Integrand size = 11, antiderivative size = 22 \[ \int \frac {1+x}{-1+x^3} \, dx=\frac {2}{3} \log (1-x)-\frac {1}{3} \log \left (1+x+x^2\right ) \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1875, 31, 642} \[ \int \frac {1+x}{-1+x^3} \, dx=\frac {2}{3} \log (1-x)-\frac {1}{3} \log \left (x^2+x+1\right ) \]
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Rule 31
Rule 642
Rule 1875
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \int \frac {-1-2 x}{1+x+x^2} \, dx-\frac {2}{3} \int \frac {1}{1-x} \, dx \\ & = \frac {2}{3} \log (1-x)-\frac {1}{3} \log \left (1+x+x^2\right ) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1+x}{-1+x^3} \, dx=\frac {2}{3} \log (1-x)-\frac {1}{3} \log \left (1+x+x^2\right ) \]
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Time = 0.05 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77
method | result | size |
default | \(\frac {2 \ln \left (-1+x \right )}{3}-\frac {\ln \left (x^{2}+x +1\right )}{3}\) | \(17\) |
norman | \(\frac {2 \ln \left (-1+x \right )}{3}-\frac {\ln \left (x^{2}+x +1\right )}{3}\) | \(17\) |
risch | \(\frac {2 \ln \left (-1+x \right )}{3}-\frac {\ln \left (x^{2}+x +1\right )}{3}\) | \(17\) |
parallelrisch | \(\frac {2 \ln \left (-1+x \right )}{3}-\frac {\ln \left (x^{2}+x +1\right )}{3}\) | \(17\) |
meijerg | \(\frac {x \left (\ln \left (1-\left (x^{3}\right )^{\frac {1}{3}}\right )-\frac {\ln \left (1+\left (x^{3}\right )^{\frac {1}{3}}+\left (x^{3}\right )^{\frac {2}{3}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{3}\right )^{\frac {1}{3}}}{2+\left (x^{3}\right )^{\frac {1}{3}}}\right )\right )}{3 \left (x^{3}\right )^{\frac {1}{3}}}+\frac {x^{2} \left (\ln \left (1-\left (x^{3}\right )^{\frac {1}{3}}\right )-\frac {\ln \left (1+\left (x^{3}\right )^{\frac {1}{3}}+\left (x^{3}\right )^{\frac {2}{3}}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{3}\right )^{\frac {1}{3}}}{2+\left (x^{3}\right )^{\frac {1}{3}}}\right )\right )}{3 \left (x^{3}\right )^{\frac {2}{3}}}\) | \(125\) |
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Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73 \[ \int \frac {1+x}{-1+x^3} \, dx=-\frac {1}{3} \, \log \left (x^{2} + x + 1\right ) + \frac {2}{3} \, \log \left (x - 1\right ) \]
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Time = 0.04 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {1+x}{-1+x^3} \, dx=\frac {2 \log {\left (x - 1 \right )}}{3} - \frac {\log {\left (x^{2} + x + 1 \right )}}{3} \]
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Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73 \[ \int \frac {1+x}{-1+x^3} \, dx=-\frac {1}{3} \, \log \left (x^{2} + x + 1\right ) + \frac {2}{3} \, \log \left (x - 1\right ) \]
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Time = 0.29 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {1+x}{-1+x^3} \, dx=-\frac {1}{3} \, \log \left (x^{2} + x + 1\right ) + \frac {2}{3} \, \log \left ({\left | x - 1 \right |}\right ) \]
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Time = 0.17 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73 \[ \int \frac {1+x}{-1+x^3} \, dx=\frac {2\,\ln \left (x-1\right )}{3}-\frac {\ln \left (x^2+x+1\right )}{3} \]
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