Integrand size = 40, antiderivative size = 18 \[ \int \frac {-2+6 x^2+\left (1-x^2\right ) \log \left (x^2-2 x^4+x^6\right )}{-x^2+x^4} \, dx=-\frac {1}{4}+\frac {\log \left (\left (-x+x^3\right )^2\right )}{x} \]
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Time = 0.14 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1607, 6857, 464, 213, 2605, 212} \[ \int \frac {-2+6 x^2+\left (1-x^2\right ) \log \left (x^2-2 x^4+x^6\right )}{-x^2+x^4} \, dx=\frac {\log \left (x^2 \left (1-x^2\right )^2\right )}{x} \]
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Rule 212
Rule 213
Rule 464
Rule 1607
Rule 2605
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \frac {-2+6 x^2+\left (1-x^2\right ) \log \left (x^2-2 x^4+x^6\right )}{x^2 \left (-1+x^2\right )} \, dx \\ & = \int \left (\frac {2 \left (-1+3 x^2\right )}{x^2 \left (-1+x^2\right )}-\frac {\log \left (x^2 \left (-1+x^2\right )^2\right )}{x^2}\right ) \, dx \\ & = 2 \int \frac {-1+3 x^2}{x^2 \left (-1+x^2\right )} \, dx-\int \frac {\log \left (x^2 \left (-1+x^2\right )^2\right )}{x^2} \, dx \\ & = -\frac {2}{x}+\frac {\log \left (x^2 \left (1-x^2\right )^2\right )}{x}+4 \int \frac {1}{-1+x^2} \, dx-\int \frac {2-6 x^2}{x^2 \left (1-x^2\right )} \, dx \\ & = -4 \text {arctanh}(x)+\frac {\log \left (x^2 \left (1-x^2\right )^2\right )}{x}+4 \int \frac {1}{1-x^2} \, dx \\ & = \frac {\log \left (x^2 \left (1-x^2\right )^2\right )}{x} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {-2+6 x^2+\left (1-x^2\right ) \log \left (x^2-2 x^4+x^6\right )}{-x^2+x^4} \, dx=\frac {\log \left (x^2 \left (-1+x^2\right )^2\right )}{x} \]
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Time = 0.14 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00
method | result | size |
default | \(\frac {\ln \left (x^{6}-2 x^{4}+x^{2}\right )}{x}\) | \(18\) |
norman | \(\frac {\ln \left (x^{6}-2 x^{4}+x^{2}\right )}{x}\) | \(18\) |
risch | \(\frac {\ln \left (x^{6}-2 x^{4}+x^{2}\right )}{x}\) | \(18\) |
parallelrisch | \(\frac {\ln \left (x^{6}-2 x^{4}+x^{2}\right )}{x}\) | \(18\) |
parts | \(\frac {\ln \left (x^{6}-2 x^{4}+x^{2}\right )}{x}\) | \(18\) |
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Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {-2+6 x^2+\left (1-x^2\right ) \log \left (x^2-2 x^4+x^6\right )}{-x^2+x^4} \, dx=\frac {\log \left (x^{6} - 2 \, x^{4} + x^{2}\right )}{x} \]
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Time = 0.07 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {-2+6 x^2+\left (1-x^2\right ) \log \left (x^2-2 x^4+x^6\right )}{-x^2+x^4} \, dx=\frac {\log {\left (x^{6} - 2 x^{4} + x^{2} \right )}}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (16) = 32\).
Time = 0.23 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.44 \[ \int \frac {-2+6 x^2+\left (1-x^2\right ) \log \left (x^2-2 x^4+x^6\right )}{-x^2+x^4} \, dx=\frac {2 \, {\left ({\left (x + 1\right )} \log \left (x + 1\right ) - {\left (x - 1\right )} \log \left (x - 1\right ) + \log \left (x\right ) + 1\right )}}{x} - \frac {2}{x} - 2 \, \log \left (x + 1\right ) + 2 \, \log \left (x - 1\right ) \]
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Time = 0.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {-2+6 x^2+\left (1-x^2\right ) \log \left (x^2-2 x^4+x^6\right )}{-x^2+x^4} \, dx=\frac {\log \left (x^{6} - 2 \, x^{4} + x^{2}\right )}{x} \]
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Time = 8.30 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {-2+6 x^2+\left (1-x^2\right ) \log \left (x^2-2 x^4+x^6\right )}{-x^2+x^4} \, dx=\frac {\ln \left (x^2\,{\left (x^2-1\right )}^2\right )}{x} \]
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