Integrand size = 65, antiderivative size = 20 \[ \int \left (-15-5 e^x+5 x^2+\left (-3-e^x+x^2\right ) \log (x)+\left (-18+e^x (-6-5 x)+16 x^2+\left (-3+e^x (-1-x)+3 x^2\right ) \log (x)\right ) \log (2 x)\right ) \, dx=x \left (-3-e^x+x^2\right ) (5+\log (x)) \log (2 x) \]
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Leaf count is larger than twice the leaf count of optimal. \(87\) vs. \(2(20)=40\).
Time = 0.62 (sec) , antiderivative size = 87, normalized size of antiderivative = 4.35, number of steps used = 46, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.246, Rules used = {2225, 2634, 2209, 2207, 6874, 2326, 2230, 2637, 6618, 6610, 14, 2332, 2341, 2408, 2413, 12} \[ \int \left (-15-5 e^x+5 x^2+\left (-3-e^x+x^2\right ) \log (x)+\left (-18+e^x (-6-5 x)+16 x^2+\left (-3+e^x (-1-x)+3 x^2\right ) \log (x)\right ) \log (2 x)\right ) \, dx=-\frac {x^3}{9}+\frac {1}{9} x^3 (1-3 \log (x))+\frac {1}{3} x^3 \log (x)+x^3 \log (x) \log (2 x)+5 x^3 \log (2 x)-5 e^x x \log (2 x)-e^x x \log (x) \log (2 x)-3 x \log (x) \log (2 x)-15 x \log (2 x) \]
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Rule 12
Rule 14
Rule 2207
Rule 2209
Rule 2225
Rule 2230
Rule 2326
Rule 2332
Rule 2341
Rule 2408
Rule 2413
Rule 2634
Rule 2637
Rule 6610
Rule 6618
Rule 6874
Rubi steps \begin{align*} \text {integral}& = -15 x+\frac {5 x^3}{3}-5 \int e^x \, dx+\int \left (-3-e^x+x^2\right ) \log (x) \, dx+\int \left (-18+e^x (-6-5 x)+16 x^2+\left (-3+e^x (-1-x)+3 x^2\right ) \log (x)\right ) \log (2 x) \, dx \\ & = -5 e^x-15 x+\frac {5 x^3}{3}-e^x \log (x)-3 x \log (x)+\frac {1}{3} x^3 \log (x)-\int \left (-3-\frac {e^x}{x}+\frac {x^2}{3}\right ) \, dx+\int \left (-e^x (6+5 x+\log (x)+x \log (x)) \log (2 x)+\left (-18+16 x^2-3 \log (x)+3 x^2 \log (x)\right ) \log (2 x)\right ) \, dx \\ & = -5 e^x-12 x+\frac {14 x^3}{9}-e^x \log (x)-3 x \log (x)+\frac {1}{3} x^3 \log (x)+\int \frac {e^x}{x} \, dx-\int e^x (6+5 x+\log (x)+x \log (x)) \log (2 x) \, dx+\int \left (-18+16 x^2-3 \log (x)+3 x^2 \log (x)\right ) \log (2 x) \, dx \\ & = -5 e^x-12 x+\frac {14 x^3}{9}+\text {Ei}(x)-e^x \log (x)-3 x \log (x)+\frac {1}{3} x^3 \log (x)-\int \left (6 e^x \log (2 x)+5 e^x x \log (2 x)+e^x \log (x) \log (2 x)+e^x x \log (x) \log (2 x)\right ) \, dx+\int \left (-18 \log (2 x)+16 x^2 \log (2 x)-3 \log (x) \log (2 x)+3 x^2 \log (x) \log (2 x)\right ) \, dx \\ & = -5 e^x-12 x+\frac {14 x^3}{9}+\text {Ei}(x)-e^x \log (x)-3 x \log (x)+\frac {1}{3} x^3 \log (x)-3 \int \log (x) \log (2 x) \, dx+3 \int x^2 \log (x) \log (2 x) \, dx-5 \int e^x x \log (2 x) \, dx-6 \int e^x \log (2 x) \, dx+16 \int x^2 \log (2 x) \, dx-18 \int \log (2 x) \, dx-\int e^x \log (x) \log (2 x) \, dx-\int e^x x \log (x) \log (2 x) \, dx \\ & = -5 e^x+6 x-\frac {2 x^3}{9}+\text {Ei}(x)-e^x \log (x)-3 x \log (x)+\frac {1}{3} x^3 \log (x)-e^x \log (2 x)-15 x \log (2 x)-5 e^x x \log (2 x)+5 x^3 \log (2 x)-3 x \log (x) \log (2 x)-e^x x \log (x) \log (2 x)+x^3 \log (x) \log (2 x)+3 \int (-1+\log (x)) \, dx-3 \int \frac {1}{9} x^2 (-1+3 \log (x)) \, dx+5 \int \frac {e^x (-1+x)}{x} \, dx+6 \int \frac {e^x}{x} \, dx+\int \frac {e^x \log (x)}{x} \, dx+\int \frac {e^x (-1+x) \log (x)}{x} \, dx+\int \frac {e^x \log (2 x)}{x} \, dx+\int \frac {e^x (-1+x) \log (2 x)}{x} \, dx \\ & = -5 e^x+3 x-\frac {2 x^3}{9}+7 \text {Ei}(x)-3 x \log (x)+\frac {1}{3} x^3 \log (x)-15 x \log (2 x)-5 e^x x \log (2 x)+5 x^3 \log (2 x)-3 x \log (x) \log (2 x)-e^x x \log (x) \log (2 x)+x^3 \log (x) \log (2 x)-\frac {1}{3} \int x^2 (-1+3 \log (x)) \, dx+3 \int \log (x) \, dx+5 \int \left (e^x-\frac {e^x}{x}\right ) \, dx-2 \int \frac {e^x-\text {Ei}(x)}{x} \, dx-2 \int \frac {\text {Ei}(x)}{x} \, dx \\ & = -5 e^x-\frac {x^3}{9}+7 \text {Ei}(x)+\frac {1}{9} x^3 (1-3 \log (x))+\frac {1}{3} x^3 \log (x)-15 x \log (2 x)-5 e^x x \log (2 x)+5 x^3 \log (2 x)-3 x \log (x) \log (2 x)-e^x x \log (x) \log (2 x)+x^3 \log (x) \log (2 x)+5 \int e^x \, dx-5 \int \frac {e^x}{x} \, dx-2 \left ((E_1(-x)+\text {Ei}(x)) \log (x)-\int \frac {E_1(-x)}{x} \, dx\right )-2 \int \left (\frac {e^x}{x}-\frac {\text {Ei}(x)}{x}\right ) \, dx \\ & = -\frac {x^3}{9}+2 \text {Ei}(x)+\frac {1}{9} x^3 (1-3 \log (x))+\frac {1}{3} x^3 \log (x)-2 \left (x \, _3F_3(1,1,1;2,2,2;x)+\frac {1}{2} \log ^2(-x)+\gamma \log (x)+(E_1(-x)+\text {Ei}(x)) \log (x)\right )-15 x \log (2 x)-5 e^x x \log (2 x)+5 x^3 \log (2 x)-3 x \log (x) \log (2 x)-e^x x \log (x) \log (2 x)+x^3 \log (x) \log (2 x)-2 \left (\int \frac {e^x}{x} \, dx-\int \frac {\text {Ei}(x)}{x} \, dx\right ) \\ & = -\frac {x^3}{9}+2 \text {Ei}(x)+\frac {1}{9} x^3 (1-3 \log (x))+\frac {1}{3} x^3 \log (x)-2 \left (x \, _3F_3(1,1,1;2,2,2;x)+\frac {1}{2} \log ^2(-x)+\gamma \log (x)+(E_1(-x)+\text {Ei}(x)) \log (x)\right )-15 x \log (2 x)-5 e^x x \log (2 x)+5 x^3 \log (2 x)-3 x \log (x) \log (2 x)-e^x x \log (x) \log (2 x)+x^3 \log (x) \log (2 x)-2 \left (\text {Ei}(x)-(E_1(-x)+\text {Ei}(x)) \log (x)+\int \frac {E_1(-x)}{x} \, dx\right ) \\ & = -\frac {x^3}{9}+2 \text {Ei}(x)+\frac {1}{9} x^3 (1-3 \log (x))+\frac {1}{3} x^3 \log (x)-2 \left (\text {Ei}(x)-x \, _3F_3(1,1,1;2,2,2;x)-\frac {1}{2} \log ^2(-x)-\gamma \log (x)-(E_1(-x)+\text {Ei}(x)) \log (x)\right )-2 \left (x \, _3F_3(1,1,1;2,2,2;x)+\frac {1}{2} \log ^2(-x)+\gamma \log (x)+(E_1(-x)+\text {Ei}(x)) \log (x)\right )-15 x \log (2 x)-5 e^x x \log (2 x)+5 x^3 \log (2 x)-3 x \log (x) \log (2 x)-e^x x \log (x) \log (2 x)+x^3 \log (x) \log (2 x) \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \left (-15-5 e^x+5 x^2+\left (-3-e^x+x^2\right ) \log (x)+\left (-18+e^x (-6-5 x)+16 x^2+\left (-3+e^x (-1-x)+3 x^2\right ) \log (x)\right ) \log (2 x)\right ) \, dx=x \left (-3-e^x+x^2\right ) (5+\log (x)) \log (2 x) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(56\) vs. \(2(19)=38\).
Time = 0.27 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.85
method | result | size |
parallelrisch | \(5 x^{3} \ln \left (2 x \right )-3 x \ln \left (x \right ) \ln \left (2 x \right )-\ln \left (x \right ) {\mathrm e}^{x} \ln \left (2 x \right ) x -5 x \,{\mathrm e}^{x} \ln \left (2 x \right )-15 x \ln \left (2 x \right )+\ln \left (x \right ) \ln \left (2 x \right ) x^{3}\) | \(57\) |
risch | \(x^{3} \ln \left (x \right )^{2}-x \,{\mathrm e}^{x} \ln \left (x \right )^{2}-3 x \ln \left (x \right )^{2}+\ln \left (2\right ) \ln \left (x \right ) x^{3}+5 x^{3} \ln \left (x \right )-3 x \ln \left (2\right ) \ln \left (x \right )-15 x \ln \left (x \right )-x \ln \left (2\right ) {\mathrm e}^{x} \ln \left (x \right )-5 x \,{\mathrm e}^{x} \ln \left (x \right )-15 x \ln \left (2\right )+5 x^{3} \ln \left (2\right )-5 x \ln \left (2\right ) {\mathrm e}^{x}\) | \(88\) |
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Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (19) = 38\).
Time = 0.26 (sec) , antiderivative size = 70, normalized size of antiderivative = 3.50 \[ \int \left (-15-5 e^x+5 x^2+\left (-3-e^x+x^2\right ) \log (x)+\left (-18+e^x (-6-5 x)+16 x^2+\left (-3+e^x (-1-x)+3 x^2\right ) \log (x)\right ) \log (2 x)\right ) \, dx=-5 \, x e^{x} \log \left (2\right ) + {\left (x^{3} - x e^{x} - 3 \, x\right )} \log \left (x\right )^{2} + 5 \, {\left (x^{3} - 3 \, x\right )} \log \left (2\right ) + {\left (5 \, x^{3} - {\left (x \log \left (2\right ) + 5 \, x\right )} e^{x} + {\left (x^{3} - 3 \, x\right )} \log \left (2\right ) - 15 \, x\right )} \log \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (19) = 38\).
Time = 0.22 (sec) , antiderivative size = 85, normalized size of antiderivative = 4.25 \[ \int \left (-15-5 e^x+5 x^2+\left (-3-e^x+x^2\right ) \log (x)+\left (-18+e^x (-6-5 x)+16 x^2+\left (-3+e^x (-1-x)+3 x^2\right ) \log (x)\right ) \log (2 x)\right ) \, dx=5 x^{3} \log {\left (2 \right )} - 15 x \log {\left (2 \right )} + \left (x^{3} - 3 x\right ) \log {\left (x \right )}^{2} + \left (- x \log {\left (x \right )}^{2} - 5 x \log {\left (x \right )} - x \log {\left (2 \right )} \log {\left (x \right )} - 5 x \log {\left (2 \right )}\right ) e^{x} + \left (x^{3} \log {\left (2 \right )} + 5 x^{3} - 15 x - 3 x \log {\left (2 \right )}\right ) \log {\left (x \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (19) = 38\).
Time = 0.35 (sec) , antiderivative size = 87, normalized size of antiderivative = 4.35 \[ \int \left (-15-5 e^x+5 x^2+\left (-3-e^x+x^2\right ) \log (x)+\left (-18+e^x (-6-5 x)+16 x^2+\left (-3+e^x (-1-x)+3 x^2\right ) \log (x)\right ) \log (2 x)\right ) \, dx=\frac {5}{3} \, x^{3} {\left (3 \, \log \left (2\right ) - 1\right )} + \frac {5}{3} \, x^{3} + {\left (x^{3} - 3 \, x\right )} \log \left (x\right )^{2} - 15 \, x {\left (\log \left (2\right ) - 1\right )} - {\left (x {\left (\log \left (2\right ) + 5\right )} \log \left (x\right ) + x \log \left (x\right )^{2} + 5 \, x \log \left (2\right ) - 5\right )} e^{x} + {\left (x^{3} {\left (\log \left (2\right ) + 5\right )} - 3 \, x {\left (\log \left (2\right ) + 5\right )}\right )} \log \left (x\right ) - 15 \, x - 5 \, e^{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (19) = 38\).
Time = 0.26 (sec) , antiderivative size = 107, normalized size of antiderivative = 5.35 \[ \int \left (-15-5 e^x+5 x^2+\left (-3-e^x+x^2\right ) \log (x)+\left (-18+e^x (-6-5 x)+16 x^2+\left (-3+e^x (-1-x)+3 x^2\right ) \log (x)\right ) \log (2 x)\right ) \, dx=x^{3} \log \left (2\right ) \log \left (x\right ) + x^{3} \log \left (x\right )^{2} + 5 \, x^{3} \log \left (2\right ) + \frac {14}{3} \, x^{3} \log \left (x\right ) - x e^{x} \log \left (2\right ) \log \left (x\right ) - x e^{x} \log \left (x\right )^{2} - 5 \, x e^{x} \log \left (2\right ) - 5 \, x e^{x} \log \left (x\right ) - 3 \, x \log \left (2\right ) \log \left (x\right ) - 3 \, x \log \left (x\right )^{2} - 15 \, x \log \left (2\right ) + \frac {1}{3} \, {\left (x^{3} - 9 \, x - 3 \, e^{x}\right )} \log \left (x\right ) - 12 \, x \log \left (x\right ) + e^{x} \log \left (x\right ) \]
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Time = 9.31 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int \left (-15-5 e^x+5 x^2+\left (-3-e^x+x^2\right ) \log (x)+\left (-18+e^x (-6-5 x)+16 x^2+\left (-3+e^x (-1-x)+3 x^2\right ) \log (x)\right ) \log (2 x)\right ) \, dx=-x\,\left (\ln \left (x\right )+5\right )\,\left (\ln \left (2\right )+\ln \left (x\right )\right )\,\left ({\mathrm {e}}^x-x^2+3\right ) \]
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