Optimal. Leaf size=30 \[ \frac {x}{2}+\left (x+x \left (5+x+\frac {1}{3} \log (2 x)\right )\right ) \log \left (\frac {2+x}{3}\right ) \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(95\) vs. \(2(30)=60\).
time = 0.24, antiderivative size = 95, normalized size of antiderivative = 3.17, number
of steps used = 24, number of rules used = 13, integrand size = 60, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules
used = {6874, 2332, 2442, 45, 2436, 2417, 2458, 2393, 2353, 2352, 712, 2354, 2438}
\begin {gather*} \frac {x}{2}+\frac {1}{36} (6 x+19)^2 \log \left (\frac {x}{3}+\frac {2}{3}\right )+\frac {2}{3} \log \left (\frac {3}{2}\right ) \log (x)-\frac {2}{3} \log \left (\frac {x}{2}+1\right ) \log (2 x)-\frac {1}{3} (x+2) \log \left (\frac {x+2}{3}\right )+\frac {1}{3} (x+2) \log (2 x) \log \left (\frac {x+2}{3}\right )-\frac {337}{36} \log (x+2) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 712
Rule 2332
Rule 2352
Rule 2353
Rule 2354
Rule 2393
Rule 2417
Rule 2436
Rule 2438
Rule 2442
Rule 2458
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {1}{3} \log \left (\frac {2}{3}+\frac {x}{3}\right ) (19+6 x+\log (2 x))+\frac {6+39 x+6 x^2+2 x \log (2 x)}{6 (2+x)}\right ) \, dx\\ &=\frac {1}{6} \int \frac {6+39 x+6 x^2+2 x \log (2 x)}{2+x} \, dx+\frac {1}{3} \int \log \left (\frac {2}{3}+\frac {x}{3}\right ) (19+6 x+\log (2 x)) \, dx\\ &=\frac {1}{6} \int \left (\frac {3 \left (2+13 x+2 x^2\right )}{2+x}+\frac {2 x \log (2 x)}{2+x}\right ) \, dx+\frac {1}{3} \int \left ((19+6 x) \log \left (\frac {2}{3}+\frac {x}{3}\right )+\log \left (\frac {2}{3}+\frac {x}{3}\right ) \log (2 x)\right ) \, dx\\ &=\frac {1}{3} \int (19+6 x) \log \left (\frac {2}{3}+\frac {x}{3}\right ) \, dx+\frac {1}{3} \int \frac {x \log (2 x)}{2+x} \, dx+\frac {1}{3} \int \log \left (\frac {2}{3}+\frac {x}{3}\right ) \log (2 x) \, dx+\frac {1}{2} \int \frac {2+13 x+2 x^2}{2+x} \, dx\\ &=\frac {1}{36} (19+6 x)^2 \log \left (\frac {2}{3}+\frac {x}{3}\right )-\frac {1}{3} x \log (2 x)+\frac {1}{3} (2+x) \log (2 x) \log \left (\frac {2+x}{3}\right )-\frac {1}{108} \int \frac {(19+6 x)^2}{\frac {2}{3}+\frac {x}{3}} \, dx+\frac {1}{3} \int \left (\log (2 x)-\frac {2 \log (2 x)}{2+x}\right ) \, dx-\frac {1}{3} \int \left (-1+\frac {(2+x) \log \left (\frac {2+x}{3}\right )}{x}\right ) \, dx+\frac {1}{2} \int \left (9+2 x-\frac {16}{2+x}\right ) \, dx\\ &=\frac {29 x}{6}+\frac {x^2}{2}+\frac {1}{36} (19+6 x)^2 \log \left (\frac {2}{3}+\frac {x}{3}\right )-\frac {1}{3} x \log (2 x)+\frac {1}{3} (2+x) \log (2 x) \log \left (\frac {2+x}{3}\right )-8 \log (2+x)-\frac {1}{108} \int \left (468+108 x+\frac {147}{2+x}\right ) \, dx+\frac {1}{3} \int \log (2 x) \, dx-\frac {1}{3} \int \frac {(2+x) \log \left (\frac {2+x}{3}\right )}{x} \, dx-\frac {2}{3} \int \frac {\log (2 x)}{2+x} \, dx\\ &=\frac {x}{6}+\frac {1}{36} (19+6 x)^2 \log \left (\frac {2}{3}+\frac {x}{3}\right )-\frac {2}{3} \log \left (1+\frac {x}{2}\right ) \log (2 x)+\frac {1}{3} (2+x) \log (2 x) \log \left (\frac {2+x}{3}\right )-\frac {337}{36} \log (2+x)-\frac {1}{3} \text {Subst}\left (\int \frac {x \log \left (\frac {x}{3}\right )}{-2+x} \, dx,x,2+x\right )+\frac {2}{3} \int \frac {\log \left (1+\frac {x}{2}\right )}{x} \, dx\\ &=\frac {x}{6}+\frac {1}{36} (19+6 x)^2 \log \left (\frac {2}{3}+\frac {x}{3}\right )-\frac {2}{3} \log \left (1+\frac {x}{2}\right ) \log (2 x)+\frac {1}{3} (2+x) \log (2 x) \log \left (\frac {2+x}{3}\right )-\frac {337}{36} \log (2+x)-\frac {2 \text {Li}_2\left (-\frac {x}{2}\right )}{3}-\frac {1}{3} \text {Subst}\left (\int \left (\log \left (\frac {x}{3}\right )+\frac {2 \log \left (\frac {x}{3}\right )}{-2+x}\right ) \, dx,x,2+x\right )\\ &=\frac {x}{6}+\frac {1}{36} (19+6 x)^2 \log \left (\frac {2}{3}+\frac {x}{3}\right )-\frac {2}{3} \log \left (1+\frac {x}{2}\right ) \log (2 x)+\frac {1}{3} (2+x) \log (2 x) \log \left (\frac {2+x}{3}\right )-\frac {337}{36} \log (2+x)-\frac {2 \text {Li}_2\left (-\frac {x}{2}\right )}{3}-\frac {1}{3} \text {Subst}\left (\int \log \left (\frac {x}{3}\right ) \, dx,x,2+x\right )-\frac {2}{3} \text {Subst}\left (\int \frac {\log \left (\frac {x}{3}\right )}{-2+x} \, dx,x,2+x\right )\\ &=\frac {x}{2}+\frac {1}{36} (19+6 x)^2 \log \left (\frac {2}{3}+\frac {x}{3}\right )+\frac {2}{3} \log \left (\frac {3}{2}\right ) \log (x)-\frac {2}{3} \log \left (1+\frac {x}{2}\right ) \log (2 x)-\frac {1}{3} (2+x) \log \left (\frac {2+x}{3}\right )+\frac {1}{3} (2+x) \log (2 x) \log \left (\frac {2+x}{3}\right )-\frac {337}{36} \log (2+x)-\frac {2 \text {Li}_2\left (-\frac {x}{2}\right )}{3}-\frac {2}{3} \text {Subst}\left (\int \frac {\log \left (\frac {x}{2}\right )}{-2+x} \, dx,x,2+x\right )\\ &=\frac {x}{2}+\frac {1}{36} (19+6 x)^2 \log \left (\frac {2}{3}+\frac {x}{3}\right )+\frac {2}{3} \log \left (\frac {3}{2}\right ) \log (x)-\frac {2}{3} \log \left (1+\frac {x}{2}\right ) \log (2 x)-\frac {1}{3} (2+x) \log \left (\frac {2+x}{3}\right )+\frac {1}{3} (2+x) \log (2 x) \log \left (\frac {2+x}{3}\right )-\frac {337}{36} \log (2+x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.05, size = 28, normalized size = 0.93 \begin {gather*} \frac {1}{6} \left (3 x+2 x (18+3 x+\log (2 x)) \log \left (\frac {2+x}{3}\right )\right ) \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. \(85\) vs.
\(2(24)=48\).
time = 2.17, size = 86, normalized size = 2.87
method | result | size |
risch | \(\left (\frac {x \ln \left (2 x \right )}{3}+x^{2}+6 x \right ) \ln \left (\frac {2}{3}+\frac {x}{3}\right )+\frac {x}{2}\) | \(26\) |
norman | \(\ln \left (\frac {2}{3}+\frac {x}{3}\right ) x^{2}+\frac {x}{2}+6 \ln \left (\frac {2}{3}+\frac {x}{3}\right ) x +\frac {\ln \left (\frac {2}{3}+\frac {x}{3}\right ) \ln \left (2 x \right ) x}{3}\) | \(37\) |
default | \(\left (2+x \right )^{2} \ln \left (2+x \right )+\frac {x}{2}-\frac {20}{3}+\frac {7 \left (2+x \right ) \ln \left (2+x \right )}{3}+\frac {x \ln \left (x \right ) \ln \left (2+x \right )}{3}-\frac {x \ln \left (2+x \right )}{3}-\frac {26 \ln \left (2+x \right )}{3}-\frac {x \ln \left (2\right ) \ln \left (3\right )}{3}+\frac {x \ln \left (2\right ) \ln \left (2+x \right )}{3}-\frac {2 \ln \left (2\right )}{3}-\frac {x \ln \left (3\right ) \ln \left (x \right )}{3}-x^{2} \ln \left (3\right )-6 x \ln \left (3\right )\) | \(86\) |
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. 101 vs.
\(2 (27) = 54\).
time = 0.53, size = 101, normalized size = 3.37 \begin {gather*} -\frac {1}{3} \, x \log \left (3\right ) \log \left (x\right ) - \frac {1}{3} \, {\left (\log \left (3\right ) \log \left (2\right ) - \log \left (3\right ) - 1\right )} x + \frac {1}{3} \, {\left (x {\left (\log \left (2\right ) - 1\right )} + x \log \left (x\right ) - 2\right )} \log \left (x + 2\right ) - \frac {38}{3} \, \log \left (3\right ) \log \left (x + 2\right ) + \frac {38}{3} \, \log \left (x + 2\right )^{2} + {\left (x^{2} - 4 \, x + 8 \, \log \left (x + 2\right )\right )} \log \left (\frac {1}{3} \, x + \frac {2}{3}\right ) + \frac {31}{3} \, {\left (x - 2 \, \log \left (x + 2\right )\right )} \log \left (\frac {1}{3} \, x + \frac {2}{3}\right ) + \frac {1}{6} \, x + \frac {2}{3} \, \log \left (x + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 31, normalized size = 1.03 \begin {gather*} \frac {1}{3} \, x \log \left (2 \, x\right ) \log \left (\frac {1}{3} \, x + \frac {2}{3}\right ) + {\left (x^{2} + 6 \, x\right )} \log \left (\frac {1}{3} \, x + \frac {2}{3}\right ) + \frac {1}{2} \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.50, size = 37, normalized size = 1.23 \begin {gather*} \frac {x}{2} + \left (x^{2} + \frac {x \log {\left (2 x \right )}}{3} + 6 x + \frac {9}{2}\right ) \log {\left (\frac {x}{3} + \frac {2}{3} \right )} - \frac {9 \log {\left (x + 2 \right )}}{2} \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. 55 vs.
\(2 (27) = 54\).
time = 0.40, size = 55, normalized size = 1.83 \begin {gather*} \frac {1}{3} \, {\left ({\left (x + 2\right )} \log \left (\frac {1}{3} \, x + \frac {2}{3}\right ) - 2 \, \log \left (\frac {1}{3} \, x + \frac {2}{3}\right )\right )} \log \left (2 \, x\right ) + {\left ({\left (x + 2\right )}^{2} + 2 \, x + 4\right )} \log \left (\frac {1}{3} \, x + \frac {2}{3}\right ) + \frac {1}{2} \, x - 8 \, \log \left (\frac {1}{3} \, x + \frac {2}{3}\right ) + 1 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.24, size = 25, normalized size = 0.83 \begin {gather*} \frac {x}{2}+\ln \left (\frac {x}{3}+\frac {2}{3}\right )\,\left (6\,x+\frac {x\,\ln \left (2\,x\right )}{3}+x^2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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