Integrand size = 147, antiderivative size = 32 \[ \int \frac {e^{\frac {-x+(-16-4 x) \left (i \pi -\log \left (\frac {5}{2}\right )\right )+(4+x) \log (x)}{-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )+\log (x)}} \left (-1-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )-16 \left (i \pi -\log \left (\frac {5}{2}\right )\right )^2+\left (1+8 \left (i \pi -\log \left (\frac {5}{2}\right )\right )\right ) \log (x)-\log ^2(x)\right )}{16 \left (i \pi -\log \left (\frac {5}{2}\right )\right )^2-8 \left (i \pi -\log \left (\frac {5}{2}\right )\right ) \log (x)+\log ^2(x)} \, dx=3-e^{4+x+\frac {x}{4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )-\log (x)}} \]
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\[ \int \frac {e^{\frac {-x+(-16-4 x) \left (i \pi -\log \left (\frac {5}{2}\right )\right )+(4+x) \log (x)}{-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )+\log (x)}} \left (-1-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )-16 \left (i \pi -\log \left (\frac {5}{2}\right )\right )^2+\left (1+8 \left (i \pi -\log \left (\frac {5}{2}\right )\right )\right ) \log (x)-\log ^2(x)\right )}{16 \left (i \pi -\log \left (\frac {5}{2}\right )\right )^2-8 \left (i \pi -\log \left (\frac {5}{2}\right )\right ) \log (x)+\log ^2(x)} \, dx=\int \frac {\exp \left (\frac {-x+(-16-4 x) \left (i \pi -\log \left (\frac {5}{2}\right )\right )+(4+x) \log (x)}{-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )+\log (x)}\right ) \left (-1-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )-16 \left (i \pi -\log \left (\frac {5}{2}\right )\right )^2+\left (1+8 \left (i \pi -\log \left (\frac {5}{2}\right )\right )\right ) \log (x)-\log ^2(x)\right )}{16 \left (i \pi -\log \left (\frac {5}{2}\right )\right )^2-8 \left (i \pi -\log \left (\frac {5}{2}\right )\right ) \log (x)+\log ^2(x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\exp \left (\frac {-x+(-16-4 x) \left (i \pi -\log \left (\frac {5}{2}\right )\right )+(4+x) \log (x)}{-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )+\log (x)}\right ) \left (1-4 \left (\pi +i \log \left (\frac {5}{2}\right )\right ) \left (-i+4 \pi +4 i \log \left (\frac {5}{2}\right )\right )-\left (1+8 \left (i \pi -\log \left (\frac {5}{2}\right )\right )\right ) \log (x)+\log ^2(x)\right )}{\left (4 \pi \left (1+\frac {i \log \left (\frac {5}{2}\right )}{\pi }\right )+i \log (x)\right )^2} \, dx \\ & = \int \left (-\exp \left (\frac {-x+(-16-4 x) \left (i \pi -\log \left (\frac {5}{2}\right )\right )+(4+x) \log (x)}{-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )+\log (x)}\right )+\frac {\exp \left (\frac {-x+(-16-4 x) \left (i \pi -\log \left (\frac {5}{2}\right )\right )+(4+x) \log (x)}{-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )+\log (x)}\right )}{\left (4 \pi \left (1+\frac {i \log \left (\frac {5}{2}\right )}{\pi }\right )+i \log (x)\right )^2}+\frac {i \exp \left (\frac {-x+(-16-4 x) \left (i \pi -\log \left (\frac {5}{2}\right )\right )+(4+x) \log (x)}{-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )+\log (x)}\right )}{4 \pi \left (1+\frac {i \log \left (\frac {5}{2}\right )}{\pi }\right )+i \log (x)}\right ) \, dx \\ & = i \int \frac {\exp \left (\frac {-x+(-16-4 x) \left (i \pi -\log \left (\frac {5}{2}\right )\right )+(4+x) \log (x)}{-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )+\log (x)}\right )}{4 \pi \left (1+\frac {i \log \left (\frac {5}{2}\right )}{\pi }\right )+i \log (x)} \, dx-\int \exp \left (\frac {-x+(-16-4 x) \left (i \pi -\log \left (\frac {5}{2}\right )\right )+(4+x) \log (x)}{-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )+\log (x)}\right ) \, dx+\int \frac {\exp \left (\frac {-x+(-16-4 x) \left (i \pi -\log \left (\frac {5}{2}\right )\right )+(4+x) \log (x)}{-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )+\log (x)}\right )}{\left (4 \pi \left (1+\frac {i \log \left (\frac {5}{2}\right )}{\pi }\right )+i \log (x)\right )^2} \, dx \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(85\) vs. \(2(32)=64\).
Time = 0.36 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.66 \[ \int \frac {e^{\frac {-x+(-16-4 x) \left (i \pi -\log \left (\frac {5}{2}\right )\right )+(4+x) \log (x)}{-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )+\log (x)}} \left (-1-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )-16 \left (i \pi -\log \left (\frac {5}{2}\right )\right )^2+\left (1+8 \left (i \pi -\log \left (\frac {5}{2}\right )\right )\right ) \log (x)-\log ^2(x)\right )}{16 \left (i \pi -\log \left (\frac {5}{2}\right )\right )^2-8 \left (i \pi -\log \left (\frac {5}{2}\right )\right ) \log (x)+\log ^2(x)} \, dx=-e^{4+x+\frac {4 \pi (4+x)+i \left (16 \log \left (\frac {5}{2}\right )+x \left (-1+4 \log \left (\frac {5}{2}\right )\right )\right )}{4 \pi +4 i \log \left (\frac {5}{2}\right )+i \log (x)}} x^{(4+x) \left (-\frac {1}{\log (x)}+\frac {1}{-4 i \pi +4 \log \left (\frac {5}{2}\right )+\log (x)}\right )} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (25 ) = 50\).
Time = 3.19 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.94
method | result | size |
risch | \(-{\mathrm e}^{\frac {4 i \pi x -x \ln \left (x \right )+16 i \pi +4 x \ln \left (2\right )-4 x \ln \left (5\right )-4 \ln \left (x \right )+16 \ln \left (2\right )-16 \ln \left (5\right )+x}{-4 \ln \left (5\right )+4 \ln \left (2\right )+4 i \pi -\ln \left (x \right )}}\) | \(62\) |
norman | \(\frac {\left (-16 \pi ^{2}-16 \ln \left (2\right )^{2}+32 \ln \left (2\right ) \ln \left (5\right )-16 \ln \left (5\right )^{2}\right ) {\mathrm e}^{\frac {\left (4+x \right ) \ln \left (x \right )+\left (-16-4 x \right ) \left (\ln \left (\frac {2}{5}\right )+i \pi \right )-x}{\ln \left (x \right )-4 \ln \left (\frac {2}{5}\right )-4 i \pi }}+\left (-8 \ln \left (5\right )+8 \ln \left (2\right )\right ) \ln \left (x \right ) {\mathrm e}^{\frac {\left (4+x \right ) \ln \left (x \right )+\left (-16-4 x \right ) \left (\ln \left (\frac {2}{5}\right )+i \pi \right )-x}{\ln \left (x \right )-4 \ln \left (\frac {2}{5}\right )-4 i \pi }}-\ln \left (x \right )^{2} {\mathrm e}^{\frac {\left (4+x \right ) \ln \left (x \right )+\left (-16-4 x \right ) \left (\ln \left (\frac {2}{5}\right )+i \pi \right )-x}{\ln \left (x \right )-4 \ln \left (\frac {2}{5}\right )-4 i \pi }}}{16 \pi ^{2}+16 \ln \left (2\right )^{2}-32 \ln \left (2\right ) \ln \left (5\right )-8 \ln \left (2\right ) \ln \left (x \right )+16 \ln \left (5\right )^{2}+8 \ln \left (5\right ) \ln \left (x \right )+\ln \left (x \right )^{2}}\) | \(202\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (50) = 100\).
Time = 0.26 (sec) , antiderivative size = 115, normalized size of antiderivative = 3.59 \[ \int \frac {e^{\frac {-x+(-16-4 x) \left (i \pi -\log \left (\frac {5}{2}\right )\right )+(4+x) \log (x)}{-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )+\log (x)}} \left (-1-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )-16 \left (i \pi -\log \left (\frac {5}{2}\right )\right )^2+\left (1+8 \left (i \pi -\log \left (\frac {5}{2}\right )\right )\right ) \log (x)-\log ^2(x)\right )}{16 \left (i \pi -\log \left (\frac {5}{2}\right )\right )^2-8 \left (i \pi -\log \left (\frac {5}{2}\right )\right ) \log (x)+\log ^2(x)} \, dx=-e^{\left (-\frac {4 i \, \pi x}{-4 i \, \pi - 4 \, \log \left (\frac {2}{5}\right ) + \log \left (x\right )} - \frac {4 \, x \log \left (\frac {2}{5}\right )}{-4 i \, \pi - 4 \, \log \left (\frac {2}{5}\right ) + \log \left (x\right )} + \frac {x \log \left (x\right )}{-4 i \, \pi - 4 \, \log \left (\frac {2}{5}\right ) + \log \left (x\right )} - \frac {16 i \, \pi }{-4 i \, \pi - 4 \, \log \left (\frac {2}{5}\right ) + \log \left (x\right )} - \frac {x}{-4 i \, \pi - 4 \, \log \left (\frac {2}{5}\right ) + \log \left (x\right )} - \frac {16 \, \log \left (\frac {2}{5}\right )}{-4 i \, \pi - 4 \, \log \left (\frac {2}{5}\right ) + \log \left (x\right )} + \frac {4 \, \log \left (x\right )}{-4 i \, \pi - 4 \, \log \left (\frac {2}{5}\right ) + \log \left (x\right )}\right )} \]
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Timed out. \[ \int \frac {e^{\frac {-x+(-16-4 x) \left (i \pi -\log \left (\frac {5}{2}\right )\right )+(4+x) \log (x)}{-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )+\log (x)}} \left (-1-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )-16 \left (i \pi -\log \left (\frac {5}{2}\right )\right )^2+\left (1+8 \left (i \pi -\log \left (\frac {5}{2}\right )\right )\right ) \log (x)-\log ^2(x)\right )}{16 \left (i \pi -\log \left (\frac {5}{2}\right )\right )^2-8 \left (i \pi -\log \left (\frac {5}{2}\right )\right ) \log (x)+\log ^2(x)} \, dx=\text {Timed out} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (50) = 100\).
Time = 2.97 (sec) , antiderivative size = 184, normalized size of antiderivative = 5.75 \[ \int \frac {e^{\frac {-x+(-16-4 x) \left (i \pi -\log \left (\frac {5}{2}\right )\right )+(4+x) \log (x)}{-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )+\log (x)}} \left (-1-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )-16 \left (i \pi -\log \left (\frac {5}{2}\right )\right )^2+\left (1+8 \left (i \pi -\log \left (\frac {5}{2}\right )\right )\right ) \log (x)-\log ^2(x)\right )}{16 \left (i \pi -\log \left (\frac {5}{2}\right )\right )^2-8 \left (i \pi -\log \left (\frac {5}{2}\right )\right ) \log (x)+\log ^2(x)} \, dx=-e^{\left (-\frac {4 i \, \pi x}{-4 i \, \pi + 4 \, \log \left (5\right ) - 4 \, \log \left (2\right ) + \log \left (x\right )} + \frac {4 \, x \log \left (5\right )}{-4 i \, \pi + 4 \, \log \left (5\right ) - 4 \, \log \left (2\right ) + \log \left (x\right )} - \frac {4 \, x \log \left (2\right )}{-4 i \, \pi + 4 \, \log \left (5\right ) - 4 \, \log \left (2\right ) + \log \left (x\right )} + \frac {x \log \left (x\right )}{-4 i \, \pi + 4 \, \log \left (5\right ) - 4 \, \log \left (2\right ) + \log \left (x\right )} - \frac {16 i \, \pi }{-4 i \, \pi + 4 \, \log \left (5\right ) - 4 \, \log \left (2\right ) + \log \left (x\right )} - \frac {x}{-4 i \, \pi + 4 \, \log \left (5\right ) - 4 \, \log \left (2\right ) + \log \left (x\right )} + \frac {16 \, \log \left (5\right )}{-4 i \, \pi + 4 \, \log \left (5\right ) - 4 \, \log \left (2\right ) + \log \left (x\right )} - \frac {16 \, \log \left (2\right )}{-4 i \, \pi + 4 \, \log \left (5\right ) - 4 \, \log \left (2\right ) + \log \left (x\right )} + \frac {4 \, \log \left (x\right )}{-4 i \, \pi + 4 \, \log \left (5\right ) - 4 \, \log \left (2\right ) + \log \left (x\right )}\right )} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (50) = 100\).
Time = 1.00 (sec) , antiderivative size = 117, normalized size of antiderivative = 3.66 \[ \int \frac {e^{\frac {-x+(-16-4 x) \left (i \pi -\log \left (\frac {5}{2}\right )\right )+(4+x) \log (x)}{-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )+\log (x)}} \left (-1-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )-16 \left (i \pi -\log \left (\frac {5}{2}\right )\right )^2+\left (1+8 \left (i \pi -\log \left (\frac {5}{2}\right )\right )\right ) \log (x)-\log ^2(x)\right )}{16 \left (i \pi -\log \left (\frac {5}{2}\right )\right )^2-8 \left (i \pi -\log \left (\frac {5}{2}\right )\right ) \log (x)+\log ^2(x)} \, dx=-e^{\left (\frac {4 \, \pi x}{4 \, \pi + 4 i \, \log \left (5\right ) - 4 i \, \log \left (2\right ) + i \, \log \left (x\right )} + \frac {4 i \, x \log \left (5\right )}{4 \, \pi + 4 i \, \log \left (5\right ) - 4 i \, \log \left (2\right ) + i \, \log \left (x\right )} - \frac {4 i \, x \log \left (2\right )}{4 \, \pi + 4 i \, \log \left (5\right ) - 4 i \, \log \left (2\right ) + i \, \log \left (x\right )} + \frac {i \, x \log \left (x\right )}{4 \, \pi + 4 i \, \log \left (5\right ) - 4 i \, \log \left (2\right ) + i \, \log \left (x\right )} - \frac {i \, x}{4 \, \pi + 4 i \, \log \left (5\right ) - 4 i \, \log \left (2\right ) + i \, \log \left (x\right )} + 4\right )} \]
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Time = 16.60 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.56 \[ \int \frac {e^{\frac {-x+(-16-4 x) \left (i \pi -\log \left (\frac {5}{2}\right )\right )+(4+x) \log (x)}{-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )+\log (x)}} \left (-1-4 \left (i \pi -\log \left (\frac {5}{2}\right )\right )-16 \left (i \pi -\log \left (\frac {5}{2}\right )\right )^2+\left (1+8 \left (i \pi -\log \left (\frac {5}{2}\right )\right )\right ) \log (x)-\log ^2(x)\right )}{16 \left (i \pi -\log \left (\frac {5}{2}\right )\right )^2-8 \left (i \pi -\log \left (\frac {5}{2}\right )\right ) \log (x)+\log ^2(x)} \, dx=-{\mathrm {e}}^{\frac {\Pi \,16{}\mathrm {i}}{-\ln \left (\frac {625\,x}{16}\right )+\Pi \,4{}\mathrm {i}}+\frac {x}{-\ln \left (\frac {625\,x}{16}\right )+\Pi \,4{}\mathrm {i}}+\frac {\Pi \,x\,4{}\mathrm {i}}{-\ln \left (\frac {625\,x}{16}\right )+\Pi \,4{}\mathrm {i}}}\,{\left (\frac {625\,x}{16}\right )}^{\frac {x\,1{}\mathrm {i}+4{}\mathrm {i}}{4\,\Pi -\ln \left (\frac {2}{5}\right )\,4{}\mathrm {i}+\ln \left (x\right )\,1{}\mathrm {i}}} \]
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