Optimal. Leaf size=23 \[ \frac {x^4 (9+x) \log (\log (x))}{3-\frac {x}{-2+\log (x)}} \]
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Rubi [F] time = 3.97, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {108 x^3+30 x^4+2 x^5+\left (-108 x^3-21 x^4-x^5\right ) \log (x)+\left (27 x^3+3 x^4\right ) \log ^2(x)+\left (\left (432 x^3+105 x^4+7 x^5\right ) \log (x)+\left (-432 x^3-87 x^4-4 x^5\right ) \log ^2(x)+\left (108 x^3+15 x^4\right ) \log ^3(x)\right ) \log (\log (x))}{\left (36+12 x+x^2\right ) \log (x)+(-36-6 x) \log ^2(x)+9 \log ^3(x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^3 \left (2 \left (54+15 x+x^2\right )+3 (36+5 x) \log ^3(x) \log (\log (x))+\log ^2(x) \left (3 (9+x)-\left (432+87 x+4 x^2\right ) \log (\log (x))\right )+\log (x) \left (-108-21 x-x^2+\left (432+105 x+7 x^2\right ) \log (\log (x))\right )\right )}{(6+x-3 \log (x))^2 \log (x)} \, dx\\ &=\int \left (-\frac {108 x^3}{(6+x-3 \log (x))^2}-\frac {21 x^4}{(6+x-3 \log (x))^2}-\frac {x^5}{(6+x-3 \log (x))^2}+\frac {2 x^3 (6+x) (9+x)}{(6+x-3 \log (x))^2 \log (x)}+\frac {3 x^3 (9+x) \log (x)}{(6+x-3 \log (x))^2}-\frac {x^3 \left (-432-105 x-7 x^2+432 \log (x)+87 x \log (x)+4 x^2 \log (x)-108 \log ^2(x)-15 x \log ^2(x)\right ) \log (\log (x))}{(6+x-3 \log (x))^2}\right ) \, dx\\ &=2 \int \frac {x^3 (6+x) (9+x)}{(6+x-3 \log (x))^2 \log (x)} \, dx+3 \int \frac {x^3 (9+x) \log (x)}{(6+x-3 \log (x))^2} \, dx-21 \int \frac {x^4}{(6+x-3 \log (x))^2} \, dx-108 \int \frac {x^3}{(6+x-3 \log (x))^2} \, dx-\int \frac {x^5}{(6+x-3 \log (x))^2} \, dx-\int \frac {x^3 \left (-432-105 x-7 x^2+432 \log (x)+87 x \log (x)+4 x^2 \log (x)-108 \log ^2(x)-15 x \log ^2(x)\right ) \log (\log (x))}{(6+x-3 \log (x))^2} \, dx\\ &=2 \int \left (\frac {3 x^3 (9+x)}{(6+x-3 \log (x))^2}+\frac {3 x^3 (9+x)}{(6+x) (6+x-3 \log (x))}+\frac {x^3 (9+x)}{(6+x) \log (x)}\right ) \, dx+3 \int \left (\frac {x^3 \left (54+15 x+x^2\right )}{3 (6+x-3 \log (x))^2}-\frac {x^3 (9+x)}{3 (6+x-3 \log (x))}\right ) \, dx-21 \int \frac {x^4}{(6+x-3 \log (x))^2} \, dx-108 \int \frac {x^3}{(6+x-3 \log (x))^2} \, dx-\int \frac {x^5}{(6+x-3 \log (x))^2} \, dx-\int \left (-\frac {432 x^3 \log (\log (x))}{(6+x-3 \log (x))^2}-\frac {105 x^4 \log (\log (x))}{(6+x-3 \log (x))^2}-\frac {7 x^5 \log (\log (x))}{(6+x-3 \log (x))^2}+\frac {432 x^3 \log (x) \log (\log (x))}{(6+x-3 \log (x))^2}+\frac {87 x^4 \log (x) \log (\log (x))}{(6+x-3 \log (x))^2}+\frac {4 x^5 \log (x) \log (\log (x))}{(6+x-3 \log (x))^2}-\frac {108 x^3 \log ^2(x) \log (\log (x))}{(6+x-3 \log (x))^2}-\frac {15 x^4 \log ^2(x) \log (\log (x))}{(6+x-3 \log (x))^2}\right ) \, dx\\ &=2 \int \frac {x^3 (9+x)}{(6+x) \log (x)} \, dx-4 \int \frac {x^5 \log (x) \log (\log (x))}{(6+x-3 \log (x))^2} \, dx+6 \int \frac {x^3 (9+x)}{(6+x-3 \log (x))^2} \, dx+6 \int \frac {x^3 (9+x)}{(6+x) (6+x-3 \log (x))} \, dx+7 \int \frac {x^5 \log (\log (x))}{(6+x-3 \log (x))^2} \, dx+15 \int \frac {x^4 \log ^2(x) \log (\log (x))}{(6+x-3 \log (x))^2} \, dx-21 \int \frac {x^4}{(6+x-3 \log (x))^2} \, dx-87 \int \frac {x^4 \log (x) \log (\log (x))}{(6+x-3 \log (x))^2} \, dx+105 \int \frac {x^4 \log (\log (x))}{(6+x-3 \log (x))^2} \, dx-108 \int \frac {x^3}{(6+x-3 \log (x))^2} \, dx+108 \int \frac {x^3 \log ^2(x) \log (\log (x))}{(6+x-3 \log (x))^2} \, dx+432 \int \frac {x^3 \log (\log (x))}{(6+x-3 \log (x))^2} \, dx-432 \int \frac {x^3 \log (x) \log (\log (x))}{(6+x-3 \log (x))^2} \, dx-\int \frac {x^5}{(6+x-3 \log (x))^2} \, dx+\int \frac {x^3 \left (54+15 x+x^2\right )}{(6+x-3 \log (x))^2} \, dx-\int \frac {x^3 (9+x)}{6+x-3 \log (x)} \, dx\\ &=2 \int \frac {x^3 (9+x)}{(6+x) \log (x)} \, dx-4 \int \frac {x^5 \log (x) \log (\log (x))}{(6+x-3 \log (x))^2} \, dx+6 \int \left (\frac {9 x^3}{(6+x-3 \log (x))^2}+\frac {x^4}{(6+x-3 \log (x))^2}\right ) \, dx+6 \int \left (\frac {108}{6+x-3 \log (x)}-\frac {18 x}{6+x-3 \log (x)}+\frac {3 x^2}{6+x-3 \log (x)}+\frac {x^3}{6+x-3 \log (x)}-\frac {648}{(6+x) (6+x-3 \log (x))}\right ) \, dx+7 \int \frac {x^5 \log (\log (x))}{(6+x-3 \log (x))^2} \, dx+15 \int \frac {x^4 \log ^2(x) \log (\log (x))}{(6+x-3 \log (x))^2} \, dx-21 \int \frac {x^4}{(6+x-3 \log (x))^2} \, dx-87 \int \frac {x^4 \log (x) \log (\log (x))}{(6+x-3 \log (x))^2} \, dx+105 \int \frac {x^4 \log (\log (x))}{(6+x-3 \log (x))^2} \, dx-108 \int \frac {x^3}{(6+x-3 \log (x))^2} \, dx+108 \int \frac {x^3 \log ^2(x) \log (\log (x))}{(6+x-3 \log (x))^2} \, dx+432 \int \frac {x^3 \log (\log (x))}{(6+x-3 \log (x))^2} \, dx-432 \int \frac {x^3 \log (x) \log (\log (x))}{(6+x-3 \log (x))^2} \, dx+\int \left (\frac {54 x^3}{(6+x-3 \log (x))^2}+\frac {15 x^4}{(6+x-3 \log (x))^2}+\frac {x^5}{(6+x-3 \log (x))^2}\right ) \, dx-\int \left (\frac {9 x^3}{6+x-3 \log (x)}+\frac {x^4}{6+x-3 \log (x)}\right ) \, dx-\int \frac {x^5}{(6+x-3 \log (x))^2} \, dx\\ &=2 \int \frac {x^3 (9+x)}{(6+x) \log (x)} \, dx-4 \int \frac {x^5 \log (x) \log (\log (x))}{(6+x-3 \log (x))^2} \, dx+6 \int \frac {x^4}{(6+x-3 \log (x))^2} \, dx+6 \int \frac {x^3}{6+x-3 \log (x)} \, dx+7 \int \frac {x^5 \log (\log (x))}{(6+x-3 \log (x))^2} \, dx-9 \int \frac {x^3}{6+x-3 \log (x)} \, dx+15 \int \frac {x^4}{(6+x-3 \log (x))^2} \, dx+15 \int \frac {x^4 \log ^2(x) \log (\log (x))}{(6+x-3 \log (x))^2} \, dx+18 \int \frac {x^2}{6+x-3 \log (x)} \, dx-21 \int \frac {x^4}{(6+x-3 \log (x))^2} \, dx+2 \left (54 \int \frac {x^3}{(6+x-3 \log (x))^2} \, dx\right )-87 \int \frac {x^4 \log (x) \log (\log (x))}{(6+x-3 \log (x))^2} \, dx+105 \int \frac {x^4 \log (\log (x))}{(6+x-3 \log (x))^2} \, dx-108 \int \frac {x^3}{(6+x-3 \log (x))^2} \, dx-108 \int \frac {x}{6+x-3 \log (x)} \, dx+108 \int \frac {x^3 \log ^2(x) \log (\log (x))}{(6+x-3 \log (x))^2} \, dx+432 \int \frac {x^3 \log (\log (x))}{(6+x-3 \log (x))^2} \, dx-432 \int \frac {x^3 \log (x) \log (\log (x))}{(6+x-3 \log (x))^2} \, dx+648 \int \frac {1}{6+x-3 \log (x)} \, dx-3888 \int \frac {1}{(6+x) (6+x-3 \log (x))} \, dx-\int \frac {x^4}{6+x-3 \log (x)} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.09, size = 24, normalized size = 1.04 \begin {gather*} -\frac {x^4 (9+x) (-2+\log (x)) \log (\log (x))}{6+x-3 \log (x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 37, normalized size = 1.61 \begin {gather*} \frac {{\left (2 \, x^{5} + 18 \, x^{4} - {\left (x^{5} + 9 \, x^{4}\right )} \log \relax (x)\right )} \log \left (\log \relax (x)\right )}{x - 3 \, \log \relax (x) + 6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 34, normalized size = 1.48 \begin {gather*} \frac {1}{3} \, {\left (x^{5} + 9 \, x^{4} - \frac {x^{6} + 9 \, x^{5}}{x - 3 \, \log \relax (x) + 6}\right )} \log \left (\log \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 31, normalized size = 1.35
method | result | size |
risch | \(-\frac {x^{4} \left (x \ln \relax (x )+9 \ln \relax (x )-2 x -18\right ) \ln \left (\ln \relax (x )\right )}{x -3 \ln \relax (x )+6}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 37, normalized size = 1.61 \begin {gather*} \frac {{\left (2 \, x^{5} + 18 \, x^{4} - {\left (x^{5} + 9 \, x^{4}\right )} \log \relax (x)\right )} \log \left (\log \relax (x)\right )}{x - 3 \, \log \relax (x) + 6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.33, size = 394, normalized size = 17.13 \begin {gather*} \frac {\ln \left (\ln \relax (x)\right )\,\left (\left (x+6\right )\,\left (\frac {\frac {25\,x^5}{4}+\frac {117\,x^4}{4}-144\,x^3}{x-3}-\frac {31\,x^5+33\,x^4-648\,x^3}{x-3}-\frac {5\,x^5+16\,x^4-\frac {339\,x^3}{2}-\frac {459\,x^2}{2}+1377\,x}{x-3}+\frac {\frac {155\,x^5}{4}+\frac {331\,x^4}{4}-\frac {1347\,x^3}{2}-\frac {459\,x^2}{2}+1377\,x}{x-3}\right )+\ln \relax (x)\,\left (\frac {3\,\left (31\,x^5+33\,x^4-648\,x^3\right )}{x-3}-\frac {3\,\left (\frac {25\,x^5}{4}+\frac {117\,x^4}{4}-144\,x^3\right )}{x-3}+\frac {3\,\left (5\,x^5+16\,x^4-\frac {339\,x^3}{2}-\frac {459\,x^2}{2}+1377\,x\right )}{x-3}-\frac {3\,\left (\frac {155\,x^5}{4}+\frac {331\,x^4}{4}-\frac {1347\,x^3}{2}-\frac {459\,x^2}{2}+1377\,x\right )}{x-3}+\left (\frac {20\,x^5+33\,x^4-432\,x^3}{x-3}-\frac {25\,x^5+69\,x^4-432\,x^3}{x-3}\right )\,\left (x+6\right )+\frac {x^4\,\left (4\,x^2+87\,x+432\right )}{x-3}\right )-{\ln \relax (x)}^2\,\left (\frac {3\,\left (20\,x^5+33\,x^4-432\,x^3\right )}{x-3}-\frac {3\,\left (25\,x^5+69\,x^4-432\,x^3\right )}{x-3}+\frac {3\,x^4\,\left (5\,x+36\right )}{x-3}\right )-\frac {x^4\,\left (7\,x^2+105\,x+432\right )}{x-3}\right )}{x-3\,\ln \relax (x)+6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.57, size = 37, normalized size = 1.61 \begin {gather*} \frac {\left (- x^{5} \log {\relax (x )} + 2 x^{5} - 9 x^{4} \log {\relax (x )} + 18 x^{4}\right ) \log {\left (\log {\relax (x )} \right )}}{x - 3 \log {\relax (x )} + 6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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