Integrand size = 137, antiderivative size = 31 \[ \int \frac {2 e^{2 x} \log ^2(x)+4 e^x \log ^4(x)+2 \log ^6(x)+e^{\frac {3 x+15 \log (x)}{\log (x)}} \left (-3 e^x x^2+3 e^x x^2 \log (x)+\left (-3 x^2+e^x \left (2 x-x^2\right )\right ) \log ^2(x)+\left (-2 x+3 x^2\right ) \log ^3(x)+2 x \log ^4(x)\right )}{2 e^{2 x} \log ^2(x)+4 e^x \log ^4(x)+2 \log ^6(x)} \, dx=x+\frac {e^{3 \left (5+\frac {x}{\log (x)}\right )} x^2}{2 \left (e^x+\log ^2(x)\right )} \]
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Leaf count is larger than twice the leaf count of optimal. \(72\) vs. \(2(31)=62\).
Time = 2.54 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.32, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {6873, 12, 6874, 2326} \[ \int \frac {2 e^{2 x} \log ^2(x)+4 e^x \log ^4(x)+2 \log ^6(x)+e^{\frac {3 x+15 \log (x)}{\log (x)}} \left (-3 e^x x^2+3 e^x x^2 \log (x)+\left (-3 x^2+e^x \left (2 x-x^2\right )\right ) \log ^2(x)+\left (-2 x+3 x^2\right ) \log ^3(x)+2 x \log ^4(x)\right )}{2 e^{2 x} \log ^2(x)+4 e^x \log ^4(x)+2 \log ^6(x)} \, dx=x+\frac {x e^{\frac {3 x}{\log (x)}+15} \left (e^x x-x \log ^3(x)+x \log ^2(x)-e^x x \log (x)\right )}{2 \left (\frac {1}{\log ^2(x)}-\frac {1}{\log (x)}\right ) \log ^2(x) \left (e^x+\log ^2(x)\right )^2} \]
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Rule 12
Rule 2326
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {2 e^{2 x} \log ^2(x)+4 e^x \log ^4(x)+2 \log ^6(x)+e^{\frac {3 x+15 \log (x)}{\log (x)}} \left (-3 e^x x^2+3 e^x x^2 \log (x)+\left (-3 x^2+e^x \left (2 x-x^2\right )\right ) \log ^2(x)+\left (-2 x+3 x^2\right ) \log ^3(x)+2 x \log ^4(x)\right )}{2 \log ^2(x) \left (e^x+\log ^2(x)\right )^2} \, dx \\ & = \frac {1}{2} \int \frac {2 e^{2 x} \log ^2(x)+4 e^x \log ^4(x)+2 \log ^6(x)+e^{\frac {3 x+15 \log (x)}{\log (x)}} \left (-3 e^x x^2+3 e^x x^2 \log (x)+\left (-3 x^2+e^x \left (2 x-x^2\right )\right ) \log ^2(x)+\left (-2 x+3 x^2\right ) \log ^3(x)+2 x \log ^4(x)\right )}{\log ^2(x) \left (e^x+\log ^2(x)\right )^2} \, dx \\ & = \frac {1}{2} \int \left (2+\frac {e^{15+\frac {3 x}{\log (x)}} x \left (-3 e^x x+3 e^x x \log (x)+2 e^x \log ^2(x)-3 x \log ^2(x)-e^x x \log ^2(x)-2 \log ^3(x)+3 x \log ^3(x)+2 \log ^4(x)\right )}{\log ^2(x) \left (e^x+\log ^2(x)\right )^2}\right ) \, dx \\ & = x+\frac {1}{2} \int \frac {e^{15+\frac {3 x}{\log (x)}} x \left (-3 e^x x+3 e^x x \log (x)+2 e^x \log ^2(x)-3 x \log ^2(x)-e^x x \log ^2(x)-2 \log ^3(x)+3 x \log ^3(x)+2 \log ^4(x)\right )}{\log ^2(x) \left (e^x+\log ^2(x)\right )^2} \, dx \\ & = x+\frac {e^{15+\frac {3 x}{\log (x)}} x \left (e^x x-e^x x \log (x)+x \log ^2(x)-x \log ^3(x)\right )}{2 \left (\frac {1}{\log ^2(x)}-\frac {1}{\log (x)}\right ) \log ^2(x) \left (e^x+\log ^2(x)\right )^2} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {2 e^{2 x} \log ^2(x)+4 e^x \log ^4(x)+2 \log ^6(x)+e^{\frac {3 x+15 \log (x)}{\log (x)}} \left (-3 e^x x^2+3 e^x x^2 \log (x)+\left (-3 x^2+e^x \left (2 x-x^2\right )\right ) \log ^2(x)+\left (-2 x+3 x^2\right ) \log ^3(x)+2 x \log ^4(x)\right )}{2 e^{2 x} \log ^2(x)+4 e^x \log ^4(x)+2 \log ^6(x)} \, dx=\frac {1}{2} \left (2 x+\frac {e^{15+\frac {3 x}{\log (x)}} x^2}{e^x+\log ^2(x)}\right ) \]
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Time = 3.74 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97
method | result | size |
risch | \(x +\frac {x^{2} {\mathrm e}^{\frac {15 \ln \left (x \right )+3 x}{\ln \left (x \right )}}}{2 \ln \left (x \right )^{2}+2 \,{\mathrm e}^{x}}\) | \(30\) |
parallelrisch | \(-\frac {-x^{2} {\mathrm e}^{\frac {15 \ln \left (x \right )+3 x}{\ln \left (x \right )}}-2 x \ln \left (x \right )^{2}-2 \,{\mathrm e}^{x} x}{2 \left (\ln \left (x \right )^{2}+{\mathrm e}^{x}\right )}\) | \(43\) |
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Time = 0.25 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.32 \[ \int \frac {2 e^{2 x} \log ^2(x)+4 e^x \log ^4(x)+2 \log ^6(x)+e^{\frac {3 x+15 \log (x)}{\log (x)}} \left (-3 e^x x^2+3 e^x x^2 \log (x)+\left (-3 x^2+e^x \left (2 x-x^2\right )\right ) \log ^2(x)+\left (-2 x+3 x^2\right ) \log ^3(x)+2 x \log ^4(x)\right )}{2 e^{2 x} \log ^2(x)+4 e^x \log ^4(x)+2 \log ^6(x)} \, dx=\frac {x^{2} e^{\left (\frac {3 \, {\left (x + 5 \, \log \left (x\right )\right )}}{\log \left (x\right )}\right )} + 2 \, x \log \left (x\right )^{2} + 2 \, x e^{x}}{2 \, {\left (\log \left (x\right )^{2} + e^{x}\right )}} \]
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Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int \frac {2 e^{2 x} \log ^2(x)+4 e^x \log ^4(x)+2 \log ^6(x)+e^{\frac {3 x+15 \log (x)}{\log (x)}} \left (-3 e^x x^2+3 e^x x^2 \log (x)+\left (-3 x^2+e^x \left (2 x-x^2\right )\right ) \log ^2(x)+\left (-2 x+3 x^2\right ) \log ^3(x)+2 x \log ^4(x)\right )}{2 e^{2 x} \log ^2(x)+4 e^x \log ^4(x)+2 \log ^6(x)} \, dx=\frac {x^{2} e^{\frac {3 x + 15 \log {\left (x \right )}}{\log {\left (x \right )}}}}{2 e^{x} + 2 \log {\left (x \right )}^{2}} + x \]
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Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.23 \[ \int \frac {2 e^{2 x} \log ^2(x)+4 e^x \log ^4(x)+2 \log ^6(x)+e^{\frac {3 x+15 \log (x)}{\log (x)}} \left (-3 e^x x^2+3 e^x x^2 \log (x)+\left (-3 x^2+e^x \left (2 x-x^2\right )\right ) \log ^2(x)+\left (-2 x+3 x^2\right ) \log ^3(x)+2 x \log ^4(x)\right )}{2 e^{2 x} \log ^2(x)+4 e^x \log ^4(x)+2 \log ^6(x)} \, dx=\frac {x^{2} e^{\left (\frac {3 \, x}{\log \left (x\right )} + 15\right )} + 2 \, x \log \left (x\right )^{2} + 2 \, x e^{x}}{2 \, {\left (\log \left (x\right )^{2} + e^{x}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (26) = 52\).
Time = 0.35 (sec) , antiderivative size = 119, normalized size of antiderivative = 3.84 \[ \int \frac {2 e^{2 x} \log ^2(x)+4 e^x \log ^4(x)+2 \log ^6(x)+e^{\frac {3 x+15 \log (x)}{\log (x)}} \left (-3 e^x x^2+3 e^x x^2 \log (x)+\left (-3 x^2+e^x \left (2 x-x^2\right )\right ) \log ^2(x)+\left (-2 x+3 x^2\right ) \log ^3(x)+2 x \log ^4(x)\right )}{2 e^{2 x} \log ^2(x)+4 e^x \log ^4(x)+2 \log ^6(x)} \, dx=\frac {2 \, x^{3} e^{x} \log \left (x\right )^{2} + x^{4} e^{\left (x + \frac {3 \, {\left (x + 5 \, \log \left (x\right )\right )}}{\log \left (x\right )}\right )} + 2 \, x^{3} e^{\left (2 \, x\right )} + 2 \, x^{2} e^{\left (2 \, x\right )} + 4 \, x^{2} e^{\left (\frac {3 \, {\left (x + 5 \, \log \left (x\right )\right )}}{\log \left (x\right )}\right )} - 4 \, x e^{x} \log \left (x\right ) + 8 \, x \log \left (x\right )^{2} + 8 \, x e^{x}}{2 \, {\left (x^{2} e^{x} \log \left (x\right )^{2} + x^{2} e^{\left (2 \, x\right )} + 4 \, \log \left (x\right )^{2} + 4 \, e^{x}\right )}} \]
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Time = 14.73 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int \frac {2 e^{2 x} \log ^2(x)+4 e^x \log ^4(x)+2 \log ^6(x)+e^{\frac {3 x+15 \log (x)}{\log (x)}} \left (-3 e^x x^2+3 e^x x^2 \log (x)+\left (-3 x^2+e^x \left (2 x-x^2\right )\right ) \log ^2(x)+\left (-2 x+3 x^2\right ) \log ^3(x)+2 x \log ^4(x)\right )}{2 e^{2 x} \log ^2(x)+4 e^x \log ^4(x)+2 \log ^6(x)} \, dx=x+\frac {x^2\,{\mathrm {e}}^{15}\,{\mathrm {e}}^{\frac {3\,x}{\ln \left (x\right )}}}{2\,\left ({\ln \left (x\right )}^2+{\mathrm {e}}^x\right )} \]
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