Optimal. Leaf size=15 \[ \log \left (\frac {3 (-2-x) \log (x)}{50 x}\right ) \]
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Rubi [A] time = 0.17, antiderivative size = 12, normalized size of antiderivative = 0.80, number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1593, 6742, 36, 29, 31, 2302} \begin {gather*} -\log (x)+\log (x+2)+\log (\log (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 1593
Rule 2302
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2+x-2 \log (x)}{x (2+x) \log (x)} \, dx\\ &=\int \left (-\frac {2}{x (2+x)}+\frac {1}{x \log (x)}\right ) \, dx\\ &=-\left (2 \int \frac {1}{x (2+x)} \, dx\right )+\int \frac {1}{x \log (x)} \, dx\\ &=-\int \frac {1}{x} \, dx+\int \frac {1}{2+x} \, dx+\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )\\ &=-\log (x)+\log (2+x)+\log (\log (x))\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.04, size = 12, normalized size = 0.80 \begin {gather*} -\log (x)+\log (2+x)+\log (\log (x)) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 12, normalized size = 0.80 \begin {gather*} \log \left (x + 2\right ) - \log \relax (x) + \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 = 0.22, size = 12, normalized size = 0.80 \begin {gather*} \log \left (x + 2\right ) - \log \relax (x) + \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 = 0.17, size = 13, normalized size = 0.87
method | result | size |
default | \(\ln \left (\ln \relax (x )\right )+\ln \left (2+x \right )-\ln \relax (x )\) | \(13\) |
norman | \(\ln \left (\ln \relax (x )\right )+\ln \left (2+x \right )-\ln \relax (x )\) | \(13\) |
risch | \(\ln \left (\ln \relax (x )\right )+\ln \left (2+x \right )-\ln \relax (x )\) | \(13\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 12, normalized size = 0.80 \begin {gather*} \log \left (x + 2\right ) - \log \relax (x) + \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 = 3.52, size = 12, normalized size = 0.80 \begin {gather*} \ln \left (x+2\right )+\ln \left (\ln \relax (x)\right )-\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 12, normalized size = 0.80 \begin {gather*} - \log {\relax (x )} + \log {\left (x + 2 \right )} + \log {\left (\log {\relax (x )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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