Optimal. Leaf size=28 \[ -\log \left (\frac {e^{2 \left (x+e^3 x+\log \left (\frac {x}{1+e^x}\right )\right )}}{x}\right ) \]
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Rubi [A] time = 0.10, antiderivative size = 22, normalized size of antiderivative = 0.79, number of steps used = 9, number of rules used = 7, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {6, 6742, 2282, 36, 29, 31, 43} \begin {gather*} -2 e^3 x-2 x+2 \log \left (e^x+1\right )-\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 29
Rule 31
Rule 36
Rule 43
Rule 2282
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-1+\left (-2-2 e^3\right ) x+e^x \left (-1-2 e^3 x\right )}{x+e^x x} \, dx\\ &=\int \left (-\frac {2}{1+e^x}+\frac {-1-2 e^3 x}{x}\right ) \, dx\\ &=-\left (2 \int \frac {1}{1+e^x} \, dx\right )+\int \frac {-1-2 e^3 x}{x} \, dx\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{x (1+x)} \, dx,x,e^x\right )\right )+\int \left (-2 e^3-\frac {1}{x}\right ) \, dx\\ &=-2 e^3 x-\log (x)-2 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^x\right )+2 \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,e^x\right )\\ &=-2 x-2 e^3 x+2 \log \left (1+e^x\right )-\log (x)\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.07, size = 21, normalized size = 0.75 \begin {gather*} -2 \left (1+e^3\right ) x+2 \log \left (1+e^x\right )-\log (x) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 20, normalized size = 0.71 \begin {gather*} -2 \, x e^{3} - 2 \, x - \log \relax (x) + 2 \, \log \left (e^{x} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.05, size = 20, normalized size = 0.71 \begin {gather*} -2 \, x e^{3} - 2 \, x - \log \relax (x) + 2 \, \log \left (e^{x} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 21, normalized size = 0.75
method | result | size |
norman | \(\left (-2 \,{\mathrm e}^{3}-2\right ) x -\ln \relax (x )+2 \ln \left ({\mathrm e}^{x}+1\right )\) | \(21\) |
risch | \(-2 x \,{\mathrm e}^{3}-\ln \relax (x )-2 x +2 \ln \left ({\mathrm e}^{x}+1\right )\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 19, normalized size = 0.68 \begin {gather*} -2 \, x {\left (e^{3} + 1\right )} - \log \relax (x) + 2 \, \log \left (e^{x} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.17, size = 21, normalized size = 0.75 \begin {gather*} 2\,\ln \left ({\mathrm {e}}^x+1\right )-\ln \relax (x)-x\,\left (2\,{\mathrm {e}}^3+2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 19, normalized size = 0.68 \begin {gather*} - x \left (2 + 2 e^{3}\right ) - \log {\relax (x )} + 2 \log {\left (e^{x} + 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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