Optimal. Leaf size=31 \[ -7-\log \left (\frac {-4+x}{e^{\frac {e^{e^3}}{3}}-\frac {e^x}{x}}\right ) \]
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Rubi [F] time = 1.05, 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 {-e^{\frac {e^{e^3}}{3}} x^2+e^x \left (-4+6 x-x^2\right )}{e^x \left (4 x-x^2\right )+e^{\frac {e^{e^3}}{3}} \left (-4 x^2+x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-e^{\frac {e^{e^3}}{3}} x^2+e^x \left (-4+6 x-x^2\right )}{(4-x) x \left (e^x-e^{\frac {e^{e^3}}{3}} x\right )} \, dx\\ &=\int \left (-\frac {e^{\frac {e^{e^3}}{3}} (-1+x)}{-e^x+e^{\frac {e^{e^3}}{3}} x}+\frac {4-6 x+x^2}{(-4+x) x}\right ) \, dx\\ &=-\left (e^{\frac {e^{e^3}}{3}} \int \frac {-1+x}{-e^x+e^{\frac {e^{e^3}}{3}} x} \, dx\right )+\int \frac {4-6 x+x^2}{(-4+x) x} \, dx\\ &=-\left (e^{\frac {e^{e^3}}{3}} \int \left (\frac {1}{e^x-e^{\frac {e^{e^3}}{3}} x}+\frac {x}{-e^x+e^{\frac {e^{e^3}}{3}} x}\right ) \, dx\right )+\int \left (1+\frac {1}{4-x}-\frac {1}{x}\right ) \, dx\\ &=x-\log (4-x)-\log (x)-e^{\frac {e^{e^3}}{3}} \int \frac {1}{e^x-e^{\frac {e^{e^3}}{3}} x} \, dx-e^{\frac {e^{e^3}}{3}} \int \frac {x}{-e^x+e^{\frac {e^{e^3}}{3}} x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.18, size = 32, normalized size = 1.03 \begin {gather*} -\log (4-x)-\log (x)+\log \left (e^x-e^{\frac {e^{e^3}}{3}} x\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 24, normalized size = 0.77 \begin {gather*} -\log \left (x^{2} - 4 \, x\right ) + \log \left (-x e^{\left (\frac {1}{3} \, e^{\left (e^{3}\right )}\right )} + e^{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.63, size = 24, normalized size = 0.77 \begin {gather*} \log \left (-x e^{\left (\frac {1}{3} \, e^{\left (e^{3}\right )}\right )} + e^{x}\right ) - \log \left (x - 4\right ) - \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 25, normalized size = 0.81
method | result | size |
risch | \(-\ln \left (x^{2}-4 x \right )+\ln \left ({\mathrm e}^{x}-x \,{\mathrm e}^{\frac {{\mathrm e}^{{\mathrm e}^{3}}}{3}}\right )\) | \(25\) |
norman | \(-\ln \relax (x )-\ln \left (x -4\right )+\ln \left (x \,{\mathrm e}^{\frac {{\mathrm e}^{{\mathrm e}^{3}}}{3}}-{\mathrm e}^{x}\right )\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 24, normalized size = 0.77 \begin {gather*} \log \left (-x e^{\left (\frac {1}{3} \, e^{\left (e^{3}\right )}\right )} + e^{x}\right ) - \log \left (x - 4\right ) - \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.40, size = 24, normalized size = 0.77 \begin {gather*} \ln \left (x-{\mathrm {e}}^{-\frac {{\mathrm {e}}^{{\mathrm {e}}^3}}{3}}\,{\mathrm {e}}^x\right )-\ln \left (x-4\right )-\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 22, normalized size = 0.71 \begin {gather*} - \log {\left (x^{2} - 4 x \right )} + \log {\left (- x e^{\frac {e^{e^{3}}}{3}} + e^{x} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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