Integrand size = 12, antiderivative size = 18 \[ \int \frac {1}{2+4 x+3 x^2} \, dx=\frac {\arctan \left (\frac {2+3 x}{\sqrt {2}}\right )}{\sqrt {2}} \]
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Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {632, 210} \[ \int \frac {1}{2+4 x+3 x^2} \, dx=\frac {\arctan \left (\frac {3 x+2}{\sqrt {2}}\right )}{\sqrt {2}} \]
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Rule 210
Rule 632
Rubi steps \begin{align*} \text {integral}& = -\left (2 \text {Subst}\left (\int \frac {1}{-8-x^2} \, dx,x,4+6 x\right )\right ) \\ & = \frac {\tan ^{-1}\left (\frac {2+3 x}{\sqrt {2}}\right )}{\sqrt {2}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{2+4 x+3 x^2} \, dx=\frac {\arctan \left (\frac {2+3 x}{\sqrt {2}}\right )}{\sqrt {2}} \]
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Time = 4.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94
method | result | size |
default | \(\frac {\sqrt {2}\, \arctan \left (\frac {\left (4+6 x \right ) \sqrt {2}}{4}\right )}{2}\) | \(17\) |
risch | \(\frac {\arctan \left (\frac {\left (2+3 x \right ) \sqrt {2}}{2}\right ) \sqrt {2}}{2}\) | \(17\) |
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none
Time = 0.74 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {1}{2+4 x+3 x^2} \, dx=\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (3 \, x + 2\right )}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {1}{2+4 x+3 x^2} \, dx=\frac {\sqrt {2} \operatorname {atan}{\left (\frac {3 \sqrt {2} x}{2} + \sqrt {2} \right )}}{2} \]
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none
Time = 0.28 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {1}{2+4 x+3 x^2} \, dx=\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (3 \, x + 2\right )}\right ) \]
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none
Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {1}{2+4 x+3 x^2} \, dx=\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (3 \, x + 2\right )}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {1}{2+4 x+3 x^2} \, dx=\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\left (3\,x+2\right )}{2}\right )}{2} \]
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