Integrand size = 22, antiderivative size = 15 \[ \int \frac {-x+2 x^3}{1-x^2+x^4} \, dx=\frac {1}{2} \log \left (1-x^2+x^4\right ) \]
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Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {1601} \[ \int \frac {-x+2 x^3}{1-x^2+x^4} \, dx=\frac {1}{2} \log \left (x^4-x^2+1\right ) \]
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Rule 1601
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \log \left (1-x^2+x^4\right ) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {-x+2 x^3}{1-x^2+x^4} \, dx=\frac {1}{2} \log \left (1-x^2+x^4\right ) \]
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Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93
method | result | size |
default | \(\frac {\ln \left (x^{4}-x^{2}+1\right )}{2}\) | \(14\) |
norman | \(\frac {\ln \left (x^{4}-x^{2}+1\right )}{2}\) | \(14\) |
risch | \(\frac {\ln \left (x^{4}-x^{2}+1\right )}{2}\) | \(14\) |
parallelrisch | \(\frac {\ln \left (x^{4}-x^{2}+1\right )}{2}\) | \(14\) |
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none
Time = 0.24 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {-x+2 x^3}{1-x^2+x^4} \, dx=\frac {1}{2} \, \log \left (x^{4} - x^{2} + 1\right ) \]
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Time = 0.04 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.67 \[ \int \frac {-x+2 x^3}{1-x^2+x^4} \, dx=\frac {\log {\left (x^{4} - x^{2} + 1 \right )}}{2} \]
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none
Time = 0.19 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {-x+2 x^3}{1-x^2+x^4} \, dx=\frac {1}{2} \, \log \left (x^{4} - x^{2} + 1\right ) \]
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none
Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {-x+2 x^3}{1-x^2+x^4} \, dx=\frac {1}{2} \, \log \left (x^{4} - x^{2} + 1\right ) \]
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Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {-x+2 x^3}{1-x^2+x^4} \, dx=\frac {\ln \left (x^4-x^2+1\right )}{2} \]
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