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 = 9.95, 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 = {6874, 2104,
814, 648, 632, 210, 642, 2603} \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 210
Rule 632
Rule 642
Rule 648
Rule 814
Rule 2104
Rule 2603
Rule 6874
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 = 0.07, 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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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.49, size = 111, normalized size = 4.83
method | result | size |
norman | \(-2 x -x \ln \left (\frac {2 x \ln \left (2\right )+x^{4}-x^{2}}{2 \ln \left (2\right )}\right )\) | \(29\) |
risch | \(-2 x -x \ln \left (\frac {2 x \ln \left (2\right )+x^{4}-x^{2}}{2 \ln \left (2\right )}\right )\) | \(29\) |
default | \(x \ln \left (2\right )-2 x +2 \left (\munderset {\textit {\_R} =\RootOf \left (2 \ln \left (2\right )+\textit {\_Z}^{3}-\textit {\_Z} \right )}{\sum }\frac {\left (-\textit {\_R} +3 \ln \left (2\right )\right ) \ln \left (x -\textit {\_R} \right )}{3 \textit {\_R}^{2}-1}\right )-x \ln \left (x \left (2 \ln \left (2\right )+x^{3}-x \right )\right )+2 \left (\munderset {\textit {\_R} =\RootOf \left (2 \ln \left (2\right )+\textit {\_Z}^{3}-\textit {\_Z} \right )}{\sum }\frac {\left (\textit {\_R} -3 \ln \left (2\right )\right ) \ln \left (x -\textit {\_R} \right )}{3 \textit {\_R}^{2}-1}\right )+x \ln \left (\ln \left (2\right )\right )\) | \(111\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 30, normalized size = 1.30 \begin {gather*} x {\left (\log \left (2\right ) + \log \left (\log \left (2\right )\right ) - 2\right )} - x \log \left (x^{3} - x + 2 \, \log \left (2\right )\right ) - x \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 28, normalized size = 1.22 \begin {gather*} -x \log \left (\frac {x^{4} - x^{2} + 2 \, x \log \left (2\right )}{2 \, \log \left (2\right )}\right ) - 2 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 26, normalized size = 1.13 \begin {gather*} - x \log {\left (\frac {\frac {x^{4}}{2} - \frac {x^{2}}{2} + x \log {\left (2 \right )}}{\log {\left (2 \right )}} \right )} - 2 x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 28, normalized size = 1.22 \begin {gather*} x {\left (\log \left (2\right ) + \log \left (\log \left (2\right )\right ) - 2\right )} - x \log \left (x^{4} - x^{2} + 2 \, x \log \left (2\right )\right ) \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 \left (2\right )-2\,x+x\,\ln \left (\ln \left (2\right )\right )-x\,\ln \left (x^4-x^2+2\,\ln \left (2\right )\,x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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