Optimal. Leaf size=20 \[ \frac {3^{-x} e x \log (2)}{(-2+\log (x)) \log (x)} \]
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Rubi [F] time = 0.83, 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 \log (2)+(-4 e \log (2)+2 e x \log (2) \log (3)) \log (x)+(e \log (2)-e x \log (2) \log (3)) \log ^2(x)}{4\ 3^x \log ^2(x)-4\ 3^x \log ^3(x)+3^x \log ^4(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 {3^{-x} e \log (2) \left (2+(-4+x \log (9)) \log (x)+(1-x \log (3)) \log ^2(x)\right )}{(2-\log (x))^2 \log ^2(x)} \, dx\\ &=(e \log (2)) \int \frac {3^{-x} \left (2+(-4+x \log (9)) \log (x)+(1-x \log (3)) \log ^2(x)\right )}{(2-\log (x))^2 \log ^2(x)} \, dx\\ &=(e \log (2)) \int \left (-\frac {3^{-x}}{2 (-2+\log (x))^2}+\frac {3^{-x} (2-x \log (9))}{4 (-2+\log (x))}+\frac {3^{-x}}{2 \log ^2(x)}+\frac {3^{-x} (-2+x \log (9))}{4 \log (x)}\right ) \, dx\\ &=\frac {1}{4} (e \log (2)) \int \frac {3^{-x} (2-x \log (9))}{-2+\log (x)} \, dx+\frac {1}{4} (e \log (2)) \int \frac {3^{-x} (-2+x \log (9))}{\log (x)} \, dx-\frac {1}{2} (e \log (2)) \int \frac {3^{-x}}{(-2+\log (x))^2} \, dx+\frac {1}{2} (e \log (2)) \int \frac {3^{-x}}{\log ^2(x)} \, dx\\ &=\frac {1}{4} (e \log (2)) \int \left (\frac {2\ 3^{-x}}{-2+\log (x)}-\frac {3^{-x} x \log (9)}{-2+\log (x)}\right ) \, dx+\frac {1}{4} (e \log (2)) \int \left (-\frac {2\ 3^{-x}}{\log (x)}+\frac {3^{-x} x \log (9)}{\log (x)}\right ) \, dx-\frac {1}{2} (e \log (2)) \int \frac {3^{-x}}{(-2+\log (x))^2} \, dx+\frac {1}{2} (e \log (2)) \int \frac {3^{-x}}{\log ^2(x)} \, dx\\ &=-\left (\frac {1}{2} (e \log (2)) \int \frac {3^{-x}}{(-2+\log (x))^2} \, dx\right )+\frac {1}{2} (e \log (2)) \int \frac {3^{-x}}{-2+\log (x)} \, dx+\frac {1}{2} (e \log (2)) \int \frac {3^{-x}}{\log ^2(x)} \, dx-\frac {1}{2} (e \log (2)) \int \frac {3^{-x}}{\log (x)} \, dx-\frac {1}{4} (e \log (2) \log (9)) \int \frac {3^{-x} x}{-2+\log (x)} \, dx+\frac {1}{4} (e \log (2) \log (9)) \int \frac {3^{-x} x}{\log (x)} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.24, size = 20, normalized size = 1.00 \begin {gather*} \frac {3^{-x} e x \log (2)}{(-2+\log (x)) \log (x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.03, size = 24, normalized size = 1.20 \begin {gather*} \frac {x e \log \relax (2)}{3^{x} \log \relax (x)^{2} - 2 \cdot 3^{x} \log \relax (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 {{\left (x e \log \relax (3) \log \relax (2) - e \log \relax (2)\right )} \log \relax (x)^{2} - 2 \, e \log \relax (2) - 2 \, {\left (x e \log \relax (3) \log \relax (2) - 2 \, e \log \relax (2)\right )} \log \relax (x)}{3^{x} \log \relax (x)^{4} - 4 \cdot 3^{x} \log \relax (x)^{3} + 4 \cdot 3^{x} \log \relax (x)^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 22, normalized size = 1.10
method | result | size |
risch | \(\frac {{\mathrm e} 3^{-x} x \ln \relax (2)}{\left (\ln \relax (x )-2\right ) \ln \relax (x )}\) | \(22\) |
norman | \(\frac {{\mathrm e} \,{\mathrm e}^{-x \ln \relax (3)} x \ln \relax (2)}{\left (\ln \relax (x )-2\right ) \ln \relax (x )}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 23, normalized size = 1.15 \begin {gather*} \frac {x e^{\left (-x \log \relax (3) + 1\right )} \log \relax (2)}{\log \relax (x)^{2} - 2 \, \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.00, size = 21, normalized size = 1.05 \begin {gather*} \frac {x\,\mathrm {e}\,\ln \relax (2)}{3^x\,\ln \relax (x)\,\left (\ln \relax (x)-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.33, size = 24, normalized size = 1.20 \begin {gather*} \frac {e x e^{- x \log {\relax (3 )}} \log {\relax (2 )}}{\log {\relax (x )}^{2} - 2 \log {\relax (x )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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