Integrand size = 15, antiderivative size = 12 \[ \int \frac {1+x^3}{-x+x^3} \, dx=x+\log (1-x)-\log (x) \]
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Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1607, 1816} \[ \int \frac {1+x^3}{-x+x^3} \, dx=x+\log (1-x)-\log (x) \]
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Rule 1607
Rule 1816
Rubi steps \begin{align*} \text {integral}& = \int \frac {1+x^3}{x \left (-1+x^2\right )} \, dx \\ & = \int \left (1+\frac {1}{-1+x}-\frac {1}{x}\right ) \, dx \\ & = x+\log (1-x)-\log (x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {1+x^3}{-x+x^3} \, dx=x+\log (1-x)-\log (x) \]
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Time = 0.86 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.92
method | result | size |
default | \(x -\ln \left (x \right )+\ln \left (x -1\right )\) | \(11\) |
norman | \(x -\ln \left (x \right )+\ln \left (x -1\right )\) | \(11\) |
risch | \(x -\ln \left (x \right )+\ln \left (x -1\right )\) | \(11\) |
parallelrisch | \(x -\ln \left (x \right )+\ln \left (x -1\right )\) | \(11\) |
meijerg | \(\frac {\ln \left (-x^{2}+1\right )}{2}-\ln \left (x \right )-\frac {i \pi }{2}-\frac {i \left (2 i x -2 i \operatorname {arctanh}\left (x \right )\right )}{2}\) | \(33\) |
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none
Time = 0.25 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {1+x^3}{-x+x^3} \, dx=x + \log \left (x - 1\right ) - \log \left (x\right ) \]
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Time = 0.05 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {1+x^3}{-x+x^3} \, dx=x - \log {\left (x \right )} + \log {\left (x - 1 \right )} \]
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none
Time = 0.21 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {1+x^3}{-x+x^3} \, dx=x + \log \left (x - 1\right ) - \log \left (x\right ) \]
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none
Time = 0.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {1+x^3}{-x+x^3} \, dx=x + \log \left ({\left | x - 1 \right |}\right ) - \log \left ({\left | x \right |}\right ) \]
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Time = 9.14 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {1+x^3}{-x+x^3} \, dx=x-2\,\mathrm {atanh}\left (2\,x-1\right ) \]
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