Integrand size = 11, antiderivative size = 38 \[ \int \frac {1}{x^4 (4+6 x)} \, dx=-\frac {1}{12 x^3}+\frac {3}{16 x^2}-\frac {9}{16 x}-\frac {27 \log (x)}{32}+\frac {27}{32} \log (2+3 x) \]
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Time = 0.01 (sec) , antiderivative size = 38, 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} \[ \int \frac {1}{x^4 (4+6 x)} \, dx=-\frac {1}{12 x^3}+\frac {3}{16 x^2}-\frac {9}{16 x}-\frac {27 \log (x)}{32}+\frac {27}{32} \log (3 x+2) \]
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Rule 46
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{4 x^4}-\frac {3}{8 x^3}+\frac {9}{16 x^2}-\frac {27}{32 x}+\frac {81}{32 (2+3 x)}\right ) \, dx \\ & = -\frac {1}{12 x^3}+\frac {3}{16 x^2}-\frac {9}{16 x}-\frac {27 \log (x)}{32}+\frac {27}{32} \log (2+3 x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^4 (4+6 x)} \, dx=-\frac {1}{12 x^3}+\frac {3}{16 x^2}-\frac {9}{16 x}-\frac {27 \log (x)}{32}+\frac {27}{32} \log (2+3 x) \]
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Time = 0.05 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.74
method | result | size |
norman | \(\frac {-\frac {1}{12}+\frac {3}{16} x -\frac {9}{16} x^{2}}{x^{3}}-\frac {27 \ln \left (x \right )}{32}+\frac {27 \ln \left (2+3 x \right )}{32}\) | \(28\) |
risch | \(\frac {-\frac {1}{12}+\frac {3}{16} x -\frac {9}{16} x^{2}}{x^{3}}-\frac {27 \ln \left (x \right )}{32}+\frac {27 \ln \left (2+3 x \right )}{32}\) | \(28\) |
default | \(-\frac {1}{12 x^{3}}+\frac {3}{16 x^{2}}-\frac {9}{16 x}-\frac {27 \ln \left (x \right )}{32}+\frac {27 \ln \left (2+3 x \right )}{32}\) | \(29\) |
parallelrisch | \(-\frac {81 \ln \left (x \right ) x^{3}-81 \ln \left (\frac {2}{3}+x \right ) x^{3}+8+54 x^{2}-18 x}{96 x^{3}}\) | \(32\) |
meijerg | \(-\frac {1}{12 x^{3}}+\frac {3}{16 x^{2}}-\frac {9}{16 x}-\frac {27 \ln \left (x \right )}{32}-\frac {27 \ln \left (3\right )}{32}+\frac {27 \ln \left (2\right )}{32}+\frac {27 \ln \left (1+\frac {3 x}{2}\right )}{32}\) | \(37\) |
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Time = 0.22 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x^4 (4+6 x)} \, dx=\frac {81 \, x^{3} \log \left (3 \, x + 2\right ) - 81 \, x^{3} \log \left (x\right ) - 54 \, x^{2} + 18 \, x - 8}{96 \, x^{3}} \]
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Time = 0.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x^4 (4+6 x)} \, dx=- \frac {27 \log {\left (x \right )}}{32} + \frac {27 \log {\left (x + \frac {2}{3} \right )}}{32} + \frac {- 27 x^{2} + 9 x - 4}{48 x^{3}} \]
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Time = 0.21 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x^4 (4+6 x)} \, dx=-\frac {27 \, x^{2} - 9 \, x + 4}{48 \, x^{3}} + \frac {27}{32} \, \log \left (3 \, x + 2\right ) - \frac {27}{32} \, \log \left (x\right ) \]
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Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.79 \[ \int \frac {1}{x^4 (4+6 x)} \, dx=-\frac {27 \, x^{2} - 9 \, x + 4}{48 \, x^{3}} + \frac {27}{32} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {27}{32} \, \log \left ({\left | x \right |}\right ) \]
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Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.63 \[ \int \frac {1}{x^4 (4+6 x)} \, dx=\frac {27\,\mathrm {atanh}\left (3\,x+1\right )}{16}-\frac {\frac {9\,x^2}{16}-\frac {3\,x}{16}+\frac {1}{12}}{x^3} \]
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