Integrand size = 11, antiderivative size = 24 \[ \int \frac {1}{x^2 (4+6 x)} \, dx=-\frac {1}{4 x}-\frac {3 \log (x)}{8}+\frac {3}{8} \log (2+3 x) \]
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Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {46} \[ \int \frac {1}{x^2 (4+6 x)} \, dx=-\frac {1}{4 x}-\frac {3 \log (x)}{8}+\frac {3}{8} \log (3 x+2) \]
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Rule 46
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{4 x^2}-\frac {3}{8 x}+\frac {9}{8 (2+3 x)}\right ) \, dx \\ & = -\frac {1}{4 x}-\frac {3 \log (x)}{8}+\frac {3}{8} \log (2+3 x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 (4+6 x)} \, dx=-\frac {1}{4 x}-\frac {3 \log (x)}{8}+\frac {3}{8} \log (2+3 x) \]
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Time = 0.05 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79
method | result | size |
default | \(-\frac {1}{4 x}-\frac {3 \ln \left (x \right )}{8}+\frac {3 \ln \left (2+3 x \right )}{8}\) | \(19\) |
norman | \(-\frac {1}{4 x}-\frac {3 \ln \left (x \right )}{8}+\frac {3 \ln \left (2+3 x \right )}{8}\) | \(19\) |
risch | \(-\frac {1}{4 x}-\frac {3 \ln \left (x \right )}{8}+\frac {3 \ln \left (2+3 x \right )}{8}\) | \(19\) |
parallelrisch | \(-\frac {3 \ln \left (x \right ) x -3 \ln \left (\frac {2}{3}+x \right ) x +2}{8 x}\) | \(20\) |
meijerg | \(-\frac {1}{4 x}-\frac {3 \ln \left (x \right )}{8}-\frac {3 \ln \left (3\right )}{8}+\frac {3 \ln \left (2\right )}{8}+\frac {3 \ln \left (1+\frac {3 x}{2}\right )}{8}\) | \(27\) |
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none
Time = 0.22 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^2 (4+6 x)} \, dx=\frac {3 \, x \log \left (3 \, x + 2\right ) - 3 \, x \log \left (x\right ) - 2}{8 \, x} \]
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Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^2 (4+6 x)} \, dx=- \frac {3 \log {\left (x \right )}}{8} + \frac {3 \log {\left (x + \frac {2}{3} \right )}}{8} - \frac {1}{4 x} \]
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none
Time = 0.21 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^2 (4+6 x)} \, dx=-\frac {1}{4 \, x} + \frac {3}{8} \, \log \left (3 \, x + 2\right ) - \frac {3}{8} \, \log \left (x\right ) \]
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none
Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^2 (4+6 x)} \, dx=-\frac {1}{4 \, x} + \frac {3}{8} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {3}{8} \, \log \left ({\left | x \right |}\right ) \]
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Time = 0.09 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^2 (4+6 x)} \, dx=-\frac {3\,\ln \left (\frac {x}{6\,x+4}\right )}{8}-\frac {1}{4\,x} \]
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