Integrand size = 9, antiderivative size = 8 \[ \int \frac {x}{-1+x^4} \, dx=-\frac {1}{2} \text {arctanh}\left (x^2\right ) \]
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Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {281, 213} \[ \int \frac {x}{-1+x^4} \, dx=-\frac {1}{2} \text {arctanh}\left (x^2\right ) \]
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Rule 213
Rule 281
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,x^2\right ) \\ & = -\frac {1}{2} \text {arctanh}\left (x^2\right ) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(23\) vs. \(2(8)=16\).
Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 2.88 \[ \int \frac {x}{-1+x^4} \, dx=\frac {1}{4} \log \left (1-x^2\right )-\frac {1}{4} \log \left (1+x^2\right ) \]
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Time = 0.10 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.88
method | result | size |
meijerg | \(-\frac {\operatorname {arctanh}\left (x^{2}\right )}{2}\) | \(7\) |
risch | \(-\frac {\ln \left (x^{2}+1\right )}{4}+\frac {\ln \left (x^{2}-1\right )}{4}\) | \(18\) |
default | \(\frac {\ln \left (-1+x \right )}{4}+\frac {\ln \left (1+x \right )}{4}-\frac {\ln \left (x^{2}+1\right )}{4}\) | \(22\) |
norman | \(\frac {\ln \left (-1+x \right )}{4}+\frac {\ln \left (1+x \right )}{4}-\frac {\ln \left (x^{2}+1\right )}{4}\) | \(22\) |
parallelrisch | \(\frac {\ln \left (-1+x \right )}{4}+\frac {\ln \left (1+x \right )}{4}-\frac {\ln \left (x^{2}+1\right )}{4}\) | \(22\) |
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Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (6) = 12\).
Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 2.12 \[ \int \frac {x}{-1+x^4} \, dx=-\frac {1}{4} \, \log \left (x^{2} + 1\right ) + \frac {1}{4} \, \log \left (x^{2} - 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 15 vs. \(2 (7) = 14\).
Time = 0.04 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.88 \[ \int \frac {x}{-1+x^4} \, dx=\frac {\log {\left (x^{2} - 1 \right )}}{4} - \frac {\log {\left (x^{2} + 1 \right )}}{4} \]
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Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (6) = 12\).
Time = 0.20 (sec) , antiderivative size = 17, normalized size of antiderivative = 2.12 \[ \int \frac {x}{-1+x^4} \, dx=-\frac {1}{4} \, \log \left (x^{2} + 1\right ) + \frac {1}{4} \, \log \left (x^{2} - 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 18 vs. \(2 (6) = 12\).
Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 2.25 \[ \int \frac {x}{-1+x^4} \, dx=-\frac {1}{4} \, \log \left (x^{2} + 1\right ) + \frac {1}{4} \, \log \left ({\left | x^{2} - 1 \right |}\right ) \]
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Time = 0.07 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \frac {x}{-1+x^4} \, dx=-\frac {\mathrm {atanh}\left (x^2\right )}{2} \]
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