Integrand size = 17, antiderivative size = 17 \[ \int \frac {1+x^3}{-x^2+x^3} \, dx=\frac {1}{x}+x+2 \log (1-x)-\log (x) \]
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Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1607, 1634} \[ \int \frac {1+x^3}{-x^2+x^3} \, dx=x+\frac {1}{x}+2 \log (1-x)-\log (x) \]
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Rule 1607
Rule 1634
Rubi steps \begin{align*} \text {integral}& = \int \frac {1+x^3}{(-1+x) x^2} \, dx \\ & = \int \left (1+\frac {2}{-1+x}-\frac {1}{x^2}-\frac {1}{x}\right ) \, dx \\ & = \frac {1}{x}+x+2 \log (1-x)-\log (x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {1+x^3}{-x^2+x^3} \, dx=\frac {1}{x}+x+2 \log (1-x)-\log (x) \]
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Time = 0.18 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94
method | result | size |
default | \(x +2 \ln \left (-1+x \right )+\frac {1}{x}-\ln \left (x \right )\) | \(16\) |
risch | \(x +2 \ln \left (-1+x \right )+\frac {1}{x}-\ln \left (x \right )\) | \(16\) |
norman | \(\frac {x^{2}+1}{x}-\ln \left (x \right )+2 \ln \left (-1+x \right )\) | \(21\) |
meijerg | \(\frac {1}{x}-\ln \left (x \right )-i \pi +2 \ln \left (1-x \right )+x\) | \(22\) |
parallelrisch | \(-\frac {x \ln \left (x \right )-2 \ln \left (-1+x \right ) x -x^{2}-1}{x}\) | \(24\) |
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none
Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.24 \[ \int \frac {1+x^3}{-x^2+x^3} \, dx=\frac {x^{2} + 2 \, x \log \left (x - 1\right ) - x \log \left (x\right ) + 1}{x} \]
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Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \frac {1+x^3}{-x^2+x^3} \, dx=x - \log {\left (x \right )} + 2 \log {\left (x - 1 \right )} + \frac {1}{x} \]
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none
Time = 0.19 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {1+x^3}{-x^2+x^3} \, dx=x + \frac {1}{x} + 2 \, \log \left (x - 1\right ) - \log \left (x\right ) \]
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Time = 0.29 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {1+x^3}{-x^2+x^3} \, dx=x + \frac {1}{x} + 2 \, \log \left ({\left | x - 1 \right |}\right ) - \log \left ({\left | x \right |}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {1+x^3}{-x^2+x^3} \, dx=x+2\,\ln \left (x-1\right )-\ln \left (x\right )+\frac {1}{x} \]
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