Integrand size = 22, antiderivative size = 14 \[ \int \frac {1+2 x^2}{1+4 x^2+4 x^4} \, dx=\frac {\arctan \left (\sqrt {2} x\right )}{\sqrt {2}} \]
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Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {28, 21, 209} \[ \int \frac {1+2 x^2}{1+4 x^2+4 x^4} \, dx=\frac {\arctan \left (\sqrt {2} x\right )}{\sqrt {2}} \]
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Rule 21
Rule 28
Rule 209
Rubi steps \begin{align*} \text {integral}& = 4 \int \frac {1+2 x^2}{\left (2+4 x^2\right )^2} \, dx \\ & = \int \frac {1}{1+2 x^2} \, dx \\ & = \frac {\tan ^{-1}\left (\sqrt {2} x\right )}{\sqrt {2}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1+2 x^2}{1+4 x^2+4 x^4} \, dx=\frac {\arctan \left (\sqrt {2} x\right )}{\sqrt {2}} \]
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Time = 0.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86
method | result | size |
default | \(\frac {\arctan \left (x \sqrt {2}\right ) \sqrt {2}}{2}\) | \(12\) |
risch | \(\frac {\arctan \left (x \sqrt {2}\right ) \sqrt {2}}{2}\) | \(12\) |
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Time = 0.24 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.79 \[ \int \frac {1+2 x^2}{1+4 x^2+4 x^4} \, dx=\frac {1}{2} \, \sqrt {2} \arctan \left (\sqrt {2} x\right ) \]
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Time = 0.04 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1+2 x^2}{1+4 x^2+4 x^4} \, dx=\frac {\sqrt {2} \operatorname {atan}{\left (\sqrt {2} x \right )}}{2} \]
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Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.79 \[ \int \frac {1+2 x^2}{1+4 x^2+4 x^4} \, dx=\frac {1}{2} \, \sqrt {2} \arctan \left (\sqrt {2} x\right ) \]
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Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.79 \[ \int \frac {1+2 x^2}{1+4 x^2+4 x^4} \, dx=\frac {1}{2} \, \sqrt {2} \arctan \left (\sqrt {2} x\right ) \]
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Time = 0.03 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.79 \[ \int \frac {1+2 x^2}{1+4 x^2+4 x^4} \, dx=\frac {\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x\right )}{2} \]
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