Optimal. Leaf size=23 \[ -5+x+\left (1+3 e^{-x}\right ) \log ^2(x)+5 \log (3 x) \]
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Rubi [A] time = 0.29, antiderivative size = 21, normalized size of antiderivative = 0.91, number of steps used = 8, number of rules used = 5, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {6742, 14, 43, 2301, 2202} \begin {gather*} x+3 e^{-x} \log ^2(x)+\log ^2(x)+5 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 43
Rule 2202
Rule 2301
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {5+x+2 \log (x)}{x}-\frac {3 e^{-x} \log (x) (-2+x \log (x))}{x}\right ) \, dx\\ &=-\left (3 \int \frac {e^{-x} \log (x) (-2+x \log (x))}{x} \, dx\right )+\int \frac {5+x+2 \log (x)}{x} \, dx\\ &=3 e^{-x} \log ^2(x)+\int \left (\frac {5+x}{x}+\frac {2 \log (x)}{x}\right ) \, dx\\ &=3 e^{-x} \log ^2(x)+2 \int \frac {\log (x)}{x} \, dx+\int \frac {5+x}{x} \, dx\\ &=\log ^2(x)+3 e^{-x} \log ^2(x)+\int \left (1+\frac {5}{x}\right ) \, dx\\ &=x+5 \log (x)+\log ^2(x)+3 e^{-x} \log ^2(x)\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.24, size = 23, normalized size = 1.00 \begin {gather*} \frac {25}{4}+x+5 \log (x)+\left (1+3 e^{-x}\right ) \log ^2(x) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 25, normalized size = 1.09 \begin {gather*} {\left ({\left (e^{x} + 3\right )} \log \relax (x)^{2} + x e^{x} + 5 \, e^{x} \log \relax (x)\right )} e^{\left (-x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 20, normalized size = 0.87 \begin {gather*} 3 \, e^{\left (-x\right )} \log \relax (x)^{2} + \log \relax (x)^{2} + x + 5 \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 20, normalized size = 0.87
method | result | size |
risch | \(\left (3+{\mathrm e}^{x}\right ) {\mathrm e}^{-x} \ln \relax (x )^{2}+x +5 \ln \relax (x )\) | \(20\) |
default | \(x +5 \ln \relax (x )+3 \ln \relax (x )^{2} {\mathrm e}^{-x}+\ln \relax (x )^{2}\) | \(21\) |
norman | \(\left ({\mathrm e}^{x} x +{\mathrm e}^{x} \ln \relax (x )^{2}+5 \,{\mathrm e}^{x} \ln \relax (x )+3 \ln \relax (x )^{2}\right ) {\mathrm e}^{-x}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 20, normalized size = 0.87 \begin {gather*} 3 \, e^{\left (-x\right )} \log \relax (x)^{2} + \log \relax (x)^{2} + x + 5 \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.58, size = 20, normalized size = 0.87 \begin {gather*} x+5\,\ln \relax (x)+{\ln \relax (x)}^2+3\,{\mathrm {e}}^{-x}\,{\ln \relax (x)}^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.33, size = 20, normalized size = 0.87 \begin {gather*} x + \log {\relax (x )}^{2} + 5 \log {\relax (x )} + 3 e^{- x} \log {\relax (x )}^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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