Optimal. Leaf size=20 \[ 3-x+\frac {(-12+x+\log (2)) (-1+\log (6+x))}{x} \]
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Rubi [B] time = 0.37, antiderivative size = 84, normalized size of antiderivative = 4.20, number of steps used = 9, number of rules used = 7, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {1593, 6742, 1620, 2395, 36, 29, 31} \begin {gather*} -x-\frac {1}{36} (72-\log (64)) \log (x)+\frac {1}{6} (12-\log (2)) \log (x)+\frac {1}{36} (108-\log (64)) \log (x+6)-\frac {1}{6} (12-\log (2)) \log (x+6)-\frac {(12-\log (2)) \log (x+6)}{x}+\frac {72-\log (64)}{6 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 1593
Rule 1620
Rule 2395
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-72-24 x-5 x^2-x^3+(6+2 x) \log (2)+(72+12 x+(-6-x) \log (2)) \log (6+x)}{x^2 (6+x)} \, dx\\ &=\int \left (\frac {-72-5 x^2-x^3-x (24-\log (4))+\log (64)}{x^2 (6+x)}-\frac {(-12+\log (2)) \log (6+x)}{x^2}\right ) \, dx\\ &=(12-\log (2)) \int \frac {\log (6+x)}{x^2} \, dx+\int \frac {-72-5 x^2-x^3-x (24-\log (4))+\log (64)}{x^2 (6+x)} \, dx\\ &=-\frac {(12-\log (2)) \log (6+x)}{x}+(12-\log (2)) \int \frac {1}{x (6+x)} \, dx+\int \left (-1+\frac {108-\log (64)}{36 (6+x)}+\frac {-72+\log (64)}{6 x^2}+\frac {-72+\log (64)}{36 x}\right ) \, dx\\ &=-x+\frac {72-\log (64)}{6 x}-\frac {1}{36} (72-\log (64)) \log (x)-\frac {(12-\log (2)) \log (6+x)}{x}+\frac {1}{36} (108-\log (64)) \log (6+x)+\frac {1}{6} (12-\log (2)) \int \frac {1}{x} \, dx+\frac {1}{6} (-12+\log (2)) \int \frac {1}{6+x} \, dx\\ &=-x+\frac {72-\log (64)}{6 x}+\frac {1}{6} (12-\log (2)) \log (x)-\frac {1}{36} (72-\log (64)) \log (x)-\frac {1}{6} (12-\log (2)) \log (6+x)-\frac {(12-\log (2)) \log (6+x)}{x}+\frac {1}{36} (108-\log (64)) \log (6+x)\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.11, size = 39, normalized size = 1.95 \begin {gather*} \frac {12}{x}-x-\frac {\log (2)}{x}+\log (6+x)-\frac {12 \log (6+x)}{x}+\frac {\log (2) \log (6+x)}{x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 23, normalized size = 1.15 \begin {gather*} -\frac {x^{2} - {\left (x + \log \relax (2) - 12\right )} \log \left (x + 6\right ) + \log \relax (2) - 12}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 29, normalized size = 1.45 \begin {gather*} -x + \frac {{\left (\log \relax (2) - 12\right )} \log \left (x + 6\right )}{x} - \frac {\log \relax (2) - 12}{x} + \log \left (x + 6\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.62, size = 31, normalized size = 1.55
method | result | size |
norman | \(\frac {\left (\ln \relax (2)-12\right ) \ln \left (x +6\right )+x \ln \left (x +6\right )-x^{2}-\ln \relax (2)+12}{x}\) | \(31\) |
risch | \(\frac {\left (\ln \relax (2)-12\right ) \ln \left (x +6\right )}{x}+\frac {x \ln \left (x +6\right )-x^{2}-\ln \relax (2)+12}{x}\) | \(35\) |
derivativedivides | \(\frac {\ln \relax (2) \ln \left (x +6\right ) \left (x +6\right )}{6 x}-\frac {\ln \relax (2)}{x}-\frac {\ln \left (x +6\right ) \ln \relax (2)}{6}-\frac {2 \ln \left (x +6\right ) \left (x +6\right )}{x}-x -6+\frac {12}{x}+3 \ln \left (x +6\right )\) | \(58\) |
default | \(\frac {\ln \relax (2) \ln \left (x +6\right ) \left (x +6\right )}{6 x}-\frac {\ln \relax (2)}{x}-\frac {\ln \left (x +6\right ) \ln \relax (2)}{6}-\frac {2 \ln \left (x +6\right ) \left (x +6\right )}{x}-x -6+\frac {12}{x}+3 \ln \left (x +6\right )\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.53, size = 79, normalized size = 3.95 \begin {gather*} -\frac {1}{6} \, {\left (\frac {6}{x} - \log \left (x + 6\right ) + \log \relax (x)\right )} \log \relax (2) - \frac {1}{3} \, {\left (\log \left (x + 6\right ) - \log \relax (x)\right )} \log \relax (2) - \frac {1}{6} \, {\left (\log \relax (2) - 12\right )} \log \relax (x) - x + \frac {{\left (x {\left (\log \relax (2) - 12\right )} + 6 \, \log \relax (2) - 72\right )} \log \left (x + 6\right )}{6 \, x} + \frac {12}{x} + 3 \, \log \left (x + 6\right ) - 2 \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.99, size = 22, normalized size = 1.10 \begin {gather*} \ln \left (x+6\right )-x+\frac {\left (\ln \left (x+6\right )-1\right )\,\left (\ln \relax (2)-12\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.34, size = 24, normalized size = 1.20 \begin {gather*} - x + \log {\left (x + 6 \right )} + \frac {\left (-12 + \log {\relax (2 )}\right ) \log {\left (x + 6 \right )}}{x} - \frac {-12 + \log {\relax (2 )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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