Optimal. Leaf size=33 \[ \log \left (\left (e^{3 \left (5+\frac {2}{\log (2)}+\frac {3 (3-x)}{\log \left (\frac {x}{2}\right )}\right )}+x\right ) \log (4)\right ) \]
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Rubi [F]
time = 13.70, 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 {x \log ^2\left (\frac {x}{2}\right )+\exp \left (\frac {(27-9 x) \log (2)+(6+15 \log (2)) \log \left (\frac {x}{2}\right )}{\log (2) \log \left (\frac {x}{2}\right )}\right ) \left (-27+9 x-9 x \log \left (\frac {x}{2}\right )\right )}{\exp \left (\frac {(27-9 x) \log (2)+(6+15 \log (2)) \log \left (\frac {x}{2}\right )}{\log (2) \log \left (\frac {x}{2}\right )}\right ) x \log ^2\left (\frac {x}{2}\right )+x^2 \log ^2\left (\frac {x}{2}\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 {x \log ^2\left (\frac {x}{2}\right )-9 e^{15+\frac {6}{\log (2)}-\frac {9 (-3+x)}{\log \left (\frac {x}{2}\right )}} \left (3-x+x \log \left (\frac {x}{2}\right )\right )}{x \left (e^{15+\frac {6}{\log (2)}-\frac {9 (-3+x)}{\log \left (\frac {x}{2}\right )}}+x\right ) \log ^2\left (\frac {x}{2}\right )} \, dx\\ &=\int \left (\frac {1}{x}+\frac {e^{15 \left (1+\frac {2}{\log (32)}\right )+\frac {27}{\log \left (\frac {x}{2}\right )}} \left (-27+9 x-9 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x \left (e^{15+\frac {6}{\log (2)}+\frac {27}{\log \left (\frac {x}{2}\right )}}+e^{\frac {9 x}{\log \left (\frac {x}{2}\right )}} x\right ) \log ^2\left (\frac {x}{2}\right )}\right ) \, dx\\ &=\log (x)+\int \frac {e^{15 \left (1+\frac {2}{\log (32)}\right )+\frac {27}{\log \left (\frac {x}{2}\right )}} \left (-27+9 x-9 x \log \left (\frac {x}{2}\right )-\log ^2\left (\frac {x}{2}\right )\right )}{x \left (e^{15+\frac {6}{\log (2)}+\frac {27}{\log \left (\frac {x}{2}\right )}}+e^{\frac {9 x}{\log \left (\frac {x}{2}\right )}} x\right ) \log ^2\left (\frac {x}{2}\right )} \, dx\\ &=\log (x)+\int \left (-\frac {e^{15 \left (1+\frac {2}{\log (32)}\right )+\frac {27}{\log \left (\frac {x}{2}\right )}}}{x \left (e^{15+\frac {6}{\log (2)}+\frac {27}{\log \left (\frac {x}{2}\right )}}+e^{\frac {9 x}{\log \left (\frac {x}{2}\right )}} x\right )}+\frac {9 e^{15 \left (1+\frac {2}{\log (32)}\right )+\frac {27}{\log \left (\frac {x}{2}\right )}}}{\left (e^{15+\frac {6}{\log (2)}+\frac {27}{\log \left (\frac {x}{2}\right )}}+e^{\frac {9 x}{\log \left (\frac {x}{2}\right )}} x\right ) \log ^2\left (\frac {x}{2}\right )}-\frac {27 e^{15 \left (1+\frac {2}{\log (32)}\right )+\frac {27}{\log \left (\frac {x}{2}\right )}}}{x \left (e^{15+\frac {6}{\log (2)}+\frac {27}{\log \left (\frac {x}{2}\right )}}+e^{\frac {9 x}{\log \left (\frac {x}{2}\right )}} x\right ) \log ^2\left (\frac {x}{2}\right )}-\frac {9 e^{15 \left (1+\frac {2}{\log (32)}\right )+\frac {27}{\log \left (\frac {x}{2}\right )}}}{\left (e^{15+\frac {6}{\log (2)}+\frac {27}{\log \left (\frac {x}{2}\right )}}+e^{\frac {9 x}{\log \left (\frac {x}{2}\right )}} x\right ) \log \left (\frac {x}{2}\right )}\right ) \, dx\\ &=\log (x)+9 \int \frac {e^{15 \left (1+\frac {2}{\log (32)}\right )+\frac {27}{\log \left (\frac {x}{2}\right )}}}{\left (e^{15+\frac {6}{\log (2)}+\frac {27}{\log \left (\frac {x}{2}\right )}}+e^{\frac {9 x}{\log \left (\frac {x}{2}\right )}} x\right ) \log ^2\left (\frac {x}{2}\right )} \, dx-9 \int \frac {e^{15 \left (1+\frac {2}{\log (32)}\right )+\frac {27}{\log \left (\frac {x}{2}\right )}}}{\left (e^{15+\frac {6}{\log (2)}+\frac {27}{\log \left (\frac {x}{2}\right )}}+e^{\frac {9 x}{\log \left (\frac {x}{2}\right )}} x\right ) \log \left (\frac {x}{2}\right )} \, dx-27 \int \frac {e^{15 \left (1+\frac {2}{\log (32)}\right )+\frac {27}{\log \left (\frac {x}{2}\right )}}}{x \left (e^{15+\frac {6}{\log (2)}+\frac {27}{\log \left (\frac {x}{2}\right )}}+e^{\frac {9 x}{\log \left (\frac {x}{2}\right )}} x\right ) \log ^2\left (\frac {x}{2}\right )} \, dx-\int \frac {e^{15 \left (1+\frac {2}{\log (32)}\right )+\frac {27}{\log \left (\frac {x}{2}\right )}}}{x \left (e^{15+\frac {6}{\log (2)}+\frac {27}{\log \left (\frac {x}{2}\right )}}+e^{\frac {9 x}{\log \left (\frac {x}{2}\right )}} x\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.55, size = 26, normalized size = 0.79 \begin {gather*} \log \left (e^{15+\frac {6}{\log (2)}-\frac {9 (-3+x)}{\log \left (\frac {x}{2}\right )}}+x\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 36, normalized size = 1.09
method | result | size |
norman | \(\ln \left ({\mathrm e}^{\frac {\left (15 \ln \left (2\right )+6\right ) \ln \left (\frac {x}{2}\right )+\left (-9 x +27\right ) \ln \left (2\right )}{\ln \left (2\right ) \ln \left (\frac {x}{2}\right )}}+x \right )\) | \(36\) |
risch | \(-\frac {9 \left (x -3\right )}{\ln \left (\frac {x}{2}\right )}-\frac {\left (15 \ln \left (2\right )+6\right ) \ln \left (\frac {x}{2}\right )+\left (-9 x +27\right ) \ln \left (2\right )}{\ln \left (2\right ) \ln \left (\frac {x}{2}\right )}+\ln \left ({\mathrm e}^{-\frac {3 \left (-5 \ln \left (\frac {x}{2}\right ) \ln \left (2\right )+3 x \ln \left (2\right )-2 \ln \left (\frac {x}{2}\right )-9 \ln \left (2\right )\right )}{\ln \left (\frac {x}{2}\right ) \ln \left (2\right )}}+x \right )\) | \(85\) |
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. 56 vs.
\(2 (27) = 54\).
time = 0.54, size = 56, normalized size = 1.70 \begin {gather*} \frac {9 \, x}{\log \left (2\right ) - \log \left (x\right )} + \log \left (x\right ) + \log \left (\frac {x e^{\left (-\frac {9 \, x}{\log \left (2\right ) - \log \left (x\right )}\right )} + e^{\left (-\frac {27}{\log \left (2\right ) - \log \left (x\right )} + \frac {6}{\log \left (2\right )} + 15\right )}}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 36, normalized size = 1.09 \begin {gather*} \log \left (x + e^{\left (-\frac {3 \, {\left (3 \, {\left (x - 3\right )} \log \left (2\right ) - {\left (5 \, \log \left (2\right ) + 2\right )} \log \left (\frac {1}{2} \, x\right )\right )}}{\log \left (2\right ) \log \left (\frac {1}{2} \, x\right )}\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.24, size = 32, normalized size = 0.97 \begin {gather*} \log {\left (x + e^{\frac {\left (27 - 9 x\right ) \log {\left (2 \right )} + \left (6 + 15 \log {\left (2 \right )}\right ) \log {\left (\frac {x}{2} \right )}}{\log {\left (2 \right )} \log {\left (\frac {x}{2} \right )}}} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.20, size = 41, normalized size = 1.24 \begin {gather*} \log \left (x + e^{\left (\frac {9 \, {\left (x \log \left (2\right ) - 3 \, \log \left (x\right )\right )}}{\log \left (2\right )^{2} - \log \left (2\right ) \log \left (x\right )} + \frac {3 \, {\left (5 \, \log \left (2\right ) - 7\right )}}{\log \left (2\right )}\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.50, size = 64, normalized size = 1.94 \begin {gather*} \ln \left (x+\frac {{\mathrm {e}}^{\frac {3\,\ln \left (x\right )\,\left (2\,\ln \left (\frac {x}{2}\right )+5\,\ln \left (2\right )\,\ln \left (x\right )-5\,{\ln \left (2\right )}^2\right )}{{\ln \left (\frac {x}{2}\right )}^2\,\ln \left (2\right )}-\frac {9\,x-21}{\ln \left (\frac {x}{2}\right )}}}{2^{\frac {15}{\ln \left (\frac {x}{2}\right )}}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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