Optimal. Leaf size=30 \[ \frac {1}{4}-\log (x) \left (e^{3 (4+x) \left (1-x^2\right )}-\log (\log (x))\right )^2 \]
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Rubi [B] time = 0.40, antiderivative size = 125, normalized size of antiderivative = 4.17, number of steps used = 10, number of rules used = 5, integrand size = 143, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.035, Rules used = {14, 2288, 2360, 2295, 2296} \begin {gather*} -\frac {e^{-6 x^3-24 x^2+6 x+24} \left (-3 x^3 \log (x)-8 x^2 \log (x)+x \log (x)\right )}{x \left (-3 x^2-8 x+1\right )}+\frac {2 e^{-3 x^3-12 x^2+3 x+12} \left (-3 x^3 \log (x) \log (\log (x))-8 x^2 \log (x) \log (\log (x))+x \log (x) \log (\log (x))\right )}{x \left (-3 x^2-8 x+1\right )}-\log (x) \log ^2(\log (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2288
Rule 2295
Rule 2296
Rule 2360
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {e^{24+6 x-24 x^2-6 x^3} \left (-1-6 x \log (x)+48 x^2 \log (x)+18 x^3 \log (x)\right )}{x}-\frac {\log (\log (x)) (2+\log (\log (x)))}{x}-\frac {2 e^{12+3 x-12 x^2-3 x^3} \left (-1-\log (\log (x))-3 x \log (x) \log (\log (x))+24 x^2 \log (x) \log (\log (x))+9 x^3 \log (x) \log (\log (x))\right )}{x}\right ) \, dx\\ &=-\left (2 \int \frac {e^{12+3 x-12 x^2-3 x^3} \left (-1-\log (\log (x))-3 x \log (x) \log (\log (x))+24 x^2 \log (x) \log (\log (x))+9 x^3 \log (x) \log (\log (x))\right )}{x} \, dx\right )+\int \frac {e^{24+6 x-24 x^2-6 x^3} \left (-1-6 x \log (x)+48 x^2 \log (x)+18 x^3 \log (x)\right )}{x} \, dx-\int \frac {\log (\log (x)) (2+\log (\log (x)))}{x} \, dx\\ &=-\frac {e^{24+6 x-24 x^2-6 x^3} \left (x \log (x)-8 x^2 \log (x)-3 x^3 \log (x)\right )}{x \left (1-8 x-3 x^2\right )}+\frac {2 e^{12+3 x-12 x^2-3 x^3} \left (x \log (x) \log (\log (x))-8 x^2 \log (x) \log (\log (x))-3 x^3 \log (x) \log (\log (x))\right )}{x \left (1-8 x-3 x^2\right )}-\operatorname {Subst}(\int \log (x) (2+\log (x)) \, dx,x,\log (x))\\ &=-\frac {e^{24+6 x-24 x^2-6 x^3} \left (x \log (x)-8 x^2 \log (x)-3 x^3 \log (x)\right )}{x \left (1-8 x-3 x^2\right )}+\frac {2 e^{12+3 x-12 x^2-3 x^3} \left (x \log (x) \log (\log (x))-8 x^2 \log (x) \log (\log (x))-3 x^3 \log (x) \log (\log (x))\right )}{x \left (1-8 x-3 x^2\right )}-\operatorname {Subst}\left (\int \left (2 \log (x)+\log ^2(x)\right ) \, dx,x,\log (x)\right )\\ &=-\frac {e^{24+6 x-24 x^2-6 x^3} \left (x \log (x)-8 x^2 \log (x)-3 x^3 \log (x)\right )}{x \left (1-8 x-3 x^2\right )}+\frac {2 e^{12+3 x-12 x^2-3 x^3} \left (x \log (x) \log (\log (x))-8 x^2 \log (x) \log (\log (x))-3 x^3 \log (x) \log (\log (x))\right )}{x \left (1-8 x-3 x^2\right )}-2 \operatorname {Subst}(\int \log (x) \, dx,x,\log (x))-\operatorname {Subst}\left (\int \log ^2(x) \, dx,x,\log (x)\right )\\ &=2 \log (x)-\frac {e^{24+6 x-24 x^2-6 x^3} \left (x \log (x)-8 x^2 \log (x)-3 x^3 \log (x)\right )}{x \left (1-8 x-3 x^2\right )}-2 \log (x) \log (\log (x))-\log (x) \log ^2(\log (x))+\frac {2 e^{12+3 x-12 x^2-3 x^3} \left (x \log (x) \log (\log (x))-8 x^2 \log (x) \log (\log (x))-3 x^3 \log (x) \log (\log (x))\right )}{x \left (1-8 x-3 x^2\right )}+2 \operatorname {Subst}(\int \log (x) \, dx,x,\log (x))\\ &=-\frac {e^{24+6 x-24 x^2-6 x^3} \left (x \log (x)-8 x^2 \log (x)-3 x^3 \log (x)\right )}{x \left (1-8 x-3 x^2\right )}-\log (x) \log ^2(\log (x))+\frac {2 e^{12+3 x-12 x^2-3 x^3} \left (x \log (x) \log (\log (x))-8 x^2 \log (x) \log (\log (x))-3 x^3 \log (x) \log (\log (x))\right )}{x \left (1-8 x-3 x^2\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.15, size = 39, normalized size = 1.30 \begin {gather*} -e^{-6 x^2 (4+x)} \log (x) \left (e^{3 (4+x)}-e^{3 x^2 (4+x)} \log (\log (x))\right )^2 \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.87, size = 53, normalized size = 1.77 \begin {gather*} 2 \, e^{\left (-3 \, x^{3} - 12 \, x^{2} + 3 \, x + 12\right )} \log \relax (x) \log \left (\log \relax (x)\right ) - \log \relax (x) \log \left (\log \relax (x)\right )^{2} - e^{\left (-6 \, x^{3} - 24 \, x^{2} + 6 \, x + 24\right )} \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.39, size = 53, normalized size = 1.77 \begin {gather*} 2 \, e^{\left (-3 \, x^{3} - 12 \, x^{2} + 3 \, x + 12\right )} \log \relax (x) \log \left (\log \relax (x)\right ) - \log \relax (x) \log \left (\log \relax (x)\right )^{2} - e^{\left (-6 \, x^{3} - 24 \, x^{2} + 6 \, x + 24\right )} \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 46, normalized size = 1.53
method | result | size |
risch | \(-{\mathrm e}^{-6 \left (x -1\right ) \left (4+x \right ) \left (x +1\right )} \ln \relax (x )+2 \ln \relax (x ) \ln \left (\ln \relax (x )\right ) {\mathrm e}^{-3 \left (x -1\right ) \left (4+x \right ) \left (x +1\right )}-\ln \relax (x ) \ln \left (\ln \relax (x )\right )^{2}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.71, size = 60, normalized size = 2.00 \begin {gather*} -{\left (e^{\left (24 \, x^{2}\right )} \log \relax (x) \log \left (\log \relax (x)\right )^{2} - 2 \, e^{\left (-3 \, x^{3} + 12 \, x^{2} + 3 \, x + 12\right )} \log \relax (x) \log \left (\log \relax (x)\right ) + e^{\left (-6 \, x^{3} + 6 \, x + 24\right )} \log \relax (x)\right )} e^{\left (-24 \, x^{2}\right )} \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 = -0.03 \begin {gather*} \int -\frac {{\ln \left (\ln \relax (x)\right )}^2+\left ({\mathrm {e}}^{-3\,x^3-12\,x^2+3\,x+12}\,\ln \relax (x)\,\left (18\,x^3+48\,x^2-6\,x\right )-2\,{\mathrm {e}}^{-3\,x^3-12\,x^2+3\,x+12}+2\right )\,\ln \left (\ln \relax (x)\right )-2\,{\mathrm {e}}^{-3\,x^3-12\,x^2+3\,x+12}+{\mathrm {e}}^{-6\,x^3-24\,x^2+6\,x+24}-{\mathrm {e}}^{-6\,x^3-24\,x^2+6\,x+24}\,\ln \relax (x)\,\left (18\,x^3+48\,x^2-6\,x\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 35.66, size = 56, normalized size = 1.87 \begin {gather*} - e^{- 6 x^{3} - 24 x^{2} + 6 x + 24} \log {\relax (x )} + 2 e^{- 3 x^{3} - 12 x^{2} + 3 x + 12} \log {\relax (x )} \log {\left (\log {\relax (x )} \right )} - \log {\relax (x )} \log {\left (\log {\relax (x )} \right )}^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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