Optimal. Leaf size=29 \[ \log (3) \left (x+3 \left (3-x+x \log \left (\frac {3}{\log \left (\frac {e^x+x}{x}\right )}\right )\right )\right ) \]
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Rubi [F]
time = 1.66, 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 {e^x (3-3 x) \log (3)+\left (-2 e^x \log (3)-2 x \log (3)\right ) \log \left (\frac {e^x+x}{x}\right )+\left (3 e^x \log (3)+3 x \log (3)\right ) \log \left (\frac {e^x+x}{x}\right ) \log \left (\frac {3}{\log \left (\frac {e^x+x}{x}\right )}\right )}{\left (e^x+x\right ) \log \left (\frac {e^x+x}{x}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \log (3) \left (-2-\frac {3 e^x (-1+x)}{\left (e^x+x\right ) \log \left (\frac {e^x+x}{x}\right )}+3 \log \left (\frac {3}{\log \left (\frac {e^x+x}{x}\right )}\right )\right ) \, dx\\ &=\log (3) \int \left (-2-\frac {3 e^x (-1+x)}{\left (e^x+x\right ) \log \left (\frac {e^x+x}{x}\right )}+3 \log \left (\frac {3}{\log \left (\frac {e^x+x}{x}\right )}\right )\right ) \, dx\\ &=-2 x \log (3)-(3 \log (3)) \int \frac {e^x (-1+x)}{\left (e^x+x\right ) \log \left (\frac {e^x+x}{x}\right )} \, dx+(3 \log (3)) \int \log \left (\frac {3}{\log \left (\frac {e^x+x}{x}\right )}\right ) \, dx\\ &=-2 x \log (3)+3 x \log (3) \log \left (\frac {3}{\log \left (\frac {e^x+x}{x}\right )}\right )-(3 \log (3)) \int \frac {e^x (1-x)}{\left (e^x+x\right ) \log \left (1+\frac {e^x}{x}\right )} \, dx-(3 \log (3)) \int \frac {e^x (-1+x)}{\left (e^x+x\right ) \log \left (1+\frac {e^x}{x}\right )} \, dx\\ &=-2 x \log (3)+3 x \log (3) \log \left (\frac {3}{\log \left (\frac {e^x+x}{x}\right )}\right )-(3 \log (3)) \int \left (\frac {e^x x}{\left (-e^x-x\right ) \log \left (1+\frac {e^x}{x}\right )}+\frac {e^x}{\left (e^x+x\right ) \log \left (1+\frac {e^x}{x}\right )}\right ) \, dx-(3 \log (3)) \int \left (\frac {e^x}{\left (-e^x-x\right ) \log \left (1+\frac {e^x}{x}\right )}+\frac {e^x x}{\left (e^x+x\right ) \log \left (1+\frac {e^x}{x}\right )}\right ) \, dx\\ &=-2 x \log (3)+3 x \log (3) \log \left (\frac {3}{\log \left (\frac {e^x+x}{x}\right )}\right )-(3 \log (3)) \int \frac {e^x}{\left (-e^x-x\right ) \log \left (1+\frac {e^x}{x}\right )} \, dx-(3 \log (3)) \int \frac {e^x x}{\left (-e^x-x\right ) \log \left (1+\frac {e^x}{x}\right )} \, dx-(3 \log (3)) \int \frac {e^x}{\left (e^x+x\right ) \log \left (1+\frac {e^x}{x}\right )} \, dx-(3 \log (3)) \int \frac {e^x x}{\left (e^x+x\right ) \log \left (1+\frac {e^x}{x}\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.11, size = 25, normalized size = 0.86 \begin {gather*} \log (3) \left (-2 x+3 x \log \left (\frac {3}{\log \left (\frac {e^x+x}{x}\right )}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 9.77, size = 797, normalized size = 27.48
method | result | size |
risch | \(\text {Expression too large to display}\) | \(797\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 30, normalized size = 1.03 \begin {gather*} -3 \, x \log \left (3\right ) \log \left (\log \left (x + e^{x}\right ) - \log \left (x\right )\right ) + {\left (3 \, \log \left (3\right )^{2} - 2 \, \log \left (3\right )\right )} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 25, normalized size = 0.86 \begin {gather*} 3 \, x \log \left (3\right ) \log \left (\frac {3}{\log \left (\frac {x + e^{x}}{x}\right )}\right ) - 2 \, x \log \left (3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.01, size = 25, normalized size = 0.86 \begin {gather*} 3\,x\,\ln \left (3\right )\,\ln \left (\frac {3}{\ln \left (\frac {x+{\mathrm {e}}^x}{x}\right )}\right )-2\,x\,\ln \left (3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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