Integrand size = 12, antiderivative size = 16 \[ \int -\frac {18 x}{-1+9 x^2} \, dx=\frac {1}{4}+\log \left (\frac {2}{-1+9 x^2}\right ) \]
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Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.62, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {12, 266} \[ \int -\frac {18 x}{-1+9 x^2} \, dx=-\log \left (1-9 x^2\right ) \]
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Rule 12
Rule 266
Rubi steps \begin{align*} \text {integral}& = -\left (18 \int \frac {x}{-1+9 x^2} \, dx\right ) \\ & = -\log \left (1-9 x^2\right ) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.62 \[ \int -\frac {18 x}{-1+9 x^2} \, dx=-\log \left (1-9 x^2\right ) \]
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Time = 0.07 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.69
method | result | size |
derivativedivides | \(-\ln \left (9 x^{2}-1\right )\) | \(11\) |
default | \(-\ln \left (9 x^{2}-1\right )\) | \(11\) |
meijerg | \(-\ln \left (-9 x^{2}+1\right )\) | \(11\) |
risch | \(-\ln \left (9 x^{2}-1\right )\) | \(11\) |
parallelrisch | \(-\ln \left (x -\frac {1}{3}\right )-\ln \left (x +\frac {1}{3}\right )\) | \(14\) |
norman | \(-\ln \left (-1+3 x \right )-\ln \left (1+3 x \right )\) | \(18\) |
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none
Time = 0.23 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.62 \[ \int -\frac {18 x}{-1+9 x^2} \, dx=-\log \left (9 \, x^{2} - 1\right ) \]
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Time = 0.03 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.50 \[ \int -\frac {18 x}{-1+9 x^2} \, dx=- \log {\left (9 x^{2} - 1 \right )} \]
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none
Time = 0.18 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.62 \[ \int -\frac {18 x}{-1+9 x^2} \, dx=-\log \left (9 \, x^{2} - 1\right ) \]
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none
Time = 0.29 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.69 \[ \int -\frac {18 x}{-1+9 x^2} \, dx=-\log \left ({\left | 9 \, x^{2} - 1 \right |}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.50 \[ \int -\frac {18 x}{-1+9 x^2} \, dx=-\ln \left (x^2-\frac {1}{9}\right ) \]
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