Optimal. Leaf size=20 \[ -\frac {x}{e^x+1}+x-\log \left (e^x+1\right ) \]
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Rubi [A] time = 0.13, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2267, 6688, 2191, 2282, 36, 29, 31} \[ -\frac {x}{e^x+1}+x-\log \left (e^x+1\right ) \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 2191
Rule 2267
Rule 2282
Rule 6688
Rubi steps
\begin {align*} \int \frac {x}{2+e^{-x}+e^x} \, dx &=\int \frac {e^x x}{1+2 e^x+e^{2 x}} \, dx\\ &=\int \frac {e^x x}{\left (1+e^x\right )^2} \, dx\\ &=-\frac {x}{1+e^x}+\int \frac {1}{1+e^x} \, dx\\ &=-\frac {x}{1+e^x}+\operatorname {Subst}\left (\int \frac {1}{x (1+x)} \, dx,x,e^x\right )\\ &=-\frac {x}{1+e^x}+\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^x\right )-\operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,e^x\right )\\ &=x-\frac {x}{1+e^x}-\log \left (1+e^x\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 20, normalized size = 1.00 \[ -\frac {x}{e^x+1}+x-\log \left (e^x+1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 23, normalized size = 1.15 \[ \frac {x e^{x} - {\left (e^{x} + 1\right )} \log \left (e^{x} + 1\right )}{e^{x} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 28, normalized size = 1.40 \[ \frac {x e^{x} - e^{x} \log \left (e^{x} + 1\right ) - \log \left (e^{x} + 1\right )}{e^{x} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 19, normalized size = 0.95 \[ \frac {x \,{\mathrm e}^{x}}{{\mathrm e}^{x}+1}-\ln \left ({\mathrm e}^{x}+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.11, size = 18, normalized size = 0.90 \[ \frac {x e^{x}}{e^{x} + 1} - \log \left (e^{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 18, normalized size = 0.90 \[ \frac {x\,{\mathrm {e}}^x}{{\mathrm {e}}^x+1}-\ln \left ({\mathrm {e}}^x+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 14, normalized size = 0.70 \[ x - \frac {x}{e^{x} + 1} - \log {\left (e^{x} + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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