Optimal. Leaf size=12 \[ x-2 \log \left (1-e^x\right ) \]
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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 = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2320, 78}
\begin {gather*} x-2 \log \left (1-e^x\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 2320
Rubi steps
\begin {align*} \int \frac {1+e^x}{1-e^x} \, dx &=\text {Subst}\left (\int \frac {1+x}{(1-x) x} \, dx,x,e^x\right )\\ &=\text {Subst}\left (\int \left (-\frac {2}{-1+x}+\frac {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 = 13, normalized size = 1.08 \begin {gather*} \log \left (e^x\right )-2 \log \left (-1+e^x\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.89, size = 10, normalized size = 0.83 \begin {gather*} x-2 \text {Log}\left [-1+E^x\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 12, normalized size = 1.00
method | result | size |
norman | \(x -2 \ln \left (-1+{\mathrm e}^{x}\right )\) | \(10\) |
risch | \(x -2 \ln \left (-1+{\mathrm e}^{x}\right )\) | \(10\) |
derivativedivides | \(-2 \ln \left (-1+{\mathrm e}^{x}\right )+\ln \left ({\mathrm e}^{x}\right )\) | \(12\) |
default | \(-2 \ln \left (-1+{\mathrm e}^{x}\right )+\ln \left ({\mathrm e}^{x}\right )\) | \(12\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 9, normalized size = 0.75 \begin {gather*} x - 2 \, \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 = 0.36, size = 9, normalized size = 0.75 \begin {gather*} x - 2 \, \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 = 0.04, size = 8, normalized size = 0.67 \begin {gather*} 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 = 0.00, size = 11, normalized size = 0.92 \begin {gather*} x-2 \ln \left |\mathrm {e}^{x}-1\right | \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 9, normalized size = 0.75 \begin {gather*} x-2\,\ln \left ({\mathrm {e}}^x-1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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