Optimal. Leaf size=17 \[ \log \left (20 x (4+x) \left (\frac {1}{3}-\frac {\log (x)}{5}\right )\right ) \]
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Rubi [A]
time = 0.19, antiderivative size = 14, normalized size of antiderivative = 0.82, number of steps
used = 7, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {6873, 6874, 78,
2339, 29} \begin {gather*} \log (x)+\log (x+4)+\log (5-3 \log (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 78
Rule 2339
Rule 6873
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8+7 x-(12+6 x) \log (x)}{x (4+x) (5-3 \log (x))} \, dx\\ &=\int \left (\frac {2 (2+x)}{x (4+x)}+\frac {3}{x (-5+3 \log (x))}\right ) \, dx\\ &=2 \int \frac {2+x}{x (4+x)} \, dx+3 \int \frac {1}{x (-5+3 \log (x))} \, dx\\ &=2 \int \left (\frac {1}{2 x}+\frac {1}{2 (4+x)}\right ) \, dx+\text {Subst}\left (\int \frac {1}{x} \, dx,x,-5+3 \log (x)\right )\\ &=\log (x)+\log (4+x)+\log (5-3 \log (x))\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.04, size = 14, normalized size = 0.82 \begin {gather*} \log (x)+\log (4+x)+\log (5-3 \log (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.31, size = 15, normalized size = 0.88
method | result | size |
default | \(\ln \left (x \right )+\ln \left (3 \ln \left (x \right )-5\right )+\ln \left (4+x \right )\) | \(15\) |
norman | \(\ln \left (x \right )+\ln \left (3 \ln \left (x \right )-5\right )+\ln \left (4+x \right )\) | \(15\) |
risch | \(\ln \left (x^{2}+4 x \right )+\ln \left (\ln \left (x \right )-\frac {5}{3}\right )\) | \(15\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 12, normalized size = 0.71 \begin {gather*} \log \left (x + 4\right ) + \log \left (x\right ) + \log \left (\log \left (x\right ) - \frac {5}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 16, normalized size = 0.94 \begin {gather*} \log \left (x^{2} + 4 \, x\right ) + \log \left (3 \, \log \left (x\right ) - 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 15, normalized size = 0.88 \begin {gather*} \log {\left (x^{2} + 4 x \right )} + \log {\left (\log {\left (x \right )} - \frac {5}{3} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.38, size = 14, normalized size = 0.82 \begin {gather*} \log \left (x + 4\right ) + \log \left (x\right ) + \log \left (3 \, \log \left (x\right ) - 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.25, size = 14, normalized size = 0.82 \begin {gather*} \ln \left (3\,\ln \left (x\right )-5\right )+\ln \left (x+4\right )+\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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