Integrand size = 140, antiderivative size = 24 \[ \int \frac {e^2 (-2-6 x)+4 x+12 x^2+\left (-6 x+6 e^2 x-30 x^2\right ) \log (4)+18 x^2 \log ^2(4)+\left (-8 x+8 e^2 x-40 x^2+48 x^2 \log (4)\right ) \log (16)+32 x^2 \log ^2(16)+\left (2+8 x-2 e^2 x+10 x^2+\left (-6 x-12 x^2\right ) \log (4)+\left (-8 x-16 x^2\right ) \log (16)\right ) \log (x)+\left (2 x+2 x^2\right ) \log ^2(x)}{x} \, dx=\left (-e^2+\log (x)+x (2-3 \log (4)-4 \log (16)+\log (x))\right )^2 \]
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Leaf count is larger than twice the leaf count of optimal. \(160\) vs. \(2(24)=48\).
Time = 0.13 (sec) , antiderivative size = 160, normalized size of antiderivative = 6.67, number of steps used = 17, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {6, 14, 77, 2404, 2338, 2341, 2332, 2367, 2333, 2342} \[ \int \frac {e^2 (-2-6 x)+4 x+12 x^2+\left (-6 x+6 e^2 x-30 x^2\right ) \log (4)+18 x^2 \log ^2(4)+\left (-8 x+8 e^2 x-40 x^2+48 x^2 \log (4)\right ) \log (16)+32 x^2 \log ^2(16)+\left (2+8 x-2 e^2 x+10 x^2+\left (-6 x-12 x^2\right ) \log (4)+\left (-8 x-16 x^2\right ) \log (16)\right ) \log (x)+\left (2 x+2 x^2\right ) \log ^2(x)}{x} \, dx=\frac {x^2}{2}+x^2 \log ^2(x)+x^2 (5-6 \log (4)-8 \log (16)) \log (x)-x^2 \log (x)+x^2 (2-4 \log (16)-\log (64)) (3-4 \log (16)-\log (64))-\frac {1}{2} x^2 (5-6 \log (4)-8 \log (16))+4 x+2 x \log ^2(x)+\log ^2(x)+2 x \left (4-e^2-\log (4194304)\right ) \log (x)-4 x \log (x)+2 x \left (2-e^2 (3-4 \log (16)-\log (64))-\log (4194304)\right )-2 x \left (4-e^2-\log (4194304)\right )-2 e^2 \log (x) \]
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Rule 6
Rule 14
Rule 77
Rule 2332
Rule 2333
Rule 2338
Rule 2341
Rule 2342
Rule 2367
Rule 2404
Rubi steps \begin{align*} \text {integral}& = \int \frac {e^2 (-2-6 x)+4 x+\left (-6 x+6 e^2 x-30 x^2\right ) \log (4)+x^2 \left (12+18 \log ^2(4)\right )+\left (-8 x+8 e^2 x-40 x^2+48 x^2 \log (4)\right ) \log (16)+32 x^2 \log ^2(16)+\left (2+8 x-2 e^2 x+10 x^2+\left (-6 x-12 x^2\right ) \log (4)+\left (-8 x-16 x^2\right ) \log (16)\right ) \log (x)+\left (2 x+2 x^2\right ) \log ^2(x)}{x} \, dx \\ & = \int \frac {e^2 (-2-6 x)+4 x+\left (-6 x+6 e^2 x-30 x^2\right ) \log (4)+\left (-8 x+8 e^2 x-40 x^2+48 x^2 \log (4)\right ) \log (16)+x^2 \left (12+18 \log ^2(4)+32 \log ^2(16)\right )+\left (2+8 x-2 e^2 x+10 x^2+\left (-6 x-12 x^2\right ) \log (4)+\left (-8 x-16 x^2\right ) \log (16)\right ) \log (x)+\left (2 x+2 x^2\right ) \log ^2(x)}{x} \, dx \\ & = \int \left (\frac {2 \left (e^2-x (2-4 \log (16)-\log (64))\right ) (-1-x (3-4 \log (16)-\log (64)))}{x}+\frac {2 \left (1+x^2 (5-6 \log (4)-8 \log (16))+x \left (4-e^2-\log (4194304)\right )\right ) \log (x)}{x}+2 (1+x) \log ^2(x)\right ) \, dx \\ & = 2 \int \frac {\left (e^2-x (2-4 \log (16)-\log (64))\right ) (-1-x (3-4 \log (16)-\log (64)))}{x} \, dx+2 \int \frac {\left (1+x^2 (5-6 \log (4)-8 \log (16))+x \left (4-e^2-\log (4194304)\right )\right ) \log (x)}{x} \, dx+2 \int (1+x) \log ^2(x) \, dx \\ & = 2 \int \left (2-\frac {e^2}{x}-e^2 (3-4 \log (16)-\log (64))+x (2-4 \log (16)-\log (64)) (3-4 \log (16)-\log (64))-\log (4194304)\right ) \, dx+2 \int \left (\frac {\log (x)}{x}+x (5-6 \log (4)-8 \log (16)) \log (x)+\left (4-e^2-\log (4194304)\right ) \log (x)\right ) \, dx+2 \int \left (\log ^2(x)+x \log ^2(x)\right ) \, dx \\ & = x^2 (2-4 \log (16)-\log (64)) (3-4 \log (16)-\log (64))+2 x \left (2-e^2 (3-4 \log (16)-\log (64))-\log (4194304)\right )-2 e^2 \log (x)+2 \int \frac {\log (x)}{x} \, dx+2 \int \log ^2(x) \, dx+2 \int x \log ^2(x) \, dx+(2 (5-6 \log (4)-8 \log (16))) \int x \log (x) \, dx+\left (2 \left (4-e^2-\log (4194304)\right )\right ) \int \log (x) \, dx \\ & = -\frac {1}{2} x^2 (5-6 \log (4)-8 \log (16))+x^2 (2-4 \log (16)-\log (64)) (3-4 \log (16)-\log (64))-2 x \left (4-e^2-\log (4194304)\right )+2 x \left (2-e^2 (3-4 \log (16)-\log (64))-\log (4194304)\right )-2 e^2 \log (x)+x^2 (5-6 \log (4)-8 \log (16)) \log (x)+2 x \left (4-e^2-\log (4194304)\right ) \log (x)+\log ^2(x)+2 x \log ^2(x)+x^2 \log ^2(x)-2 \int x \log (x) \, dx-4 \int \log (x) \, dx \\ & = 4 x+\frac {x^2}{2}-\frac {1}{2} x^2 (5-6 \log (4)-8 \log (16))+x^2 (2-4 \log (16)-\log (64)) (3-4 \log (16)-\log (64))-2 x \left (4-e^2-\log (4194304)\right )+2 x \left (2-e^2 (3-4 \log (16)-\log (64))-\log (4194304)\right )-2 e^2 \log (x)-4 x \log (x)-x^2 \log (x)+x^2 (5-6 \log (4)-8 \log (16)) \log (x)+2 x \left (4-e^2-\log (4194304)\right ) \log (x)+\log ^2(x)+2 x \log ^2(x)+x^2 \log ^2(x) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(152\) vs. \(2(24)=48\).
Time = 0.17 (sec) , antiderivative size = 152, normalized size of antiderivative = 6.33 \[ \int \frac {e^2 (-2-6 x)+4 x+12 x^2+\left (-6 x+6 e^2 x-30 x^2\right ) \log (4)+18 x^2 \log ^2(4)+\left (-8 x+8 e^2 x-40 x^2+48 x^2 \log (4)\right ) \log (16)+32 x^2 \log ^2(16)+\left (2+8 x-2 e^2 x+10 x^2+\left (-6 x-12 x^2\right ) \log (4)+\left (-8 x-16 x^2\right ) \log (16)\right ) \log (x)+\left (2 x+2 x^2\right ) \log ^2(x)}{x} \, dx=2 \left (-2 e^2 x+2 x^2+\frac {1}{4} x^2 \log (16)+\frac {5}{4} x^2 \log (256)+2 e^2 x \log (2048)-\frac {5}{2} x^2 \log (4194304)+\frac {1}{2} x^2 \log (4) \log (4194304)+\frac {5}{2} x^2 \log (16) \log (4194304)-e^2 \log (x)+2 x \log (x)-e^2 x \log (x)+2 x^2 \log (x)-\frac {1}{2} x^2 \log (16) \log (x)-\frac {5}{2} x^2 \log (256) \log (x)-2 x \log (2048) \log (x)+\frac {\log ^2(x)}{2}+x \log ^2(x)+\frac {1}{2} x^2 \log ^2(x)\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(81\) vs. \(2(19)=38\).
Time = 0.96 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.42
method | result | size |
norman | \(\ln \left (x \right )^{2}+x^{2} \ln \left (x \right )^{2}-2 \,{\mathrm e}^{2} \ln \left (x \right )+\left (-4 \,{\mathrm e}^{2}+44 \,{\mathrm e}^{2} \ln \left (2\right )\right ) x +\left (4-88 \ln \left (2\right )+484 \ln \left (2\right )^{2}\right ) x^{2}+\left (4-44 \ln \left (2\right )\right ) x^{2} \ln \left (x \right )+\left (4-2 \,{\mathrm e}^{2}-44 \ln \left (2\right )\right ) x \ln \left (x \right )+2 x \ln \left (x \right )^{2}\) | \(82\) |
risch | \(\left (x^{2}+2 x +1\right ) \ln \left (x \right )^{2}+\left (-44 x^{2} \ln \left (2\right )-2 \,{\mathrm e}^{2} x -44 x \ln \left (2\right )+4 x^{2}+4 x \right ) \ln \left (x \right )+484 x^{2} \ln \left (2\right )^{2}+44 x \,{\mathrm e}^{2} \ln \left (2\right )-88 x^{2} \ln \left (2\right )-2 \,{\mathrm e}^{2} \ln \left (x \right )-4 \,{\mathrm e}^{2} x +4 x^{2}\) | \(83\) |
parallelrisch | \(x^{2} \ln \left (x \right )^{2}-44 x^{2} \ln \left (2\right ) \ln \left (x \right )+484 x^{2} \ln \left (2\right )^{2}-2 x \,{\mathrm e}^{2} \ln \left (x \right )+44 x \,{\mathrm e}^{2} \ln \left (2\right )+2 x \ln \left (x \right )^{2}-44 x \ln \left (2\right ) \ln \left (x \right )+4 x^{2} \ln \left (x \right )-88 x^{2} \ln \left (2\right )-2 \,{\mathrm e}^{2} \ln \left (x \right )-4 \,{\mathrm e}^{2} x +\ln \left (x \right )^{2}+4 x \ln \left (x \right )+4 x^{2}\) | \(95\) |
default | \(x^{2} \ln \left (x \right )^{2}+4 x^{2} \ln \left (x \right )+4 x^{2}-88 \ln \left (2\right ) \left (\frac {x^{2} \ln \left (x \right )}{2}-\frac {x^{2}}{4}\right )+484 x^{2} \ln \left (2\right )^{2}-2 \,{\mathrm e}^{2} \left (x \ln \left (x \right )-x \right )+44 x \,{\mathrm e}^{2} \ln \left (2\right )+2 x \ln \left (x \right )^{2}+4 x \ln \left (x \right )-44 \ln \left (2\right ) \left (x \ln \left (x \right )-x \right )-110 x^{2} \ln \left (2\right )-6 \,{\mathrm e}^{2} x -44 x \ln \left (2\right )-2 \,{\mathrm e}^{2} \ln \left (x \right )+\ln \left (x \right )^{2}\) | \(118\) |
parts | \(x^{2} \ln \left (x \right )^{2}+4 x^{2} \ln \left (x \right )+4 x^{2}-88 \ln \left (2\right ) \left (\frac {x^{2} \ln \left (x \right )}{2}-\frac {x^{2}}{4}\right )+484 x^{2} \ln \left (2\right )^{2}-2 \,{\mathrm e}^{2} \left (x \ln \left (x \right )-x \right )+44 x \,{\mathrm e}^{2} \ln \left (2\right )+2 x \ln \left (x \right )^{2}+4 x \ln \left (x \right )-44 \ln \left (2\right ) \left (x \ln \left (x \right )-x \right )-110 x^{2} \ln \left (2\right )-6 \,{\mathrm e}^{2} x -44 x \ln \left (2\right )-2 \,{\mathrm e}^{2} \ln \left (x \right )+\ln \left (x \right )^{2}\) | \(118\) |
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Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (21) = 42\).
Time = 0.27 (sec) , antiderivative size = 77, normalized size of antiderivative = 3.21 \[ \int \frac {e^2 (-2-6 x)+4 x+12 x^2+\left (-6 x+6 e^2 x-30 x^2\right ) \log (4)+18 x^2 \log ^2(4)+\left (-8 x+8 e^2 x-40 x^2+48 x^2 \log (4)\right ) \log (16)+32 x^2 \log ^2(16)+\left (2+8 x-2 e^2 x+10 x^2+\left (-6 x-12 x^2\right ) \log (4)+\left (-8 x-16 x^2\right ) \log (16)\right ) \log (x)+\left (2 x+2 x^2\right ) \log ^2(x)}{x} \, dx=484 \, x^{2} \log \left (2\right )^{2} + {\left (x^{2} + 2 \, x + 1\right )} \log \left (x\right )^{2} + 4 \, x^{2} - 4 \, x e^{2} - 44 \, {\left (2 \, x^{2} - x e^{2}\right )} \log \left (2\right ) + 2 \, {\left (2 \, x^{2} - {\left (x + 1\right )} e^{2} - 22 \, {\left (x^{2} + x\right )} \log \left (2\right ) + 2 \, x\right )} \log \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (19) = 38\).
Time = 0.16 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.62 \[ \int \frac {e^2 (-2-6 x)+4 x+12 x^2+\left (-6 x+6 e^2 x-30 x^2\right ) \log (4)+18 x^2 \log ^2(4)+\left (-8 x+8 e^2 x-40 x^2+48 x^2 \log (4)\right ) \log (16)+32 x^2 \log ^2(16)+\left (2+8 x-2 e^2 x+10 x^2+\left (-6 x-12 x^2\right ) \log (4)+\left (-8 x-16 x^2\right ) \log (16)\right ) \log (x)+\left (2 x+2 x^2\right ) \log ^2(x)}{x} \, dx=x^{2} \left (- 88 \log {\left (2 \right )} + 4 + 484 \log {\left (2 \right )}^{2}\right ) + x \left (- 4 e^{2} + 44 e^{2} \log {\left (2 \right )}\right ) + \left (x^{2} + 2 x + 1\right ) \log {\left (x \right )}^{2} + \left (- 44 x^{2} \log {\left (2 \right )} + 4 x^{2} - 44 x \log {\left (2 \right )} - 2 x e^{2} + 4 x\right ) \log {\left (x \right )} - 2 e^{2} \log {\left (x \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 135 vs. \(2 (21) = 42\).
Time = 0.19 (sec) , antiderivative size = 135, normalized size of antiderivative = 5.62 \[ \int \frac {e^2 (-2-6 x)+4 x+12 x^2+\left (-6 x+6 e^2 x-30 x^2\right ) \log (4)+18 x^2 \log ^2(4)+\left (-8 x+8 e^2 x-40 x^2+48 x^2 \log (4)\right ) \log (16)+32 x^2 \log ^2(16)+\left (2+8 x-2 e^2 x+10 x^2+\left (-6 x-12 x^2\right ) \log (4)+\left (-8 x-16 x^2\right ) \log (16)\right ) \log (x)+\left (2 x+2 x^2\right ) \log ^2(x)}{x} \, dx=484 \, x^{2} \log \left (2\right )^{2} + \frac {1}{2} \, {\left (2 \, \log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 1\right )} x^{2} - 110 \, x^{2} \log \left (2\right ) + 44 \, x e^{2} \log \left (2\right ) + 5 \, x^{2} \log \left (x\right ) + 2 \, {\left (\log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 2\right )} x + \frac {7}{2} \, x^{2} - 2 \, {\left (x \log \left (x\right ) - x\right )} e^{2} - 6 \, x e^{2} - 22 \, {\left (2 \, x^{2} \log \left (x\right ) - x^{2}\right )} \log \left (2\right ) - 44 \, {\left (x \log \left (x\right ) - x\right )} \log \left (2\right ) - 44 \, x \log \left (2\right ) + 8 \, x \log \left (x\right ) - 2 \, e^{2} \log \left (x\right ) + \log \left (x\right )^{2} - 4 \, x \]
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Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (21) = 42\).
Time = 0.26 (sec) , antiderivative size = 94, normalized size of antiderivative = 3.92 \[ \int \frac {e^2 (-2-6 x)+4 x+12 x^2+\left (-6 x+6 e^2 x-30 x^2\right ) \log (4)+18 x^2 \log ^2(4)+\left (-8 x+8 e^2 x-40 x^2+48 x^2 \log (4)\right ) \log (16)+32 x^2 \log ^2(16)+\left (2+8 x-2 e^2 x+10 x^2+\left (-6 x-12 x^2\right ) \log (4)+\left (-8 x-16 x^2\right ) \log (16)\right ) \log (x)+\left (2 x+2 x^2\right ) \log ^2(x)}{x} \, dx=484 \, x^{2} \log \left (2\right )^{2} - 44 \, x^{2} \log \left (2\right ) \log \left (x\right ) + x^{2} \log \left (x\right )^{2} - 88 \, x^{2} \log \left (2\right ) + 44 \, x e^{2} \log \left (2\right ) + 4 \, x^{2} \log \left (x\right ) - 2 \, x e^{2} \log \left (x\right ) - 44 \, x \log \left (2\right ) \log \left (x\right ) + 2 \, x \log \left (x\right )^{2} + 4 \, x^{2} - 4 \, x e^{2} + 4 \, x \log \left (x\right ) - 2 \, e^{2} \log \left (x\right ) + \log \left (x\right )^{2} \]
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Time = 10.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46 \[ \int \frac {e^2 (-2-6 x)+4 x+12 x^2+\left (-6 x+6 e^2 x-30 x^2\right ) \log (4)+18 x^2 \log ^2(4)+\left (-8 x+8 e^2 x-40 x^2+48 x^2 \log (4)\right ) \log (16)+32 x^2 \log ^2(16)+\left (2+8 x-2 e^2 x+10 x^2+\left (-6 x-12 x^2\right ) \log (4)+\left (-8 x-16 x^2\right ) \log (16)\right ) \log (x)+\left (2 x+2 x^2\right ) \log ^2(x)}{x} \, dx=\left (2\,x+\ln \left (x\right )-22\,x\,\ln \left (2\right )+x\,\ln \left (x\right )\right )\,\left (2\,x-2\,{\mathrm {e}}^2+\ln \left (x\right )-22\,x\,\ln \left (2\right )+x\,\ln \left (x\right )\right ) \]
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