Integrand size = 11, antiderivative size = 31 \[ \int \frac {1}{-2 x^3+x^4} \, dx=\frac {1}{4 x^2}+\frac {1}{4 x}+\frac {1}{8} \log (2-x)-\frac {\log (x)}{8} \]
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Time = 0.01 (sec) , antiderivative size = 31, 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} \[ \int \frac {1}{-2 x^3+x^4} \, dx=\frac {1}{4 x^2}+\frac {1}{4 x}+\frac {1}{8} \log (2-x)-\frac {\log (x)}{8} \]
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Rule 46
Rule 1607
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(-2+x) x^3} \, dx \\ & = \int \left (\frac {1}{8 (-2+x)}-\frac {1}{2 x^3}-\frac {1}{4 x^2}-\frac {1}{8 x}\right ) \, dx \\ & = \frac {1}{4 x^2}+\frac {1}{4 x}+\frac {1}{8} \log (2-x)-\frac {\log (x)}{8} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {1}{-2 x^3+x^4} \, dx=\frac {1}{4 x^2}+\frac {1}{4 x}+\frac {1}{8} \log (2-x)-\frac {\log (x)}{8} \]
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Time = 0.05 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.68
method | result | size |
norman | \(\frac {\frac {1}{4}+\frac {x}{4}}{x^{2}}-\frac {\ln \left (x \right )}{8}+\frac {\ln \left (-2+x \right )}{8}\) | \(21\) |
risch | \(\frac {\frac {1}{4}+\frac {x}{4}}{x^{2}}-\frac {\ln \left (x \right )}{8}+\frac {\ln \left (-2+x \right )}{8}\) | \(21\) |
default | \(\frac {1}{4 x^{2}}+\frac {1}{4 x}-\frac {\ln \left (x \right )}{8}+\frac {\ln \left (-2+x \right )}{8}\) | \(22\) |
parallelrisch | \(-\frac {x^{2} \ln \left (x \right )-\ln \left (-2+x \right ) x^{2}-2-2 x}{8 x^{2}}\) | \(26\) |
meijerg | \(\frac {1}{4 x^{2}}+\frac {1}{4 x}-\frac {\ln \left (x \right )}{8}+\frac {\ln \left (2\right )}{8}-\frac {i \pi }{8}+\frac {\ln \left (1-\frac {x}{2}\right )}{8}\) | \(32\) |
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Time = 0.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81 \[ \int \frac {1}{-2 x^3+x^4} \, dx=\frac {x^{2} \log \left (x - 2\right ) - x^{2} \log \left (x\right ) + 2 \, x + 2}{8 \, x^{2}} \]
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Time = 0.05 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.61 \[ \int \frac {1}{-2 x^3+x^4} \, dx=- \frac {\log {\left (x \right )}}{8} + \frac {\log {\left (x - 2 \right )}}{8} + \frac {x + 1}{4 x^{2}} \]
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Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.61 \[ \int \frac {1}{-2 x^3+x^4} \, dx=\frac {x + 1}{4 \, x^{2}} + \frac {1}{8} \, \log \left (x - 2\right ) - \frac {1}{8} \, \log \left (x\right ) \]
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Time = 0.32 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.68 \[ \int \frac {1}{-2 x^3+x^4} \, dx=\frac {x + 1}{4 \, x^{2}} + \frac {1}{8} \, \log \left ({\left | x - 2 \right |}\right ) - \frac {1}{8} \, \log \left ({\left | x \right |}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.52 \[ \int \frac {1}{-2 x^3+x^4} \, dx=\frac {\frac {x}{4}+\frac {1}{4}}{x^2}-\frac {\mathrm {atanh}\left (x-1\right )}{4} \]
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