Integrand size = 189, antiderivative size = 34 \[ \int \frac {\left (e^3 \left (-4 x^3+4 x^4\right )+e^6 \left (2 x^3-6 x^4+4 x^5\right )\right ) \log \left (\frac {-1-2 e^3 x+e^6 \left (x-x^2\right )}{e^6}\right )+\left (-3 x^2+2 x^3+e^3 \left (-6 x^3+4 x^4\right )+e^6 \left (3 x^3-5 x^4+2 x^5\right )\right ) \log ^2\left (\frac {-1-2 e^3 x+e^6 \left (x-x^2\right )}{e^6}\right )}{20-40 x+20 x^2+e^3 \left (40 x-80 x^2+40 x^3\right )+e^6 \left (-20 x+60 x^2-60 x^3+20 x^4\right )} \, dx=\frac {1}{4} \left (2-\frac {x^2 \log ^2\left (x-\left (\frac {1}{e^3}+x\right )^2\right )}{-5+\frac {5}{x}}\right ) \]
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Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 8.29 (sec) , antiderivative size = 3703, normalized size of antiderivative = 108.91, number of steps used = 155, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.106, Rules used = {6820, 12, 6860, 1642, 648, 632, 212, 642, 2608, 2603, 787, 2605, 814, 2604, 2465, 2441, 2440, 2438, 2437, 2338} \[ \int \frac {\left (e^3 \left (-4 x^3+4 x^4\right )+e^6 \left (2 x^3-6 x^4+4 x^5\right )\right ) \log \left (\frac {-1-2 e^3 x+e^6 \left (x-x^2\right )}{e^6}\right )+\left (-3 x^2+2 x^3+e^3 \left (-6 x^3+4 x^4\right )+e^6 \left (3 x^3-5 x^4+2 x^5\right )\right ) \log ^2\left (\frac {-1-2 e^3 x+e^6 \left (x-x^2\right )}{e^6}\right )}{20-40 x+20 x^2+e^3 \left (40 x-80 x^2+40 x^3\right )+e^6 \left (-20 x+60 x^2-60 x^3+20 x^4\right )} \, dx =\text {Too large to display} \]
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Rule 12
Rule 212
Rule 632
Rule 642
Rule 648
Rule 787
Rule 814
Rule 1642
Rule 2338
Rule 2437
Rule 2438
Rule 2440
Rule 2441
Rule 2465
Rule 2603
Rule 2604
Rule 2605
Rule 2608
Rule 6820
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^2 \log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right ) \left (4 e^3 (-1+x) x+2 e^6 x \left (1-3 x+2 x^2\right )+(-3+2 x) \left (1+2 e^3 x+e^6 (-1+x) x\right ) \log \left (-\frac {1}{e^6}+x-\frac {2 x}{e^3}-x^2\right )\right )}{20 (1-x)^2 \left (1+e^3 \left (2-e^3\right ) x+e^6 x^2\right )} \, dx \\ & = \frac {1}{20} \int \frac {x^2 \log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right ) \left (4 e^3 (-1+x) x+2 e^6 x \left (1-3 x+2 x^2\right )+(-3+2 x) \left (1+2 e^3 x+e^6 (-1+x) x\right ) \log \left (-\frac {1}{e^6}+x-\frac {2 x}{e^3}-x^2\right )\right )}{(1-x)^2 \left (1+e^3 \left (2-e^3\right ) x+e^6 x^2\right )} \, dx \\ & = \frac {1}{20} \int \left (\frac {2 e^3 x^3 \left (-2+e^3-2 e^3 x\right ) \log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{(1-x) \left (1+e^3 \left (2-e^3\right ) x+e^6 x^2\right )}+\frac {x^2 (-3+2 x) \log ^2\left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{(1-x)^2}\right ) \, dx \\ & = \frac {1}{20} \int \frac {x^2 (-3+2 x) \log ^2\left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{(1-x)^2} \, dx+\frac {1}{10} e^3 \int \frac {x^3 \left (-2+e^3-2 e^3 x\right ) \log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{(1-x) \left (1+e^3 \left (2-e^3\right ) x+e^6 x^2\right )} \, dx \\ & = \frac {1}{20} \int \left (\log ^2\left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )-\frac {\log ^2\left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{(-1+x)^2}+2 x \log ^2\left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )\right ) \, dx+\frac {1}{10} e^3 \int \left (\frac {\left (-2+3 e^3\right ) \log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{e^6}+\frac {\left (2+e^3\right ) \log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{\left (1+2 e^3\right ) (-1+x)}+\frac {2 x \log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{e^3}+\frac {\left (2+e^3-4 e^6+e^9+e^3 \left (2-2 e^3-8 e^6+6 e^9-e^{12}\right ) x\right ) \log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{e^6 \left (1+2 e^3\right ) \left (1+e^3 \left (2-e^3\right ) x+e^6 x^2\right )}\right ) \, dx \\ & = \frac {1}{20} \int \log ^2\left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right ) \, dx-\frac {1}{20} \int \frac {\log ^2\left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{(-1+x)^2} \, dx+\frac {1}{10} \int x \log ^2\left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right ) \, dx+\frac {1}{5} \int x \log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right ) \, dx-\frac {\left (2-3 e^3\right ) \int \log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right ) \, dx}{10 e^3}+\frac {\int \frac {\left (2+e^3-4 e^6+e^9+e^3 \left (2-2 e^3-8 e^6+6 e^9-e^{12}\right ) x\right ) \log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{1+e^3 \left (2-e^3\right ) x+e^6 x^2} \, dx}{10 e^3 \left (1+2 e^3\right )}+\frac {\left (e^3 \left (2+e^3\right )\right ) \int \frac {\log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{-1+x} \, dx}{10 \left (1+2 e^3\right )} \\ & = -\frac {\left (2-3 e^3\right ) x \log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{10 e^3}+\frac {1}{10} x^2 \log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )+\frac {e^3 \left (2+e^3\right ) \log (-1+x) \log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{10 \left (1+2 e^3\right )}-\frac {\log ^2\left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{20 (1-x)}+\frac {1}{20} x \log ^2\left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )+\frac {1}{20} x^2 \log ^2\left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )-\frac {1}{10} \int \frac {e^3 x^2 \left (2-e^3+2 e^3 x\right )}{1+e^3 \left (2-e^3\right ) x+e^6 x^2} \, dx-\frac {1}{10} \int \frac {e^3 \left (-2+e^3-2 e^3 x\right ) \log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{(1-x) \left (1+e^3 \left (2-e^3\right ) x+e^6 x^2\right )} \, dx-\frac {1}{10} \int \frac {e^3 x \left (2-e^3+2 e^3 x\right ) \log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{1+e^3 \left (2-e^3\right ) x+e^6 x^2} \, dx-\frac {1}{10} \int \frac {e^3 x^2 \left (2-e^3+2 e^3 x\right ) \log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{1+e^3 \left (2-e^3\right ) x+e^6 x^2} \, dx+\frac {\left (2-3 e^3\right ) \int \frac {e^3 x \left (2-e^3+2 e^3 x\right )}{1+e^3 \left (2-e^3\right ) x+e^6 x^2} \, dx}{10 e^3}+\frac {\int \left (\frac {\left (e^3 \left (2-2 e^3-8 e^6+6 e^9-e^{12}\right )-\frac {e^{9/2} \left (-8-6 e^3+18 e^6-8 e^9+e^{12}\right )}{\sqrt {-4+e^3}}\right ) \log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{e^3 \left (2-e^3\right )-e^{9/2} \sqrt {-4+e^3}+2 e^6 x}+\frac {\left (e^3 \left (2-2 e^3-8 e^6+6 e^9-e^{12}\right )+\frac {e^{9/2} \left (-8-6 e^3+18 e^6-8 e^9+e^{12}\right )}{\sqrt {-4+e^3}}\right ) \log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{e^3 \left (2-e^3\right )+e^{9/2} \sqrt {-4+e^3}+2 e^6 x}\right ) \, dx}{10 e^3 \left (1+2 e^3\right )}-\frac {\left (e^3 \left (2+e^3\right )\right ) \int \frac {\left (1-\frac {2}{e^3}-2 x\right ) \log (-1+x)}{-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2} \, dx}{10 \left (1+2 e^3\right )} \\ & = -\frac {\left (2-3 e^3\right ) x \log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{10 e^3}+\frac {1}{10} x^2 \log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )+\frac {e^3 \left (2+e^3\right ) \log (-1+x) \log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{10 \left (1+2 e^3\right )}-\frac {\log ^2\left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{20 (1-x)}+\frac {1}{20} x \log ^2\left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )+\frac {1}{20} x^2 \log ^2\left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )-\frac {1}{10} e^3 \int \frac {x^2 \left (2-e^3+2 e^3 x\right )}{1+e^3 \left (2-e^3\right ) x+e^6 x^2} \, dx-\frac {1}{10} e^3 \int \frac {\left (-2+e^3-2 e^3 x\right ) \log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{(1-x) \left (1+e^3 \left (2-e^3\right ) x+e^6 x^2\right )} \, dx-\frac {1}{10} e^3 \int \frac {x \left (2-e^3+2 e^3 x\right ) \log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{1+e^3 \left (2-e^3\right ) x+e^6 x^2} \, dx-\frac {1}{10} e^3 \int \frac {x^2 \left (2-e^3+2 e^3 x\right ) \log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{1+e^3 \left (2-e^3\right ) x+e^6 x^2} \, dx+\frac {1}{10} \left (2-3 e^3\right ) \int \frac {x \left (2-e^3+2 e^3 x\right )}{1+e^3 \left (2-e^3\right ) x+e^6 x^2} \, dx-\frac {\left (e^3 \left (2+e^3\right )\right ) \int \left (-\frac {2 \log (-1+x)}{1-\frac {2}{e^3}-\frac {\sqrt {-4+e^3}}{e^{3/2}}-2 x}-\frac {2 \log (-1+x)}{1-\frac {2}{e^3}+\frac {\sqrt {-4+e^3}}{e^{3/2}}-2 x}\right ) \, dx}{10 \left (1+2 e^3\right )}+\frac {\left (2-2 e^3-8 e^6+6 e^9-e^{12}-\frac {e^{3/2} \left (8+6 e^3-18 e^6+8 e^9-e^{12}\right )}{\sqrt {-4+e^3}}\right ) \int \frac {\log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{e^3 \left (2-e^3\right )+e^{9/2} \sqrt {-4+e^3}+2 e^6 x} \, dx}{10 \left (1+2 e^3\right )}+\frac {\left (2-2 e^3-8 e^6+6 e^9-e^{12}+\frac {e^{3/2} \left (8+6 e^3-18 e^6+8 e^9-e^{12}\right )}{\sqrt {-4+e^3}}\right ) \int \frac {\log \left (-\frac {1}{e^6}+\left (1-\frac {2}{e^3}\right ) x-x^2\right )}{e^3 \left (2-e^3\right )-e^{9/2} \sqrt {-4+e^3}+2 e^6 x} \, dx}{10 \left (1+2 e^3\right )} \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.97 \[ \int \frac {\left (e^3 \left (-4 x^3+4 x^4\right )+e^6 \left (2 x^3-6 x^4+4 x^5\right )\right ) \log \left (\frac {-1-2 e^3 x+e^6 \left (x-x^2\right )}{e^6}\right )+\left (-3 x^2+2 x^3+e^3 \left (-6 x^3+4 x^4\right )+e^6 \left (3 x^3-5 x^4+2 x^5\right )\right ) \log ^2\left (\frac {-1-2 e^3 x+e^6 \left (x-x^2\right )}{e^6}\right )}{20-40 x+20 x^2+e^3 \left (40 x-80 x^2+40 x^3\right )+e^6 \left (-20 x+60 x^2-60 x^3+20 x^4\right )} \, dx=\frac {x^3 \log ^2\left (-\frac {1}{e^6}+x-\frac {2 x}{e^3}-x^2\right )}{20 (-1+x)} \]
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Time = 3.32 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00
method | result | size |
risch | \(\frac {{\ln \left (\left (\left (-x^{2}+x \right ) {\mathrm e}^{6}-2 x \,{\mathrm e}^{3}-1\right ) {\mathrm e}^{-6}\right )}^{2} x^{3}}{20 x -20}\) | \(34\) |
norman | \(\frac {{\ln \left (\left (\left (-x^{2}+x \right ) {\mathrm e}^{6}-2 x \,{\mathrm e}^{3}-1\right ) {\mathrm e}^{-6}\right )}^{2} x^{3}}{20 x -20}\) | \(38\) |
parallelrisch | \(\frac {{\ln \left (\left (\left (-x^{2}+x \right ) {\mathrm e}^{6}-2 x \,{\mathrm e}^{3}-1\right ) {\mathrm e}^{-6}\right )}^{2} x^{3}}{20 x -20}\) | \(38\) |
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Time = 0.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {\left (e^3 \left (-4 x^3+4 x^4\right )+e^6 \left (2 x^3-6 x^4+4 x^5\right )\right ) \log \left (\frac {-1-2 e^3 x+e^6 \left (x-x^2\right )}{e^6}\right )+\left (-3 x^2+2 x^3+e^3 \left (-6 x^3+4 x^4\right )+e^6 \left (3 x^3-5 x^4+2 x^5\right )\right ) \log ^2\left (\frac {-1-2 e^3 x+e^6 \left (x-x^2\right )}{e^6}\right )}{20-40 x+20 x^2+e^3 \left (40 x-80 x^2+40 x^3\right )+e^6 \left (-20 x+60 x^2-60 x^3+20 x^4\right )} \, dx=\frac {x^{3} \log \left (-{\left ({\left (x^{2} - x\right )} e^{6} + 2 \, x e^{3} + 1\right )} e^{\left (-6\right )}\right )^{2}}{20 \, {\left (x - 1\right )}} \]
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Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91 \[ \int \frac {\left (e^3 \left (-4 x^3+4 x^4\right )+e^6 \left (2 x^3-6 x^4+4 x^5\right )\right ) \log \left (\frac {-1-2 e^3 x+e^6 \left (x-x^2\right )}{e^6}\right )+\left (-3 x^2+2 x^3+e^3 \left (-6 x^3+4 x^4\right )+e^6 \left (3 x^3-5 x^4+2 x^5\right )\right ) \log ^2\left (\frac {-1-2 e^3 x+e^6 \left (x-x^2\right )}{e^6}\right )}{20-40 x+20 x^2+e^3 \left (40 x-80 x^2+40 x^3\right )+e^6 \left (-20 x+60 x^2-60 x^3+20 x^4\right )} \, dx=\frac {x^{3} \log {\left (\frac {- 2 x e^{3} + \left (- x^{2} + x\right ) e^{6} - 1}{e^{6}} \right )}^{2}}{20 x - 20} \]
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Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (27) = 54\).
Time = 0.41 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.94 \[ \int \frac {\left (e^3 \left (-4 x^3+4 x^4\right )+e^6 \left (2 x^3-6 x^4+4 x^5\right )\right ) \log \left (\frac {-1-2 e^3 x+e^6 \left (x-x^2\right )}{e^6}\right )+\left (-3 x^2+2 x^3+e^3 \left (-6 x^3+4 x^4\right )+e^6 \left (3 x^3-5 x^4+2 x^5\right )\right ) \log ^2\left (\frac {-1-2 e^3 x+e^6 \left (x-x^2\right )}{e^6}\right )}{20-40 x+20 x^2+e^3 \left (40 x-80 x^2+40 x^3\right )+e^6 \left (-20 x+60 x^2-60 x^3+20 x^4\right )} \, dx=\frac {x^{3} \log \left (-x^{2} e^{6} + x {\left (e^{6} - 2 \, e^{3}\right )} - 1\right )^{2} - 12 \, x^{3} \log \left (-x^{2} e^{6} + x {\left (e^{6} - 2 \, e^{3}\right )} - 1\right ) + 36 \, x^{3} - 36 \, x + 36}{20 \, {\left (x - 1\right )}} \]
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\[ \int \frac {\left (e^3 \left (-4 x^3+4 x^4\right )+e^6 \left (2 x^3-6 x^4+4 x^5\right )\right ) \log \left (\frac {-1-2 e^3 x+e^6 \left (x-x^2\right )}{e^6}\right )+\left (-3 x^2+2 x^3+e^3 \left (-6 x^3+4 x^4\right )+e^6 \left (3 x^3-5 x^4+2 x^5\right )\right ) \log ^2\left (\frac {-1-2 e^3 x+e^6 \left (x-x^2\right )}{e^6}\right )}{20-40 x+20 x^2+e^3 \left (40 x-80 x^2+40 x^3\right )+e^6 \left (-20 x+60 x^2-60 x^3+20 x^4\right )} \, dx=\int { \frac {{\left (2 \, x^{3} - 3 \, x^{2} + {\left (2 \, x^{5} - 5 \, x^{4} + 3 \, x^{3}\right )} e^{6} + 2 \, {\left (2 \, x^{4} - 3 \, x^{3}\right )} e^{3}\right )} \log \left (-{\left ({\left (x^{2} - x\right )} e^{6} + 2 \, x e^{3} + 1\right )} e^{\left (-6\right )}\right )^{2} + 2 \, {\left ({\left (2 \, x^{5} - 3 \, x^{4} + x^{3}\right )} e^{6} + 2 \, {\left (x^{4} - x^{3}\right )} e^{3}\right )} \log \left (-{\left ({\left (x^{2} - x\right )} e^{6} + 2 \, x e^{3} + 1\right )} e^{\left (-6\right )}\right )}{20 \, {\left (x^{2} + {\left (x^{4} - 3 \, x^{3} + 3 \, x^{2} - x\right )} e^{6} + 2 \, {\left (x^{3} - 2 \, x^{2} + x\right )} e^{3} - 2 \, x + 1\right )}} \,d x } \]
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Time = 18.23 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88 \[ \int \frac {\left (e^3 \left (-4 x^3+4 x^4\right )+e^6 \left (2 x^3-6 x^4+4 x^5\right )\right ) \log \left (\frac {-1-2 e^3 x+e^6 \left (x-x^2\right )}{e^6}\right )+\left (-3 x^2+2 x^3+e^3 \left (-6 x^3+4 x^4\right )+e^6 \left (3 x^3-5 x^4+2 x^5\right )\right ) \log ^2\left (\frac {-1-2 e^3 x+e^6 \left (x-x^2\right )}{e^6}\right )}{20-40 x+20 x^2+e^3 \left (40 x-80 x^2+40 x^3\right )+e^6 \left (-20 x+60 x^2-60 x^3+20 x^4\right )} \, dx=\frac {x^3\,{\ln \left (x-{\mathrm {e}}^{-6}-2\,x\,{\mathrm {e}}^{-3}-x^2\right )}^2}{20\,\left (x-1\right )} \]
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