Optimal. Leaf size=18 \[ \frac {\log \left (\frac {3}{x}\right )}{12 x^3 (2+x)} \]
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Rubi [B] time = 0.26, antiderivative size = 62, normalized size of antiderivative = 3.44, number of steps used = 14, number of rules used = 9, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1594, 27, 12, 6742, 44, 2357, 2304, 2314, 31} \begin {gather*} \frac {\log \left (\frac {3}{x}\right )}{24 x^3}-\frac {\log \left (\frac {3}{x}\right )}{48 x^2}+\frac {\log \left (\frac {3}{x}\right )}{96 x}+\frac {x \log \left (\frac {3}{x}\right )}{192 (x+2)}+\frac {\log (x)}{192} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 31
Rule 44
Rule 1594
Rule 2304
Rule 2314
Rule 2357
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2-x+(-6-4 x) \log \left (\frac {3}{x}\right )}{x^4 \left (48+48 x+12 x^2\right )} \, dx\\ &=\int \frac {-2-x+(-6-4 x) \log \left (\frac {3}{x}\right )}{12 x^4 (2+x)^2} \, dx\\ &=\frac {1}{12} \int \frac {-2-x+(-6-4 x) \log \left (\frac {3}{x}\right )}{x^4 (2+x)^2} \, dx\\ &=\frac {1}{12} \int \left (-\frac {1}{x^4 (2+x)}-\frac {2 (3+2 x) \log \left (\frac {3}{x}\right )}{x^4 (2+x)^2}\right ) \, dx\\ &=-\left (\frac {1}{12} \int \frac {1}{x^4 (2+x)} \, dx\right )-\frac {1}{6} \int \frac {(3+2 x) \log \left (\frac {3}{x}\right )}{x^4 (2+x)^2} \, dx\\ &=-\left (\frac {1}{12} \int \left (\frac {1}{2 x^4}-\frac {1}{4 x^3}+\frac {1}{8 x^2}-\frac {1}{16 x}+\frac {1}{16 (2+x)}\right ) \, dx\right )-\frac {1}{6} \int \left (\frac {3 \log \left (\frac {3}{x}\right )}{4 x^4}-\frac {\log \left (\frac {3}{x}\right )}{4 x^3}+\frac {\log \left (\frac {3}{x}\right )}{16 x^2}-\frac {\log \left (\frac {3}{x}\right )}{16 (2+x)^2}\right ) \, dx\\ &=\frac {1}{72 x^3}-\frac {1}{96 x^2}+\frac {1}{96 x}+\frac {\log (x)}{192}-\frac {1}{192} \log (2+x)-\frac {1}{96} \int \frac {\log \left (\frac {3}{x}\right )}{x^2} \, dx+\frac {1}{96} \int \frac {\log \left (\frac {3}{x}\right )}{(2+x)^2} \, dx+\frac {1}{24} \int \frac {\log \left (\frac {3}{x}\right )}{x^3} \, dx-\frac {1}{8} \int \frac {\log \left (\frac {3}{x}\right )}{x^4} \, dx\\ &=\frac {\log \left (\frac {3}{x}\right )}{24 x^3}-\frac {\log \left (\frac {3}{x}\right )}{48 x^2}+\frac {\log \left (\frac {3}{x}\right )}{96 x}+\frac {x \log \left (\frac {3}{x}\right )}{192 (2+x)}+\frac {\log (x)}{192}-\frac {1}{192} \log (2+x)+\frac {1}{192} \int \frac {1}{2+x} \, dx\\ &=\frac {\log \left (\frac {3}{x}\right )}{24 x^3}-\frac {\log \left (\frac {3}{x}\right )}{48 x^2}+\frac {\log \left (\frac {3}{x}\right )}{96 x}+\frac {x \log \left (\frac {3}{x}\right )}{192 (2+x)}+\frac {\log (x)}{192}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.07, size = 21, normalized size = 1.17 \begin {gather*} \frac {\log (6561)+8 \log \left (\frac {1}{x}\right )}{96 x^3 (2+x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 19, normalized size = 1.06 \begin {gather*} \frac {\log \left (\frac {3}{x}\right )}{12 \, {\left (x^{4} + 2 \, x^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 42, normalized size = 2.33 \begin {gather*} \frac {1}{192} \, {\left (\frac {2}{x} + \frac {1}{\frac {2}{x} + 1} - \frac {4}{x^{2}} + \frac {8}{x^{3}}\right )} \log \left (\frac {3}{x}\right ) - \frac {1}{192} \, \log \left (\frac {3}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 17, normalized size = 0.94
method | result | size |
norman | \(\frac {\ln \left (\frac {3}{x}\right )}{12 x^{3} \left (2+x \right )}\) | \(17\) |
risch | \(\frac {\ln \left (\frac {3}{x}\right )}{12 x^{3} \left (2+x \right )}\) | \(17\) |
derivativedivides | \(\frac {\ln \left (\frac {3}{x}\right )}{24 x^{3}}-\frac {\ln \left (\frac {3}{x}\right )}{48 x^{2}}+\frac {\ln \left (\frac {3}{x}\right )}{96 x}-\frac {\ln \left (\frac {3}{x}\right )}{32 x \left (\frac {6}{x}+3\right )}\) | \(55\) |
default | \(\frac {\ln \left (\frac {3}{x}\right )}{24 x^{3}}-\frac {\ln \left (\frac {3}{x}\right )}{48 x^{2}}+\frac {\ln \left (\frac {3}{x}\right )}{96 x}-\frac {\ln \left (\frac {3}{x}\right )}{32 x \left (\frac {6}{x}+3\right )}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.90, size = 79, normalized size = 4.39 \begin {gather*} -\frac {6 \, x^{3} + 6 \, x^{2} + 3 \, {\left (x^{4} + 2 \, x^{3} + 4\right )} \log \relax (x) - 4 \, x - 12 \, \log \relax (3) + 4}{144 \, {\left (x^{4} + 2 \, x^{3}\right )}} + \frac {3 \, x^{3} + 3 \, x^{2} - 2 \, x + 2}{72 \, {\left (x^{4} + 2 \, x^{3}\right )}} + \frac {1}{48} \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.04, size = 22, normalized size = 1.22 \begin {gather*} \frac {x\,\left (\ln \left (\frac {1}{x}\right )+\ln \relax (3)\right )}{12\,\left (x^5+2\,x^4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 14, normalized size = 0.78 \begin {gather*} \frac {\log {\left (\frac {3}{x} \right )}}{12 x^{4} + 24 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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