Optimal. Leaf size=33 \[ e^{\frac {2 x \left (\frac {4}{x}+\frac {2}{\log (x)}\right )}{\left (1-e^x\right ) (4+x)}}+x \]
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Rubi [F] time = 41.19, 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 {\left (16+8 x+x^2+e^x \left (-32-16 x-2 x^2\right )+e^{2 x} \left (16+8 x+x^2\right )\right ) \log ^2(x)+\exp \left (\frac {-4 x-8 \log (x)}{\left (-4-x+e^x (4+x)\right ) \log (x)}\right ) \left (-16-4 x+e^x (16+4 x)+\left (16+e^x \left (-16+16 x+4 x^2\right )\right ) \log (x)+\left (-8+e^x (40+8 x)\right ) \log ^2(x)\right )}{\left (16+8 x+x^2+e^x \left (-32-16 x-2 x^2\right )+e^{2 x} \left (16+8 x+x^2\right )\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 (1+\frac {4 \exp \left (-\frac {4 (x+2 \log (x))}{\left (-1+e^x\right ) (4+x) \log (x)}\right ) \left (\left (-1+e^x\right ) (4+x)+\left (4+e^x \left (-4+4 x+x^2\right )\right ) \log (x)+2 \left (-1+e^x (5+x)\right ) \log ^2(x)\right )}{\left (-1+e^x\right )^2 (4+x)^2 \log ^2(x)}\right ) \, dx\\ &=x+4 \int \frac {\exp \left (-\frac {4 (x+2 \log (x))}{\left (-1+e^x\right ) (4+x) \log (x)}\right ) \left (\left (-1+e^x\right ) (4+x)+\left (4+e^x \left (-4+4 x+x^2\right )\right ) \log (x)+2 \left (-1+e^x (5+x)\right ) \log ^2(x)\right )}{\left (-1+e^x\right )^2 (4+x)^2 \log ^2(x)} \, dx\\ &=x+4 \int \left (\frac {\exp \left (-\frac {4 (x+2 \log (x))}{\left (-1+e^x\right ) (4+x) \log (x)}\right ) (x+2 \log (x))}{\left (-1+e^x\right )^2 (4+x) \log (x)}+\frac {\exp \left (-\frac {4 (x+2 \log (x))}{\left (-1+e^x\right ) (4+x) \log (x)}\right ) \left (4+x-4 \log (x)+4 x \log (x)+x^2 \log (x)+10 \log ^2(x)+2 x \log ^2(x)\right )}{\left (-1+e^x\right ) (4+x)^2 \log ^2(x)}\right ) \, dx\\ &=x+4 \int \frac {\exp \left (-\frac {4 (x+2 \log (x))}{\left (-1+e^x\right ) (4+x) \log (x)}\right ) (x+2 \log (x))}{\left (-1+e^x\right )^2 (4+x) \log (x)} \, dx+4 \int \frac {\exp \left (-\frac {4 (x+2 \log (x))}{\left (-1+e^x\right ) (4+x) \log (x)}\right ) \left (4+x-4 \log (x)+4 x \log (x)+x^2 \log (x)+10 \log ^2(x)+2 x \log ^2(x)\right )}{\left (-1+e^x\right ) (4+x)^2 \log ^2(x)} \, dx\\ &=x+4 \int \left (\frac {10 \exp \left (-\frac {4 (x+2 \log (x))}{\left (-1+e^x\right ) (4+x) \log (x)}\right )}{\left (-1+e^x\right ) (4+x)^2}+\frac {2 \exp \left (-\frac {4 (x+2 \log (x))}{\left (-1+e^x\right ) (4+x) \log (x)}\right ) x}{\left (-1+e^x\right ) (4+x)^2}+\frac {4 \exp \left (-\frac {4 (x+2 \log (x))}{\left (-1+e^x\right ) (4+x) \log (x)}\right )}{\left (-1+e^x\right ) (4+x)^2 \log ^2(x)}+\frac {\exp \left (-\frac {4 (x+2 \log (x))}{\left (-1+e^x\right ) (4+x) \log (x)}\right ) x}{\left (-1+e^x\right ) (4+x)^2 \log ^2(x)}-\frac {4 \exp \left (-\frac {4 (x+2 \log (x))}{\left (-1+e^x\right ) (4+x) \log (x)}\right )}{\left (-1+e^x\right ) (4+x)^2 \log (x)}+\frac {4 \exp \left (-\frac {4 (x+2 \log (x))}{\left (-1+e^x\right ) (4+x) \log (x)}\right ) x}{\left (-1+e^x\right ) (4+x)^2 \log (x)}+\frac {\exp \left (-\frac {4 (x+2 \log (x))}{\left (-1+e^x\right ) (4+x) \log (x)}\right ) x^2}{\left (-1+e^x\right ) (4+x)^2 \log (x)}\right ) \, dx+4 \int \left (\frac {2 \exp \left (-\frac {4 (x+2 \log (x))}{\left (-1+e^x\right ) (4+x) \log (x)}\right )}{\left (-1+e^x\right )^2 (4+x)}+\frac {\exp \left (-\frac {4 (x+2 \log (x))}{\left (-1+e^x\right ) (4+x) \log (x)}\right ) x}{\left (-1+e^x\right )^2 (4+x) \log (x)}\right ) \, dx\\ &=x+4 \int \frac {\exp \left (-\frac {4 (x+2 \log (x))}{\left (-1+e^x\right ) (4+x) \log (x)}\right ) x}{\left (-1+e^x\right ) (4+x)^2 \log ^2(x)} \, dx+4 \int \frac {\exp \left (-\frac {4 (x+2 \log (x))}{\left (-1+e^x\right ) (4+x) \log (x)}\right ) x^2}{\left (-1+e^x\right ) (4+x)^2 \log (x)} \, dx+4 \int \frac {\exp \left (-\frac {4 (x+2 \log (x))}{\left (-1+e^x\right ) (4+x) \log (x)}\right ) x}{\left (-1+e^x\right )^2 (4+x) \log (x)} \, dx+8 \int \frac {\exp \left (-\frac {4 (x+2 \log (x))}{\left (-1+e^x\right ) (4+x) \log (x)}\right ) x}{\left (-1+e^x\right ) (4+x)^2} \, dx+8 \int \frac {\exp \left (-\frac {4 (x+2 \log (x))}{\left (-1+e^x\right ) (4+x) \log (x)}\right )}{\left (-1+e^x\right )^2 (4+x)} \, dx+16 \int \frac {\exp \left (-\frac {4 (x+2 \log (x))}{\left (-1+e^x\right ) (4+x) \log (x)}\right )}{\left (-1+e^x\right ) (4+x)^2 \log ^2(x)} \, dx-16 \int \frac {\exp \left (-\frac {4 (x+2 \log (x))}{\left (-1+e^x\right ) (4+x) \log (x)}\right )}{\left (-1+e^x\right ) (4+x)^2 \log (x)} \, dx+16 \int \frac {\exp \left (-\frac {4 (x+2 \log (x))}{\left (-1+e^x\right ) (4+x) \log (x)}\right ) x}{\left (-1+e^x\right ) (4+x)^2 \log (x)} \, dx+40 \int \frac {\exp \left (-\frac {4 (x+2 \log (x))}{\left (-1+e^x\right ) (4+x) \log (x)}\right )}{\left (-1+e^x\right ) (4+x)^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A] time = 0.35, size = 38, normalized size = 1.15 \begin {gather*} e^{-\frac {8}{\left (-1+e^x\right ) (4+x)}-\frac {4 x}{\left (-1+e^x\right ) (4+x) \log (x)}}+x \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 28, normalized size = 0.85 \begin {gather*} x + e^{\left (-\frac {4 \, {\left (x + 2 \, \log \relax (x)\right )}}{{\left ({\left (x + 4\right )} e^{x} - x - 4\right )} \log \relax (x)}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 27, normalized size = 0.82
method | result | size |
risch | \(x +{\mathrm e}^{-\frac {4 \left (2 \ln \relax (x )+x \right )}{\left (4+x \right ) \left ({\mathrm e}^{x}-1\right ) \ln \relax (x )}}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.54, size = 82, normalized size = 2.48 \begin {gather*} {\left (x e^{\left (\frac {8}{{\left (x + 4\right )} e^{x} - x - 4} + \frac {4}{{\left (e^{x} - 1\right )} \log \relax (x)}\right )} + e^{\left (\frac {16}{{\left ({\left (x + 4\right )} e^{x} - x - 4\right )} \log \relax (x)}\right )}\right )} e^{\left (-\frac {8}{{\left (x + 4\right )} e^{x} - x - 4} - \frac {4}{{\left (e^{x} - 1\right )} \log \relax (x)}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.69, size = 59, normalized size = 1.79 \begin {gather*} x+x^{\frac {8}{4\,\ln \relax (x)-4\,{\mathrm {e}}^x\,\ln \relax (x)+x\,\ln \relax (x)-x\,{\mathrm {e}}^x\,\ln \relax (x)}}\,{\mathrm {e}}^{\frac {4\,x}{4\,\ln \relax (x)-4\,{\mathrm {e}}^x\,\ln \relax (x)+x\,\ln \relax (x)-x\,{\mathrm {e}}^x\,\ln \relax (x)}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.64, size = 26, normalized size = 0.79 \begin {gather*} x + e^{\frac {- 4 x - 8 \log {\relax (x )}}{\left (- x + \left (x + 4\right ) e^{x} - 4\right ) \log {\relax (x )}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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