Optimal. Leaf size=12 \[ e^x-\log \left (1+e^x\right ) \]
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Rubi [A]
time = 0.01, 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 = {2280, 45}
\begin {gather*} e^x-\log \left (e^x+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2280
Rubi steps
\begin {align*} \int \frac {e^{2 x}}{1+e^x} \, dx &=\text {Subst}\left (\int \frac {x}{1+x} \, dx,x,e^x\right )\\ &=\text {Subst}\left (\int \left (1+\frac {1}{-1-x}\right ) \, dx,x,e^x\right )\\ &=e^x-\log \left (1+e^x\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} e^x-\log \left (1+e^x\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 11, normalized size = 0.92
method | result | size |
default | \({\mathrm e}^{x}-\ln \left (1+{\mathrm e}^{x}\right )\) | \(11\) |
norman | \({\mathrm e}^{x}-\ln \left (1+{\mathrm e}^{x}\right )\) | \(11\) |
risch | \({\mathrm e}^{x}-\ln \left (1+{\mathrm e}^{x}\right )\) | \(11\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.93, size = 10, normalized size = 0.83 \begin {gather*} e^{x} - \log \left (e^{x} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 10, normalized size = 0.83 \begin {gather*} e^{x} - \log \left (e^{x} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 8, normalized size = 0.67 \begin {gather*} e^{x} - \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 = 0.69, size = 10, normalized size = 0.83 \begin {gather*} e^{x} - \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 = 0.05, size = 10, normalized size = 0.83 \begin {gather*} {\mathrm {e}}^x-\ln \left ({\mathrm {e}}^x+1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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