 \[ \int \frac {x^2}{1+2 x^2+x^4} \, dx=-\frac {x}{2 \left (1+x^2\right )}+\frac {\arctan (x)}{2} \]
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Time = 0.00 (sec) , antiderivative size = 19, 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 = {28, 294, 209} \[ \int \frac {x^2}{1+2 x^2+x^4} \, dx=\frac {\arctan (x)}{2}-\frac {x}{2 \left (x^2+1\right )} \]
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Rule 28
Rule 209
Rule 294
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^2}{\left (1+x^2\right )^2} \, dx \\ & = -\frac {x}{2 \left (1+x^2\right )}+\frac {1}{2} \int \frac {1}{1+x^2} \, dx \\ & = -\frac {x}{2 \left (1+x^2\right )}+\frac {1}{2} \tan ^{-1}(x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{1+2 x^2+x^4} \, dx=-\frac {x}{2 \left (1+x^2\right )}+\frac {\arctan (x)}{2} \]
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Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84
method | result | size |
default | \(-\frac {x}{2 \left (x^{2}+1\right )}+\frac {\arctan \left (x \right )}{2}\) | \(16\) |
risch | \(-\frac {x}{2 \left (x^{2}+1\right )}+\frac {\arctan \left (x \right )}{2}\) | \(16\) |
parallelrisch | \(-\frac {i \ln \left (x -i\right ) x^{2}-i \ln \left (x +i\right ) x^{2}+i \ln \left (x -i\right )-i \ln \left (x +i\right )+2 x}{4 \left (x^{2}+1\right )}\) | \(52\) |
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none
Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {x^2}{1+2 x^2+x^4} \, dx=\frac {{\left (x^{2} + 1\right )} \arctan \left (x\right ) - x}{2 \, {\left (x^{2} + 1\right )}} \]
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Time = 0.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.63 \[ \int \frac {x^2}{1+2 x^2+x^4} \, dx=- \frac {x}{2 x^{2} + 2} + \frac {\operatorname {atan}{\left (x \right )}}{2} \]
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none
Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {x^2}{1+2 x^2+x^4} \, dx=-\frac {x}{2 \, {\left (x^{2} + 1\right )}} + \frac {1}{2} \, \arctan \left (x\right ) \]
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none
Time = 0.28 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {x^2}{1+2 x^2+x^4} \, dx=-\frac {x}{2 \, {\left (x^{2} + 1\right )}} + \frac {1}{2} \, \arctan \left (x\right ) \]
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Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {x^2}{1+2 x^2+x^4} \, dx=\frac {\mathrm {atan}\left (x\right )}{2}-\frac {x}{2\,\left (x^2+1\right )} \]
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