Integrand size = 12, antiderivative size = 22 \[ \int \frac {b x}{2+3 x^4} \, dx=\frac {b \arctan \left (\sqrt {\frac {3}{2}} x^2\right )}{2 \sqrt {6}} \]
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Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {12, 281, 209} \[ \int \frac {b x}{2+3 x^4} \, dx=\frac {b \arctan \left (\sqrt {\frac {3}{2}} x^2\right )}{2 \sqrt {6}} \]
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Rule 12
Rule 209
Rule 281
Rubi steps \begin{align*} \text {integral}& = b \int \frac {x}{2+3 x^4} \, dx \\ & = \frac {1}{2} b \text {Subst}\left (\int \frac {1}{2+3 x^2} \, dx,x,x^2\right ) \\ & = \frac {b \tan ^{-1}\left (\sqrt {\frac {3}{2}} x^2\right )}{2 \sqrt {6}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {b x}{2+3 x^4} \, dx=\frac {b \arctan \left (\sqrt {\frac {3}{2}} x^2\right )}{2 \sqrt {6}} \]
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Time = 1.47 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73
method | result | size |
default | \(\frac {b \arctan \left (\frac {x^{2} \sqrt {6}}{2}\right ) \sqrt {6}}{12}\) | \(16\) |
risch | \(\frac {b \arctan \left (\frac {x^{2} \sqrt {6}}{2}\right ) \sqrt {6}}{12}\) | \(16\) |
meijerg | \(\frac {\sqrt {6}\, b \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, x^{2}}{2}\right )}{12}\) | \(19\) |
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Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int \frac {b x}{2+3 x^4} \, dx=\frac {1}{12} \, \sqrt {6} b \arctan \left (\frac {1}{2} \, \sqrt {6} x^{2}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {b x}{2+3 x^4} \, dx=\frac {\sqrt {6} b \operatorname {atan}{\left (\frac {\sqrt {6} x^{2}}{2} \right )}}{12} \]
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Time = 0.28 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int \frac {b x}{2+3 x^4} \, dx=\frac {1}{12} \, \sqrt {6} b \arctan \left (\frac {1}{2} \, \sqrt {6} x^{2}\right ) \]
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Time = 0.27 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int \frac {b x}{2+3 x^4} \, dx=\frac {1}{12} \, \sqrt {6} b \arctan \left (\frac {1}{2} \, \sqrt {6} x^{2}\right ) \]
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Time = 9.06 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int \frac {b x}{2+3 x^4} \, dx=\frac {\sqrt {6}\,b\,\mathrm {atan}\left (\frac {\sqrt {6}\,x^2}{2}\right )}{12} \]
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