Optimal. Leaf size=27 \[ \log \left (\frac {(-4+x)^2 \log \left (x^2\right )}{1-\frac {4}{2-x}+2 x}\right ) \]
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Rubi [A]
time = 0.82, antiderivative size = 33, normalized size of antiderivative = 1.22, number of steps
used = 8, number of rules used = 6, integrand size = 77, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.078, Rules used = {6873, 6874,
6860, 642, 2339, 29} \begin {gather*} -\log \left (2 x^2-3 x+2\right )+\log \left (\log \left (x^2\right )\right )+\log (2-x)+2 \log (4-x) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 642
Rule 2339
Rule 6860
Rule 6873
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {32-72 x+72 x^2-30 x^3+4 x^4+\left (8 x-20 x^2+2 x^3+2 x^4\right ) \log \left (x^2\right )}{x \left (16-36 x+36 x^2-15 x^3+2 x^4\right ) \log \left (x^2\right )} \, dx\\ &=\int \left (\frac {2 \left (4-10 x+x^2+x^3\right )}{(-4+x) (-2+x) \left (2-3 x+2 x^2\right )}+\frac {2}{x \log \left (x^2\right )}\right ) \, dx\\ &=2 \int \frac {4-10 x+x^2+x^3}{(-4+x) (-2+x) \left (2-3 x+2 x^2\right )} \, dx+2 \int \frac {1}{x \log \left (x^2\right )} \, dx\\ &=2 \int \left (\frac {1}{-4+x}+\frac {1}{2 (-2+x)}+\frac {3-4 x}{2 \left (2-3 x+2 x^2\right )}\right ) \, dx+\text {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (x^2\right )\right )\\ &=\log (2-x)+2 \log (4-x)+\log \left (\log \left (x^2\right )\right )+\int \frac {3-4 x}{2-3 x+2 x^2} \, dx\\ &=\log (2-x)+2 \log (4-x)-\log \left (2-3 x+2 x^2\right )+\log \left (\log \left (x^2\right )\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.07, size = 33, normalized size = 1.22 \begin {gather*} \log (2-x)+2 \log (4-x)-\log \left (2-3 x+2 x^2\right )+\log \left (\log \left (x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 30, normalized size = 1.11
method | result | size |
default | \(\ln \left (\ln \left (x^{2}\right )\right )+\ln \left (x -2\right )-\ln \left (2 x^{2}-3 x +2\right )+2 \ln \left (x -4\right )\) | \(30\) |
norman | \(\ln \left (\ln \left (x^{2}\right )\right )+\ln \left (x -2\right )-\ln \left (2 x^{2}-3 x +2\right )+2 \ln \left (x -4\right )\) | \(30\) |
risch | \(\ln \left (\ln \left (x^{2}\right )\right )+\ln \left (x -2\right )-\ln \left (2 x^{2}-3 x +2\right )+2 \ln \left (x -4\right )\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 27, normalized size = 1.00 \begin {gather*} -\log \left (2 \, x^{2} - 3 \, x + 2\right ) + \log \left (x - 2\right ) + 2 \, \log \left (x - 4\right ) + \log \left (\log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 29, normalized size = 1.07 \begin {gather*} -\log \left (2 \, x^{2} - 3 \, x + 2\right ) + \log \left (x - 2\right ) + 2 \, \log \left (x - 4\right ) + \log \left (\log \left (x^{2}\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 29, normalized size = 1.07 \begin {gather*} 2 \log {\left (x - 4 \right )} + \log {\left (x - 2 \right )} - \log {\left (2 x^{2} - 3 x + 2 \right )} + \log {\left (\log {\left (x^{2} \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 29, normalized size = 1.07 \begin {gather*} -\log \left (2 \, x^{2} - 3 \, x + 2\right ) + \log \left (x - 2\right ) + 2 \, \log \left (x - 4\right ) + \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 = 3.42, size = 27, normalized size = 1.00 \begin {gather*} \ln \left (x-2\right )+2\,\ln \left (x-4\right )-\ln \left (x^2-\frac {3\,x}{2}+1\right )+\ln \left (\ln \left (x^2\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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