Optimal. Leaf size=12 \[ 2 \log \left (e^x+1\right )-x \]
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Rubi [A] time = 0.02, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2282, 72} \[ 2 \log \left (e^x+1\right )-x \]
Antiderivative was successfully verified.
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Rule 72
Rule 2282
Rubi steps
\begin {align*} \int \frac {-1+e^x}{1+e^x} \, dx &=\operatorname {Subst}\left (\int \frac {-1+x}{x (1+x)} \, dx,x,e^x\right )\\ &=\operatorname {Subst}\left (\int \left (-\frac {1}{x}+\frac {2}{1+x}\right ) \, dx,x,e^x\right )\\ &=-x+2 \log \left (1+e^x\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 12, normalized size = 1.00 \[ 2 \log \left (e^x+1\right )-x \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {-1+e^x}{1+e^x} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.01, size = 11, normalized size = 0.92 \[ -x + 2 \, \log \left (e^{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.57, size = 11, normalized size = 0.92 \[ -x + 2 \, \log \left (e^{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 12, normalized size = 1.00
method | result | size |
norman | \(-x +2 \ln \left (1+{\mathrm e}^{x}\right )\) | \(12\) |
risch | \(-x +2 \ln \left (1+{\mathrm e}^{x}\right )\) | \(12\) |
derivativedivides | \(2 \ln \left (1+{\mathrm e}^{x}\right )-\ln \left ({\mathrm e}^{x}\right )\) | \(14\) |
default | \(2 \ln \left (1+{\mathrm e}^{x}\right )-\ln \left ({\mathrm e}^{x}\right )\) | \(14\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 11, normalized size = 0.92 \[ -x + 2 \, \log \left (e^{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 11, normalized size = 0.92 \[ 2\,\ln \left ({\mathrm {e}}^x+1\right )-x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 8, normalized size = 0.67 \[ - x + 2 \log {\left (e^{x} + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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