Optimal. Leaf size=22 \[ \left (-3-e^3\right ) \left (\frac {4}{3}+\log \left (x+e^{-x} x\right )\right ) \]
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Rubi [A] time = 0.29, antiderivative size = 30, normalized size of antiderivative = 1.36, number of steps used = 8, number of rules used = 7, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {6688, 12, 6742, 2282, 36, 29, 31} \begin {gather*} \left (3+e^3\right ) x-\left (3+e^3\right ) \log \left (e^x+1\right )-\left (3+e^3\right ) \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 29
Rule 31
Rule 36
Rule 2282
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (3+e^3\right ) \left (-1-e^x+x\right )}{\left (1+e^x\right ) x} \, dx\\ &=\left (3+e^3\right ) \int \frac {-1-e^x+x}{\left (1+e^x\right ) x} \, dx\\ &=\left (3+e^3\right ) \int \left (\frac {1}{1+e^x}-\frac {1}{x}\right ) \, dx\\ &=-\left (\left (3+e^3\right ) \log (x)\right )+\left (3+e^3\right ) \int \frac {1}{1+e^x} \, dx\\ &=-\left (\left (3+e^3\right ) \log (x)\right )+\left (3+e^3\right ) \operatorname {Subst}\left (\int \frac {1}{x (1+x)} \, dx,x,e^x\right )\\ &=-\left (\left (3+e^3\right ) \log (x)\right )+\left (-3-e^3\right ) \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,e^x\right )+\left (3+e^3\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^x\right )\\ &=\left (3+e^3\right ) x-\left (3+e^3\right ) \log \left (1+e^x\right )-\left (3+e^3\right ) \log (x)\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.02, size = 19, normalized size = 0.86 \begin {gather*} -\left (\left (3+e^3\right ) \left (-x+\log \left (1+e^x\right )+\log (x)\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 27, normalized size = 1.23 \begin {gather*} x e^{3} - {\left (e^{3} + 3\right )} \log \relax (x) - {\left (e^{3} + 3\right )} \log \left (e^{x} + 1\right ) + 3 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 34, normalized size = 1.55 \begin {gather*} x e^{3} - e^{3} \log \relax (x) - e^{3} \log \left (e^{x} + 1\right ) + 3 \, x - 3 \, \log \relax (x) - 3 \, \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 = 0.07, size = 29, normalized size = 1.32
method | result | size |
norman | \(\left ({\mathrm e}^{3}+3\right ) x +\left (-{\mathrm e}^{3}-3\right ) \ln \relax (x )+\left (-{\mathrm e}^{3}-3\right ) \ln \left ({\mathrm e}^{x}+1\right )\) | \(29\) |
risch | \(-\ln \relax (x ) {\mathrm e}^{3}-3 \ln \relax (x )+x \,{\mathrm e}^{3}+3 x -{\mathrm e}^{3} \ln \left ({\mathrm e}^{x}+1\right )-3 \ln \left ({\mathrm e}^{x}+1\right )\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.39, size = 26, normalized size = 1.18 \begin {gather*} x {\left (e^{3} + 3\right )} - {\left (e^{3} + 3\right )} \log \relax (x) - {\left (e^{3} + 3\right )} \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 = 0.09, size = 16, normalized size = 0.73 \begin {gather*} \left (x-\ln \left (x\,\left ({\mathrm {e}}^x+1\right )\right )\right )\,\left ({\mathrm {e}}^3+3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 29, normalized size = 1.32 \begin {gather*} x \left (3 + e^{3}\right ) + \left (- e^{3} - 3\right ) \log {\relax (x )} + \left (- e^{3} - 3\right ) \log {\left (e^{x} + 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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