Optimal. Leaf size=29 \[ -1+\frac {4}{x}+4 \left (e^{3 e^{-x} x (1+\log (x))^2}+x\right )^2 \]
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Rubi [F] time = 8.51, 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 {e^{-x} \left (e^x \left (-4+8 x^3\right )+e^{2 e^{-x} \left (3 x+6 x \log (x)+3 x \log ^2(x)\right )} \left (72 x^2-24 x^3+\left (96 x^2-48 x^3\right ) \log (x)+\left (24 x^2-24 x^3\right ) \log ^2(x)\right )+e^{e^{-x} \left (3 x+6 x \log (x)+3 x \log ^2(x)\right )} \left (8 e^x x^2+72 x^3-24 x^4+\left (96 x^3-48 x^4\right ) \log (x)+\left (24 x^3-24 x^4\right ) \log ^2(x)\right )\right )}{x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {4}{x^2}+8 x-24 e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} x^{12 e^{-x} x} (1+\log (x)) (-3+x+(-1+x) \log (x))+8 e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} x^{6 e^{-x} x} \left (e^x-3 (-3+x) x-6 (-2+x) x \log (x)-3 (-1+x) x \log ^2(x)\right )\right ) \, dx\\ &=\frac {4}{x}+4 x^2+8 \int e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} x^{6 e^{-x} x} \left (e^x-3 (-3+x) x-6 (-2+x) x \log (x)-3 (-1+x) x \log ^2(x)\right ) \, dx-24 \int e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} x^{12 e^{-x} x} (1+\log (x)) (-3+x+(-1+x) \log (x)) \, dx\\ &=\frac {4}{x}+4 x^2+8 \int \left (e^{x-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} x^{6 e^{-x} x}-3 e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} (-3+x) x^{1+6 e^{-x} x}-6 e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} (-2+x) x^{1+6 e^{-x} x} \log (x)-3 e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} (-1+x) x^{1+6 e^{-x} x} \log ^2(x)\right ) \, dx-24 \int \left (-3 e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} x^{12 e^{-x} x}+e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} x^{1+12 e^{-x} x}+2 e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} (-2+x) x^{12 e^{-x} x} \log (x)+e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} (-1+x) x^{12 e^{-x} x} \log ^2(x)\right ) \, dx\\ &=\frac {4}{x}+4 x^2+8 \int e^{x-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} x^{6 e^{-x} x} \, dx-24 \int e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} (-3+x) x^{1+6 e^{-x} x} \, dx-24 \int e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} x^{1+12 e^{-x} x} \, dx-24 \int e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} (-1+x) x^{12 e^{-x} x} \log ^2(x) \, dx-24 \int e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} (-1+x) x^{1+6 e^{-x} x} \log ^2(x) \, dx-48 \int e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} (-2+x) x^{12 e^{-x} x} \log (x) \, dx-48 \int e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} (-2+x) x^{1+6 e^{-x} x} \log (x) \, dx+72 \int e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} x^{12 e^{-x} x} \, dx\\ &=\frac {4}{x}+4 x^2+8 \int e^{3 e^{-x} x \left (1+\log ^2(x)\right )} x^{6 e^{-x} x} \, dx-24 \int e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} x^{1+12 e^{-x} x} \, dx-24 \int \left (-3 e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} x^{1+6 e^{-x} x}+e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} x^{2+6 e^{-x} x}\right ) \, dx-24 \int \left (-e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} x^{1+6 e^{-x} x} \log ^2(x)+e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} x^{2+6 e^{-x} x} \log ^2(x)\right ) \, dx-24 \int \left (-e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} x^{12 e^{-x} x} \log ^2(x)+e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} x^{1+12 e^{-x} x} \log ^2(x)\right ) \, dx-48 \int \left (-2 e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} x^{1+6 e^{-x} x} \log (x)+e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} x^{2+6 e^{-x} x} \log (x)\right ) \, dx-48 \int \left (-2 e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} x^{12 e^{-x} x} \log (x)+e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} x^{1+12 e^{-x} x} \log (x)\right ) \, dx+72 \int e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} x^{12 e^{-x} x} \, dx\\ &=\frac {4}{x}+4 x^2+8 \int e^{3 e^{-x} x \left (1+\log ^2(x)\right )} x^{6 e^{-x} x} \, dx-24 \int e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} x^{2+6 e^{-x} x} \, dx-24 \int e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} x^{1+12 e^{-x} x} \, dx+24 \int e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} x^{12 e^{-x} x} \log ^2(x) \, dx+24 \int e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} x^{1+6 e^{-x} x} \log ^2(x) \, dx-24 \int e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} x^{2+6 e^{-x} x} \log ^2(x) \, dx-24 \int e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} x^{1+12 e^{-x} x} \log ^2(x) \, dx-48 \int e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} x^{2+6 e^{-x} x} \log (x) \, dx-48 \int e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} x^{1+12 e^{-x} x} \log (x) \, dx+72 \int e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} x^{12 e^{-x} x} \, dx+72 \int e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} x^{1+6 e^{-x} x} \, dx+96 \int e^{-e^{-x} x \left (-6+e^x-6 \log ^2(x)\right )} x^{12 e^{-x} x} \log (x) \, dx+96 \int e^{-e^{-x} x \left (-3+e^x-3 \log ^2(x)\right )} x^{1+6 e^{-x} x} \log (x) \, dx\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 4.97, size = 83, normalized size = 2.86 \begin {gather*} \frac {4}{x}+4 x^2+4 e^{6 e^{-x} x+6 e^{-x} x \log ^2(x)} x^{12 e^{-x} x}+8 e^{3 e^{-x} x+3 e^{-x} x \log ^2(x)} x^{1+6 e^{-x} x} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.65, size = 57, normalized size = 1.97 \begin {gather*} \frac {4 \, {\left (x^{3} + 2 \, x^{2} e^{\left (3 \, {\left (x \log \relax (x)^{2} + 2 \, x \log \relax (x) + x\right )} e^{\left (-x\right )}\right )} + x e^{\left (6 \, {\left (x \log \relax (x)^{2} + 2 \, x \log \relax (x) + x\right )} e^{\left (-x\right )}\right )} + 1\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {4 \, {\left (6 \, {\left (x^{3} + {\left (x^{3} - x^{2}\right )} \log \relax (x)^{2} - 3 \, x^{2} + 2 \, {\left (x^{3} - 2 \, x^{2}\right )} \log \relax (x)\right )} e^{\left (6 \, {\left (x \log \relax (x)^{2} + 2 \, x \log \relax (x) + x\right )} e^{\left (-x\right )}\right )} + 2 \, {\left (3 \, x^{4} - 9 \, x^{3} - x^{2} e^{x} + 3 \, {\left (x^{4} - x^{3}\right )} \log \relax (x)^{2} + 6 \, {\left (x^{4} - 2 \, x^{3}\right )} \log \relax (x)\right )} e^{\left (3 \, {\left (x \log \relax (x)^{2} + 2 \, x \log \relax (x) + x\right )} e^{\left (-x\right )}\right )} - {\left (2 \, x^{3} - 1\right )} e^{x}\right )} e^{\left (-x\right )}}{x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 45, normalized size = 1.55
method | result | size |
risch | \(4 x^{2}+\frac {4}{x}+4 \,{\mathrm e}^{6 x \left (\ln \relax (x )+1\right )^{2} {\mathrm e}^{-x}}+8 x \,{\mathrm e}^{3 x \left (\ln \relax (x )+1\right )^{2} {\mathrm e}^{-x}}\) | \(45\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 4 \, x^{2} + \frac {4}{x} + 4 \, e^{\left (6 \, x e^{\left (-x\right )} \log \relax (x)^{2} + 12 \, x e^{\left (-x\right )} \log \relax (x) + 6 \, x e^{\left (-x\right )}\right )} + 4 \, \int -2 \, {\left (3 \, {\left (x^{2} - x\right )} \log \relax (x)^{2} + 3 \, x^{2} + 6 \, {\left (x^{2} - 2 \, x\right )} \log \relax (x) - 9 \, x - e^{x}\right )} e^{\left (3 \, x e^{\left (-x\right )} \log \relax (x)^{2} + 6 \, x e^{\left (-x\right )} \log \relax (x) + 3 \, x e^{\left (-x\right )} - x\right )}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.22, size = 74, normalized size = 2.55 \begin {gather*} 4\,x^{12\,x\,{\mathrm {e}}^{-x}}\,{\mathrm {e}}^{6\,x\,{\mathrm {e}}^{-x}\,{\ln \relax (x)}^2+6\,x\,{\mathrm {e}}^{-x}}+\frac {4}{x}+4\,x^2+8\,x\,x^{6\,x\,{\mathrm {e}}^{-x}}\,{\mathrm {e}}^{3\,x\,{\mathrm {e}}^{-x}\,{\ln \relax (x)}^2+3\,x\,{\mathrm {e}}^{-x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 176.54, size = 60, normalized size = 2.07 \begin {gather*} 4 x^{2} + 8 x e^{\left (3 x \log {\relax (x )}^{2} + 6 x \log {\relax (x )} + 3 x\right ) e^{- x}} + 4 e^{2 \left (3 x \log {\relax (x )}^{2} + 6 x \log {\relax (x )} + 3 x\right ) e^{- x}} + \frac {4}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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