Optimal. Leaf size=27 \[ -e^{-4+e^{2 x}} x \log (x \log (5))+\log \left (-e^x+\log (x)\right ) \]
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Rubi [F] time = 1.93, 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 {1-e^x x+e^{-4+e^{2 x}} \left (e^x x-x \log (x)+\left (e^x x+2 e^{3 x} x^2+\left (-x-2 e^{2 x} x^2\right ) \log (x)\right ) \log (x \log (5))\right )}{-e^x x+x \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 {-1+e^x x-e^{-4+e^{2 x}} x \left (e^x-\log (x)\right ) \left (1+\left (1+2 e^{2 x} x\right ) \log (x \log (5))\right )}{x \left (e^x-\log (x)\right )} \, dx\\ &=\int \left (\frac {-1+x \log (x)}{x \left (e^x-\log (x)\right )}-2 e^{-4+e^{2 x}+2 x} x \log (x \log (5))+\frac {e^4-e^{e^{2 x}}-e^{e^{2 x}} \log (x \log (5))}{e^4}\right ) \, dx\\ &=-\left (2 \int e^{-4+e^{2 x}+2 x} x \log (x \log (5)) \, dx\right )+\frac {\int \left (e^4-e^{e^{2 x}}-e^{e^{2 x}} \log (x \log (5))\right ) \, dx}{e^4}+\int \frac {-1+x \log (x)}{x \left (e^x-\log (x)\right )} \, dx\\ &=x+2 \int \frac {\int e^{-4+e^{2 x}+2 x} x \, dx}{x} \, dx-\frac {\int e^{e^{2 x}} \, dx}{e^4}-\frac {\int e^{e^{2 x}} \log (x \log (5)) \, dx}{e^4}-(2 \log (x \log (5))) \int e^{-4+e^{2 x}+2 x} x \, dx+\int \left (-\frac {1}{x \left (e^x-\log (x)\right )}+\frac {\log (x)}{e^x-\log (x)}\right ) \, dx\\ &=x-\frac {\text {Ei}\left (e^{2 x}\right ) \log (x \log (5))}{2 e^4}+2 \int \frac {\int e^{-4+e^{2 x}+2 x} x \, dx}{x} \, dx-\frac {\operatorname {Subst}\left (\int \frac {e^x}{x} \, dx,x,e^{2 x}\right )}{2 e^4}+\frac {\int \frac {\text {Ei}\left (e^{2 x}\right )}{2 x} \, dx}{e^4}-(2 \log (x \log (5))) \int e^{-4+e^{2 x}+2 x} x \, dx-\int \frac {1}{x \left (e^x-\log (x)\right )} \, dx+\int \frac {\log (x)}{e^x-\log (x)} \, dx\\ &=x-\frac {\text {Ei}\left (e^{2 x}\right )}{2 e^4}-\frac {\text {Ei}\left (e^{2 x}\right ) \log (x \log (5))}{2 e^4}+2 \int \frac {\int e^{-4+e^{2 x}+2 x} x \, dx}{x} \, dx+\frac {\int \frac {\text {Ei}\left (e^{2 x}\right )}{x} \, dx}{2 e^4}-(2 \log (x \log (5))) \int e^{-4+e^{2 x}+2 x} x \, dx-\int \frac {1}{x \left (e^x-\log (x)\right )} \, dx+\int \frac {\log (x)}{e^x-\log (x)} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.57, size = 27, normalized size = 1.00 \begin {gather*} -e^{-4+e^{2 x}} x \log (x \log (5))+\log \left (e^x-\log (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 28, normalized size = 1.04 \begin {gather*} -{\left (x \log \relax (x) + x \log \left (\log \relax (5)\right )\right )} e^{\left (e^{\left (2 \, x\right )} - 4\right )} + \log \left (-e^{x} + \log \relax (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 {x e^{x} - {\left (x e^{x} + {\left (2 \, x^{2} e^{\left (3 \, x\right )} + x e^{x} - {\left (2 \, x^{2} e^{\left (2 \, x\right )} + x\right )} \log \relax (x)\right )} \log \left (x \log \relax (5)\right ) - x \log \relax (x)\right )} e^{\left (e^{\left (2 \, x\right )} - 4\right )} - 1}{x e^{x} - x \log \relax (x)}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 30, normalized size = 1.11
method | result | size |
risch | \(\ln \left (\ln \relax (x )-{\mathrm e}^{x}\right )+\left (-x \ln \left (\ln \relax (5)\right )-x \ln \relax (x )\right ) {\mathrm e}^{{\mathrm e}^{2 x}-4}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.75, size = 28, normalized size = 1.04 \begin {gather*} -{\left (x \log \relax (x) + x \log \left (\log \relax (5)\right )\right )} e^{\left (e^{\left (2 \, x\right )} - 4\right )} + \log \left (e^{x} - \log \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.55, size = 28, normalized size = 1.04 \begin {gather*} \ln \left (\ln \relax (x)-{\mathrm {e}}^x\right )-{\mathrm {e}}^{{\mathrm {e}}^{2\,x}-4}\,\left (x\,\ln \left (\ln \relax (5)\right )+x\,\ln \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 158.64, size = 29, normalized size = 1.07 \begin {gather*} \left (- x \log {\relax (x )} - x \log {\left (\log {\relax (5 )} \right )}\right ) e^{e^{2 x} - 4} + \log {\left (e^{x} - \log {\relax (x )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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