Optimal. Leaf size=30 \[ \log \left (x^2 \log (4)+\frac {\left (x-x^3\right ) \left (x-\log \left (2+\log \left (x^2\right )\right )\right )}{x}\right ) \]
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Rubi [A] time = 0.49, antiderivative size = 34, normalized size of antiderivative = 1.13, number of steps used = 3, number of rules used = 3, integrand size = 134, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {6, 6741, 6684} \begin {gather*} \log \left (-x^3+x^2 \log \left (\log \left (x^2\right )+2\right )+x^2 \log (4)-\log \left (\log \left (x^2\right )+2\right )+x\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 6684
Rule 6741
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2+2 x-6 x^3+x^2 (2+4 \log (4))+\left (x-3 x^3+2 x^2 \log (4)\right ) \log \left (x^2\right )+\left (4 x^2+2 x^2 \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )}{2 x^2-2 x^4+2 x^3 \log (4)+\left (x^2-x^4+x^3 \log (4)\right ) \log \left (x^2\right )+\left (-2 x+2 x^3+\left (-x+x^3\right ) \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )} \, dx\\ &=\int \frac {-2+2 x-6 x^3+x^2 (2+4 \log (4))+\left (x-3 x^3+2 x^2 \log (4)\right ) \log \left (x^2\right )+\left (4 x^2+2 x^2 \log \left (x^2\right )\right ) \log \left (2+\log \left (x^2\right )\right )}{x \left (2+\log \left (x^2\right )\right ) \left (x-x^3+x^2 \log (4)-\log \left (2+\log \left (x^2\right )\right )+x^2 \log \left (2+\log \left (x^2\right )\right )\right )} \, dx\\ &=\log \left (x-x^3+x^2 \log (4)-\log \left (2+\log \left (x^2\right )\right )+x^2 \log \left (2+\log \left (x^2\right )\right )\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.10, size = 34, normalized size = 1.13 \begin {gather*} \log \left (-x+x^3-x^2 \log (4)+\log \left (2+\log \left (x^2\right )\right )-x^2 \log \left (2+\log \left (x^2\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.16, size = 45, normalized size = 1.50 \begin {gather*} \log \left (x^{2} - 1\right ) + \log \left (-\frac {x^{3} - 2 \, x^{2} \log \relax (2) - {\left (x^{2} - 1\right )} \log \left (\log \left (x^{2}\right ) + 2\right ) - x}{x^{2} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.50, size = 35, normalized size = 1.17 \begin {gather*} \log \left (-x^{3} + 2 \, x^{2} \log \relax (2) + x^{2} \log \left (\log \left (x^{2}\right ) + 2\right ) + x - \log \left (\log \left (x^{2}\right ) + 2\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (2 x^{2} \ln \left (x^{2}\right )+4 x^{2}\right ) \ln \left (2+\ln \left (x^{2}\right )\right )+\left (4 x^{2} \ln \relax (2)-3 x^{3}+x \right ) \ln \left (x^{2}\right )+8 x^{2} \ln \relax (2)-6 x^{3}+2 x^{2}+2 x -2}{\left (\left (x^{3}-x \right ) \ln \left (x^{2}\right )+2 x^{3}-2 x \right ) \ln \left (2+\ln \left (x^{2}\right )\right )+\left (2 x^{3} \ln \relax (2)-x^{4}+x^{2}\right ) \ln \left (x^{2}\right )+4 x^{3} \ln \relax (2)-2 x^{4}+2 x^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 47, normalized size = 1.57 \begin {gather*} \log \left (x + 1\right ) + \log \left (x - 1\right ) + \log \left (-\frac {x^{3} - 3 \, x^{2} \log \relax (2) - {\left (x^{2} - 1\right )} \log \left (\log \relax (x) + 1\right ) - x + \log \relax (2)}{x^{2} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {2\,x+\ln \left (x^2\right )\,\left (-3\,x^3+4\,\ln \relax (2)\,x^2+x\right )+\ln \left (\ln \left (x^2\right )+2\right )\,\left (2\,x^2\,\ln \left (x^2\right )+4\,x^2\right )+8\,x^2\,\ln \relax (2)+2\,x^2-6\,x^3-2}{4\,x^3\,\ln \relax (2)-\ln \left (\ln \left (x^2\right )+2\right )\,\left (2\,x+\ln \left (x^2\right )\,\left (x-x^3\right )-2\,x^3\right )+\ln \left (x^2\right )\,\left (-x^4+2\,\ln \relax (2)\,x^3+x^2\right )+2\,x^2-2\,x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.43, size = 34, normalized size = 1.13 \begin {gather*} \log {\left (x^{2} - 1 \right )} + \log {\left (\log {\left (\log {\left (x^{2} \right )} + 2 \right )} + \frac {- x^{3} + 2 x^{2} \log {\relax (2 )} + x}{x^{2} - 1} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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