Integrand size = 9, antiderivative size = 17 \[ \int \frac {1}{-x+x^3} \, dx=-\log (x)+\frac {1}{2} \log \left (1-x^2\right ) \]
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Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {1607, 272, 36, 31, 29} \[ \int \frac {1}{-x+x^3} \, dx=\frac {1}{2} \log \left (1-x^2\right )-\log (x) \]
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Rule 29
Rule 31
Rule 36
Rule 272
Rule 1607
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x \left (-1+x^2\right )} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{(-1+x) x} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{-1+x} \, dx,x,x^2\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right ) \\ & = -\log (x)+\frac {1}{2} \log \left (1-x^2\right ) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {1}{-x+x^3} \, dx=-\log (x)+\frac {1}{2} \log \left (1-x^2\right ) \]
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Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82
method | result | size |
risch | \(-\ln \left (x \right )+\frac {\ln \left (x^{2}-1\right )}{2}\) | \(14\) |
default | \(\frac {\ln \left (-1+x \right )}{2}-\ln \left (x \right )+\frac {\ln \left (1+x \right )}{2}\) | \(18\) |
norman | \(\frac {\ln \left (-1+x \right )}{2}-\ln \left (x \right )+\frac {\ln \left (1+x \right )}{2}\) | \(18\) |
parallelrisch | \(\frac {\ln \left (-1+x \right )}{2}-\ln \left (x \right )+\frac {\ln \left (1+x \right )}{2}\) | \(18\) |
meijerg | \(-\ln \left (x \right )-\frac {i \pi }{2}+\frac {\ln \left (-x^{2}+1\right )}{2}\) | \(20\) |
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none
Time = 0.24 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {1}{-x+x^3} \, dx=\frac {1}{2} \, \log \left (x^{2} - 1\right ) - \log \left (x\right ) \]
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Time = 0.04 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.59 \[ \int \frac {1}{-x+x^3} \, dx=- \log {\left (x \right )} + \frac {\log {\left (x^{2} - 1 \right )}}{2} \]
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none
Time = 0.19 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {1}{-x+x^3} \, dx=\frac {1}{2} \, \log \left (x + 1\right ) + \frac {1}{2} \, \log \left (x - 1\right ) - \log \left (x\right ) \]
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none
Time = 0.28 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int \frac {1}{-x+x^3} \, dx=-\frac {1}{2} \, \log \left (x^{2}\right ) + \frac {1}{2} \, \log \left ({\left | x^{2} - 1 \right |}\right ) \]
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Time = 0.12 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {1}{-x+x^3} \, dx=\frac {\ln \left (x^2-1\right )}{2}-\ln \left (x\right ) \]
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