Optimal. Leaf size=31 \[ 2+2 x-\left (-2+5 e^{-\frac {e^x}{x}}\right )^2 (5+5 \log (x))^2 \]
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Rubi [F] time = 14.27, 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^{-\frac {2 e^x}{x}} \left (-1250 x+e^x (-1250+1250 x)+\left (-1250 x+e^x (-2500+2500 x)\right ) \log (x)+e^x (-1250+1250 x) \log ^2(x)+e^{\frac {2 e^x}{x}} \left (-200 x+2 x^2-200 x \log (x)\right )+e^{\frac {e^x}{x}} \left (e^x (500-500 x)+1000 x+\left (e^x (1000-1000 x)+1000 x\right ) \log (x)+e^x (500-500 x) \log ^2(x)\right )\right )}{x^2} \, 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 {250 e^{-\frac {2 e^x}{x}+x} \left (-5+2 e^{\frac {e^x}{x}}\right ) (-1+x) (1+\log (x))^2}{x^2}+\frac {2 e^{-\frac {2 e^x}{x}} \left (-625+500 e^{\frac {e^x}{x}}-100 e^{\frac {2 e^x}{x}}+e^{\frac {2 e^x}{x}} x-625 \log (x)+500 e^{\frac {e^x}{x}} \log (x)-100 e^{\frac {2 e^x}{x}} \log (x)\right )}{x}\right ) \, dx\\ &=2 \int \frac {e^{-\frac {2 e^x}{x}} \left (-625+500 e^{\frac {e^x}{x}}-100 e^{\frac {2 e^x}{x}}+e^{\frac {2 e^x}{x}} x-625 \log (x)+500 e^{\frac {e^x}{x}} \log (x)-100 e^{\frac {2 e^x}{x}} \log (x)\right )}{x} \, dx-250 \int \frac {e^{-\frac {2 e^x}{x}+x} \left (-5+2 e^{\frac {e^x}{x}}\right ) (-1+x) (1+\log (x))^2}{x^2} \, dx\\ &=2 \int \frac {e^{-\frac {2 e^x}{x}} \left (-625+500 e^{\frac {e^x}{x}}+e^{\frac {2 e^x}{x}} (-100+x)-25 \left (5-2 e^{\frac {e^x}{x}}\right )^2 \log (x)\right )}{x} \, dx-250 \int \left (-\frac {5 e^{-\frac {2 e^x}{x}+x} (-1+x) (1+\log (x))^2}{x^2}+\frac {2 e^{-\frac {e^x}{x}+x} (-1+x) (1+\log (x))^2}{x^2}\right ) \, dx\\ &=2 \int \left (\frac {-100+x-100 \log (x)}{x}-\frac {625 e^{-\frac {2 e^x}{x}} (1+\log (x))}{x}+\frac {500 e^{-\frac {e^x}{x}} (1+\log (x))}{x}\right ) \, dx-500 \int \frac {e^{-\frac {e^x}{x}+x} (-1+x) (1+\log (x))^2}{x^2} \, dx+1250 \int \frac {e^{-\frac {2 e^x}{x}+x} (-1+x) (1+\log (x))^2}{x^2} \, dx\\ &=2 \int \frac {-100+x-100 \log (x)}{x} \, dx-500 \int \left (\frac {e^{-\frac {e^x}{x}+x} (-1+x)}{x^2}+\frac {2 e^{-\frac {e^x}{x}+x} (-1+x) \log (x)}{x^2}+\frac {e^{-\frac {e^x}{x}+x} (-1+x) \log ^2(x)}{x^2}\right ) \, dx+1000 \int \frac {e^{-\frac {e^x}{x}} (1+\log (x))}{x} \, dx-1250 \int \frac {e^{-\frac {2 e^x}{x}} (1+\log (x))}{x} \, dx+1250 \int \left (\frac {e^{-\frac {2 e^x}{x}+x} (-1+x)}{x^2}+\frac {2 e^{-\frac {2 e^x}{x}+x} (-1+x) \log (x)}{x^2}+\frac {e^{-\frac {2 e^x}{x}+x} (-1+x) \log ^2(x)}{x^2}\right ) \, dx\\ &=2 \int \left (\frac {-100+x}{x}-\frac {100 \log (x)}{x}\right ) \, dx-500 \int \frac {e^{-\frac {e^x}{x}+x} (-1+x)}{x^2} \, dx-500 \int \frac {e^{-\frac {e^x}{x}+x} (-1+x) \log ^2(x)}{x^2} \, dx-1000 \int \frac {e^{-\frac {e^x}{x}+x} (-1+x) \log (x)}{x^2} \, dx+1000 \int \left (\frac {e^{-\frac {e^x}{x}}}{x}+\frac {e^{-\frac {e^x}{x}} \log (x)}{x}\right ) \, dx+1250 \int \frac {e^{-\frac {2 e^x}{x}+x} (-1+x)}{x^2} \, dx+1250 \int \frac {e^{-\frac {2 e^x}{x}+x} (-1+x) \log ^2(x)}{x^2} \, dx-1250 \int \left (\frac {e^{-\frac {2 e^x}{x}}}{x}+\frac {e^{-\frac {2 e^x}{x}} \log (x)}{x}\right ) \, dx+2500 \int \frac {e^{-\frac {2 e^x}{x}+x} (-1+x) \log (x)}{x^2} \, dx\\ &=2 \int \frac {-100+x}{x} \, dx-200 \int \frac {\log (x)}{x} \, dx-500 \int \left (-\frac {e^{-\frac {e^x}{x}+x}}{x^2}+\frac {e^{-\frac {e^x}{x}+x}}{x}\right ) \, dx-500 \int \left (-\frac {e^{-\frac {e^x}{x}+x} \log ^2(x)}{x^2}+\frac {e^{-\frac {e^x}{x}+x} \log ^2(x)}{x}\right ) \, dx+1000 \int \frac {e^{-\frac {e^x}{x}}}{x} \, dx+1000 \int \frac {e^{-\frac {e^x}{x}} \log (x)}{x} \, dx+1000 \int \frac {-\int \frac {e^{-\frac {e^x}{x}+x}}{x^2} \, dx+\int \frac {e^{-\frac {e^x}{x}+x}}{x} \, dx}{x} \, dx+1250 \int \left (-\frac {e^{-\frac {2 e^x}{x}+x}}{x^2}+\frac {e^{-\frac {2 e^x}{x}+x}}{x}\right ) \, dx-1250 \int \frac {e^{-\frac {2 e^x}{x}}}{x} \, dx-1250 \int \frac {e^{-\frac {2 e^x}{x}} \log (x)}{x} \, dx+1250 \int \left (-\frac {e^{-\frac {2 e^x}{x}+x} \log ^2(x)}{x^2}+\frac {e^{-\frac {2 e^x}{x}+x} \log ^2(x)}{x}\right ) \, dx-2500 \int \frac {-\int \frac {e^{-\frac {2 e^x}{x}+x}}{x^2} \, dx+\int \frac {e^{-\frac {2 e^x}{x}+x}}{x} \, dx}{x} \, dx+(1000 \log (x)) \int \frac {e^{-\frac {e^x}{x}+x}}{x^2} \, dx-(1000 \log (x)) \int \frac {e^{-\frac {e^x}{x}+x}}{x} \, dx-(2500 \log (x)) \int \frac {e^{-\frac {2 e^x}{x}+x}}{x^2} \, dx+(2500 \log (x)) \int \frac {e^{-\frac {2 e^x}{x}+x}}{x} \, dx\\ &=-100 \log ^2(x)+2 \int \left (1-\frac {100}{x}\right ) \, dx+500 \int \frac {e^{-\frac {e^x}{x}+x}}{x^2} \, dx-500 \int \frac {e^{-\frac {e^x}{x}+x}}{x} \, dx+500 \int \frac {e^{-\frac {e^x}{x}+x} \log ^2(x)}{x^2} \, dx-500 \int \frac {e^{-\frac {e^x}{x}+x} \log ^2(x)}{x} \, dx+1000 \int \frac {e^{-\frac {e^x}{x}}}{x} \, dx-1000 \int \frac {\int \frac {e^{-\frac {e^x}{x}}}{x} \, dx}{x} \, dx+1000 \int \left (-\frac {\int \frac {e^{-\frac {e^x}{x}+x}}{x^2} \, dx}{x}+\frac {\int \frac {e^{-\frac {e^x}{x}+x}}{x} \, dx}{x}\right ) \, dx-1250 \int \frac {e^{-\frac {2 e^x}{x}+x}}{x^2} \, dx-1250 \int \frac {e^{-\frac {2 e^x}{x}}}{x} \, dx+1250 \int \frac {e^{-\frac {2 e^x}{x}+x}}{x} \, dx-1250 \int \frac {e^{-\frac {2 e^x}{x}+x} \log ^2(x)}{x^2} \, dx+1250 \int \frac {e^{-\frac {2 e^x}{x}+x} \log ^2(x)}{x} \, dx+1250 \int \frac {\int \frac {e^{-\frac {2 e^x}{x}}}{x} \, dx}{x} \, dx-2500 \int \left (-\frac {\int \frac {e^{-\frac {2 e^x}{x}+x}}{x^2} \, dx}{x}+\frac {\int \frac {e^{-\frac {2 e^x}{x}+x}}{x} \, dx}{x}\right ) \, dx+(1000 \log (x)) \int \frac {e^{-\frac {e^x}{x}+x}}{x^2} \, dx+(1000 \log (x)) \int \frac {e^{-\frac {e^x}{x}}}{x} \, dx-(1000 \log (x)) \int \frac {e^{-\frac {e^x}{x}+x}}{x} \, dx-(1250 \log (x)) \int \frac {e^{-\frac {2 e^x}{x}}}{x} \, dx-(2500 \log (x)) \int \frac {e^{-\frac {2 e^x}{x}+x}}{x^2} \, dx+(2500 \log (x)) \int \frac {e^{-\frac {2 e^x}{x}+x}}{x} \, dx\\ &=2 x-200 \log (x)-100 \log ^2(x)+500 \int \frac {e^{-\frac {e^x}{x}+x}}{x^2} \, dx-500 \int \frac {e^{-\frac {e^x}{x}+x}}{x} \, dx+500 \int \frac {e^{-\frac {e^x}{x}+x} \log ^2(x)}{x^2} \, dx-500 \int \frac {e^{-\frac {e^x}{x}+x} \log ^2(x)}{x} \, dx+1000 \int \frac {e^{-\frac {e^x}{x}}}{x} \, dx-1000 \int \frac {\int \frac {e^{-\frac {e^x}{x}+x}}{x^2} \, dx}{x} \, dx-1000 \int \frac {\int \frac {e^{-\frac {e^x}{x}}}{x} \, dx}{x} \, dx+1000 \int \frac {\int \frac {e^{-\frac {e^x}{x}+x}}{x} \, dx}{x} \, dx-1250 \int \frac {e^{-\frac {2 e^x}{x}+x}}{x^2} \, dx-1250 \int \frac {e^{-\frac {2 e^x}{x}}}{x} \, dx+1250 \int \frac {e^{-\frac {2 e^x}{x}+x}}{x} \, dx-1250 \int \frac {e^{-\frac {2 e^x}{x}+x} \log ^2(x)}{x^2} \, dx+1250 \int \frac {e^{-\frac {2 e^x}{x}+x} \log ^2(x)}{x} \, dx+1250 \int \frac {\int \frac {e^{-\frac {2 e^x}{x}}}{x} \, dx}{x} \, dx+2500 \int \frac {\int \frac {e^{-\frac {2 e^x}{x}+x}}{x^2} \, dx}{x} \, dx-2500 \int \frac {\int \frac {e^{-\frac {2 e^x}{x}+x}}{x} \, dx}{x} \, dx+(1000 \log (x)) \int \frac {e^{-\frac {e^x}{x}+x}}{x^2} \, dx+(1000 \log (x)) \int \frac {e^{-\frac {e^x}{x}}}{x} \, dx-(1000 \log (x)) \int \frac {e^{-\frac {e^x}{x}+x}}{x} \, dx-(1250 \log (x)) \int \frac {e^{-\frac {2 e^x}{x}}}{x} \, dx-(2500 \log (x)) \int \frac {e^{-\frac {2 e^x}{x}+x}}{x^2} \, dx+(2500 \log (x)) \int \frac {e^{-\frac {2 e^x}{x}+x}}{x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.16, size = 52, normalized size = 1.68 \begin {gather*} 2 \left (x-100 \log (x)-50 \log ^2(x)-\frac {625}{2} e^{-\frac {2 e^x}{x}} (1+\log (x))^2+250 e^{-\frac {e^x}{x}} (1+\log (x))^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 65, normalized size = 2.10 \begin {gather*} -{\left (2 \, {\left (50 \, \log \relax (x)^{2} - x + 100 \, \log \relax (x)\right )} e^{\left (\frac {2 \, e^{x}}{x}\right )} - 500 \, {\left (\log \relax (x)^{2} + 2 \, \log \relax (x) + 1\right )} e^{\left (\frac {e^{x}}{x}\right )} + 625 \, \log \relax (x)^{2} + 1250 \, \log \relax (x) + 625\right )} e^{\left (-\frac {2 \, e^{x}}{x}\right )} \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 {2 \, {\left (625 \, {\left (x - 1\right )} e^{x} \log \relax (x)^{2} + 625 \, {\left (x - 1\right )} e^{x} + {\left (x^{2} - 100 \, x \log \relax (x) - 100 \, x\right )} e^{\left (\frac {2 \, e^{x}}{x}\right )} - 250 \, {\left ({\left (x - 1\right )} e^{x} \log \relax (x)^{2} + {\left (x - 1\right )} e^{x} + 2 \, {\left ({\left (x - 1\right )} e^{x} - x\right )} \log \relax (x) - 2 \, x\right )} e^{\left (\frac {e^{x}}{x}\right )} + 625 \, {\left (2 \, {\left (x - 1\right )} e^{x} - x\right )} \log \relax (x) - 625 \, x\right )} e^{\left (-\frac {2 \, e^{x}}{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 = 0.11, size = 57, normalized size = 1.84
method | result | size |
risch | \(-100 \ln \relax (x )^{2}+2 x -200 \ln \relax (x )+\left (500 \ln \relax (x )^{2}+1000 \ln \relax (x )+500\right ) {\mathrm e}^{-\frac {{\mathrm e}^{x}}{x}}+\left (-625 \ln \relax (x )^{2}-1250 \ln \relax (x )-625\right ) {\mathrm e}^{-\frac {2 \,{\mathrm e}^{x}}{x}}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.70, size = 54, normalized size = 1.74 \begin {gather*} 500 \, {\left (\log \relax (x)^{2} + 2 \, \log \relax (x) + 1\right )} e^{\left (-\frac {e^{x}}{x}\right )} - 625 \, {\left (\log \relax (x)^{2} + 2 \, \log \relax (x) + 1\right )} e^{\left (-\frac {2 \, e^{x}}{x}\right )} - 100 \, \log \relax (x)^{2} + 2 \, x - 200 \, \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 = -0.03 \begin {gather*} \int -\frac {{\mathrm {e}}^{-\frac {2\,{\mathrm {e}}^x}{x}}\,\left (1250\,x+{\mathrm {e}}^{\frac {2\,{\mathrm {e}}^x}{x}}\,\left (200\,x+200\,x\,\ln \relax (x)-2\,x^2\right )-{\mathrm {e}}^{\frac {{\mathrm {e}}^x}{x}}\,\left (-{\mathrm {e}}^x\,\left (500\,x-500\right )\,{\ln \relax (x)}^2+\left (1000\,x-{\mathrm {e}}^x\,\left (1000\,x-1000\right )\right )\,\ln \relax (x)+1000\,x-{\mathrm {e}}^x\,\left (500\,x-500\right )\right )-{\mathrm {e}}^x\,\left (1250\,x-1250\right )+\ln \relax (x)\,\left (1250\,x-{\mathrm {e}}^x\,\left (2500\,x-2500\right )\right )-{\mathrm {e}}^x\,{\ln \relax (x)}^2\,\left (1250\,x-1250\right )\right )}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 24.30, size = 58, normalized size = 1.87 \begin {gather*} 2 x + \left (- 625 \log {\relax (x )}^{2} - 1250 \log {\relax (x )} - 625\right ) e^{- \frac {2 e^{x}}{x}} + \left (500 \log {\relax (x )}^{2} + 1000 \log {\relax (x )} + 500\right ) e^{- \frac {e^{x}}{x}} - 100 \log {\relax (x )}^{2} - 200 \log {\relax (x )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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