Optimal. Leaf size=23 \[ \frac {4}{-2+x}+2 x+\log (2 (x-x (5+\log (x)))) \]
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Rubi [A]
time = 0.31, antiderivative size = 20, normalized size of antiderivative = 0.87, number of steps
used = 7, number of rules used = 5, integrand size = 62, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {6820, 6874,
1634, 2339, 29} \begin {gather*} 2 x-\frac {4}{2-x}+\log (x)+\log (\log (x)+4) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 1634
Rule 2339
Rule 6820
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {20-4 x-27 x^2+8 x^3+\left (4-7 x^2+2 x^3\right ) \log (x)}{(2-x)^2 x (4+\log (x))} \, dx\\ &=\int \left (\frac {4-7 x^2+2 x^3}{(-2+x)^2 x}+\frac {1}{x (4+\log (x))}\right ) \, dx\\ &=\int \frac {4-7 x^2+2 x^3}{(-2+x)^2 x} \, dx+\int \frac {1}{x (4+\log (x))} \, dx\\ &=\int \left (2-\frac {4}{(-2+x)^2}+\frac {1}{x}\right ) \, dx+\text {Subst}\left (\int \frac {1}{x} \, dx,x,4+\log (x)\right )\\ &=-\frac {4}{2-x}+2 x+\log (x)+\log (4+\log (x))\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.05, size = 18, normalized size = 0.78 \begin {gather*} \frac {4}{-2+x}+2 x+\log (x)+\log (4+\log (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 28, normalized size = 1.22
method | result | size |
norman | \(\frac {2 x^{2}-4}{x -2}+\ln \left (x \right )+\ln \left (\ln \left (x \right )+4\right )\) | \(22\) |
default | \(\frac {x \ln \left (x \right )-2 \ln \left (x \right )+2 x^{2}-4}{x -2}+\ln \left (\ln \left (x \right )+4\right )\) | \(28\) |
risch | \(\frac {x \ln \left (x \right )+2 x^{2}-2 \ln \left (x \right )-4 x +4}{x -2}+\ln \left (\ln \left (x \right )+4\right )\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 23, normalized size = 1.00 \begin {gather*} \frac {2 \, {\left (x^{2} - 2 \, x + 2\right )}}{x - 2} + \log \left (x\right ) + \log \left (\log \left (x\right ) + 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 31, normalized size = 1.35 \begin {gather*} \frac {2 \, x^{2} + {\left (x - 2\right )} \log \left (x\right ) + {\left (x - 2\right )} \log \left (\log \left (x\right ) + 4\right ) - 4 \, x + 4}{x - 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 17, normalized size = 0.74 \begin {gather*} 2 x + \log {\left (x \right )} + \log {\left (\log {\left (x \right )} + 4 \right )} + \frac {4}{x - 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 18, normalized size = 0.78 \begin {gather*} 2 \, x + \frac {4}{x - 2} + \log \left (x\right ) + \log \left (\log \left (x\right ) + 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.33, size = 18, normalized size = 0.78 \begin {gather*} 2\,x+\ln \left (\ln \left (x\right )+4\right )+\ln \left (x\right )+\frac {4}{x-2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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