Optimal. Leaf size=31 \[ x \left (\frac {x+\frac {(-1+\log (x))^2}{x}}{3+2 x \log (x)}-\log (2 x)\right ) \]
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Rubi [F] time = 2.08, 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 {-6-11 x+6 x^2-2 x^3+\left (6-2 x-12 x^2+2 x^3\right ) \log (x)+\left (6 x-4 x^3\right ) \log ^2(x)-2 x \log ^3(x)+\left (-9 x-12 x^2 \log (x)-4 x^3 \log ^2(x)\right ) \log (2 x)}{9 x+12 x^2 \log (x)+4 x^3 \log ^2(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 {-6-11 x+6 x^2-2 x^3+\left (6-2 x-12 x^2+2 x^3\right ) \log (x)+\left (6 x-4 x^3\right ) \log ^2(x)-2 x \log ^3(x)+\left (-9 x-12 x^2 \log (x)-4 x^3 \log ^2(x)\right ) \log (2 x)}{x (3+2 x \log (x))^2} \, dx\\ &=\int \left (-\frac {11}{(3+2 x \log (x))^2}-\frac {6}{x (3+2 x \log (x))^2}+\frac {6 x}{(3+2 x \log (x))^2}-\frac {2 x^2}{(3+2 x \log (x))^2}+\frac {2 \left (3-x-6 x^2+x^3\right ) \log (x)}{x (3+2 x \log (x))^2}-\frac {2 \left (-3+2 x^2\right ) \log ^2(x)}{(3+2 x \log (x))^2}-\frac {2 \log ^3(x)}{(3+2 x \log (x))^2}-\log (2 x)\right ) \, dx\\ &=-\left (2 \int \frac {x^2}{(3+2 x \log (x))^2} \, dx\right )+2 \int \frac {\left (3-x-6 x^2+x^3\right ) \log (x)}{x (3+2 x \log (x))^2} \, dx-2 \int \frac {\left (-3+2 x^2\right ) \log ^2(x)}{(3+2 x \log (x))^2} \, dx-2 \int \frac {\log ^3(x)}{(3+2 x \log (x))^2} \, dx-6 \int \frac {1}{x (3+2 x \log (x))^2} \, dx+6 \int \frac {x}{(3+2 x \log (x))^2} \, dx-11 \int \frac {1}{(3+2 x \log (x))^2} \, dx-\int \log (2 x) \, dx\\ &=x-x \log (2 x)-2 \int \frac {x^2}{(3+2 x \log (x))^2} \, dx-2 \int \left (-\frac {3}{4 x^3}+\frac {\log (x)}{4 x^2}-\frac {27}{8 x^3 (3+2 x \log (x))^2}+\frac {27}{8 x^3 (3+2 x \log (x))}\right ) \, dx-2 \int \left (\frac {-3+2 x^2}{4 x^2}+\frac {9 \left (-3+2 x^2\right )}{4 x^2 (3+2 x \log (x))^2}-\frac {3 \left (-3+2 x^2\right )}{2 x^2 (3+2 x \log (x))}\right ) \, dx+2 \int \left (-\frac {3 \left (3-x-6 x^2+x^3\right )}{2 x^2 (3+2 x \log (x))^2}+\frac {3-x-6 x^2+x^3}{2 x^2 (3+2 x \log (x))}\right ) \, dx-6 \int \frac {1}{x (3+2 x \log (x))^2} \, dx+6 \int \frac {x}{(3+2 x \log (x))^2} \, dx-11 \int \frac {1}{(3+2 x \log (x))^2} \, dx\\ &=-\frac {3}{4 x^2}+x-x \log (2 x)-\frac {1}{2} \int \frac {-3+2 x^2}{x^2} \, dx-\frac {1}{2} \int \frac {\log (x)}{x^2} \, dx-2 \int \frac {x^2}{(3+2 x \log (x))^2} \, dx-3 \int \frac {3-x-6 x^2+x^3}{x^2 (3+2 x \log (x))^2} \, dx+3 \int \frac {-3+2 x^2}{x^2 (3+2 x \log (x))} \, dx-\frac {9}{2} \int \frac {-3+2 x^2}{x^2 (3+2 x \log (x))^2} \, dx-6 \int \frac {1}{x (3+2 x \log (x))^2} \, dx+6 \int \frac {x}{(3+2 x \log (x))^2} \, dx+\frac {27}{4} \int \frac {1}{x^3 (3+2 x \log (x))^2} \, dx-\frac {27}{4} \int \frac {1}{x^3 (3+2 x \log (x))} \, dx-11 \int \frac {1}{(3+2 x \log (x))^2} \, dx+\int \frac {3-x-6 x^2+x^3}{x^2 (3+2 x \log (x))} \, dx\\ &=-\frac {3}{4 x^2}+\frac {1}{2 x}+x+\frac {\log (x)}{2 x}-x \log (2 x)-\frac {1}{2} \int \left (2-\frac {3}{x^2}\right ) \, dx-2 \int \frac {x^2}{(3+2 x \log (x))^2} \, dx-3 \int \left (-\frac {6}{(3+2 x \log (x))^2}+\frac {3}{x^2 (3+2 x \log (x))^2}-\frac {1}{x (3+2 x \log (x))^2}+\frac {x}{(3+2 x \log (x))^2}\right ) \, dx+3 \int \left (\frac {2}{3+2 x \log (x)}-\frac {3}{x^2 (3+2 x \log (x))}\right ) \, dx-\frac {9}{2} \int \left (\frac {2}{(3+2 x \log (x))^2}-\frac {3}{x^2 (3+2 x \log (x))^2}\right ) \, dx-6 \int \frac {1}{x (3+2 x \log (x))^2} \, dx+6 \int \frac {x}{(3+2 x \log (x))^2} \, dx+\frac {27}{4} \int \frac {1}{x^3 (3+2 x \log (x))^2} \, dx-\frac {27}{4} \int \frac {1}{x^3 (3+2 x \log (x))} \, dx-11 \int \frac {1}{(3+2 x \log (x))^2} \, dx+\int \left (-\frac {6}{3+2 x \log (x)}+\frac {3}{x^2 (3+2 x \log (x))}-\frac {1}{x (3+2 x \log (x))}+\frac {x}{3+2 x \log (x)}\right ) \, dx\\ &=-\frac {3}{4 x^2}-\frac {1}{x}+\frac {\log (x)}{2 x}-x \log (2 x)-2 \int \frac {x^2}{(3+2 x \log (x))^2} \, dx+3 \int \frac {1}{x (3+2 x \log (x))^2} \, dx-3 \int \frac {x}{(3+2 x \log (x))^2} \, dx+3 \int \frac {1}{x^2 (3+2 x \log (x))} \, dx-6 \int \frac {1}{x (3+2 x \log (x))^2} \, dx+6 \int \frac {x}{(3+2 x \log (x))^2} \, dx+\frac {27}{4} \int \frac {1}{x^3 (3+2 x \log (x))^2} \, dx-\frac {27}{4} \int \frac {1}{x^3 (3+2 x \log (x))} \, dx-9 \int \frac {1}{(3+2 x \log (x))^2} \, dx-9 \int \frac {1}{x^2 (3+2 x \log (x))^2} \, dx-9 \int \frac {1}{x^2 (3+2 x \log (x))} \, dx-11 \int \frac {1}{(3+2 x \log (x))^2} \, dx+\frac {27}{2} \int \frac {1}{x^2 (3+2 x \log (x))^2} \, dx+18 \int \frac {1}{(3+2 x \log (x))^2} \, dx-\int \frac {1}{x (3+2 x \log (x))} \, dx+\int \frac {x}{3+2 x \log (x)} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.06, size = 40, normalized size = 1.29 \begin {gather*} \frac {1+x^2+\log ^2(x)-3 x \log (2 x)-2 \log (x) \left (1+x^2 \log (2 x)\right )}{3+2 x \log (x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 50, normalized size = 1.61 \begin {gather*} -\frac {{\left (2 \, x^{2} - 1\right )} \log \relax (x)^{2} - x^{2} + 3 \, x \log \relax (2) + {\left (2 \, x^{2} \log \relax (2) + 3 \, x + 2\right )} \log \relax (x) - 1}{2 \, x \log \relax (x) + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 61, normalized size = 1.97 \begin {gather*} -x \log \relax (2) - \frac {1}{2} \, {\left (2 \, x - \frac {1}{x}\right )} \log \relax (x) + \frac {4 \, x^{4} + 4 \, x^{2} + 12 \, x + 9}{4 \, {\left (2 \, x^{3} \log \relax (x) + 3 \, x^{2}\right )}} - \frac {4 \, x + 3}{4 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 43, normalized size = 1.39
method | result | size |
default | \(\frac {x^{2}+\ln \relax (x )^{2}-2 \ln \relax (x )-\frac {11 x \ln \relax (x )}{3}-2 x^{2} \ln \relax (x )^{2}}{3+2 x \ln \relax (x )}-x \ln \relax (2)\) | \(43\) |
risch | \(-\frac {\left (2 x^{2}-1\right ) \ln \relax (x )}{2 x}-\frac {3+4 x^{3} \ln \relax (2)+4 x}{4 x^{2}}+\frac {4 x^{4}+4 x^{2}+12 x +9}{4 x^{2} \left (3+2 x \ln \relax (x )\right )}\) | \(62\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 50, normalized size = 1.61 \begin {gather*} -\frac {{\left (2 \, x^{2} - 1\right )} \log \relax (x)^{2} - x^{2} + 3 \, x \log \relax (2) + {\left (2 \, x^{2} \log \relax (2) + 3 \, x + 2\right )} \log \relax (x) - 1}{2 \, x \log \relax (x) + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.41, size = 85, normalized size = 2.74 \begin {gather*} -\frac {2\,x^2\,\ln \relax (x)+\frac {11\,x^3\,\ln \relax (x)}{3}+3\,x^3\,\left (\ln \left (2\,x\right )-\ln \relax (x)\right )-x^2\,{\ln \relax (x)}^2+2\,x^4\,{\ln \relax (x)}^2-x^4+2\,x^4\,\ln \relax (x)\,\left (\ln \left (2\,x\right )-\ln \relax (x)\right )}{2\,x^3\,\ln \relax (x)+3\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.36, size = 56, normalized size = 1.81 \begin {gather*} - x \log {\relax (2 )} + \frac {4 x^{4} + 4 x^{2} + 12 x + 9}{8 x^{3} \log {\relax (x )} + 12 x^{2}} + \frac {\left (1 - 2 x^{2}\right ) \log {\relax (x )}}{2 x} - \frac {4 x + 3}{4 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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