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 = 0.50, 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 = {6742, 6684, 2302, 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 2302
Rule 6684
Rule 6742
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 \operatorname {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 = 0.19, 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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fricas [A] time = 0.70, 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 \relax (2) + \log \left (2 \, x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.50, size = 26, normalized size = 1.04 \begin {gather*} x^{2} + 2 \, \log \left (-x + \log \relax (2) + \log \relax (x) - 1\right ) + 10 \, \log \left (2 \, \log \relax (2) + \log \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.12, size = 36, normalized size = 1.44
method | result | size |
risch | \(x^{2}+2 \ln \left (\ln \relax (x )-\frac {i \left (2 i \ln \relax (2)-2 i x -2 i\right )}{2}\right )+10 \ln \left (\ln \relax (x )+2 \ln \relax (2)\right )\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.94, size = 26, normalized size = 1.04 \begin {gather*} x^{2} + 2 \, \log \left (-x + \log \relax (2) + \log \relax (x) - 1\right ) + 10 \, \log \left (2 \, \log \relax (2) + \log \relax (x)\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 \relax (x)-10\,\ln \left (x-1\right )+2\,\ln \left (\frac {8\,x-\ln \left (256\right )-8\,\ln \relax (x)+8}{x}\right )+10\,\ln \left (\frac {\left (16\,\ln \relax (2)+8\,\ln \relax (x)\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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sympy [A] time = 0.62, size = 26, normalized size = 1.04 \begin {gather*} x^{2} + 10 \log {\left (\log {\left (2 x \right )} + \log {\relax (2 )} \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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