Integrand size = 100, antiderivative size = 29 \[ \int \frac {e^x \left (-8+2 e^3\right ) (i \pi +\log (10))+e^x \left (-4 x+e^3 x\right ) (i \pi +\log (10)) \log \left (\frac {5}{x^2}\right )+\left (-4 x+e^3 x\right ) (i \pi +\log (10)) \log ^2\left (\frac {5}{x^2}\right )}{e^x x \log \left (\frac {5}{x^2}\right )+x^2 \log ^2\left (\frac {5}{x^2}\right )} \, dx=\left (-4+e^3\right ) (i \pi +\log (10)) \log \left (x+\frac {e^x}{\log \left (\frac {5}{x^2}\right )}\right ) \]
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\[ \int \frac {e^x \left (-8+2 e^3\right ) (i \pi +\log (10))+e^x \left (-4 x+e^3 x\right ) (i \pi +\log (10)) \log \left (\frac {5}{x^2}\right )+\left (-4 x+e^3 x\right ) (i \pi +\log (10)) \log ^2\left (\frac {5}{x^2}\right )}{e^x x \log \left (\frac {5}{x^2}\right )+x^2 \log ^2\left (\frac {5}{x^2}\right )} \, dx=\int \frac {e^x \left (-8+2 e^3\right ) (i \pi +\log (10))+e^x \left (-4 x+e^3 x\right ) (i \pi +\log (10)) \log \left (\frac {5}{x^2}\right )+\left (-4 x+e^3 x\right ) (i \pi +\log (10)) \log ^2\left (\frac {5}{x^2}\right )}{e^x x \log \left (\frac {5}{x^2}\right )+x^2 \log ^2\left (\frac {5}{x^2}\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (4-e^3\right ) (i \pi +\log (10)) \left (-2 e^x-e^x x \log \left (\frac {5}{x^2}\right )-x \log ^2\left (\frac {5}{x^2}\right )\right )}{x \log \left (\frac {5}{x^2}\right ) \left (e^x+x \log \left (\frac {5}{x^2}\right )\right )} \, dx \\ & = \left (\left (4-e^3\right ) (i \pi +\log (10))\right ) \int \frac {-2 e^x-e^x x \log \left (\frac {5}{x^2}\right )-x \log ^2\left (\frac {5}{x^2}\right )}{x \log \left (\frac {5}{x^2}\right ) \left (e^x+x \log \left (\frac {5}{x^2}\right )\right )} \, dx \\ & = \left (\left (4-e^3\right ) (i \pi +\log (10))\right ) \int \left (\frac {-2-x \log \left (\frac {5}{x^2}\right )}{x \log \left (\frac {5}{x^2}\right )}+\frac {2-\log \left (\frac {5}{x^2}\right )+x \log \left (\frac {5}{x^2}\right )}{e^x+x \log \left (\frac {5}{x^2}\right )}\right ) \, dx \\ & = \left (\left (4-e^3\right ) (i \pi +\log (10))\right ) \int \frac {-2-x \log \left (\frac {5}{x^2}\right )}{x \log \left (\frac {5}{x^2}\right )} \, dx+\left (\left (4-e^3\right ) (i \pi +\log (10))\right ) \int \frac {2-\log \left (\frac {5}{x^2}\right )+x \log \left (\frac {5}{x^2}\right )}{e^x+x \log \left (\frac {5}{x^2}\right )} \, dx \\ & = \left (\left (4-e^3\right ) (i \pi +\log (10))\right ) \int \left (-1-\frac {2}{x \log \left (\frac {5}{x^2}\right )}\right ) \, dx+\left (\left (4-e^3\right ) (i \pi +\log (10))\right ) \int \frac {2+(-1+x) \log \left (\frac {5}{x^2}\right )}{e^x+x \log \left (\frac {5}{x^2}\right )} \, dx \\ & = -\left (\left (4-e^3\right ) x (i \pi +\log (10))\right )+\left (\left (4-e^3\right ) (i \pi +\log (10))\right ) \int \left (\frac {2}{e^x+x \log \left (\frac {5}{x^2}\right )}-\frac {\log \left (\frac {5}{x^2}\right )}{e^x+x \log \left (\frac {5}{x^2}\right )}+\frac {x \log \left (\frac {5}{x^2}\right )}{e^x+x \log \left (\frac {5}{x^2}\right )}\right ) \, dx-\left (2 \left (4-e^3\right ) (i \pi +\log (10))\right ) \int \frac {1}{x \log \left (\frac {5}{x^2}\right )} \, dx \\ & = -\left (\left (4-e^3\right ) x (i \pi +\log (10))\right )-\left (\left (4-e^3\right ) (i \pi +\log (10))\right ) \int \frac {\log \left (\frac {5}{x^2}\right )}{e^x+x \log \left (\frac {5}{x^2}\right )} \, dx+\left (\left (4-e^3\right ) (i \pi +\log (10))\right ) \int \frac {x \log \left (\frac {5}{x^2}\right )}{e^x+x \log \left (\frac {5}{x^2}\right )} \, dx+\left (\left (4-e^3\right ) (i \pi +\log (10))\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (\frac {5}{x^2}\right )\right )+\left (2 \left (4-e^3\right ) (i \pi +\log (10))\right ) \int \frac {1}{e^x+x \log \left (\frac {5}{x^2}\right )} \, dx \\ & = -\left (\left (4-e^3\right ) x (i \pi +\log (10))\right )+\left (4-e^3\right ) (i \pi +\log (10)) \log \left (\log \left (\frac {5}{x^2}\right )\right )-\left (\left (4-e^3\right ) (i \pi +\log (10))\right ) \int \frac {\log \left (\frac {5}{x^2}\right )}{e^x+x \log \left (\frac {5}{x^2}\right )} \, dx+\left (\left (4-e^3\right ) (i \pi +\log (10))\right ) \int \frac {x \log \left (\frac {5}{x^2}\right )}{e^x+x \log \left (\frac {5}{x^2}\right )} \, dx+\left (2 \left (4-e^3\right ) (i \pi +\log (10))\right ) \int \frac {1}{e^x+x \log \left (\frac {5}{x^2}\right )} \, dx \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.28 \[ \int \frac {e^x \left (-8+2 e^3\right ) (i \pi +\log (10))+e^x \left (-4 x+e^3 x\right ) (i \pi +\log (10)) \log \left (\frac {5}{x^2}\right )+\left (-4 x+e^3 x\right ) (i \pi +\log (10)) \log ^2\left (\frac {5}{x^2}\right )}{e^x x \log \left (\frac {5}{x^2}\right )+x^2 \log ^2\left (\frac {5}{x^2}\right )} \, dx=\left (-4+e^3\right ) (i \pi +\log (10)) \left (-\log \left (\log \left (\frac {5}{x^2}\right )\right )+\log \left (e^x+x \log \left (\frac {5}{x^2}\right )\right )\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (26 ) = 52\).
Time = 0.69 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.21
method | result | size |
norman | \(\left (-i \pi \,{\mathrm e}^{3}-\ln \left (10\right ) {\mathrm e}^{3}+4 i \pi +4 \ln \left (10\right )\right ) \ln \left (\ln \left (\frac {5}{x^{2}}\right )\right )+\left (i \pi \,{\mathrm e}^{3}+\ln \left (10\right ) {\mathrm e}^{3}-4 i \pi -4 \ln \left (10\right )\right ) \ln \left (x \ln \left (\frac {5}{x^{2}}\right )+{\mathrm e}^{x}\right )\) | \(64\) |
default | \(\left (-i \pi \,{\mathrm e}^{3}-\ln \left (10\right ) {\mathrm e}^{3}+4 i \pi +4 \ln \left (10\right )\right ) \ln \left (-\ln \left (\frac {5}{x^{2}}\right )\right )+\left (i \pi \,{\mathrm e}^{3}+\ln \left (10\right ) {\mathrm e}^{3}-4 i \pi -4 \ln \left (10\right )\right ) \ln \left (2 x \ln \left (x \right )-x \left (\ln \left (\frac {5}{x^{2}}\right )+2 \ln \left (x \right )\right )-{\mathrm e}^{x}\right )\) | \(79\) |
parallelrisch | \(-i \pi \,{\mathrm e}^{3} \ln \left (\ln \left (\frac {5}{x^{2}}\right )\right )-\ln \left (10\right ) {\mathrm e}^{3} \ln \left (\ln \left (\frac {5}{x^{2}}\right )\right )+4 i \pi \ln \left (\ln \left (\frac {5}{x^{2}}\right )\right )+4 \ln \left (10\right ) \ln \left (\ln \left (\frac {5}{x^{2}}\right )\right )+i \pi \,{\mathrm e}^{3} \ln \left (x \ln \left (\frac {5}{x^{2}}\right )+{\mathrm e}^{x}\right )+\ln \left (10\right ) {\mathrm e}^{3} \ln \left (x \ln \left (\frac {5}{x^{2}}\right )+{\mathrm e}^{x}\right )-4 i \pi \ln \left (x \ln \left (\frac {5}{x^{2}}\right )+{\mathrm e}^{x}\right )-4 \ln \left (10\right ) \ln \left (x \ln \left (\frac {5}{x^{2}}\right )+{\mathrm e}^{x}\right )\) | \(117\) |
risch | \(\text {Expression too large to display}\) | \(856\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (25) = 50\).
Time = 0.27 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.97 \[ \int \frac {e^x \left (-8+2 e^3\right ) (i \pi +\log (10))+e^x \left (-4 x+e^3 x\right ) (i \pi +\log (10)) \log \left (\frac {5}{x^2}\right )+\left (-4 x+e^3 x\right ) (i \pi +\log (10)) \log ^2\left (\frac {5}{x^2}\right )}{e^x x \log \left (\frac {5}{x^2}\right )+x^2 \log ^2\left (\frac {5}{x^2}\right )} \, dx=-{\left (4 i \, \pi - i \, \pi e^{3} - {\left (e^{3} - 4\right )} \log \left (10\right )\right )} \log \left (\frac {x \log \left (\frac {5}{x^{2}}\right ) + e^{x}}{x}\right ) + \frac {1}{2} \, {\left (4 i \, \pi - i \, \pi e^{3} - {\left (e^{3} - 4\right )} \log \left (10\right )\right )} \log \left (\frac {5}{x^{2}}\right ) - {\left (-4 i \, \pi + i \, \pi e^{3} + {\left (e^{3} - 4\right )} \log \left (10\right )\right )} \log \left (\log \left (\frac {5}{x^{2}}\right )\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (24) = 48\).
Time = 0.88 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.83 \[ \int \frac {e^x \left (-8+2 e^3\right ) (i \pi +\log (10))+e^x \left (-4 x+e^3 x\right ) (i \pi +\log (10)) \log \left (\frac {5}{x^2}\right )+\left (-4 x+e^3 x\right ) (i \pi +\log (10)) \log ^2\left (\frac {5}{x^2}\right )}{e^x x \log \left (\frac {5}{x^2}\right )+x^2 \log ^2\left (\frac {5}{x^2}\right )} \, dx=- \left (-4 + e^{3}\right ) \left (\log {\left (10 \right )} + i \pi \right ) \log {\left (\log {\left (\frac {1}{x^{2}} \right )} + \log {\left (5 \right )} \right )} + \left (-4 + e^{3}\right ) \left (\log {\left (10 \right )} + i \pi \right ) \log {\left (x \log {\left (\frac {1}{x^{2}} \right )} + x \log {\left (5 \right )} + e^{x} \right )} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (25) = 50\).
Time = 0.31 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.55 \[ \int \frac {e^x \left (-8+2 e^3\right ) (i \pi +\log (10))+e^x \left (-4 x+e^3 x\right ) (i \pi +\log (10)) \log \left (\frac {5}{x^2}\right )+\left (-4 x+e^3 x\right ) (i \pi +\log (10)) \log ^2\left (\frac {5}{x^2}\right )}{e^x x \log \left (\frac {5}{x^2}\right )+x^2 \log ^2\left (\frac {5}{x^2}\right )} \, dx={\left (-4 i \, \pi + {\left (i \, \pi + \log \left (5\right ) + \log \left (2\right )\right )} e^{3} - 4 \, \log \left (5\right ) - 4 \, \log \left (2\right )\right )} \log \left (x \log \left (5\right ) - 2 \, x \log \left (x\right ) + e^{x}\right ) + {\left (4 i \, \pi + {\left (-i \, \pi - \log \left (5\right ) - \log \left (2\right )\right )} e^{3} + 4 \, \log \left (5\right ) + 4 \, \log \left (2\right )\right )} \log \left (-\frac {1}{2} \, \log \left (5\right ) + \log \left (x\right )\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 205 vs. \(2 (25) = 50\).
Time = 0.38 (sec) , antiderivative size = 205, normalized size of antiderivative = 7.07 \[ \int \frac {e^x \left (-8+2 e^3\right ) (i \pi +\log (10))+e^x \left (-4 x+e^3 x\right ) (i \pi +\log (10)) \log \left (\frac {5}{x^2}\right )+\left (-4 x+e^3 x\right ) (i \pi +\log (10)) \log ^2\left (\frac {5}{x^2}\right )}{e^x x \log \left (\frac {5}{x^2}\right )+x^2 \log ^2\left (\frac {5}{x^2}\right )} \, dx=i \, \pi e^{3} \log \left (x \log \left (5\right ) - x \log \left (x^{2}\right ) + e^{x}\right ) + e^{3} \log \left (5\right ) \log \left (x \log \left (5\right ) - x \log \left (x^{2}\right ) + e^{x}\right ) + e^{3} \log \left (2\right ) \log \left (x \log \left (5\right ) - x \log \left (x^{2}\right ) + e^{x}\right ) - i \, \pi e^{3} \log \left (\log \left (5\right ) - \log \left (x^{2}\right )\right ) - e^{3} \log \left (5\right ) \log \left (\log \left (5\right ) - \log \left (x^{2}\right )\right ) - e^{3} \log \left (2\right ) \log \left (\log \left (5\right ) - \log \left (x^{2}\right )\right ) - 4 i \, \pi \log \left (x \log \left (5\right ) - x \log \left (x^{2}\right ) + e^{x}\right ) - 4 \, \log \left (5\right ) \log \left (x \log \left (5\right ) - x \log \left (x^{2}\right ) + e^{x}\right ) - 4 \, \log \left (2\right ) \log \left (x \log \left (5\right ) - x \log \left (x^{2}\right ) + e^{x}\right ) + 4 i \, \pi \log \left (\log \left (5\right ) - \log \left (x^{2}\right )\right ) + 4 \, \log \left (5\right ) \log \left (\log \left (5\right ) - \log \left (x^{2}\right )\right ) + 4 \, \log \left (2\right ) \log \left (\log \left (5\right ) - \log \left (x^{2}\right )\right ) \]
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Time = 12.51 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.21 \[ \int \frac {e^x \left (-8+2 e^3\right ) (i \pi +\log (10))+e^x \left (-4 x+e^3 x\right ) (i \pi +\log (10)) \log \left (\frac {5}{x^2}\right )+\left (-4 x+e^3 x\right ) (i \pi +\log (10)) \log ^2\left (\frac {5}{x^2}\right )}{e^x x \log \left (\frac {5}{x^2}\right )+x^2 \log ^2\left (\frac {5}{x^2}\right )} \, dx=-\left (\ln \left (10\right )+\Pi \,1{}\mathrm {i}\right )\,\left ({\mathrm {e}}^3-4\right )\,\left (\ln \left (\ln \left (\frac {5}{x^2}\right )\right )-\ln \left ({\mathrm {e}}^x+x\,\ln \left (\frac {5}{x^2}\right )\right )\right ) \]
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