Integrand size = 14, antiderivative size = 49 \[ \int \frac {x}{3+6 x+2 x^2} \, dx=\frac {1}{4} \left (1-\sqrt {3}\right ) \log \left (3-\sqrt {3}+2 x\right )+\frac {1}{4} \left (1+\sqrt {3}\right ) \log \left (3+\sqrt {3}+2 x\right ) \]
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Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {646, 31} \[ \int \frac {x}{3+6 x+2 x^2} \, dx=\frac {1}{4} \left (1-\sqrt {3}\right ) \log \left (2 x-\sqrt {3}+3\right )+\frac {1}{4} \left (1+\sqrt {3}\right ) \log \left (2 x+\sqrt {3}+3\right ) \]
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Rule 31
Rule 646
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \left (1-\sqrt {3}\right ) \int \frac {1}{3-\sqrt {3}+2 x} \, dx+\frac {1}{2} \left (1+\sqrt {3}\right ) \int \frac {1}{3+\sqrt {3}+2 x} \, dx \\ & = \frac {1}{4} \left (1-\sqrt {3}\right ) \log \left (3-\sqrt {3}+2 x\right )+\frac {1}{4} \left (1+\sqrt {3}\right ) \log \left (3+\sqrt {3}+2 x\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.90 \[ \int \frac {x}{3+6 x+2 x^2} \, dx=\frac {1}{4} \left (-\left (\left (-1+\sqrt {3}\right ) \log \left (-3+\sqrt {3}-2 x\right )\right )+\left (1+\sqrt {3}\right ) \log \left (3+\sqrt {3}+2 x\right )\right ) \]
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Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.63
method | result | size |
default | \(\frac {\ln \left (2 x^{2}+6 x +3\right )}{4}+\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (4 x +6\right ) \sqrt {3}}{6}\right )}{2}\) | \(31\) |
risch | \(\frac {\ln \left (3+2 x +\sqrt {3}\right )}{4}+\frac {\ln \left (3+2 x +\sqrt {3}\right ) \sqrt {3}}{4}+\frac {\ln \left (3+2 x -\sqrt {3}\right )}{4}-\frac {\ln \left (3+2 x -\sqrt {3}\right ) \sqrt {3}}{4}\) | \(56\) |
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Time = 0.23 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.06 \[ \int \frac {x}{3+6 x+2 x^2} \, dx=\frac {1}{4} \, \sqrt {3} \log \left (\frac {2 \, x^{2} + \sqrt {3} {\left (2 \, x + 3\right )} + 6 \, x + 6}{2 \, x^{2} + 6 \, x + 3}\right ) + \frac {1}{4} \, \log \left (2 \, x^{2} + 6 \, x + 3\right ) \]
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Time = 0.04 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.94 \[ \int \frac {x}{3+6 x+2 x^2} \, dx=\left (\frac {1}{4} - \frac {\sqrt {3}}{4}\right ) \log {\left (x - \frac {\sqrt {3}}{2} + \frac {3}{2} \right )} + \left (\frac {1}{4} + \frac {\sqrt {3}}{4}\right ) \log {\left (x + \frac {\sqrt {3}}{2} + \frac {3}{2} \right )} \]
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Time = 0.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.84 \[ \int \frac {x}{3+6 x+2 x^2} \, dx=-\frac {1}{4} \, \sqrt {3} \log \left (\frac {2 \, x - \sqrt {3} + 3}{2 \, x + \sqrt {3} + 3}\right ) + \frac {1}{4} \, \log \left (2 \, x^{2} + 6 \, x + 3\right ) \]
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Time = 0.29 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.94 \[ \int \frac {x}{3+6 x+2 x^2} \, dx=-\frac {1}{4} \, \sqrt {3} \log \left (\frac {{\left | 4 \, x - 2 \, \sqrt {3} + 6 \right |}}{{\left | 4 \, x + 2 \, \sqrt {3} + 6 \right |}}\right ) + \frac {1}{4} \, \log \left ({\left | 2 \, x^{2} + 6 \, x + 3 \right |}\right ) \]
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Time = 0.18 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.73 \[ \int \frac {x}{3+6 x+2 x^2} \, dx=\ln \left (x+\frac {\sqrt {3}}{2}+\frac {3}{2}\right )\,\left (\frac {\sqrt {3}}{4}+\frac {1}{4}\right )-\ln \left (x-\frac {\sqrt {3}}{2}+\frac {3}{2}\right )\,\left (\frac {\sqrt {3}}{4}-\frac {1}{4}\right ) \]
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