Integrand size = 14, antiderivative size = 32 \[ \int \frac {-1+x}{1-x+x^2} \, dx=\frac {\arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (1-x+x^2\right ) \]
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Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {648, 632, 210, 642} \[ \int \frac {-1+x}{1-x+x^2} \, dx=\frac {\arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (x^2-x+1\right ) \]
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Rule 210
Rule 632
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{2} \int \frac {1}{1-x+x^2} \, dx\right )+\frac {1}{2} \int \frac {-1+2 x}{1-x+x^2} \, dx \\ & = \frac {1}{2} \log \left (1-x+x^2\right )+\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right ) \\ & = -\frac {\tan ^{-1}\left (\frac {-1+2 x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (1-x+x^2\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03 \[ \int \frac {-1+x}{1-x+x^2} \, dx=-\frac {\arctan \left (\frac {-1+2 x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (1-x+x^2\right ) \]
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Time = 1.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91
method | result | size |
default | \(\frac {\ln \left (x^{2}-x +1\right )}{2}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{3}\) | \(29\) |
risch | \(\frac {\ln \left (4 x^{2}-4 x +4\right )}{2}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{3}\) | \(31\) |
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Time = 0.32 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \frac {-1+x}{1-x+x^2} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{2} \, \log \left (x^{2} - x + 1\right ) \]
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Time = 0.05 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {-1+x}{1-x+x^2} \, dx=\frac {\log {\left (x^{2} - x + 1 \right )}}{2} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3} - \frac {\sqrt {3}}{3} \right )}}{3} \]
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Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \frac {-1+x}{1-x+x^2} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{2} \, \log \left (x^{2} - x + 1\right ) \]
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Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \frac {-1+x}{1-x+x^2} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {1}{2} \, \log \left (x^{2} - x + 1\right ) \]
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Time = 0.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \frac {-1+x}{1-x+x^2} \, dx=\frac {\ln \left (x^2-x+1\right )}{2}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {2\,\sqrt {3}\,x}{3}-\frac {\sqrt {3}}{3}\right )}{3} \]
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