Integrand size = 13, antiderivative size = 6 \[ \int \frac {x^3}{x-x^3} \, dx=-x+\text {arctanh}(x) \]
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Time = 0.01 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1598, 327, 212} \[ \int \frac {x^3}{x-x^3} \, dx=\text {arctanh}(x)-x \]
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Rule 212
Rule 327
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^2}{1-x^2} \, dx \\ & = -x+\int \frac {1}{1-x^2} \, dx \\ & = -x+\tanh ^{-1}(x) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(22\) vs. \(2(6)=12\).
Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 3.67 \[ \int \frac {x^3}{x-x^3} \, dx=-x-\frac {1}{2} \log (1-x)+\frac {1}{2} \log (1+x) \]
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Result contains complex when optimal does not.
Time = 2.12 (sec) , antiderivative size = 14, normalized size of antiderivative = 2.33
method | result | size |
meijerg | \(\frac {i \left (2 i x -2 i \operatorname {arctanh}\left (x \right )\right )}{2}\) | \(14\) |
default | \(-x -\frac {\ln \left (-1+x \right )}{2}+\frac {\ln \left (1+x \right )}{2}\) | \(17\) |
norman | \(-x -\frac {\ln \left (-1+x \right )}{2}+\frac {\ln \left (1+x \right )}{2}\) | \(17\) |
risch | \(-x -\frac {\ln \left (-1+x \right )}{2}+\frac {\ln \left (1+x \right )}{2}\) | \(17\) |
parallelrisch | \(-x -\frac {\ln \left (-1+x \right )}{2}+\frac {\ln \left (1+x \right )}{2}\) | \(17\) |
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Leaf count of result is larger than twice the leaf count of optimal. 16 vs. \(2 (6) = 12\).
Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 2.67 \[ \int \frac {x^3}{x-x^3} \, dx=-x + \frac {1}{2} \, \log \left (x + 1\right ) - \frac {1}{2} \, \log \left (x - 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 14 vs. \(2 (3) = 6\).
Time = 0.04 (sec) , antiderivative size = 14, normalized size of antiderivative = 2.33 \[ \int \frac {x^3}{x-x^3} \, dx=- x - \frac {\log {\left (x - 1 \right )}}{2} + \frac {\log {\left (x + 1 \right )}}{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 16 vs. \(2 (6) = 12\).
Time = 0.19 (sec) , antiderivative size = 16, normalized size of antiderivative = 2.67 \[ \int \frac {x^3}{x-x^3} \, dx=-x + \frac {1}{2} \, \log \left (x + 1\right ) - \frac {1}{2} \, \log \left (x - 1\right ) \]
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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 = 3.00 \[ \int \frac {x^3}{x-x^3} \, dx=-x + \frac {1}{2} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{2} \, \log \left ({\left | x - 1 \right |}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \frac {x^3}{x-x^3} \, dx=\mathrm {atanh}\left (x\right )-x \]
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