Integrand size = 78, antiderivative size = 30 \[ \int \left (1-e^8+e^4 (-10-4 x)-8 x-3 x^2+\left (40+8 e^4+16 x\right ) \log \left (\frac {2}{x}\right )-16 \log ^2\left (\frac {2}{x}\right )+\left (-10-2 e^4-4 x+8 \log \left (\frac {2}{x}\right )\right ) \log (x)-\log ^2(x)\right ) \, dx=x \left (1+x-\left (-e^4-x+4 \log \left (\frac {2}{x}\right )-\log (x)\right )^2\right ) \]
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Leaf count is larger than twice the leaf count of optimal. \(140\) vs. \(2(30)=60\).
Time = 0.11 (sec) , antiderivative size = 140, normalized size of antiderivative = 4.67, number of steps used = 16, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2350, 9, 2333, 2332, 6873, 12, 6874, 2341, 2408} \[ \int \left (1-e^8+e^4 (-10-4 x)-8 x-3 x^2+\left (40+8 e^4+16 x\right ) \log \left (\frac {2}{x}\right )-16 \log ^2\left (\frac {2}{x}\right )+\left (-10-2 e^4-4 x+8 \log \left (\frac {2}{x}\right )\right ) \log (x)-\log ^2(x)\right ) \, dx=-x^3-3 x^2+8 x^2 \log \left (\frac {2}{x}\right )-2 x^2 \log (x)+\left (1-e^8\right ) x+2 \left (5+e^4\right ) x-50 x+4 \left (x+e^4+5\right )^2-\frac {1}{2} e^4 (2 x+5)^2-16 x \log ^2\left (\frac {2}{x}\right )-x \log ^2(x)+8 \left (5+e^4\right ) x \log \left (\frac {2}{x}\right )-40 x \log \left (\frac {2}{x}\right )+8 x \log \left (\frac {2}{x}\right ) \log (x)-2 \left (5+e^4\right ) x \log (x)+10 x \log (x) \]
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Rule 9
Rule 12
Rule 2332
Rule 2333
Rule 2341
Rule 2350
Rule 2408
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \left (1-e^8\right ) x-4 x^2-x^3-\frac {1}{2} e^4 (5+2 x)^2-16 \int \log ^2\left (\frac {2}{x}\right ) \, dx+\int \left (40+8 e^4+16 x\right ) \log \left (\frac {2}{x}\right ) \, dx+\int \left (-10-2 e^4-4 x+8 \log \left (\frac {2}{x}\right )\right ) \log (x) \, dx-\int \log ^2(x) \, dx \\ & = \left (1-e^8\right ) x-4 x^2-x^3-\frac {1}{2} e^4 (5+2 x)^2+8 \left (5+e^4\right ) x \log \left (\frac {2}{x}\right )+8 x^2 \log \left (\frac {2}{x}\right )-16 x \log ^2\left (\frac {2}{x}\right )-x \log ^2(x)+2 \int \log (x) \, dx-32 \int \log \left (\frac {2}{x}\right ) \, dx+\int 8 \left (5+e^4+x\right ) \, dx+\int 2 \left (-5 \left (1+\frac {e^4}{5}\right )-2 x+4 \log \left (\frac {2}{x}\right )\right ) \log (x) \, dx \\ & = -34 x+\left (1-e^8\right ) x-4 x^2-x^3+4 \left (5+e^4+x\right )^2-\frac {1}{2} e^4 (5+2 x)^2-32 x \log \left (\frac {2}{x}\right )+8 \left (5+e^4\right ) x \log \left (\frac {2}{x}\right )+8 x^2 \log \left (\frac {2}{x}\right )-16 x \log ^2\left (\frac {2}{x}\right )+2 x \log (x)-x \log ^2(x)+2 \int \left (-5 \left (1+\frac {e^4}{5}\right )-2 x+4 \log \left (\frac {2}{x}\right )\right ) \log (x) \, dx \\ & = -34 x+\left (1-e^8\right ) x-4 x^2-x^3+4 \left (5+e^4+x\right )^2-\frac {1}{2} e^4 (5+2 x)^2-32 x \log \left (\frac {2}{x}\right )+8 \left (5+e^4\right ) x \log \left (\frac {2}{x}\right )+8 x^2 \log \left (\frac {2}{x}\right )-16 x \log ^2\left (\frac {2}{x}\right )+2 x \log (x)-x \log ^2(x)+2 \int \left (-\left (\left (5+e^4\right ) \log (x)\right )-2 x \log (x)+4 \log \left (\frac {2}{x}\right ) \log (x)\right ) \, dx \\ & = -34 x+\left (1-e^8\right ) x-4 x^2-x^3+4 \left (5+e^4+x\right )^2-\frac {1}{2} e^4 (5+2 x)^2-32 x \log \left (\frac {2}{x}\right )+8 \left (5+e^4\right ) x \log \left (\frac {2}{x}\right )+8 x^2 \log \left (\frac {2}{x}\right )-16 x \log ^2\left (\frac {2}{x}\right )+2 x \log (x)-x \log ^2(x)-4 \int x \log (x) \, dx+8 \int \log \left (\frac {2}{x}\right ) \log (x) \, dx-\left (2 \left (5+e^4\right )\right ) \int \log (x) \, dx \\ & = -34 x+2 \left (5+e^4\right ) x+\left (1-e^8\right ) x-3 x^2-x^3+4 \left (5+e^4+x\right )^2-\frac {1}{2} e^4 (5+2 x)^2-32 x \log \left (\frac {2}{x}\right )+8 \left (5+e^4\right ) x \log \left (\frac {2}{x}\right )+8 x^2 \log \left (\frac {2}{x}\right )-16 x \log ^2\left (\frac {2}{x}\right )+10 x \log (x)-2 \left (5+e^4\right ) x \log (x)-2 x^2 \log (x)+8 x \log \left (\frac {2}{x}\right ) \log (x)-x \log ^2(x)-8 \int \left (1+\log \left (\frac {2}{x}\right )\right ) \, dx \\ & = -42 x+2 \left (5+e^4\right ) x+\left (1-e^8\right ) x-3 x^2-x^3+4 \left (5+e^4+x\right )^2-\frac {1}{2} e^4 (5+2 x)^2-32 x \log \left (\frac {2}{x}\right )+8 \left (5+e^4\right ) x \log \left (\frac {2}{x}\right )+8 x^2 \log \left (\frac {2}{x}\right )-16 x \log ^2\left (\frac {2}{x}\right )+10 x \log (x)-2 \left (5+e^4\right ) x \log (x)-2 x^2 \log (x)+8 x \log \left (\frac {2}{x}\right ) \log (x)-x \log ^2(x)-8 \int \log \left (\frac {2}{x}\right ) \, dx \\ & = -50 x+2 \left (5+e^4\right ) x+\left (1-e^8\right ) x-3 x^2-x^3+4 \left (5+e^4+x\right )^2-\frac {1}{2} e^4 (5+2 x)^2-40 x \log \left (\frac {2}{x}\right )+8 \left (5+e^4\right ) x \log \left (\frac {2}{x}\right )+8 x^2 \log \left (\frac {2}{x}\right )-16 x \log ^2\left (\frac {2}{x}\right )+10 x \log (x)-2 \left (5+e^4\right ) x \log (x)-2 x^2 \log (x)+8 x \log \left (\frac {2}{x}\right ) \log (x)-x \log ^2(x) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.93 \[ \int \left (1-e^8+e^4 (-10-4 x)-8 x-3 x^2+\left (40+8 e^4+16 x\right ) \log \left (\frac {2}{x}\right )-16 \log ^2\left (\frac {2}{x}\right )+\left (-10-2 e^4-4 x+8 \log \left (\frac {2}{x}\right )\right ) \log (x)-\log ^2(x)\right ) \, dx=-x \left (-1+e^8-x+2 e^4 x+x^2+16 \log ^2\left (\frac {2}{x}\right )+2 \left (e^4+x\right ) \log (x)+\log ^2(x)-8 \log \left (\frac {2}{x}\right ) \left (e^4+x+\log (x)\right )\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(91\) vs. \(2(29)=58\).
Time = 0.71 (sec) , antiderivative size = 92, normalized size of antiderivative = 3.07
method | result | size |
parallelrisch | \(-2 x^{2} {\mathrm e}^{4}-2 x \,{\mathrm e}^{4} \ln \left (x \right )+8 \,{\mathrm e}^{4} x \ln \left (\frac {2}{x}\right )-x^{3}-2 x^{2} \ln \left (x \right )+8 x^{2} \ln \left (\frac {2}{x}\right )-x \ln \left (x \right )^{2}+8 x \ln \left (\frac {2}{x}\right ) \ln \left (x \right )-16 x \ln \left (\frac {2}{x}\right )^{2}+x^{2}+\left (1-{\mathrm e}^{8}\right ) x\) | \(92\) |
risch | \(-16 x \ln \left (\frac {2}{x}\right )^{2}+8 \,{\mathrm e}^{4} x \ln \left (\frac {2}{x}\right )+8 x^{2} \ln \left (\frac {2}{x}\right )-9 x \ln \left (x \right )^{2}+8 x \ln \left (2\right ) \ln \left (x \right )-2 x \,{\mathrm e}^{4} \ln \left (x \right )-2 x^{2} \ln \left (x \right )-x \,{\mathrm e}^{8}-2 x^{2} {\mathrm e}^{4}-x^{3}+8 x \ln \left (\frac {2}{x}\right )+8 x \ln \left (x \right )-8 x \ln \left (2\right )+x^{2}+x\) | \(103\) |
parts | \(x +8 \,{\mathrm e}^{4} x \ln \left (\frac {2}{x}\right )-2 x \,{\mathrm e}^{4}+8 x \ln \left (\frac {2}{x}\right )+8 x^{2} \ln \left (\frac {2}{x}\right )+x^{2}+8 \ln \left (2\right ) \left (x \ln \left (x \right )-x \right )-2 x^{2} \ln \left (x \right )-2 \,{\mathrm e}^{4} \left (x \ln \left (x \right )-x \right )-8 \left (\ln \left (\frac {1}{x}\right )+\ln \left (x \right )\right ) \left (-x \ln \left (\frac {1}{x}\right )-x \right )-8 x \ln \left (\frac {1}{x}\right )^{2}-16 x \ln \left (\frac {1}{x}\right )-8 x \ln \left (x \right )-x^{3}-x \,{\mathrm e}^{8}-x \ln \left (x \right )^{2}-16 x \ln \left (\frac {2}{x}\right )^{2}-2 x^{2} {\mathrm e}^{4}\) | \(151\) |
default | \(x +2 \,{\mathrm e}^{4} \left (-x^{2}-5 x \right )+8 \,{\mathrm e}^{4} x \ln \left (\frac {2}{x}\right )+8 x \,{\mathrm e}^{4}+8 x \ln \left (\frac {2}{x}\right )+8 x^{2} \ln \left (\frac {2}{x}\right )+x^{2}+8 \ln \left (2\right ) \left (x \ln \left (x \right )-x \right )-2 x^{2} \ln \left (x \right )-2 \,{\mathrm e}^{4} \left (x \ln \left (x \right )-x \right )-8 \left (\ln \left (\frac {1}{x}\right )+\ln \left (x \right )\right ) \left (-x \ln \left (\frac {1}{x}\right )-x \right )-8 x \ln \left (\frac {1}{x}\right )^{2}-16 x \ln \left (\frac {1}{x}\right )-8 x \ln \left (x \right )-x^{3}-x \,{\mathrm e}^{8}-x \ln \left (x \right )^{2}-16 x \ln \left (\frac {2}{x}\right )^{2}\) | \(157\) |
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Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (24) = 48\).
Time = 0.28 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.40 \[ \int \left (1-e^8+e^4 (-10-4 x)-8 x-3 x^2+\left (40+8 e^4+16 x\right ) \log \left (\frac {2}{x}\right )-16 \log ^2\left (\frac {2}{x}\right )+\left (-10-2 e^4-4 x+8 \log \left (\frac {2}{x}\right )\right ) \log (x)-\log ^2(x)\right ) \, dx=-x^{3} - 2 \, x^{2} e^{4} - x \log \left (2\right )^{2} - 25 \, x \log \left (\frac {2}{x}\right )^{2} + x^{2} - x e^{8} - 2 \, {\left (x^{2} + x e^{4}\right )} \log \left (2\right ) + 10 \, {\left (x^{2} + x e^{4} + x \log \left (2\right )\right )} \log \left (\frac {2}{x}\right ) + x \]
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Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (20) = 40\).
Time = 0.10 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.33 \[ \int \left (1-e^8+e^4 (-10-4 x)-8 x-3 x^2+\left (40+8 e^4+16 x\right ) \log \left (\frac {2}{x}\right )-16 \log ^2\left (\frac {2}{x}\right )+\left (-10-2 e^4-4 x+8 \log \left (\frac {2}{x}\right )\right ) \log (x)-\log ^2(x)\right ) \, dx=- x^{3} + x^{2} \left (- 2 e^{4} + 1 + 8 \log {\left (2 \right )}\right ) - 25 x \log {\left (x \right )}^{2} + x \left (- e^{8} - 16 \log {\left (2 \right )}^{2} + 1 + 8 e^{4} \log {\left (2 \right )}\right ) + \left (- 10 x^{2} - 10 x e^{4} + 40 x \log {\left (2 \right )}\right ) \log {\left (x \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (24) = 48\).
Time = 0.29 (sec) , antiderivative size = 125, normalized size of antiderivative = 4.17 \[ \int \left (1-e^8+e^4 (-10-4 x)-8 x-3 x^2+\left (40+8 e^4+16 x\right ) \log \left (\frac {2}{x}\right )-16 \log ^2\left (\frac {2}{x}\right )+\left (-10-2 e^4-4 x+8 \log \left (\frac {2}{x}\right )\right ) \log (x)-\log ^2(x)\right ) \, dx=-x^{3} - 16 \, x \log \left (\frac {2}{x}\right )^{2} - {\left (\log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 2\right )} x + x^{2} + 2 \, x {\left (e^{4} - 4 \, \log \left (2\right ) - 3\right )} + 8 \, x {\left (e^{4} + 5\right )} - x e^{8} - 2 \, {\left (x^{2} + 5 \, x\right )} e^{4} - 2 \, {\left (x^{2} + x e^{4} - 4 \, x \log \left (\frac {2}{x}\right ) + x\right )} \log \left (x\right ) + 8 \, x \log \left (x\right ) + 8 \, {\left (x^{2} + x e^{4} + 5 \, x\right )} \log \left (\frac {2}{x}\right ) - 32 \, x \log \left (\frac {2}{x}\right ) - 31 \, x \]
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Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (24) = 48\).
Time = 0.28 (sec) , antiderivative size = 131, normalized size of antiderivative = 4.37 \[ \int \left (1-e^8+e^4 (-10-4 x)-8 x-3 x^2+\left (40+8 e^4+16 x\right ) \log \left (\frac {2}{x}\right )-16 \log ^2\left (\frac {2}{x}\right )+\left (-10-2 e^4-4 x+8 \log \left (\frac {2}{x}\right )\right ) \log (x)-\log ^2(x)\right ) \, dx=-x^{3} + 4 \, x^{2} {\left (\frac {2 \, e^{4}}{x} + \frac {10}{x} + 1\right )} - 2 \, x^{2} \log \left (x\right ) - 2 \, x e^{4} \log \left (x\right ) + 8 \, x \log \left (2\right ) \log \left (x\right ) - 9 \, x \log \left (x\right )^{2} - 16 \, x \log \left (\frac {2}{x}\right )^{2} - 3 \, x^{2} - x e^{8} - 2 \, {\left (x^{2} + 5 \, x\right )} e^{4} + 2 \, x e^{4} - 8 \, x \log \left (2\right ) + 8 \, x \log \left (x\right ) + 8 \, {\left (x^{2} + x e^{4} + 5 \, x\right )} \log \left (\frac {2}{x}\right ) - 32 \, x \log \left (\frac {2}{x}\right ) - 39 \, x \]
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Time = 11.63 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.40 \[ \int \left (1-e^8+e^4 (-10-4 x)-8 x-3 x^2+\left (40+8 e^4+16 x\right ) \log \left (\frac {2}{x}\right )-16 \log ^2\left (\frac {2}{x}\right )+\left (-10-2 e^4-4 x+8 \log \left (\frac {2}{x}\right )\right ) \log (x)-\log ^2(x)\right ) \, dx=-x\,\left ({\mathrm {e}}^8-x+16\,{\ln \left (\frac {2}{x}\right )}^2+2\,x\,{\mathrm {e}}^4+{\ln \left (x\right )}^2-8\,{\mathrm {e}}^4\,\ln \left (\frac {2}{x}\right )-8\,x\,\ln \left (\frac {2}{x}\right )+2\,{\mathrm {e}}^4\,\ln \left (x\right )+2\,x\,\ln \left (x\right )+x^2-8\,\ln \left (\frac {2}{x}\right )\,\ln \left (x\right )-1\right ) \]
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