Optimal. Leaf size=23 \[ e^{-5+e^{5 e^4 x} x-\frac {x \log (9)}{\log (x)}} \]
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Rubi [F] time = 1.58, 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 {\exp \left (\frac {-x \log (9)+\left (-5+e^{5 e^4 x} x\right ) \log (x)}{\log (x)}\right ) \left (\log (9)-\log (9) \log (x)+e^{5 e^4 x} \left (1+5 e^4 x\right ) \log ^2(x)\right )}{\log ^2(x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\exp \left (5 e^4 x+\frac {-x \log (9)+\left (-5+e^{5 e^4 x} x\right ) \log (x)}{\log (x)}\right ) \left (1+5 e^4 x\right )-\frac {\exp \left (\frac {-x \log (9)+\left (-5+e^{5 e^4 x} x\right ) \log (x)}{\log (x)}\right ) \log (9) (-1+\log (x))}{\log ^2(x)}\right ) \, dx\\ &=-\left (\log (9) \int \frac {\exp \left (\frac {-x \log (9)+\left (-5+e^{5 e^4 x} x\right ) \log (x)}{\log (x)}\right ) (-1+\log (x))}{\log ^2(x)} \, dx\right )+\int \exp \left (5 e^4 x+\frac {-x \log (9)+\left (-5+e^{5 e^4 x} x\right ) \log (x)}{\log (x)}\right ) \left (1+5 e^4 x\right ) \, dx\\ &=-\left (\log (9) \int \left (-\frac {\exp \left (\frac {-x \log (9)+\left (-5+e^{5 e^4 x} x\right ) \log (x)}{\log (x)}\right )}{\log ^2(x)}+\frac {\exp \left (\frac {-x \log (9)+\left (-5+e^{5 e^4 x} x\right ) \log (x)}{\log (x)}\right )}{\log (x)}\right ) \, dx\right )+\int \exp \left (\frac {-x \log (9)-5 \log (x)+5 e^4 x \log (x)+e^{5 e^4 x} x \log (x)}{\log (x)}\right ) \left (1+5 e^4 x\right ) \, dx\\ &=\log (9) \int \frac {\exp \left (\frac {-x \log (9)+\left (-5+e^{5 e^4 x} x\right ) \log (x)}{\log (x)}\right )}{\log ^2(x)} \, dx-\log (9) \int \frac {\exp \left (\frac {-x \log (9)+\left (-5+e^{5 e^4 x} x\right ) \log (x)}{\log (x)}\right )}{\log (x)} \, dx+\int \left (\exp \left (\frac {-x \log (9)-5 \log (x)+5 e^4 x \log (x)+e^{5 e^4 x} x \log (x)}{\log (x)}\right )+5 \exp \left (4+\frac {-x \log (9)-5 \log (x)+5 e^4 x \log (x)+e^{5 e^4 x} x \log (x)}{\log (x)}\right ) x\right ) \, dx\\ &=5 \int \exp \left (4+\frac {-x \log (9)-5 \log (x)+5 e^4 x \log (x)+e^{5 e^4 x} x \log (x)}{\log (x)}\right ) x \, dx+\log (9) \int \frac {\exp \left (\frac {-x \log (9)+\left (-5+e^{5 e^4 x} x\right ) \log (x)}{\log (x)}\right )}{\log ^2(x)} \, dx-\log (9) \int \frac {\exp \left (\frac {-x \log (9)+\left (-5+e^{5 e^4 x} x\right ) \log (x)}{\log (x)}\right )}{\log (x)} \, dx+\int \exp \left (\frac {-x \log (9)-5 \log (x)+5 e^4 x \log (x)+e^{5 e^4 x} x \log (x)}{\log (x)}\right ) \, dx\\ &=5 \int \exp \left (\frac {-x \log (9)-\log (x)+5 e^4 x \log (x)+e^{5 e^4 x} x \log (x)}{\log (x)}\right ) x \, dx+\log (9) \int \frac {\exp \left (\frac {-x \log (9)+\left (-5+e^{5 e^4 x} x\right ) \log (x)}{\log (x)}\right )}{\log ^2(x)} \, dx-\log (9) \int \frac {\exp \left (\frac {-x \log (9)+\left (-5+e^{5 e^4 x} x\right ) \log (x)}{\log (x)}\right )}{\log (x)} \, dx+\int \exp \left (\frac {-x \log (9)-5 \log (x)+5 e^4 x \log (x)+e^{5 e^4 x} x \log (x)}{\log (x)}\right ) \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.12, size = 24, normalized size = 1.04 \begin {gather*} 9^{-\frac {x}{\log (x)}} e^{-5+e^{5 e^4 x} x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.11, size = 27, normalized size = 1.17 \begin {gather*} e^{\left (-\frac {2 \, x \log \relax (3) - {\left (x e^{\left (5 \, x e^{4}\right )} - 5\right )} \log \relax (x)}{\log \relax (x)}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 20, normalized size = 0.87 \begin {gather*} e^{\left (x e^{\left (5 \, x e^{4}\right )} - \frac {2 \, x \log \relax (3)}{\log \relax (x)} - 5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 21, normalized size = 0.91
method | result | size |
risch | \(\left (\frac {1}{9}\right )^{\frac {x}{\ln \relax (x )}} {\mathrm e}^{x \,{\mathrm e}^{5 x \,{\mathrm e}^{4}}-5}\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 20, normalized size = 0.87 \begin {gather*} e^{\left (x e^{\left (5 \, x e^{4}\right )} - \frac {2 \, x \log \relax (3)}{\log \relax (x)} - 5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.73, size = 23, normalized size = 1.00 \begin {gather*} \frac {{\mathrm {e}}^{-5}\,{\mathrm {e}}^{x\,{\mathrm {e}}^{5\,x\,{\mathrm {e}}^4}}}{3^{\frac {2\,x}{\ln \relax (x)}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.60, size = 26, normalized size = 1.13 \begin {gather*} e^{\frac {- 2 x \log {\relax (3 )} + \left (x e^{5 x e^{4}} - 5\right ) \log {\relax (x )}}{\log {\relax (x )}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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