Optimal. Leaf size=22 \[ \left (x-\frac {x}{2-\frac {x^2 \log (x)}{\log (3)}}\right )^2 \]
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Rubi [F] time = 0.89, 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 {2 x^3 \log ^2(3)-4 x \log ^3(3)+\left (-2 x^5 \log (3)+14 x^3 \log ^2(3)\right ) \log (x)-12 x^5 \log (3) \log ^2(x)+2 x^7 \log ^3(x)}{-8 \log ^3(3)+12 x^2 \log ^2(3) \log (x)-6 x^4 \log (3) \log ^2(x)+x^6 \log ^3(x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (x \log ^2(3) \left (-x^2+\log (9)\right )+x^3 \left (x^2-7 \log (3)\right ) \log (3) \log (x)+6 x^5 \log (3) \log ^2(x)-x^7 \log ^3(x)\right )}{\left (\log (9)-x^2 \log (x)\right )^3} \, dx\\ &=2 \int \frac {x \log ^2(3) \left (-x^2+\log (9)\right )+x^3 \left (x^2-7 \log (3)\right ) \log (3) \log (x)+6 x^5 \log (3) \log ^2(x)-x^7 \log ^3(x)}{\left (\log (9)-x^2 \log (x)\right )^3} \, dx\\ &=2 \int \left (x+\frac {x \left (x^2 \log ^2(3)-\log (9) \left (6 \log ^2(3)-6 \log (3) \log (9)+\log ^2(9)\right )\right )}{\left (\log (9)-x^2 \log (x)\right )^3}-\frac {x \left (x^2 \log (3)-7 \log ^2(3)+12 \log (3) \log (9)-3 \log ^2(9)\right )}{\left (\log (9)-x^2 \log (x)\right )^2}\right ) \, dx\\ &=x^2+2 \int \frac {x \left (x^2 \log ^2(3)-\log (9) \left (6 \log ^2(3)-6 \log (3) \log (9)+\log ^2(9)\right )\right )}{\left (\log (9)-x^2 \log (x)\right )^3} \, dx-2 \int \frac {x \left (x^2 \log (3)-7 \log ^2(3)+12 \log (3) \log (9)-3 \log ^2(9)\right )}{\left (\log (9)-x^2 \log (x)\right )^2} \, dx\\ &=x^2+2 \int \left (\frac {x^3 \log ^2(3)}{\left (\log (9)-x^2 \log (x)\right )^3}-\frac {x \log (9) \left (6 \log ^2(3)-6 \log (3) \log (9)+\log ^2(9)\right )}{\left (\log (9)-x^2 \log (x)\right )^3}\right ) \, dx-2 \int \left (\frac {x^3 \log (3)}{\left (\log (9)-x^2 \log (x)\right )^2}-\frac {x \left (7 \log ^2(3)-12 \log (3) \log (9)+3 \log ^2(9)\right )}{\left (\log (9)-x^2 \log (x)\right )^2}\right ) \, dx\\ &=x^2-(2 \log (3)) \int \frac {x^3}{\left (\log (9)-x^2 \log (x)\right )^2} \, dx+\left (2 \log ^2(3)\right ) \int \frac {x^3}{\left (\log (9)-x^2 \log (x)\right )^3} \, dx-\left (2 \log (9) \left (6 \log ^2(3)-6 \log (3) \log (9)+\log ^2(9)\right )\right ) \int \frac {x}{\left (\log (9)-x^2 \log (x)\right )^3} \, dx+\left (2 \left (7 \log ^2(3)-12 \log (3) \log (9)+3 \log ^2(9)\right )\right ) \int \frac {x}{\left (\log (9)-x^2 \log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.58, size = 182, normalized size = 8.27 \begin {gather*} \frac {x^2 \left (8 \log ^2(3) \log ^3(9)-x^6 \left (\log ^2(3)+\log (3) \log (9)-\log ^2(9)\right )+4 x^2 \log ^2(9) \left (\log ^2(3)-\log (3) \log (9)+\log ^2(9)\right )+x^4 \left (\log ^3(9)+\log ^2(3) \log (81)\right )-\left (x^8 \log (9)+32 x^2 \log ^2(3) \log ^2(9)+6 x^6 \left (2 \log ^2(3)-\log (3) \log (9)+\log ^2(9)\right )+8 x^4 \log (9) \left (4 \log ^2(3)-\log (3) \log (9)+\log ^2(9)\right )\right ) \log (x)+x^4 \left (x^2+\log (81)\right )^3 \log ^2(x)\right )}{\left (x^2+\log (81)\right )^3 \left (\log (9)-x^2 \log (x)\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.58, size = 53, normalized size = 2.41 \begin {gather*} \frac {x^{6} \log \relax (x)^{2} - 2 \, x^{4} \log \relax (3) \log \relax (x) + x^{2} \log \relax (3)^{2}}{x^{4} \log \relax (x)^{2} - 4 \, x^{2} \log \relax (3) \log \relax (x) + 4 \, \log \relax (3)^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 50, normalized size = 2.27 \begin {gather*} x^{2} + \frac {2 \, x^{4} \log \relax (3) \log \relax (x) - 3 \, x^{2} \log \relax (3)^{2}}{x^{4} \log \relax (x)^{2} - 4 \, x^{2} \log \relax (3) \log \relax (x) + 4 \, \log \relax (3)^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 38, normalized size = 1.73
method | result | size |
risch | \(x^{2}-\frac {\left (-2 x^{2} \ln \relax (x )+3 \ln \relax (3)\right ) \ln \relax (3) x^{2}}{\left (-x^{2} \ln \relax (x )+2 \ln \relax (3)\right )^{2}}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.50, size = 53, normalized size = 2.41 \begin {gather*} \frac {x^{6} \log \relax (x)^{2} - 2 \, x^{4} \log \relax (3) \log \relax (x) + x^{2} \log \relax (3)^{2}}{x^{4} \log \relax (x)^{2} - 4 \, x^{2} \log \relax (3) \log \relax (x) + 4 \, \log \relax (3)^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.79, size = 229, normalized size = 10.41 \begin {gather*} \frac {\frac {52\,x^2\,{\ln \relax (3)}^3+11\,x^4\,{\ln \relax (3)}^2-8\,{\ln \relax (3)}^3\,\ln \relax (9)+x^6\,\ln \relax (3)+64\,{\ln \relax (3)}^4-4\,x^2\,{\ln \relax (3)}^2\,\ln \relax (9)}{{\left (x^2+4\,\ln \relax (3)\right )}^3}-\frac {2\,x^4\,{\ln \relax (3)}^2\,\ln \relax (x)}{{\left (x^2+4\,\ln \relax (3)\right )}^3}}{\ln \relax (x)-\frac {2\,\ln \relax (3)}{x^2}}+x^2-\frac {\frac {{\ln \relax (3)}^2\,x^2-{\ln \relax (3)}^2\,\ln \relax (9)+8\,{\ln \relax (3)}^3}{x^2\,\left (x^2+4\,\ln \relax (3)\right )}-\frac {\ln \relax (x)\,\left (\ln \relax (3)\,x^2+5\,{\ln \relax (3)}^2\right )}{x^2+4\,\ln \relax (3)}}{\frac {4\,{\ln \relax (3)}^2}{x^4}+{\ln \relax (x)}^2-\frac {4\,\ln \relax (3)\,\ln \relax (x)}{x^2}}+\frac {2\,x^4\,{\ln \relax (3)}^2}{x^6+12\,\ln \relax (3)\,x^4+48\,{\ln \relax (3)}^2\,x^2+64\,{\ln \relax (3)}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.17, size = 51, normalized size = 2.32 \begin {gather*} x^{2} + \frac {2 x^{4} \log {\relax (3 )} \log {\relax (x )} - 3 x^{2} \log {\relax (3 )}^{2}}{x^{4} \log {\relax (x )}^{2} - 4 x^{2} \log {\relax (3 )} \log {\relax (x )} + 4 \log {\relax (3 )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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