Optimal. Leaf size=26 \[ -x+\frac {x}{\left (-5+3 x+\log \left (\frac {1+\frac {6}{x}}{\log (x)}\right )\right )^2} \]
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Rubi [F]
time = 1.82, 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 {12+2 x+\left (732-1248 x+582 x^2-27 x^3-27 x^4\right ) \log (x)+\left (-444+466 x-72 x^2-27 x^3\right ) \log (x) \log \left (\frac {6+x}{x \log (x)}\right )+\left (90-39 x-9 x^2\right ) \log (x) \log ^2\left (\frac {6+x}{x \log (x)}\right )+(-6-x) \log (x) \log ^3\left (\frac {6+x}{x \log (x)}\right )}{\left (-750+1225 x-585 x^2+27 x^3+27 x^4\right ) \log (x)+\left (450-465 x+72 x^2+27 x^3\right ) \log (x) \log \left (\frac {6+x}{x \log (x)}\right )+\left (-90+39 x+9 x^2\right ) \log (x) \log ^2\left (\frac {6+x}{x \log (x)}\right )+(6+x) \log (x) \log ^3\left (\frac {6+x}{x \log (x)}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2 (6+x)+\log (x) \left (3 \left (-244+416 x-194 x^2+9 x^3+9 x^4\right )+\left (444-466 x+72 x^2+27 x^3\right ) \log \left (\frac {6+x}{x \log (x)}\right )+\left (-90+39 x+9 x^2\right ) \log ^2\left (\frac {6+x}{x \log (x)}\right )+(6+x) \log ^3\left (\frac {6+x}{x \log (x)}\right )\right )}{(6+x) \log (x) \left (5-3 x-\log \left (\frac {6+x}{x \log (x)}\right )\right )^3} \, dx\\ &=\int \left (-1-\frac {2 \left (-6-x-6 \log (x)+18 x \log (x)+3 x^2 \log (x)\right )}{(6+x) \log (x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3}+\frac {1}{\left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^2}\right ) \, dx\\ &=-x-2 \int \frac {-6-x-6 \log (x)+18 x \log (x)+3 x^2 \log (x)}{(6+x) \log (x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3} \, dx+\int \frac {1}{\left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^2} \, dx\\ &=-x-2 \int \left (-\frac {6}{(6+x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3}+\frac {18 x}{(6+x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3}+\frac {3 x^2}{(6+x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3}-\frac {6}{(6+x) \log (x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3}-\frac {x}{(6+x) \log (x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3}\right ) \, dx+\int \frac {1}{\left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^2} \, dx\\ &=-x+2 \int \frac {x}{(6+x) \log (x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3} \, dx-6 \int \frac {x^2}{(6+x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3} \, dx+12 \int \frac {1}{(6+x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3} \, dx+12 \int \frac {1}{(6+x) \log (x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3} \, dx-36 \int \frac {x}{(6+x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3} \, dx+\int \frac {1}{\left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^2} \, dx\\ &=-x+2 \int \left (\frac {1}{\log (x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3}-\frac {6}{(6+x) \log (x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3}\right ) \, dx-6 \int \left (-\frac {6}{\left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3}+\frac {x}{\left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3}+\frac {36}{(6+x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3}\right ) \, dx+12 \int \frac {1}{(6+x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3} \, dx+12 \int \frac {1}{(6+x) \log (x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3} \, dx-36 \int \left (\frac {1}{\left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3}-\frac {6}{(6+x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3}\right ) \, dx+\int \frac {1}{\left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^2} \, dx\\ &=-x+2 \int \frac {1}{\log (x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3} \, dx-6 \int \frac {x}{\left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3} \, dx+12 \int \frac {1}{(6+x) \left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^3} \, dx+\int \frac {1}{\left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.08, size = 25, normalized size = 0.96 \begin {gather*} -x+\frac {x}{\left (-5+3 x+\log \left (\frac {6+x}{x \log (x)}\right )\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(83\) vs.
\(2(26)=52\).
time = 17.96, size = 84, normalized size = 3.23
method | result | size |
norman | \(\frac {-24 x +30 x^{2}+10 \ln \left (\frac {x +6}{x \ln \left (x \right )}\right ) x -9 x^{3}-6 \ln \left (\frac {x +6}{x \ln \left (x \right )}\right ) x^{2}-\ln \left (\frac {x +6}{x \ln \left (x \right )}\right )^{2} x}{\left (3 x +\ln \left (\frac {x +6}{x \ln \left (x \right )}\right )-5\right )^{2}}\) | \(84\) |
risch | \(-x +\frac {4 x}{\left (2 \ln \left (x +6\right )-2 \ln \left (\ln \left (x \right )\right )+6 x -2 \ln \left (x \right )+i \pi \,\mathrm {csgn}\left (i \left (x +6\right )\right ) \mathrm {csgn}\left (\frac {i \left (x +6\right )}{\ln \left (x \right )}\right )^{2}+i \pi \,\mathrm {csgn}\left (\frac {i \left (x +6\right )}{\ln \left (x \right )}\right ) \mathrm {csgn}\left (\frac {i \left (x +6\right )}{\ln \left (x \right ) x}\right )^{2}+i \pi \mathrm {csgn}\left (\frac {i \left (x +6\right )}{\ln \left (x \right ) x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right )+i \pi \,\mathrm {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \mathrm {csgn}\left (\frac {i \left (x +6\right )}{\ln \left (x \right )}\right )^{2}-i \pi \mathrm {csgn}\left (\frac {i \left (x +6\right )}{\ln \left (x \right )}\right )^{3}-i \pi \mathrm {csgn}\left (\frac {i \left (x +6\right )}{\ln \left (x \right ) x}\right )^{3}-10-i \pi \,\mathrm {csgn}\left (\frac {i \left (x +6\right )}{\ln \left (x \right )}\right ) \mathrm {csgn}\left (\frac {i \left (x +6\right )}{\ln \left (x \right ) x}\right ) \mathrm {csgn}\left (\frac {i}{x}\right )-i \pi \,\mathrm {csgn}\left (\frac {i}{\ln \left (x \right )}\right ) \mathrm {csgn}\left (i \left (x +6\right )\right ) \mathrm {csgn}\left (\frac {i \left (x +6\right )}{\ln \left (x \right )}\right )\right )^{2}}\) | \(240\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 165 vs.
\(2 (26) = 52\).
time = 0.40, size = 165, normalized size = 6.35 \begin {gather*} -\frac {9 \, x^{3} + x \log \left (x + 6\right )^{2} + x \log \left (x\right )^{2} + x \log \left (\log \left (x\right )\right )^{2} - 30 \, x^{2} + 2 \, {\left (3 \, x^{2} - x \log \left (x\right ) - x \log \left (\log \left (x\right )\right ) - 5 \, x\right )} \log \left (x + 6\right ) - 2 \, {\left (3 \, x^{2} - 5 \, x\right )} \log \left (x\right ) - 2 \, {\left (3 \, x^{2} - x \log \left (x\right ) - 5 \, x\right )} \log \left (\log \left (x\right )\right ) + 24 \, x}{9 \, x^{2} + 2 \, {\left (3 \, x - \log \left (x\right ) - \log \left (\log \left (x\right )\right ) - 5\right )} \log \left (x + 6\right ) + \log \left (x + 6\right )^{2} - 2 \, {\left (3 \, x - 5\right )} \log \left (x\right ) + \log \left (x\right )^{2} - 2 \, {\left (3 \, x - \log \left (x\right ) - 5\right )} \log \left (\log \left (x\right )\right ) + \log \left (\log \left (x\right )\right )^{2} - 30 \, x + 25} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 100 vs.
\(2 (26) = 52\).
time = 0.35, size = 100, normalized size = 3.85 \begin {gather*} -\frac {9 \, x^{3} + x \log \left (\frac {x + 6}{x \log \left (x\right )}\right )^{2} - 30 \, x^{2} + 2 \, {\left (3 \, x^{2} - 5 \, x\right )} \log \left (\frac {x + 6}{x \log \left (x\right )}\right ) + 24 \, x}{9 \, x^{2} + 2 \, {\left (3 \, x - 5\right )} \log \left (\frac {x + 6}{x \log \left (x\right )}\right ) + \log \left (\frac {x + 6}{x \log \left (x\right )}\right )^{2} - 30 \, x + 25} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 39 vs.
\(2 (19) = 38\).
time = 0.16, size = 39, normalized size = 1.50 \begin {gather*} - x + \frac {x}{9 x^{2} - 30 x + \left (6 x - 10\right ) \log {\left (\frac {x + 6}{x \log {\left (x \right )}} \right )} + \log {\left (\frac {x + 6}{x \log {\left (x \right )}} \right )}^{2} + 25} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 466 vs.
\(2 (26) = 52\).
time = 1.86, size = 466, normalized size = 17.92 \begin {gather*} -x + \frac {3 \, x^{3} \log \left (x\right ) + 18 \, x^{2} \log \left (x\right ) - x^{2} - 6 \, x \log \left (x\right ) - 6 \, x}{27 \, x^{4} \log \left (x\right ) + 18 \, x^{3} \log \left (x + 6\right ) \log \left (x\right ) + 3 \, x^{2} \log \left (x + 6\right )^{2} \log \left (x\right ) - 18 \, x^{3} \log \left (x\right )^{2} - 6 \, x^{2} \log \left (x + 6\right ) \log \left (x\right )^{2} + 3 \, x^{2} \log \left (x\right )^{3} - 18 \, x^{3} \log \left (x\right ) \log \left (\log \left (x\right )\right ) - 6 \, x^{2} \log \left (x + 6\right ) \log \left (x\right ) \log \left (\log \left (x\right )\right ) + 6 \, x^{2} \log \left (x\right )^{2} \log \left (\log \left (x\right )\right ) + 3 \, x^{2} \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} + 72 \, x^{3} \log \left (x\right ) + 78 \, x^{2} \log \left (x + 6\right ) \log \left (x\right ) + 18 \, x \log \left (x + 6\right )^{2} \log \left (x\right ) - 78 \, x^{2} \log \left (x\right )^{2} - 36 \, x \log \left (x + 6\right ) \log \left (x\right )^{2} + 18 \, x \log \left (x\right )^{3} - 78 \, x^{2} \log \left (x\right ) \log \left (\log \left (x\right )\right ) - 36 \, x \log \left (x + 6\right ) \log \left (x\right ) \log \left (\log \left (x\right )\right ) + 36 \, x \log \left (x\right )^{2} \log \left (\log \left (x\right )\right ) + 18 \, x \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} - 9 \, x^{3} - 6 \, x^{2} \log \left (x + 6\right ) - x \log \left (x + 6\right )^{2} - 513 \, x^{2} \log \left (x\right ) - 214 \, x \log \left (x + 6\right ) \log \left (x\right ) - 6 \, \log \left (x + 6\right )^{2} \log \left (x\right ) + 215 \, x \log \left (x\right )^{2} + 12 \, \log \left (x + 6\right ) \log \left (x\right )^{2} - 6 \, \log \left (x\right )^{3} + 6 \, x^{2} \log \left (\log \left (x\right )\right ) + 2 \, x \log \left (x + 6\right ) \log \left (\log \left (x\right )\right ) + 214 \, x \log \left (x\right ) \log \left (\log \left (x\right )\right ) + 12 \, \log \left (x + 6\right ) \log \left (x\right ) \log \left (\log \left (x\right )\right ) - 12 \, \log \left (x\right )^{2} \log \left (\log \left (x\right )\right ) - x \log \left (\log \left (x\right )\right )^{2} - 6 \, \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} - 24 \, x^{2} - 26 \, x \log \left (x + 6\right ) - 6 \, \log \left (x + 6\right )^{2} + 656 \, x \log \left (x\right ) + 72 \, \log \left (x + 6\right ) \log \left (x\right ) - 66 \, \log \left (x\right )^{2} + 26 \, x \log \left (\log \left (x\right )\right ) + 12 \, \log \left (x + 6\right ) \log \left (\log \left (x\right )\right ) - 72 \, \log \left (x\right ) \log \left (\log \left (x\right )\right ) - 6 \, \log \left (\log \left (x\right )\right )^{2} + 155 \, x + 60 \, \log \left (x + 6\right ) - 210 \, \log \left (x\right ) - 60 \, \log \left (\log \left (x\right )\right ) - 150} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {\ln \left (x\right )\,\left (x+6\right )\,{\ln \left (\frac {x+6}{x\,\ln \left (x\right )}\right )}^3+\ln \left (x\right )\,\left (9\,x^2+39\,x-90\right )\,{\ln \left (\frac {x+6}{x\,\ln \left (x\right )}\right )}^2+\ln \left (x\right )\,\left (27\,x^3+72\,x^2-466\,x+444\right )\,\ln \left (\frac {x+6}{x\,\ln \left (x\right )}\right )-2\,x+\ln \left (x\right )\,\left (27\,x^4+27\,x^3-582\,x^2+1248\,x-732\right )-12}{\ln \left (x\right )\,\left (x+6\right )\,{\ln \left (\frac {x+6}{x\,\ln \left (x\right )}\right )}^3+\ln \left (x\right )\,\left (9\,x^2+39\,x-90\right )\,{\ln \left (\frac {x+6}{x\,\ln \left (x\right )}\right )}^2+\ln \left (x\right )\,\left (27\,x^3+72\,x^2-465\,x+450\right )\,\ln \left (\frac {x+6}{x\,\ln \left (x\right )}\right )+\ln \left (x\right )\,\left (27\,x^4+27\,x^3-585\,x^2+1225\,x-750\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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