Optimal. Leaf size=23 \[ x-x \left (3+\log \left (x+\frac {x \left (-x+x^3\right )}{\log (4)}\right )\right ) \]
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Rubi [A] time = 8.92, antiderivative size = 27, normalized size of antiderivative = 1.17, number of steps used = 21, number of rules used = 8, integrand size = 56, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6742, 2079, 800, 634, 618, 204, 628, 2523} \begin {gather*} x \left (-\log \left (-x \left (-x^3+x-\log (4)\right )\right )\right )-2 x+x \log (\log (4)) \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 800
Rule 2079
Rule 2523
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {4 x-6 x^3-3 \log (4)}{-x+x^3+\log (4)}+\log (\log (4))-\log \left (x \left (-x+x^3+\log (4)\right )\right )\right ) \, dx\\ &=x \log (\log (4))+\int \frac {4 x-6 x^3-3 \log (4)}{-x+x^3+\log (4)} \, dx-\int \log \left (x \left (-x+x^3+\log (4)\right )\right ) \, dx\\ &=-x \log \left (-x \left (x-x^3-\log (4)\right )\right )+x \log (\log (4))+\int \frac {2 x-4 x^3-\log (4)}{x-x^3-\log (4)} \, dx+\int \left (-6+\frac {-2 x+\log (64)}{-x+x^3+\log (4)}\right ) \, dx\\ &=-6 x-x \log \left (-x \left (x-x^3-\log (4)\right )\right )+x \log (\log (4))+\int \frac {-2 x+\log (64)}{-x+x^3+\log (4)} \, dx+\int \left (4+\frac {-2 x+\log (64)}{x-x^3-\log (4)}\right ) \, dx\\ &=-2 x-x \log \left (-x \left (x-x^3-\log (4)\right )\right )+x \log (\log (4))+\int \frac {-2 x+\log (64)}{x-x^3-\log (4)} \, dx+\int \frac {-2 x+\log (64)}{\left (x+\frac {2 \sqrt [3]{\frac {3}{9 \log (4)-\sqrt {-12+81 \log ^2(4)}}}+\sqrt [3]{2 \left (9 \log (4)-\sqrt {-12+81 \log ^2(4)}\right )}}{6^{2/3}}\right ) \left (x^2-\frac {1}{3} x \left (3^{2/3} \sqrt [3]{\frac {2}{9 \log (4)-\sqrt {-12+81 \log ^2(4)}}}+\sqrt [3]{\frac {3}{2} \left (9 \log (4)-\sqrt {-12+81 \log ^2(4)}\right )}\right )+\frac {1}{18} \left (-6+6 \sqrt [3]{3} \left (\frac {2}{9 \log (4)-\sqrt {-12+81 \log ^2(4)}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (9 \log (4)-\sqrt {-12+81 \log ^2(4)}\right )\right )^{2/3}\right )\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A] time = 0.12, size = 23, normalized size = 1.00 \begin {gather*} -2 x-x \log \left (\frac {x \left (-x+x^3+\log (4)\right )}{\log (4)}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 28, normalized size = 1.22 \begin {gather*} -x \log \left (\frac {x^{4} - x^{2} + 2 \, x \log \relax (2)}{2 \, \log \relax (2)}\right ) - 2 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 28, normalized size = 1.22 \begin {gather*} x {\left (\log \relax (2) + \log \left (\log \relax (2)\right ) - 2\right )} - x \log \left (x^{4} - x^{2} + 2 \, x \log \relax (2)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 29, normalized size = 1.26
method | result | size |
norman | \(-2 x -x \ln \left (\frac {2 x \ln \relax (2)+x^{4}-x^{2}}{2 \ln \relax (2)}\right )\) | \(29\) |
risch | \(-2 x -x \ln \left (\frac {2 x \ln \relax (2)+x^{4}-x^{2}}{2 \ln \relax (2)}\right )\) | \(29\) |
default | \(x \ln \relax (2)-2 x +2 \left (\munderset {\textit {\_R} =\RootOf \left (2 \ln \relax (2)+\textit {\_Z}^{3}-\textit {\_Z} \right )}{\sum }\frac {\left (-\textit {\_R} +3 \ln \relax (2)\right ) \ln \left (x -\textit {\_R} \right )}{3 \textit {\_R}^{2}-1}\right )-x \ln \left (x \left (2 \ln \relax (2)+x^{3}-x \right )\right )+2 \left (\munderset {\textit {\_R} =\RootOf \left (2 \ln \relax (2)+\textit {\_Z}^{3}-\textit {\_Z} \right )}{\sum }\frac {\left (\textit {\_R} -3 \ln \relax (2)\right ) \ln \left (x -\textit {\_R} \right )}{3 \textit {\_R}^{2}-1}\right )+x \ln \left (\ln \relax (2)\right )\) | \(111\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 30, normalized size = 1.30 \begin {gather*} x {\left (\log \relax (2) + \log \left (\log \relax (2)\right ) - 2\right )} - x \log \left (x^{3} - x + 2 \, \log \relax (2)\right ) - x \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.43, size = 31, normalized size = 1.35 \begin {gather*} x\,\ln \relax (2)-2\,x+x\,\ln \left (\ln \relax (2)\right )-x\,\ln \left (x^4-x^2+2\,\ln \relax (2)\,x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 26, normalized size = 1.13 \begin {gather*} - x \log {\left (\frac {\frac {x^{4}}{2} - \frac {x^{2}}{2} + x \log {\relax (2 )}}{\log {\relax (2 )}} \right )} - 2 x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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