Optimal. Leaf size=23 \[ 4 \left (-5-x+\log (5)+\log ^2(25)+\log \left (x \left (-e^x+x\right )\right )\right ) \]
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Rubi [F]
time = 0.21, 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-8 x+4 x^2}{e^x x-x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 \left (e^x-2 x+x^2\right )}{e^x x-x^2} \, dx\\ &=4 \int \frac {e^x-2 x+x^2}{e^x x-x^2} \, dx\\ &=4 \int \left (\frac {-1+x}{e^x-x}+\frac {1}{x}\right ) \, dx\\ &=4 \log (x)+4 \int \frac {-1+x}{e^x-x} \, dx\\ &=4 \log (x)+4 \int \left (-\frac {1}{e^x-x}+\frac {x}{e^x-x}\right ) \, dx\\ &=4 \log (x)-4 \int \frac {1}{e^x-x} \, dx+4 \int \frac {x}{e^x-x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.11, size = 16, normalized size = 0.70 \begin {gather*} 4 \left (-x+\log \left (e^x-x\right )+\log (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 18, normalized size = 0.78
method | result | size |
norman | \(-4 x +4 \ln \left (x \right )+4 \ln \left (x -{\mathrm e}^{x}\right )\) | \(18\) |
risch | \(4 \ln \left (x \right )-4 x +4 \ln \left ({\mathrm e}^{x}-x \right )\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 17, normalized size = 0.74 \begin {gather*} -4 \, x + 4 \, \log \left (x\right ) + 4 \, \log \left (-x + e^{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 17, normalized size = 0.74 \begin {gather*} -4 \, x + 4 \, \log \left (x\right ) + 4 \, \log \left (-x + e^{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 15, normalized size = 0.65 \begin {gather*} - 4 x + 4 \log {\left (x \right )} + 4 \log {\left (- x + e^{x} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 17, normalized size = 0.74 \begin {gather*} -4 \, x + 4 \, \log \left (x - e^{x}\right ) + 4 \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.29, size = 17, normalized size = 0.74 \begin {gather*} 4\,\ln \left (x-{\mathrm {e}}^x\right )-4\,x+4\,\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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