Integrand size = 94, antiderivative size = 28 \[ \int \frac {8 e^{2 x} x+12 x^2+e^x \left (-8 x-8 x^2\right )+\left (-10+6 x^2+e^{2 x} (-2+8 x)+e^x \left (-4 x-8 x^2\right )\right ) \log (x)+\left (-5+x^2-2 e^x x^2+e^{2 x} (-1+2 x)\right ) \log ^2(x)}{x^2} \, dx=4 x+\frac {x+\left (5+\left (e^x-x\right )^2\right ) (2+\log (x))^2}{x} \]
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Leaf count is larger than twice the leaf count of optimal. \(77\) vs. \(2(28)=56\).
Time = 0.17 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.75, number of steps used = 14, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.085, Rules used = {14, 2326, 2372, 2395, 2333, 2332, 2342, 2341} \[ \int \frac {8 e^{2 x} x+12 x^2+e^x \left (-8 x-8 x^2\right )+\left (-10+6 x^2+e^{2 x} (-2+8 x)+e^x \left (-4 x-8 x^2\right )\right ) \log (x)+\left (-5+x^2-2 e^x x^2+e^{2 x} (-1+2 x)\right ) \log ^2(x)}{x^2} \, dx=\frac {e^{2 x} (\log (x)+2) (2 x+x \log (x))}{x^2}+8 x+\frac {20}{x}+x \log ^2(x)+\frac {5 \log ^2(x)}{x}+4 x \log (x)+\frac {20 \log (x)}{x}-\frac {2 e^x (\log (x)+2) (2 x+x \log (x))}{x} \]
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Rule 14
Rule 2326
Rule 2332
Rule 2333
Rule 2341
Rule 2342
Rule 2372
Rule 2395
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2 e^x (2+\log (x)) (2+2 x+x \log (x))}{x}+\frac {e^{2 x} (2+\log (x)) (4 x-\log (x)+2 x \log (x))}{x^2}+\frac {12 x^2-10 \log (x)+6 x^2 \log (x)-5 \log ^2(x)+x^2 \log ^2(x)}{x^2}\right ) \, dx \\ & = -\left (2 \int \frac {e^x (2+\log (x)) (2+2 x+x \log (x))}{x} \, dx\right )+\int \frac {e^{2 x} (2+\log (x)) (4 x-\log (x)+2 x \log (x))}{x^2} \, dx+\int \frac {12 x^2-10 \log (x)+6 x^2 \log (x)-5 \log ^2(x)+x^2 \log ^2(x)}{x^2} \, dx \\ & = \frac {e^{2 x} (2+\log (x)) (2 x+x \log (x))}{x^2}-\frac {2 e^x (2+\log (x)) (2 x+x \log (x))}{x}+\int \left (12+\frac {2 \left (-5+3 x^2\right ) \log (x)}{x^2}+\frac {\left (-5+x^2\right ) \log ^2(x)}{x^2}\right ) \, dx \\ & = 12 x+\frac {e^{2 x} (2+\log (x)) (2 x+x \log (x))}{x^2}-\frac {2 e^x (2+\log (x)) (2 x+x \log (x))}{x}+2 \int \frac {\left (-5+3 x^2\right ) \log (x)}{x^2} \, dx+\int \frac {\left (-5+x^2\right ) \log ^2(x)}{x^2} \, dx \\ & = 12 x+\frac {10 \log (x)}{x}+6 x \log (x)+\frac {e^{2 x} (2+\log (x)) (2 x+x \log (x))}{x^2}-\frac {2 e^x (2+\log (x)) (2 x+x \log (x))}{x}-2 \int \left (3+\frac {5}{x^2}\right ) \, dx+\int \left (\log ^2(x)-\frac {5 \log ^2(x)}{x^2}\right ) \, dx \\ & = \frac {10}{x}+6 x+\frac {10 \log (x)}{x}+6 x \log (x)+\frac {e^{2 x} (2+\log (x)) (2 x+x \log (x))}{x^2}-\frac {2 e^x (2+\log (x)) (2 x+x \log (x))}{x}-5 \int \frac {\log ^2(x)}{x^2} \, dx+\int \log ^2(x) \, dx \\ & = \frac {10}{x}+6 x+\frac {10 \log (x)}{x}+6 x \log (x)+\frac {5 \log ^2(x)}{x}+x \log ^2(x)+\frac {e^{2 x} (2+\log (x)) (2 x+x \log (x))}{x^2}-\frac {2 e^x (2+\log (x)) (2 x+x \log (x))}{x}-2 \int \log (x) \, dx-10 \int \frac {\log (x)}{x^2} \, dx \\ & = \frac {20}{x}+8 x+\frac {20 \log (x)}{x}+4 x \log (x)+\frac {5 \log ^2(x)}{x}+x \log ^2(x)+\frac {e^{2 x} (2+\log (x)) (2 x+x \log (x))}{x^2}-\frac {2 e^x (2+\log (x)) (2 x+x \log (x))}{x} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(66\) vs. \(2(28)=56\).
Time = 0.09 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.36 \[ \int \frac {8 e^{2 x} x+12 x^2+e^x \left (-8 x-8 x^2\right )+\left (-10+6 x^2+e^{2 x} (-2+8 x)+e^x \left (-4 x-8 x^2\right )\right ) \log (x)+\left (-5+x^2-2 e^x x^2+e^{2 x} (-1+2 x)\right ) \log ^2(x)}{x^2} \, dx=\frac {4 \left (5+e^{2 x}-2 e^x x+2 x^2\right )+4 \left (5+e^{2 x}-2 e^x x+x^2\right ) \log (x)+\left (5+e^{2 x}-2 e^x x+x^2\right ) \log ^2(x)}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(65\) vs. \(2(27)=54\).
Time = 0.18 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.36
method | result | size |
risch | \(\frac {\left (x^{2}-2 \,{\mathrm e}^{x} x +{\mathrm e}^{2 x}+5\right ) \ln \left (x \right )^{2}}{x}+\frac {4 \left (x^{2}-2 \,{\mathrm e}^{x} x +{\mathrm e}^{2 x}+5\right ) \ln \left (x \right )}{x}+\frac {8 x^{2}-8 \,{\mathrm e}^{x} x +4 \,{\mathrm e}^{2 x}+20}{x}\) | \(66\) |
parallelrisch | \(\frac {x^{2} \ln \left (x \right )^{2}-2 x \,{\mathrm e}^{x} \ln \left (x \right )^{2}+\ln \left (x \right )^{2} {\mathrm e}^{2 x}+4 x^{2} \ln \left (x \right )-8 x \,{\mathrm e}^{x} \ln \left (x \right )+4 \ln \left (x \right ) {\mathrm e}^{2 x}+8 x^{2}-8 \,{\mathrm e}^{x} x +20+5 \ln \left (x \right )^{2}+4 \,{\mathrm e}^{2 x}+20 \ln \left (x \right )}{x}\) | \(81\) |
default | \(8 x -8 \,{\mathrm e}^{x} \ln \left (x \right )-2 \,{\mathrm e}^{x} \ln \left (x \right )^{2}-8 \,{\mathrm e}^{x}+\frac {\ln \left (x \right )^{2} {\mathrm e}^{2 x}+4 \ln \left (x \right ) {\mathrm e}^{2 x}+4 \,{\mathrm e}^{2 x}}{x}+x \ln \left (x \right )^{2}+4 x \ln \left (x \right )+\frac {5 \ln \left (x \right )^{2}}{x}+\frac {20 \ln \left (x \right )}{x}+\frac {20}{x}\) | \(83\) |
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Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (27) = 54\).
Time = 0.27 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.11 \[ \int \frac {8 e^{2 x} x+12 x^2+e^x \left (-8 x-8 x^2\right )+\left (-10+6 x^2+e^{2 x} (-2+8 x)+e^x \left (-4 x-8 x^2\right )\right ) \log (x)+\left (-5+x^2-2 e^x x^2+e^{2 x} (-1+2 x)\right ) \log ^2(x)}{x^2} \, dx=\frac {{\left (x^{2} - 2 \, x e^{x} + e^{\left (2 \, x\right )} + 5\right )} \log \left (x\right )^{2} + 8 \, x^{2} - 8 \, x e^{x} + 4 \, {\left (x^{2} - 2 \, x e^{x} + e^{\left (2 \, x\right )} + 5\right )} \log \left (x\right ) + 4 \, e^{\left (2 \, x\right )} + 20}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (20) = 40\).
Time = 0.16 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.54 \[ \int \frac {8 e^{2 x} x+12 x^2+e^x \left (-8 x-8 x^2\right )+\left (-10+6 x^2+e^{2 x} (-2+8 x)+e^x \left (-4 x-8 x^2\right )\right ) \log (x)+\left (-5+x^2-2 e^x x^2+e^{2 x} (-1+2 x)\right ) \log ^2(x)}{x^2} \, dx=8 x + \frac {\left (x^{2} + 5\right ) \log {\left (x \right )}^{2}}{x} + \frac {\left (4 x^{2} + 20\right ) \log {\left (x \right )}}{x} + \frac {\left (- 2 x \log {\left (x \right )}^{2} - 8 x \log {\left (x \right )} - 8 x\right ) e^{x} + \left (\log {\left (x \right )}^{2} + 4 \log {\left (x \right )} + 4\right ) e^{2 x}}{x} + \frac {20}{x} \]
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\[ \int \frac {8 e^{2 x} x+12 x^2+e^x \left (-8 x-8 x^2\right )+\left (-10+6 x^2+e^{2 x} (-2+8 x)+e^x \left (-4 x-8 x^2\right )\right ) \log (x)+\left (-5+x^2-2 e^x x^2+e^{2 x} (-1+2 x)\right ) \log ^2(x)}{x^2} \, dx=\int { -\frac {{\left (2 \, x^{2} e^{x} - x^{2} - {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} + 5\right )} \log \left (x\right )^{2} - 12 \, x^{2} - 8 \, x e^{\left (2 \, x\right )} + 8 \, {\left (x^{2} + x\right )} e^{x} - 2 \, {\left (3 \, x^{2} + {\left (4 \, x - 1\right )} e^{\left (2 \, x\right )} - 2 \, {\left (2 \, x^{2} + x\right )} e^{x} - 5\right )} \log \left (x\right )}{x^{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (27) = 54\).
Time = 0.30 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.86 \[ \int \frac {8 e^{2 x} x+12 x^2+e^x \left (-8 x-8 x^2\right )+\left (-10+6 x^2+e^{2 x} (-2+8 x)+e^x \left (-4 x-8 x^2\right )\right ) \log (x)+\left (-5+x^2-2 e^x x^2+e^{2 x} (-1+2 x)\right ) \log ^2(x)}{x^2} \, dx=\frac {x^{2} \log \left (x\right )^{2} - 2 \, x e^{x} \log \left (x\right )^{2} + 4 \, x^{2} \log \left (x\right ) - 8 \, x e^{x} \log \left (x\right ) + e^{\left (2 \, x\right )} \log \left (x\right )^{2} + 8 \, x^{2} - 8 \, x e^{x} + 4 \, e^{\left (2 \, x\right )} \log \left (x\right ) + 5 \, \log \left (x\right )^{2} + 4 \, e^{\left (2 \, x\right )} + 20 \, \log \left (x\right ) + 20}{x} \]
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Time = 13.40 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.50 \[ \int \frac {8 e^{2 x} x+12 x^2+e^x \left (-8 x-8 x^2\right )+\left (-10+6 x^2+e^{2 x} (-2+8 x)+e^x \left (-4 x-8 x^2\right )\right ) \log (x)+\left (-5+x^2-2 e^x x^2+e^{2 x} (-1+2 x)\right ) \log ^2(x)}{x^2} \, dx=\frac {4\,{\mathrm {e}}^{2\,x}+20\,\ln \left (x\right )+5\,{\ln \left (x\right )}^2+4\,{\mathrm {e}}^{2\,x}\,\ln \left (x\right )+{\mathrm {e}}^{2\,x}\,{\ln \left (x\right )}^2+20}{x}-8\,{\mathrm {e}}^x\,\ln \left (x\right )-8\,{\mathrm {e}}^x-2\,{\mathrm {e}}^x\,{\ln \left (x\right )}^2+x\,\left ({\ln \left (x\right )}^2+4\,\ln \left (x\right )+8\right ) \]
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