Integrand size = 26, antiderivative size = 25 \[ \int \frac {1-x-x^2+x^3+x^4}{-x+x^3} \, dx=x+\frac {x^2}{2}-\log (x)+\frac {1}{2} \log \left (1-x^2\right ) \]
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Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1607, 1816, 266} \[ \int \frac {1-x-x^2+x^3+x^4}{-x+x^3} \, dx=\frac {x^2}{2}+\frac {1}{2} \log \left (1-x^2\right )+x-\log (x) \]
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Rule 266
Rule 1607
Rule 1816
Rubi steps \begin{align*} \text {integral}& = \int \frac {1-x-x^2+x^3+x^4}{x \left (-1+x^2\right )} \, dx \\ & = \int \left (1-\frac {1}{x}+x+\frac {x}{-1+x^2}\right ) \, dx \\ & = x+\frac {x^2}{2}-\log (x)+\int \frac {x}{-1+x^2} \, dx \\ & = x+\frac {x^2}{2}-\log (x)+\frac {1}{2} \log \left (1-x^2\right ) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {1-x-x^2+x^3+x^4}{-x+x^3} \, dx=x+\frac {x^2}{2}-\log (x)+\frac {1}{2} \log \left (1-x^2\right ) \]
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Time = 0.84 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80
method | result | size |
risch | \(\frac {x^{2}}{2}+x -\ln \left (x \right )+\frac {\ln \left (x^{2}-1\right )}{2}\) | \(20\) |
default | \(\frac {x^{2}}{2}+x -\ln \left (x \right )+\frac {\ln \left (x +1\right )}{2}+\frac {\ln \left (x -1\right )}{2}\) | \(24\) |
norman | \(\frac {x^{2}}{2}+x -\ln \left (x \right )+\frac {\ln \left (x +1\right )}{2}+\frac {\ln \left (x -1\right )}{2}\) | \(24\) |
parallelrisch | \(\frac {x^{2}}{2}+x -\ln \left (x \right )+\frac {\ln \left (x +1\right )}{2}+\frac {\ln \left (x -1\right )}{2}\) | \(24\) |
meijerg | \(\frac {\ln \left (-x^{2}+1\right )}{2}-\ln \left (x \right )-\frac {i \pi }{2}+\frac {x^{2}}{2}-\frac {i \left (2 i x -2 i \operatorname {arctanh}\left (x \right )\right )}{2}+\operatorname {arctanh}\left (x \right )\) | \(40\) |
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Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {1-x-x^2+x^3+x^4}{-x+x^3} \, dx=\frac {1}{2} \, x^{2} + x + \frac {1}{2} \, \log \left (x^{2} - 1\right ) - \log \left (x\right ) \]
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Time = 0.04 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.68 \[ \int \frac {1-x-x^2+x^3+x^4}{-x+x^3} \, dx=\frac {x^{2}}{2} + x - \log {\left (x \right )} + \frac {\log {\left (x^{2} - 1 \right )}}{2} \]
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Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {1-x-x^2+x^3+x^4}{-x+x^3} \, dx=\frac {1}{2} \, x^{2} + x + \frac {1}{2} \, \log \left (x + 1\right ) + \frac {1}{2} \, \log \left (x - 1\right ) - \log \left (x\right ) \]
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Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {1-x-x^2+x^3+x^4}{-x+x^3} \, dx=\frac {1}{2} \, x^{2} + x + \frac {1}{2} \, \log \left ({\left | x + 1 \right |}\right ) + \frac {1}{2} \, \log \left ({\left | x - 1 \right |}\right ) - \log \left ({\left | x \right |}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {1-x-x^2+x^3+x^4}{-x+x^3} \, dx=x+\frac {\ln \left (x^2-1\right )}{2}-\ln \left (x\right )+\frac {x^2}{2} \]
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