Integrand size = 11, antiderivative size = 21 \[ \int \frac {1}{-x^3+x^4} \, dx=\frac {1}{2 x^2}+\frac {1}{x}+\log (1-x)-\log (x) \]
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Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1607, 46} \[ \int \frac {1}{-x^3+x^4} \, dx=\frac {1}{2 x^2}+\frac {1}{x}+\log (1-x)-\log (x) \]
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Rule 46
Rule 1607
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(-1+x) x^3} \, dx \\ & = \int \left (\frac {1}{-1+x}-\frac {1}{x^3}-\frac {1}{x^2}-\frac {1}{x}\right ) \, dx \\ & = \frac {1}{2 x^2}+\frac {1}{x}+\log (1-x)-\log (x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {1}{-x^3+x^4} \, dx=\frac {1}{2 x^2}+\frac {1}{x}+\log (1-x)-\log (x) \]
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Time = 0.15 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81
method | result | size |
norman | \(\frac {x +\frac {1}{2}}{x^{2}}-\ln \left (x \right )+\ln \left (-1+x \right )\) | \(17\) |
risch | \(\frac {x +\frac {1}{2}}{x^{2}}-\ln \left (x \right )+\ln \left (-1+x \right )\) | \(17\) |
default | \(\ln \left (-1+x \right )+\frac {1}{2 x^{2}}+\frac {1}{x}-\ln \left (x \right )\) | \(18\) |
meijerg | \(\frac {1}{2 x^{2}}+\frac {1}{x}-\ln \left (x \right )-i \pi +\ln \left (1-x \right )\) | \(24\) |
parallelrisch | \(-\frac {2 x^{2} \ln \left (x \right )-2 \ln \left (-1+x \right ) x^{2}-1-2 x}{2 x^{2}}\) | \(27\) |
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Time = 0.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24 \[ \int \frac {1}{-x^3+x^4} \, dx=\frac {2 \, x^{2} \log \left (x - 1\right ) - 2 \, x^{2} \log \left (x\right ) + 2 \, x + 1}{2 \, x^{2}} \]
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Time = 0.04 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {1}{-x^3+x^4} \, dx=- \log {\left (x \right )} + \log {\left (x - 1 \right )} + \frac {2 x + 1}{2 x^{2}} \]
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Time = 0.21 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {1}{-x^3+x^4} \, dx=\frac {2 \, x + 1}{2 \, x^{2}} + \log \left (x - 1\right ) - \log \left (x\right ) \]
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Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {1}{-x^3+x^4} \, dx=\frac {2 \, x + 1}{2 \, x^{2}} + \log \left ({\left | x - 1 \right |}\right ) - \log \left ({\left | x \right |}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76 \[ \int \frac {1}{-x^3+x^4} \, dx=\frac {x+\frac {1}{2}}{x^2}-2\,\mathrm {atanh}\left (2\,x-1\right ) \]
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