Optimal. Leaf size=31 \[ -\frac {4+x}{x \left (\log (x)+\frac {-x+\log \left (\frac {x^2}{3}\right )}{-1+x}\right )} \]
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Rubi [F] time = 2.16, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-4+7 x+x^2+x^3+\left (4-8 x+4 x^2\right ) \log (x)+\left (-4-x^2\right ) \log \left (\frac {x^2}{3}\right )}{x^4+\left (2 x^3-2 x^4\right ) \log (x)+\left (x^2-2 x^3+x^4\right ) \log ^2(x)+\left (-2 x^3+\left (-2 x^2+2 x^3\right ) \log (x)\right ) \log \left (\frac {x^2}{3}\right )+x^2 \log ^2\left (\frac {x^2}{3}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {7 x+x^2+x^3-4 (1-\log (3))+4 (-1+x)^2 \log (x)-x^2 \log \left (\frac {x^2}{3}\right )-4 \log \left (x^2\right )}{x^2 \left (x-(-1+x) \log (x)-\log \left (\frac {x^2}{3}\right )\right )^2} \, dx\\ &=\int \left (\frac {1}{\left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2}+\frac {7}{x \left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2}+\frac {x}{\left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2}+\frac {-4+\log (81)}{x^2 \left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2}+\frac {4 (-1+x)^2 \log (x)}{x^2 \left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2}+\frac {\log (3)-\log \left (x^2\right )}{\left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2}-\frac {4 \log \left (x^2\right )}{x^2 \left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2}\right ) \, dx\\ &=4 \int \frac {(-1+x)^2 \log (x)}{x^2 \left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx-4 \int \frac {\log \left (x^2\right )}{x^2 \left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx+7 \int \frac {1}{x \left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx+(-4+\log (81)) \int \frac {1}{x^2 \left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx+\int \frac {1}{\left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx+\int \frac {x}{\left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx+\int \frac {\log (3)-\log \left (x^2\right )}{\left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx\\ &=4 \int \left (\frac {\log (x)}{\left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2}+\frac {\log (x)}{x^2 \left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2}-\frac {2 \log (x)}{x \left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2}\right ) \, dx-4 \int \frac {\log \left (x^2\right )}{x^2 \left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx+7 \int \frac {1}{x \left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx+(-4+\log (81)) \int \frac {1}{x^2 \left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx+\int \frac {1}{\left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx+\int \frac {x}{\left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx+\int \left (\frac {\log (3)}{\left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2}-\frac {\log \left (x^2\right )}{\left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2}\right ) \, dx\\ &=4 \int \frac {\log (x)}{\left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx+4 \int \frac {\log (x)}{x^2 \left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx-4 \int \frac {\log \left (x^2\right )}{x^2 \left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx+7 \int \frac {1}{x \left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx-8 \int \frac {\log (x)}{x \left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx+\log (3) \int \frac {1}{\left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx+(-4+\log (81)) \int \frac {1}{x^2 \left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx+\int \frac {1}{\left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx+\int \frac {x}{\left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx-\int \frac {\log \left (x^2\right )}{\left (-x-\log (x)+x \log (x)+\log \left (\frac {x^2}{3}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 2.93, size = 104, normalized size = 3.35 \begin {gather*} \frac {4+x-4 x^3-x^4-x \log (81)-2 x \left (-4+3 x+x^2\right ) \log (x)+x \left (-8+3 x+x^2\right ) \log \left (\frac {x^2}{3}\right )+4 x \log \left (x^2\right )}{x \left (-x+(-1+x) \log (x)+\log \left (\frac {x^2}{3}\right )\right ) \left (1+x+x^2+x \log (3)+2 x \log (x)-x \log \left (x^2\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 28, normalized size = 0.90 \begin {gather*} \frac {x^{2} + 3 \, x - 4}{x^{2} + x \log \relax (3) - {\left (x^{2} + x\right )} \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.83, size = 33, normalized size = 1.06 \begin {gather*} -\frac {x^{2} + 3 \, x - 4}{x^{2} \log \relax (x) - x^{2} - x \log \relax (3) + x \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.15, size = 82, normalized size = 2.65
method | result | size |
risch | \(-\frac {2 \left (x^{2}+3 x -4\right )}{x \left (-i \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+2 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+2 x \ln \relax (x )-2 \ln \relax (3)-2 x +2 \ln \relax (x )\right )}\) | \(82\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 28, normalized size = 0.90 \begin {gather*} \frac {x^{2} + 3 \, x - 4}{x^{2} + x \log \relax (3) - {\left (x^{2} + x\right )} \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.87, size = 101, normalized size = 3.26 \begin {gather*} -\frac {x+\ln \left (\frac {x^2}{3}\right )\,\left (x^3+3\,x^2-4\,x\right )-4\,x^3-x^4-\ln \relax (x)\,\left (2\,x^3+6\,x^2-8\,x\right )+4}{\left (x-x^2\,\left (\ln \left (\frac {x^2}{3}\right )-2\,\ln \relax (x)\right )+x^2+x^3\right )\,\left (x-\ln \left (\frac {x^2}{3}\right )+2\,\ln \relax (x)-\ln \relax (x)\,\left (x+1\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.38, size = 24, normalized size = 0.77 \begin {gather*} \frac {- x^{2} - 3 x + 4}{- x^{2} - x \log {\relax (3 )} + \left (x^{2} + x\right ) \log {\relax (x )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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