Optimal. Leaf size=22 \[ \frac {1}{16} \left (x \left (10-\frac {\log ^2(x)}{4}\right )-\log (\log (x))\right ) \]
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Rubi [A]
time = 0.08, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {12, 6820, 2339,
29, 2332, 2333} \begin {gather*} \frac {5 x}{8}-\frac {1}{64} x \log ^2(x)-\frac {1}{16} \log (\log (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 29
Rule 2332
Rule 2333
Rule 2339
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{64} \int \frac {-4+40 x \log (x)-2 x \log ^2(x)-x \log ^3(x)}{x \log (x)} \, dx\\ &=\frac {1}{64} \int \left (40-\frac {4}{x \log (x)}-2 \log (x)-\log ^2(x)\right ) \, dx\\ &=\frac {5 x}{8}-\frac {1}{64} \int \log ^2(x) \, dx-\frac {1}{32} \int \log (x) \, dx-\frac {1}{16} \int \frac {1}{x \log (x)} \, dx\\ &=\frac {21 x}{32}-\frac {1}{32} x \log (x)-\frac {1}{64} x \log ^2(x)+\frac {1}{32} \int \log (x) \, dx-\frac {1}{16} \text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )\\ &=\frac {5 x}{8}-\frac {1}{64} x \log ^2(x)-\frac {1}{16} \log (\log (x))\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 22, normalized size = 1.00 \begin {gather*} \frac {5 x}{8}-\frac {1}{64} x \log ^2(x)-\frac {1}{16} \log (\log (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 17, normalized size = 0.77
method | result | size |
default | \(-\frac {x \ln \left (x \right )^{2}}{64}+\frac {5 x}{8}-\frac {\ln \left (\ln \left (x \right )\right )}{16}\) | \(17\) |
norman | \(-\frac {x \ln \left (x \right )^{2}}{64}+\frac {5 x}{8}-\frac {\ln \left (\ln \left (x \right )\right )}{16}\) | \(17\) |
risch | \(-\frac {x \ln \left (x \right )^{2}}{64}+\frac {5 x}{8}-\frac {\ln \left (\ln \left (x \right )\right )}{16}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 27, normalized size = 1.23 \begin {gather*} -\frac {1}{64} \, {\left (\log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 2\right )} x - \frac {1}{32} \, x \log \left (x\right ) + \frac {21}{32} \, x - \frac {1}{16} \, \log \left (\log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 16, normalized size = 0.73 \begin {gather*} -\frac {1}{64} \, x \log \left (x\right )^{2} + \frac {5}{8} \, x - \frac {1}{16} \, \log \left (\log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 19, normalized size = 0.86 \begin {gather*} - \frac {x \log {\left (x \right )}^{2}}{64} + \frac {5 x}{8} - \frac {\log {\left (\log {\left (x \right )} \right )}}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 16, normalized size = 0.73 \begin {gather*} -\frac {1}{64} \, x \log \left (x\right )^{2} + \frac {5}{8} \, x - \frac {1}{16} \, \log \left (\log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.28, size = 16, normalized size = 0.73 \begin {gather*} \frac {5\,x}{8}-\frac {\ln \left (\ln \left (x\right )\right )}{16}-\frac {x\,{\ln \left (x\right )}^2}{64} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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