Optimal. Leaf size=16 \[ 1+x-\log \left (\left (-1-e^x+x\right )^2\right ) \]
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Rubi [F] time = 0.16, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {3-e^x-x}{1+e^x-x} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-1+\frac {2 (-2+x)}{-1-e^x+x}\right ) \, dx\\ &=-x+2 \int \frac {-2+x}{-1-e^x+x} \, dx\\ &=-x+2 \int \left (\frac {2}{1+e^x-x}+\frac {x}{-1-e^x+x}\right ) \, dx\\ &=-x+2 \int \frac {x}{-1-e^x+x} \, dx+4 \int \frac {1}{1+e^x-x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.05, size = 13, normalized size = 0.81 \begin {gather*} x-2 \log \left (1+e^x-x\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 12, normalized size = 0.75 \begin {gather*} x - 2 \, \log \left (-x + e^{x} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 12, normalized size = 0.75 \begin {gather*} x - 2 \, \log \left (x - e^{x} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 13, normalized size = 0.81
method | result | size |
norman | \(x -2 \ln \left (x -{\mathrm e}^{x}-1\right )\) | \(13\) |
risch | \(x -2 \ln \left (1+{\mathrm e}^{x}-x \right )\) | \(13\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 12, normalized size = 0.75 \begin {gather*} x - 2 \, \log \left (-x + 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 = 12, normalized size = 0.75 \begin {gather*} x-2\,\ln \left (x-{\mathrm {e}}^x-1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 10, normalized size = 0.62 \begin {gather*} x - 2 \log {\left (- x + e^{x} + 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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