Optimal. Leaf size=18 \[ \frac {x \log \left (4 x^2\right )}{5 (12+2 x)} \]
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Rubi [A]
time = 0.03, antiderivative size = 16, normalized size of antiderivative = 0.89, number of steps
used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {27, 12, 6874,
2351, 31} \begin {gather*} \frac {x \log \left (4 x^2\right )}{10 (x+6)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 31
Rule 2351
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {6+x+3 \log \left (4 x^2\right )}{5 (6+x)^2} \, dx\\ &=\frac {1}{5} \int \frac {6+x+3 \log \left (4 x^2\right )}{(6+x)^2} \, dx\\ &=\frac {1}{5} \int \left (\frac {1}{6+x}+\frac {3 \log \left (4 x^2\right )}{(6+x)^2}\right ) \, dx\\ &=\frac {1}{5} \log (6+x)+\frac {3}{5} \int \frac {\log \left (4 x^2\right )}{(6+x)^2} \, dx\\ &=\frac {x \log \left (4 x^2\right )}{10 (6+x)}+\frac {1}{5} \log (6+x)-\frac {1}{5} \int \frac {1}{6+x} \, dx\\ &=\frac {x \log \left (4 x^2\right )}{10 (6+x)}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 20, normalized size = 1.11 \begin {gather*} \frac {1}{5} \left (\log (x)-\frac {3 \log \left (4 x^2\right )}{6+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 23, normalized size = 1.28
method | result | size |
norman | \(\frac {x \ln \left (4 x^{2}\right )}{10 x +60}\) | \(15\) |
risch | \(-\frac {3 \ln \left (4 x^{2}\right )}{5 \left (x +6\right )}+\frac {\ln \left (x \right )}{5}\) | \(19\) |
default | \(-\frac {6 \ln \left (2\right )}{5 \left (x +6\right )}+\frac {x \ln \left (x^{2}\right )}{10 x +60}\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 18, normalized size = 1.00 \begin {gather*} -\frac {3 \, \log \left (4 \, x^{2}\right )}{5 \, {\left (x + 6\right )}} + \frac {1}{5} \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 14, normalized size = 0.78 \begin {gather*} \frac {x \log \left (4 \, x^{2}\right )}{10 \, {\left (x + 6\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 17, normalized size = 0.94 \begin {gather*} \frac {\log {\left (x \right )}}{5} - \frac {3 \log {\left (4 x^{2} \right )}}{5 x + 30} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.38, size = 18, normalized size = 1.00 \begin {gather*} -\frac {3 \, \log \left (4 \, x^{2}\right )}{5 \, {\left (x + 6\right )}} + \frac {1}{5} \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.80, size = 15, normalized size = 0.83 \begin {gather*} \frac {x\,\ln \left (4\,x^2\right )}{10\,\left (x+6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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