Optimal. Leaf size=25 \[ \frac {e^{-x} x}{\log \left (\frac {1}{256} e^{4+4 x}-x\right )} \]
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Rubi [F] time = 4.12, 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 {256 x-4 e^{4+4 x} x+\left (e^{4+4 x} (1-x)-256 x+256 x^2\right ) \log \left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )}{\left (e^{4+5 x}-256 e^x x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\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^{-x} \left (256 x-4 e^{4+4 x} x+\left (e^{4+4 x} (1-x)-256 x+256 x^2\right ) \log \left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )\right )}{\left (e^{4+4 x}-256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx\\ &=\int \frac {e^{-x} \left (-\frac {4 \left (-64+e^{4+4 x}\right ) x}{e^{4+4 x}-256 x}-(-1+x) \log \left (\frac {1}{256} e^{4+4 x}-x\right )\right )}{\log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx\\ &=\int \left (\frac {256 e^{-x} (1-4 x) x}{\left (e^{4+4 x}-256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )}+\frac {e^{-x} \left (-4 x+\log \left (\frac {1}{256} e^{4+4 x}-x\right )-x \log \left (\frac {1}{256} e^{4+4 x}-x\right )\right )}{\log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )}\right ) \, dx\\ &=256 \int \frac {e^{-x} (1-4 x) x}{\left (e^{4+4 x}-256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx+\int \frac {e^{-x} \left (-4 x+\log \left (\frac {1}{256} e^{4+4 x}-x\right )-x \log \left (\frac {1}{256} e^{4+4 x}-x\right )\right )}{\log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx\\ &=256 \int \left (\frac {e^{-x} x}{\left (e^{4+4 x}-256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )}+\frac {4 e^{-x} x^2}{\left (-e^{4+4 x}+256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )}\right ) \, dx+\int \frac {e^{-x} \left (-4 x-(-1+x) \log \left (\frac {1}{256} e^{4+4 x}-x\right )\right )}{\log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx\\ &=256 \int \frac {e^{-x} x}{\left (e^{4+4 x}-256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx+1024 \int \frac {e^{-x} x^2}{\left (-e^{4+4 x}+256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx+\int \left (-\frac {4 e^{-x} x}{\log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )}+\frac {e^{-x} (1-x)}{\log \left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )}\right ) \, dx\\ &=-\left (4 \int \frac {e^{-x} x}{\log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx\right )+256 \int \frac {e^{-x} x}{\left (e^{4+4 x}-256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx+1024 \int \frac {e^{-x} x^2}{\left (-e^{4+4 x}+256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx+\int \frac {e^{-x} (1-x)}{\log \left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx\\ &=-\left (4 \int \frac {e^{-x} x}{\log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx\right )+256 \int \frac {e^{-x} x}{\left (e^{4+4 x}-256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx+1024 \int \frac {e^{-x} x^2}{\left (-e^{4+4 x}+256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx+\int \left (\frac {e^{-x}}{\log \left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )}-\frac {e^{-x} x}{\log \left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )}\right ) \, dx\\ &=-\left (4 \int \frac {e^{-x} x}{\log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx\right )+256 \int \frac {e^{-x} x}{\left (e^{4+4 x}-256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx+1024 \int \frac {e^{-x} x^2}{\left (-e^{4+4 x}+256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx+\int \frac {e^{-x}}{\log \left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx-\int \frac {e^{-x} x}{\log \left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.95, size = 25, normalized size = 1.00 \begin {gather*} \frac {e^{-x} x}{\log \left (\frac {1}{256} e^{4+4 x}-x\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 21, normalized size = 0.84 \begin {gather*} \frac {x e^{\left (-x\right )}}{\log \left (-x + \frac {1}{256} \, e^{\left (4 \, x + 4\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 27, normalized size = 1.08 \begin {gather*} -\frac {x}{8 \, e^{x} \log \relax (2) - e^{x} \log \left (-256 \, x + e^{\left (4 \, x + 4\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 22, normalized size = 0.88
method | result | size |
risch | \(\frac {x \,{\mathrm e}^{-x}}{\ln \left (\frac {{\mathrm e}^{4 x +4}}{256}-x \right )}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.74, size = 27, normalized size = 1.08 \begin {gather*} -\frac {x}{8 \, e^{x} \log \relax (2) - e^{x} \log \left (-256 \, x + e^{\left (4 \, x + 4\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.92, size = 106, normalized size = 4.24 \begin {gather*} \frac {x\,{\mathrm {e}}^{-x}-\frac {{\mathrm {e}}^{-x}\,\ln \left (\frac {{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^4}{256}-x\right )\,\left (256\,x-{\mathrm {e}}^{4\,x+4}\right )\,\left (x-1\right )}{4\,\left ({\mathrm {e}}^{4\,x+4}-64\right )}}{\ln \left (\frac {{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^4}{256}-x\right )}+{\mathrm {e}}^{-x}\,\left (x-x^2\right )+\frac {{\mathrm {e}}^{3\,x}\,\left (x^2-\frac {5\,x}{4}+\frac {1}{4}\right )}{{\mathrm {e}}^{4\,x}-64\,{\mathrm {e}}^{-4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 17, normalized size = 0.68 \begin {gather*} \frac {x e^{- x}}{\log {\left (- x + \frac {e^{4} e^{4 x}}{256} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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