Optimal. Leaf size=17 \[ 2 \log \left (\frac {5 (4+x) \log ^2(x)}{4 x}\right ) \]
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Rubi [A] time = 0.31, antiderivative size = 16, normalized size of antiderivative = 0.94, number of steps used = 10, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {1593, 6741, 12, 6742, 36, 29, 31, 2302} \begin {gather*} -2 \log (x)+2 \log (x+4)+4 \log (\log (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 29
Rule 31
Rule 36
Rule 1593
Rule 2302
Rule 6741
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {16+4 x-8 \log (x)}{x (4+x) \log (x)} \, dx\\ &=\int \frac {4 (4+x-2 \log (x))}{x (4+x) \log (x)} \, dx\\ &=4 \int \frac {4+x-2 \log (x)}{x (4+x) \log (x)} \, dx\\ &=4 \int \left (-\frac {2}{x (4+x)}+\frac {1}{x \log (x)}\right ) \, dx\\ &=4 \int \frac {1}{x \log (x)} \, dx-8 \int \frac {1}{x (4+x)} \, dx\\ &=-\left (2 \int \frac {1}{x} \, dx\right )+2 \int \frac {1}{4+x} \, dx+4 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )\\ &=-2 \log (x)+2 \log (4+x)+4 \log (\log (x))\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.04, size = 20, normalized size = 1.18 \begin {gather*} 4 \left (-\frac {\log (x)}{2}+\frac {1}{2} \log (4+x)+\log (\log (x))\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 16, normalized size = 0.94 \begin {gather*} 2 \, \log \left (x + 4\right ) - 2 \, \log \relax (x) + 4 \, \log \left (\log \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 16, normalized size = 0.94 \begin {gather*} 2 \, \log \left (x + 4\right ) - 2 \, \log \relax (x) + 4 \, \log \left (\log \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 17, normalized size = 1.00
method | result | size |
default | \(4 \ln \left (\ln \relax (x )\right )-2 \ln \relax (x )+2 \ln \left (4+x \right )\) | \(17\) |
norman | \(4 \ln \left (\ln \relax (x )\right )-2 \ln \relax (x )+2 \ln \left (4+x \right )\) | \(17\) |
risch | \(4 \ln \left (\ln \relax (x )\right )-2 \ln \relax (x )+2 \ln \left (4+x \right )\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 16, normalized size = 0.94 \begin {gather*} 2 \, \log \left (x + 4\right ) - 2 \, \log \relax (x) + 4 \, \log \left (\log \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.12, size = 16, normalized size = 0.94 \begin {gather*} 2\,\ln \left (x+4\right )+4\,\ln \left (\ln \relax (x)\right )-2\,\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 17, normalized size = 1.00 \begin {gather*} - 2 \log {\relax (x )} + 2 \log {\left (x + 4 \right )} + 4 \log {\left (\log {\relax (x )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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