Integrand size = 13, antiderivative size = 8 \[ \int \frac {1}{x \left (x-x^3\right )} \, dx=-\frac {1}{x}+\text {arctanh}(x) \]
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Time = 0.01 (sec) , antiderivative size = 8, 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, 331, 212} \[ \int \frac {1}{x \left (x-x^3\right )} \, dx=\text {arctanh}(x)-\frac {1}{x} \]
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Rule 212
Rule 331
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x^2 \left (1-x^2\right )} \, dx \\ & = -\frac {1}{x}+\int \frac {1}{1-x^2} \, dx \\ & = -\frac {1}{x}+\tanh ^{-1}(x) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(24\) vs. \(2(8)=16\).
Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 3.00 \[ \int \frac {1}{x \left (x-x^3\right )} \, dx=-\frac {1}{x}-\frac {1}{2} \log (1-x)+\frac {1}{2} \log (1+x) \]
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Result contains complex when optimal does not.
Time = 2.06 (sec) , antiderivative size = 16, normalized size of antiderivative = 2.00
method | result | size |
meijerg | \(\frac {i \left (\frac {2 i}{x}-2 i \operatorname {arctanh}\left (x \right )\right )}{2}\) | \(16\) |
default | \(-\frac {\ln \left (-1+x \right )}{2}+\frac {\ln \left (1+x \right )}{2}-\frac {1}{x}\) | \(19\) |
norman | \(-\frac {\ln \left (-1+x \right )}{2}+\frac {\ln \left (1+x \right )}{2}-\frac {1}{x}\) | \(19\) |
risch | \(-\frac {\ln \left (-1+x \right )}{2}+\frac {\ln \left (1+x \right )}{2}-\frac {1}{x}\) | \(19\) |
parallelrisch | \(\frac {\ln \left (1+x \right ) x -\ln \left (-1+x \right ) x -2}{2 x}\) | \(21\) |
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Leaf count of result is larger than twice the leaf count of optimal. 20 vs. \(2 (8) = 16\).
Time = 0.35 (sec) , antiderivative size = 20, normalized size of antiderivative = 2.50 \[ \int \frac {1}{x \left (x-x^3\right )} \, dx=\frac {x \log \left (x + 1\right ) - x \log \left (x - 1\right ) - 2}{2 \, x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 15 vs. \(2 (5) = 10\).
Time = 0.05 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.88 \[ \int \frac {1}{x \left (x-x^3\right )} \, dx=- \frac {\log {\left (x - 1 \right )}}{2} + \frac {\log {\left (x + 1 \right )}}{2} - \frac {1}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 18 vs. \(2 (8) = 16\).
Time = 0.19 (sec) , antiderivative size = 18, normalized size of antiderivative = 2.25 \[ \int \frac {1}{x \left (x-x^3\right )} \, dx=-\frac {1}{x} + \frac {1}{2} \, \log \left (x + 1\right ) - \frac {1}{2} \, \log \left (x - 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 20 vs. \(2 (8) = 16\).
Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 2.50 \[ \int \frac {1}{x \left (x-x^3\right )} \, dx=-\frac {1}{x} + \frac {1}{2} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{2} \, \log \left ({\left | x - 1 \right |}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (x-x^3\right )} \, dx=\mathrm {atanh}\left (x\right )-\frac {1}{x} \]
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