Optimal. Leaf size=32 \[ 3-\log \left (\frac {e^{2/x} (3+x)^2 \log ^2(2-x)}{4 x^2}\right ) \]
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Rubi [A] time = 0.37, antiderivative size = 25, normalized size of antiderivative = 0.78, number of steps used = 8, number of rules used = 6, integrand size = 51, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1594, 6728, 77, 2390, 2302, 29} \begin {gather*} -\frac {2}{x}+2 \log (x)-2 \log (x+3)-2 \log (\log (2-x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 77
Rule 1594
Rule 2302
Rule 2390
Rule 6728
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-6 x^2-2 x^3+\left (-12-10 x+8 x^2\right ) \log (2-x)}{x^2 \left (-6+x+x^2\right ) \log (2-x)} \, dx\\ &=\int \left (\frac {2 (3+4 x)}{x^2 (3+x)}-\frac {2}{(-2+x) \log (2-x)}\right ) \, dx\\ &=2 \int \frac {3+4 x}{x^2 (3+x)} \, dx-2 \int \frac {1}{(-2+x) \log (2-x)} \, dx\\ &=2 \int \left (\frac {1}{-3-x}+\frac {1}{x^2}+\frac {1}{x}\right ) \, dx-2 \operatorname {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,2-x\right )\\ &=-\frac {2}{x}+2 \log (x)-2 \log (3+x)-2 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (2-x)\right )\\ &=-\frac {2}{x}+2 \log (x)-2 \log (3+x)-2 \log (\log (2-x))\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.06, size = 21, normalized size = 0.66 \begin {gather*} -2 \left (\frac {1}{x}-\log (x)+\log (3+x)+\log (\log (2-x))\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 27, normalized size = 0.84 \begin {gather*} -\frac {2 \, {\left (x \log \left (x + 3\right ) - x \log \relax (x) + x \log \left (\log \left (-x + 2\right )\right ) + 1\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 25, normalized size = 0.78 \begin {gather*} -\frac {2}{x} - 2 \, \log \left (x + 3\right ) + 2 \, \log \relax (x) - 2 \, \log \left (\log \left (-x + 2\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 26, normalized size = 0.81
method | result | size |
norman | \(-\frac {2}{x}+2 \ln \relax (x )-2 \ln \left (\ln \left (2-x \right )\right )-2 \ln \left (3+x \right )\) | \(26\) |
risch | \(\frac {2 x \ln \relax (x )-2 x \ln \left (3+x \right )-2}{x}-2 \ln \left (\ln \left (2-x \right )\right )\) | \(29\) |
derivativedivides | \(-2 \ln \left (\ln \left (2-x \right )\right )-\frac {2}{x}+2 \ln \left (-x \right )-2 \ln \left (-3-x \right )\) | \(30\) |
default | \(-2 \ln \left (\ln \left (2-x \right )\right )-\frac {2}{x}+2 \ln \left (-x \right )-2 \ln \left (-3-x \right )\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 25, normalized size = 0.78 \begin {gather*} -\frac {2}{x} - 2 \, \log \left (x + 3\right ) + 2 \, \log \relax (x) - 2 \, \log \left (\log \left (-x + 2\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 25, normalized size = 0.78 \begin {gather*} 2\,\ln \relax (x)-2\,\ln \left (x+3\right )-2\,\ln \left (\ln \left (2-x\right )\right )-\frac {2}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 22, normalized size = 0.69 \begin {gather*} 2 \log {\relax (x )} - 2 \log {\left (x + 3 \right )} - 2 \log {\left (\log {\left (2 - x \right )} \right )} - \frac {2}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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