Optimal. Leaf size=17 \[ 1-x+\log \left (4-e x+e^x x\right ) \]
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Rubi [F] time = 0.45, 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 {-4+e^x+e (-1+x)}{4-e x+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 (\frac {1}{x}+\frac {-4-4 x+e x^2}{x \left (4-e x+e^x x\right )}\right ) \, dx\\ &=\log (x)+\int \frac {-4-4 x+e x^2}{x \left (4-e x+e^x x\right )} \, dx\\ &=\log (x)+\int \left (-\frac {e x}{-4+e x-e^x x}-\frac {4}{4-e x+e^x x}-\frac {4}{x \left (4-e x+e^x x\right )}\right ) \, dx\\ &=\log (x)-4 \int \frac {1}{4-e x+e^x x} \, dx-4 \int \frac {1}{x \left (4-e x+e^x x\right )} \, dx-e \int \frac {x}{-4+e x-e^x x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.12, size = 16, normalized size = 0.94 \begin {gather*} -x+\log \left (4-e x+e^x x\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 23, normalized size = 1.35 \begin {gather*} -x + \log \relax (x) + \log \left (-\frac {x e - x e^{x} - 4}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 16, normalized size = 0.94 \begin {gather*} -x + \log \left (x e - x e^{x} - 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 17, normalized size = 1.00
method | result | size |
norman | \(-x +\ln \left (x \,{\mathrm e}-{\mathrm e}^{x} x -4\right )\) | \(17\) |
risch | \(\ln \relax (x )-x +\ln \left ({\mathrm e}^{x}-\frac {x \,{\mathrm e}-4}{x}\right )\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.78, size = 23, normalized size = 1.35 \begin {gather*} -x + \log \relax (x) + \log \left (-\frac {x e - x e^{x} - 4}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 16, normalized size = 0.94 \begin {gather*} \ln \left (x\,{\mathrm {e}}^x-x\,\mathrm {e}+4\right )-x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 17, normalized size = 1.00 \begin {gather*} - x + \log {\relax (x )} + \log {\left (e^{x} + \frac {- e x + 4}{x} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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