Optimal. Leaf size=20 \[ 4+\frac {x}{\log \left (x+\frac {1}{2} x (x+\log (x))^2\right )} \]
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Rubi [F] time = 1.87, 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 {-2-2 x-3 x^2+(-2-4 x) \log (x)-\log ^2(x)+\left (2+x^2+2 x \log (x)+\log ^2(x)\right ) \log \left (\frac {1}{2} \left (2 x+x^3+2 x^2 \log (x)+x \log ^2(x)\right )\right )}{\left (2+x^2+2 x \log (x)+\log ^2(x)\right ) \log ^2\left (\frac {1}{2} \left (2 x+x^3+2 x^2 \log (x)+x \log ^2(x)\right )\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {-2-2 x-3 x^2-2 \log (x)-4 x \log (x)-\log ^2(x)}{\left (2+x^2+2 x \log (x)+\log ^2(x)\right ) \log ^2\left (\frac {1}{2} x \left (2+x^2+2 x \log (x)+\log ^2(x)\right )\right )}+\frac {1}{\log \left (\frac {1}{2} x \left (2+x^2+2 x \log (x)+\log ^2(x)\right )\right )}\right ) \, dx\\ &=\int \frac {-2-2 x-3 x^2-2 \log (x)-4 x \log (x)-\log ^2(x)}{\left (2+x^2+2 x \log (x)+\log ^2(x)\right ) \log ^2\left (\frac {1}{2} x \left (2+x^2+2 x \log (x)+\log ^2(x)\right )\right )} \, dx+\int \frac {1}{\log \left (\frac {1}{2} x \left (2+x^2+2 x \log (x)+\log ^2(x)\right )\right )} \, dx\\ &=\int \left (-\frac {2}{\left (2+x^2+2 x \log (x)+\log ^2(x)\right ) \log ^2\left (\frac {1}{2} x \left (2+x^2+2 x \log (x)+\log ^2(x)\right )\right )}-\frac {2 x}{\left (2+x^2+2 x \log (x)+\log ^2(x)\right ) \log ^2\left (\frac {1}{2} x \left (2+x^2+2 x \log (x)+\log ^2(x)\right )\right )}-\frac {3 x^2}{\left (2+x^2+2 x \log (x)+\log ^2(x)\right ) \log ^2\left (\frac {1}{2} x \left (2+x^2+2 x \log (x)+\log ^2(x)\right )\right )}-\frac {2 \log (x)}{\left (2+x^2+2 x \log (x)+\log ^2(x)\right ) \log ^2\left (\frac {1}{2} x \left (2+x^2+2 x \log (x)+\log ^2(x)\right )\right )}-\frac {4 x \log (x)}{\left (2+x^2+2 x \log (x)+\log ^2(x)\right ) \log ^2\left (\frac {1}{2} x \left (2+x^2+2 x \log (x)+\log ^2(x)\right )\right )}-\frac {\log ^2(x)}{\left (2+x^2+2 x \log (x)+\log ^2(x)\right ) \log ^2\left (\frac {1}{2} x \left (2+x^2+2 x \log (x)+\log ^2(x)\right )\right )}\right ) \, dx+\int \frac {1}{\log \left (\frac {1}{2} x \left (2+x^2+2 x \log (x)+\log ^2(x)\right )\right )} \, dx\\ &=-\left (2 \int \frac {1}{\left (2+x^2+2 x \log (x)+\log ^2(x)\right ) \log ^2\left (\frac {1}{2} x \left (2+x^2+2 x \log (x)+\log ^2(x)\right )\right )} \, dx\right )-2 \int \frac {x}{\left (2+x^2+2 x \log (x)+\log ^2(x)\right ) \log ^2\left (\frac {1}{2} x \left (2+x^2+2 x \log (x)+\log ^2(x)\right )\right )} \, dx-2 \int \frac {\log (x)}{\left (2+x^2+2 x \log (x)+\log ^2(x)\right ) \log ^2\left (\frac {1}{2} x \left (2+x^2+2 x \log (x)+\log ^2(x)\right )\right )} \, dx-3 \int \frac {x^2}{\left (2+x^2+2 x \log (x)+\log ^2(x)\right ) \log ^2\left (\frac {1}{2} x \left (2+x^2+2 x \log (x)+\log ^2(x)\right )\right )} \, dx-4 \int \frac {x \log (x)}{\left (2+x^2+2 x \log (x)+\log ^2(x)\right ) \log ^2\left (\frac {1}{2} x \left (2+x^2+2 x \log (x)+\log ^2(x)\right )\right )} \, dx-\int \frac {\log ^2(x)}{\left (2+x^2+2 x \log (x)+\log ^2(x)\right ) \log ^2\left (\frac {1}{2} x \left (2+x^2+2 x \log (x)+\log ^2(x)\right )\right )} \, dx+\int \frac {1}{\log \left (\frac {1}{2} x \left (2+x^2+2 x \log (x)+\log ^2(x)\right )\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.06, size = 24, normalized size = 1.20 \begin {gather*} \frac {x}{\log \left (\frac {1}{2} x \left (2+x^2+2 x \log (x)+\log ^2(x)\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 25, normalized size = 1.25 \begin {gather*} \frac {x}{\log \left (\frac {1}{2} \, x^{3} + x^{2} \log \relax (x) + \frac {1}{2} \, x \log \relax (x)^{2} + x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 29, normalized size = 1.45 \begin {gather*} -\frac {x}{\log \relax (2) - \log \left (x^{2} + 2 \, x \log \relax (x) + \log \relax (x)^{2} + 2\right ) - \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.21, size = 174, normalized size = 8.70
method | result | size |
risch | \(\frac {2 i x}{\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (\ln \relax (x )^{2}+2 x \ln \relax (x )+x^{2}+2\right )\right ) \mathrm {csgn}\left (i x \left (\ln \relax (x )^{2}+2 x \ln \relax (x )+x^{2}+2\right )\right )-\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (\ln \relax (x )^{2}+2 x \ln \relax (x )+x^{2}+2\right )\right )^{2}-\pi \,\mathrm {csgn}\left (i \left (\ln \relax (x )^{2}+2 x \ln \relax (x )+x^{2}+2\right )\right ) \mathrm {csgn}\left (i x \left (\ln \relax (x )^{2}+2 x \ln \relax (x )+x^{2}+2\right )\right )^{2}+\pi \mathrm {csgn}\left (i x \left (\ln \relax (x )^{2}+2 x \ln \relax (x )+x^{2}+2\right )\right )^{3}-2 i \ln \relax (2)+2 i \ln \relax (x )+2 i \ln \left (\ln \relax (x )^{2}+2 x \ln \relax (x )+x^{2}+2\right )}\) | \(174\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 29, normalized size = 1.45 \begin {gather*} -\frac {x}{\log \relax (2) - \log \left (x^{2} + 2 \, x \log \relax (x) + \log \relax (x)^{2} + 2\right ) - \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int -\frac {2\,x+{\ln \relax (x)}^2-\ln \left (\frac {x^3}{2}+x^2\,\ln \relax (x)+\frac {x\,{\ln \relax (x)}^2}{2}+x\right )\,\left (x^2+2\,x\,\ln \relax (x)+{\ln \relax (x)}^2+2\right )+\ln \relax (x)\,\left (4\,x+2\right )+3\,x^2+2}{{\ln \left (\frac {x^3}{2}+x^2\,\ln \relax (x)+\frac {x\,{\ln \relax (x)}^2}{2}+x\right )}^2\,\left (x^2+2\,x\,\ln \relax (x)+{\ln \relax (x)}^2+2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.42, size = 24, normalized size = 1.20 \begin {gather*} \frac {x}{\log {\left (\frac {x^{3}}{2} + x^{2} \log {\relax (x )} + \frac {x \log {\relax (x )}^{2}}{2} + x \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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