Optimal. Leaf size=21 \[ \frac {x+\frac {\log ^2(x)}{-4+e^{-x}}}{\log (x)} \]
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Rubi [F] time = 0.62, 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 {-x+8 e^x x-16 e^{2 x} x+\left (x-8 e^x x+16 e^{2 x} x\right ) \log (x)+\left (e^x-4 e^{2 x}\right ) \log ^2(x)+e^x x \log ^3(x)}{\left (x-8 e^x x+16 e^{2 x} x\right ) \log ^2(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 {e^x}{x-4 e^x x}-\frac {1}{\log ^2(x)}+\frac {1}{\log (x)}+\frac {e^x \log (x)}{\left (1-4 e^x\right )^2}\right ) \, dx\\ &=\int \frac {e^x}{x-4 e^x x} \, dx-\int \frac {1}{\log ^2(x)} \, dx+\int \frac {1}{\log (x)} \, dx+\int \frac {e^x \log (x)}{\left (1-4 e^x\right )^2} \, dx\\ &=\frac {x}{\log (x)}+\frac {\log (x)}{4 \left (1-4 e^x\right )}+\text {li}(x)-\int \frac {1}{4 \left (1-4 e^x\right ) x} \, dx+\int \frac {e^x}{x-4 e^x x} \, dx-\int \frac {1}{\log (x)} \, dx\\ &=\frac {x}{\log (x)}+\frac {\log (x)}{4 \left (1-4 e^x\right )}-\frac {1}{4} \int \frac {1}{\left (1-4 e^x\right ) x} \, dx+\int \frac {e^x}{x-4 e^x x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.11, size = 22, normalized size = 1.05 \begin {gather*} \frac {x}{\log (x)}+\frac {e^x \log (x)}{1-4 e^x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 28, normalized size = 1.33 \begin {gather*} -\frac {e^{x} \log \relax (x)^{2} - 4 \, x e^{x} + x}{{\left (4 \, e^{x} - 1\right )} \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 29, normalized size = 1.38 \begin {gather*} -\frac {e^{x} \log \relax (x)^{2} - 4 \, x e^{x} + x}{4 \, e^{x} \log \relax (x) - \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 24, normalized size = 1.14
method | result | size |
risch | \(-\frac {\ln \relax (x )}{4 \left (4 \,{\mathrm e}^{x}-1\right )}-\frac {\ln \relax (x )}{4}+\frac {x}{\ln \relax (x )}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 35, normalized size = 1.67 \begin {gather*} \frac {16 \, x e^{x} - \log \relax (x)^{2} - 4 \, x}{4 \, {\left (4 \, e^{x} \log \relax (x) - \log \relax (x)\right )}} - \frac {1}{4} \, \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 = 28, normalized size = 1.33 \begin {gather*} -\frac {{\mathrm {e}}^x\,{\ln \relax (x)}^2+x-4\,x\,{\mathrm {e}}^x}{\ln \relax (x)\,\left (4\,{\mathrm {e}}^x-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.29, size = 19, normalized size = 0.90 \begin {gather*} \frac {x}{\log {\relax (x )}} - \frac {\log {\relax (x )}}{4} - \frac {\log {\relax (x )}}{16 e^{x} - 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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