Optimal. Leaf size=31 \[ \log \left (\frac {3}{2} \left (-e^{\frac {e^4}{x}-x}+\frac {x}{2}\right ) x^2 \log (x)\right ) \]
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Rubi [F] time = 1.94, 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 e^{\frac {e^4-x^2}{x}} x-x^2+\left (-3 x^2+e^{\frac {e^4-x^2}{x}} \left (-2 e^4+4 x-2 x^2\right )\right ) \log (x)}{\left (2 e^{\frac {e^4-x^2}{x}} x^2-x^3\right ) \log (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 {\frac {2 e^{4+\frac {e^4}{x}}+2 e^{\frac {e^4}{x}} (-2+x) x+3 e^x x^2}{-2 e^{\frac {e^4}{x}}+e^x x}+\frac {x}{\log (x)}}{x^2} \, dx\\ &=\int \left (-\frac {2 e^{\frac {e^4}{x}} \left (e^4+x+x^2\right )}{x^2 \left (2 e^{\frac {e^4}{x}}-e^x x\right )}+\frac {1+3 \log (x)}{x \log (x)}\right ) \, dx\\ &=-\left (2 \int \frac {e^{\frac {e^4}{x}} \left (e^4+x+x^2\right )}{x^2 \left (2 e^{\frac {e^4}{x}}-e^x x\right )} \, dx\right )+\int \frac {1+3 \log (x)}{x \log (x)} \, dx\\ &=-\left (2 \int \left (\frac {e^{\frac {e^4}{x}}}{2 e^{\frac {e^4}{x}}-e^x x}-\frac {e^{4+\frac {e^4}{x}}}{x^2 \left (-2 e^{\frac {e^4}{x}}+e^x x\right )}-\frac {e^{\frac {e^4}{x}}}{x \left (-2 e^{\frac {e^4}{x}}+e^x x\right )}\right ) \, dx\right )+\operatorname {Subst}\left (\int \frac {1+3 x}{x} \, dx,x,\log (x)\right )\\ &=-\left (2 \int \frac {e^{\frac {e^4}{x}}}{2 e^{\frac {e^4}{x}}-e^x x} \, dx\right )+2 \int \frac {e^{4+\frac {e^4}{x}}}{x^2 \left (-2 e^{\frac {e^4}{x}}+e^x x\right )} \, dx+2 \int \frac {e^{\frac {e^4}{x}}}{x \left (-2 e^{\frac {e^4}{x}}+e^x x\right )} \, dx+\operatorname {Subst}\left (\int \left (3+\frac {1}{x}\right ) \, dx,x,\log (x)\right )\\ &=3 \log (x)+\log (\log (x))-2 \int \frac {e^{\frac {e^4}{x}}}{2 e^{\frac {e^4}{x}}-e^x x} \, dx+2 \int \frac {e^{4+\frac {e^4}{x}}}{x^2 \left (-2 e^{\frac {e^4}{x}}+e^x x\right )} \, dx+e^{-\frac {e^4}{x}} \operatorname {Subst}\left (\int \frac {e^{\frac {1}{2} e^{4-\frac {e^4}{x}+x}-\frac {1}{2} e^{4-\frac {e^4}{x}} x}}{x} \, dx,x,\frac {-2 e^{\frac {e^4}{x}}+e^x x}{x}\right )\\ &=3 \log (x)+\log (\log (x))-2 \int \frac {e^{\frac {e^4}{x}}}{2 e^{\frac {e^4}{x}}-e^x x} \, dx+2 \int \frac {e^{4+\frac {e^4}{x}}}{x^2 \left (-2 e^{\frac {e^4}{x}}+e^x x\right )} \, dx+e^{-\frac {e^4}{x}} \operatorname {Subst}\left (\int \frac {e^{\frac {1}{2} e^{4-\frac {e^4}{x}} \left (e^x-x\right )}}{x} \, dx,x,\frac {-2 e^{\frac {e^4}{x}}+e^x x}{x}\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.45, size = 30, normalized size = 0.97 \begin {gather*} -x+2 \log (x)+\log \left (2 e^{\frac {e^4}{x}}-e^x x\right )+\log (\log (x)) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 29, normalized size = 0.94 \begin {gather*} 2 \, \log \relax (x) + \log \left (-x + 2 \, e^{\left (-\frac {x^{2} - e^{4}}{x}\right )}\right ) + \log \left (\log \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 27, normalized size = 0.87 \begin {gather*} \log \left (x - 2 \, e^{\left (-\frac {x^{2} - e^{4}}{x}\right )}\right ) + 2 \, \log \relax (x) + \log \left (\log \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 27, normalized size = 0.87
method | result | size |
norman | \(2 \ln \relax (x )+\ln \left (\ln \relax (x )\right )+\ln \left (x -2 \,{\mathrm e}^{\frac {{\mathrm e}^{4}-x^{2}}{x}}\right )\) | \(27\) |
risch | \(\ln \left (-\frac {x}{2}+{\mathrm e}^{\frac {{\mathrm e}^{4}-x^{2}}{x}}\right )+2 \ln \relax (x )+\ln \left (\ln \relax (x )\right )\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 25, normalized size = 0.81 \begin {gather*} -x + \log \left (-\frac {1}{2} \, x e^{x} + e^{\left (\frac {e^{4}}{x}\right )}\right ) + 2 \, \log \relax (x) + \log \left (\log \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.60, size = 24, normalized size = 0.77 \begin {gather*} \ln \left (\ln \relax (x)\right )+\ln \left (x-2\,{\mathrm {e}}^{\frac {{\mathrm {e}}^4}{x}-x}\right )+2\,\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.37, size = 24, normalized size = 0.77 \begin {gather*} 2 \log {\relax (x )} + \log {\left (- \frac {x}{2} + e^{\frac {- x^{2} + e^{4}}{x}} \right )} + \log {\left (\log {\relax (x )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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