Optimal. Leaf size=25 \[ x^2+\log \left ((1+x-\log (2 x))^2\right )+5 \log \left (\log ^2(4 x)\right ) \]
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Rubi [A]
time = 0.31, antiderivative size = 23, normalized size of antiderivative = 0.92, number of steps
used = 7, number of rules used = 4, integrand size = 64, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {6874, 6816,
2339, 29} \begin {gather*} x^2+2 \log (x-\log (2 x)+1)+10 \log (\log (4 x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 2339
Rule 6816
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {2 \left (-1+x+x^2+x^3-x^2 \log (2 x)\right )}{x (1+x-\log (2 x))}+\frac {10}{x \log (4 x)}\right ) \, dx\\ &=2 \int \frac {-1+x+x^2+x^3-x^2 \log (2 x)}{x (1+x-\log (2 x))} \, dx+10 \int \frac {1}{x \log (4 x)} \, dx\\ &=2 \int \left (x+\frac {-1+x}{x (1+x-\log (2 x))}\right ) \, dx+10 \text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (4 x)\right )\\ &=x^2+10 \log (\log (4 x))+2 \int \frac {-1+x}{x (1+x-\log (2 x))} \, dx\\ &=x^2+2 \log (1+x-\log (2 x))+10 \log (\log (4 x))\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.07, size = 27, normalized size = 1.08 \begin {gather*} 2 \left (\frac {x^2}{2}+\log (1+x-\log (2 x))+5 \log (\log (4 x))\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.59, size = 27, normalized size = 1.08
method | result | size |
default | \(x^{2}+10 \ln \left (\ln \left (x \right )+2 \ln \left (2\right )\right )+2 \ln \left (-x +\ln \left (2\right )+\ln \left (x \right )-1\right )\) | \(27\) |
risch | \(x^{2}+2 \ln \left (\ln \left (x \right )-\frac {i \left (2 i \ln \left (2\right )-2 i x -2 i\right )}{2}\right )+10 \ln \left (\ln \left (x \right )+2 \ln \left (2\right )\right )\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 26, normalized size = 1.04 \begin {gather*} x^{2} + 2 \, \log \left (-x + \log \left (2\right ) + \log \left (x\right ) - 1\right ) + 10 \, \log \left (2 \, \log \left (2\right ) + \log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 26, normalized size = 1.04 \begin {gather*} x^{2} + 2 \, \log \left (-x + \log \left (2 \, x\right ) - 1\right ) + 10 \, \log \left (\log \left (2\right ) + \log \left (2 \, x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.23, size = 26, normalized size = 1.04 \begin {gather*} x^{2} + 10 \log {\left (\log {\left (2 x \right )} + \log {\left (2 \right )} \right )} + 2 \log {\left (- x + \log {\left (2 x \right )} - 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 28, normalized size = 1.12 \begin {gather*} x^{2} + 2 \, \log \left (2 \, x - 2 \, \log \left (2 \, x\right ) + 2\right ) + 10 \, \log \left (\log \left (2\right ) + \log \left (2 \, x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.64, size = 53, normalized size = 2.12 \begin {gather*} 12\,\ln \left (x\right )-10\,\ln \left (x-1\right )+2\,\ln \left (\frac {8\,x-\ln \left (256\right )-8\,\ln \left (x\right )+8}{x}\right )+10\,\ln \left (\frac {\left (16\,\ln \left (2\right )+8\,\ln \left (x\right )\right )\,\left (x-1\right )}{x}\right )+x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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