Integrand size = 138, antiderivative size = 31 \[ \int \frac {392-49 x+9 x^3 \log ^2(3)+e^{2 x} \left (72-81 x+18 x^2\right ) \log ^2(3)+e^x \left (\left (336-210 x+42 x^2\right ) \log (3)+\left (72-45 x+9 x^2\right ) \log ^2(3)\right )}{-196 x+49 x^2+9 x^4 \log ^2(3)+e^{2 x} \left (-36 x+9 x^2\right ) \log ^2(3)+e^x \left (\left (-168 x+42 x^2\right ) \log (3)+\left (-36 x+9 x^2\right ) \log ^2(3)\right )} \, dx=\log \left (x-\frac {(4-x) \left (e^x+\left (e^x+\frac {7}{3 \log (3)}\right )^2\right )}{x^2}\right ) \]
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\[ \int \frac {392-49 x+9 x^3 \log ^2(3)+e^{2 x} \left (72-81 x+18 x^2\right ) \log ^2(3)+e^x \left (\left (336-210 x+42 x^2\right ) \log (3)+\left (72-45 x+9 x^2\right ) \log ^2(3)\right )}{-196 x+49 x^2+9 x^4 \log ^2(3)+e^{2 x} \left (-36 x+9 x^2\right ) \log ^2(3)+e^x \left (\left (-168 x+42 x^2\right ) \log (3)+\left (-36 x+9 x^2\right ) \log ^2(3)\right )} \, dx=\int \frac {392-49 x+9 x^3 \log ^2(3)+e^{2 x} \left (72-81 x+18 x^2\right ) \log ^2(3)+e^x \left (\left (336-210 x+42 x^2\right ) \log (3)+\left (72-45 x+9 x^2\right ) \log ^2(3)\right )}{-196 x+49 x^2+9 x^4 \log ^2(3)+e^{2 x} \left (-36 x+9 x^2\right ) \log ^2(3)+e^x \left (\left (-168 x+42 x^2\right ) \log (3)+\left (-36 x+9 x^2\right ) \log ^2(3)\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {-392+49 x-9 x^3 \log ^2(3)-9 e^{2 x} \left (8-9 x+2 x^2\right ) \log ^2(3)-3 e^x \left (8-5 x+x^2\right ) \log (3) (14+\log (27))}{x \left (196-49 x-9 e^{2 x} (-4+x) \log ^2(3)-9 x^3 \log ^2(3)-3 e^x (-4+x) \log (3) (14+\log (27))\right )} \, dx \\ & = \int \left (\frac {8-9 x+2 x^2}{(-4+x) x}+\frac {-1568+784 x+90 x^3 \log ^2(3)-18 x^4 \log ^2(3)-98 x^2 \left (1+\frac {54 \log ^2(3)}{49}\right )-672 e^x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )+336 e^x x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )-42 e^x x^2 \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )}{(4-x) \left (196-49 x+36 e^{2 x} \log ^2(3)-9 e^{2 x} x \log ^2(3)-9 x^3 \log ^2(3)+168 e^x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )-42 e^x x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )\right )}\right ) \, dx \\ & = \int \frac {8-9 x+2 x^2}{(-4+x) x} \, dx+\int \frac {-1568+784 x+90 x^3 \log ^2(3)-18 x^4 \log ^2(3)-98 x^2 \left (1+\frac {54 \log ^2(3)}{49}\right )-672 e^x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )+336 e^x x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )-42 e^x x^2 \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )}{(4-x) \left (196-49 x+36 e^{2 x} \log ^2(3)-9 e^{2 x} x \log ^2(3)-9 x^3 \log ^2(3)+168 e^x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )-42 e^x x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )\right )} \, dx \\ & = \int \left (2+\frac {1}{-4+x}-\frac {2}{x}\right ) \, dx+\int \left (\frac {1568}{(-4+x) \left (196-49 x+36 e^{2 x} \log ^2(3)-9 e^{2 x} x \log ^2(3)-9 x^3 \log ^2(3)+168 e^x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )-42 e^x x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )\right )}+\frac {784 x}{(4-x) \left (196-49 x+36 e^{2 x} \log ^2(3)-9 e^{2 x} x \log ^2(3)-9 x^3 \log ^2(3)+168 e^x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )-42 e^x x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )\right )}+\frac {90 x^3 \log ^2(3)}{(4-x) \left (196-49 x+36 e^{2 x} \log ^2(3)-9 e^{2 x} x \log ^2(3)-9 x^3 \log ^2(3)+168 e^x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )-42 e^x x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )\right )}+\frac {18 x^4 \log ^2(3)}{(-4+x) \left (196-49 x+36 e^{2 x} \log ^2(3)-9 e^{2 x} x \log ^2(3)-9 x^3 \log ^2(3)+168 e^x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )-42 e^x x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )\right )}+\frac {2 x^2 \left (-49-54 \log ^2(3)\right )}{(4-x) \left (196-49 x+36 e^{2 x} \log ^2(3)-9 e^{2 x} x \log ^2(3)-9 x^3 \log ^2(3)+168 e^x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )-42 e^x x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )\right )}+\frac {16 e^x \left (-42 \log (3)-\log ^2(27)\right )}{(4-x) \left (196-49 x+36 e^{2 x} \log ^2(3)-9 e^{2 x} x \log ^2(3)-9 x^3 \log ^2(3)+168 e^x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )-42 e^x x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )\right )}+\frac {e^x x^2 \left (-42 \log (3)-\log ^2(27)\right )}{(4-x) \left (196-49 x+36 e^{2 x} \log ^2(3)-9 e^{2 x} x \log ^2(3)-9 x^3 \log ^2(3)+168 e^x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )-42 e^x x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )\right )}+\frac {8 e^x x \left (42 \log (3)+\log ^2(27)\right )}{(4-x) \left (196-49 x+36 e^{2 x} \log ^2(3)-9 e^{2 x} x \log ^2(3)-9 x^3 \log ^2(3)+168 e^x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )-42 e^x x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )\right )}\right ) \, dx \\ & = 2 x+\log (4-x)-2 \log (x)+784 \int \frac {x}{(4-x) \left (196-49 x+36 e^{2 x} \log ^2(3)-9 e^{2 x} x \log ^2(3)-9 x^3 \log ^2(3)+168 e^x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )-42 e^x x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )\right )} \, dx+1568 \int \frac {1}{(-4+x) \left (196-49 x+36 e^{2 x} \log ^2(3)-9 e^{2 x} x \log ^2(3)-9 x^3 \log ^2(3)+168 e^x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )-42 e^x x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )\right )} \, dx+\left (18 \log ^2(3)\right ) \int \frac {x^4}{(-4+x) \left (196-49 x+36 e^{2 x} \log ^2(3)-9 e^{2 x} x \log ^2(3)-9 x^3 \log ^2(3)+168 e^x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )-42 e^x x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )\right )} \, dx+\left (90 \log ^2(3)\right ) \int \frac {x^3}{(4-x) \left (196-49 x+36 e^{2 x} \log ^2(3)-9 e^{2 x} x \log ^2(3)-9 x^3 \log ^2(3)+168 e^x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )-42 e^x x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )\right )} \, dx-\left (2 \left (49+54 \log ^2(3)\right )\right ) \int \frac {x^2}{(4-x) \left (196-49 x+36 e^{2 x} \log ^2(3)-9 e^{2 x} x \log ^2(3)-9 x^3 \log ^2(3)+168 e^x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )-42 e^x x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )\right )} \, dx+\left (-42 \log (3)-\log ^2(27)\right ) \int \frac {e^x x^2}{(4-x) \left (196-49 x+36 e^{2 x} \log ^2(3)-9 e^{2 x} x \log ^2(3)-9 x^3 \log ^2(3)+168 e^x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )-42 e^x x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )\right )} \, dx+\left (8 \left (42 \log (3)+\log ^2(27)\right )\right ) \int \frac {e^x x}{(4-x) \left (196-49 x+36 e^{2 x} \log ^2(3)-9 e^{2 x} x \log ^2(3)-9 x^3 \log ^2(3)+168 e^x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )-42 e^x x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )\right )} \, dx-\left (16 \left (42 \log (3)+\log ^2(27)\right )\right ) \int \frac {e^x}{(4-x) \left (196-49 x+36 e^{2 x} \log ^2(3)-9 e^{2 x} x \log ^2(3)-9 x^3 \log ^2(3)+168 e^x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )-42 e^x x \log (3) \left (1+\frac {\log ^2(27)}{42 \log (3)}\right )\right )} \, dx \\ & = \text {Too large to display} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(77\) vs. \(2(31)=62\).
Time = 0.23 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.48 \[ \int \frac {392-49 x+9 x^3 \log ^2(3)+e^{2 x} \left (72-81 x+18 x^2\right ) \log ^2(3)+e^x \left (\left (336-210 x+42 x^2\right ) \log (3)+\left (72-45 x+9 x^2\right ) \log ^2(3)\right )}{-196 x+49 x^2+9 x^4 \log ^2(3)+e^{2 x} \left (-36 x+9 x^2\right ) \log ^2(3)+e^x \left (\left (-168 x+42 x^2\right ) \log (3)+\left (-36 x+9 x^2\right ) \log ^2(3)\right )} \, dx=-2 \log (x)+\log \left (196-49 x+168 e^x \log (3)-42 e^x x \log (3)+36 e^{2 x} \log ^2(3)-9 e^{2 x} x \log ^2(3)-9 x^3 \log ^2(3)+12 e^x \log (3) \log (27)-3 e^x x \log (3) \log (27)\right ) \]
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Time = 0.30 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.77
method | result | size |
risch | \(-2 \ln \left (x \right )+\ln \left (x -4\right )+\ln \left ({\mathrm e}^{2 x}+\frac {\left (3 \ln \left (3\right )+14\right ) {\mathrm e}^{x}}{3 \ln \left (3\right )}+\frac {9 x^{3} \ln \left (3\right )^{2}+49 x -196}{9 \ln \left (3\right )^{2} \left (x -4\right )}\right )\) | \(55\) |
norman | \(-2 \ln \left (x \right )+\ln \left (9 x^{3} \ln \left (3\right )^{2}+9 x \ln \left (3\right )^{2} {\mathrm e}^{x}+9 \ln \left (3\right )^{2} {\mathrm e}^{2 x} x -36 \ln \left (3\right )^{2} {\mathrm e}^{x}-36 \ln \left (3\right )^{2} {\mathrm e}^{2 x}+42 x \ln \left (3\right ) {\mathrm e}^{x}-168 \ln \left (3\right ) {\mathrm e}^{x}+49 x -196\right )\) | \(72\) |
parallelrisch | \(-2 \ln \left (x \right )+\ln \left (\frac {9 x^{3} \ln \left (3\right )^{2}+9 x \ln \left (3\right )^{2} {\mathrm e}^{x}+9 \ln \left (3\right )^{2} {\mathrm e}^{2 x} x -36 \ln \left (3\right )^{2} {\mathrm e}^{x}-36 \ln \left (3\right )^{2} {\mathrm e}^{2 x}+42 x \ln \left (3\right ) {\mathrm e}^{x}-168 \ln \left (3\right ) {\mathrm e}^{x}+49 x -196}{9 \ln \left (3\right )^{2}}\right )\) | \(78\) |
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Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (29) = 58\).
Time = 0.27 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.06 \[ \int \frac {392-49 x+9 x^3 \log ^2(3)+e^{2 x} \left (72-81 x+18 x^2\right ) \log ^2(3)+e^x \left (\left (336-210 x+42 x^2\right ) \log (3)+\left (72-45 x+9 x^2\right ) \log ^2(3)\right )}{-196 x+49 x^2+9 x^4 \log ^2(3)+e^{2 x} \left (-36 x+9 x^2\right ) \log ^2(3)+e^x \left (\left (-168 x+42 x^2\right ) \log (3)+\left (-36 x+9 x^2\right ) \log ^2(3)\right )} \, dx=\log \left (x - 4\right ) - 2 \, \log \left (x\right ) + \log \left (\frac {9 \, x^{3} \log \left (3\right )^{2} + 9 \, {\left (x - 4\right )} e^{\left (2 \, x\right )} \log \left (3\right )^{2} + 3 \, {\left (3 \, {\left (x - 4\right )} \log \left (3\right )^{2} + 14 \, {\left (x - 4\right )} \log \left (3\right )\right )} e^{x} + 49 \, x - 196}{x - 4}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (24) = 48\).
Time = 0.64 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.97 \[ \int \frac {392-49 x+9 x^3 \log ^2(3)+e^{2 x} \left (72-81 x+18 x^2\right ) \log ^2(3)+e^x \left (\left (336-210 x+42 x^2\right ) \log (3)+\left (72-45 x+9 x^2\right ) \log ^2(3)\right )}{-196 x+49 x^2+9 x^4 \log ^2(3)+e^{2 x} \left (-36 x+9 x^2\right ) \log ^2(3)+e^x \left (\left (-168 x+42 x^2\right ) \log (3)+\left (-36 x+9 x^2\right ) \log ^2(3)\right )} \, dx=- 2 \log {\left (x \right )} + \log {\left (x - 4 \right )} + \log {\left (e^{2 x} + \frac {\left (3 \log {\left (3 \right )} + 14\right ) e^{x}}{3 \log {\left (3 \right )}} + \frac {9 x^{3} \log {\left (3 \right )}^{2} + 49 x - 196}{9 x \log {\left (3 \right )}^{2} - 36 \log {\left (3 \right )}^{2}} \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (29) = 58\).
Time = 0.33 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.84 \[ \int \frac {392-49 x+9 x^3 \log ^2(3)+e^{2 x} \left (72-81 x+18 x^2\right ) \log ^2(3)+e^x \left (\left (336-210 x+42 x^2\right ) \log (3)+\left (72-45 x+9 x^2\right ) \log ^2(3)\right )}{-196 x+49 x^2+9 x^4 \log ^2(3)+e^{2 x} \left (-36 x+9 x^2\right ) \log ^2(3)+e^x \left (\left (-168 x+42 x^2\right ) \log (3)+\left (-36 x+9 x^2\right ) \log ^2(3)\right )} \, dx=\log \left (x - 4\right ) - 2 \, \log \left (x\right ) + \log \left (\frac {9 \, x^{3} \log \left (3\right )^{2} + 9 \, {\left (x \log \left (3\right )^{2} - 4 \, \log \left (3\right )^{2}\right )} e^{\left (2 \, x\right )} + 3 \, {\left ({\left (3 \, \log \left (3\right )^{2} + 14 \, \log \left (3\right )\right )} x - 12 \, \log \left (3\right )^{2} - 56 \, \log \left (3\right )\right )} e^{x} + 49 \, x - 196}{9 \, {\left (x \log \left (3\right )^{2} - 4 \, \log \left (3\right )^{2}\right )}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (29) = 58\).
Time = 0.38 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.29 \[ \int \frac {392-49 x+9 x^3 \log ^2(3)+e^{2 x} \left (72-81 x+18 x^2\right ) \log ^2(3)+e^x \left (\left (336-210 x+42 x^2\right ) \log (3)+\left (72-45 x+9 x^2\right ) \log ^2(3)\right )}{-196 x+49 x^2+9 x^4 \log ^2(3)+e^{2 x} \left (-36 x+9 x^2\right ) \log ^2(3)+e^x \left (\left (-168 x+42 x^2\right ) \log (3)+\left (-36 x+9 x^2\right ) \log ^2(3)\right )} \, dx=\log \left (9 \, x^{3} \log \left (3\right )^{2} + 9 \, x e^{\left (2 \, x\right )} \log \left (3\right )^{2} + 9 \, x e^{x} \log \left (3\right )^{2} + 42 \, x e^{x} \log \left (3\right ) - 36 \, e^{\left (2 \, x\right )} \log \left (3\right )^{2} - 36 \, e^{x} \log \left (3\right )^{2} - 168 \, e^{x} \log \left (3\right ) + 49 \, x - 196\right ) - 2 \, \log \left (x\right ) \]
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Timed out. \[ \int \frac {392-49 x+9 x^3 \log ^2(3)+e^{2 x} \left (72-81 x+18 x^2\right ) \log ^2(3)+e^x \left (\left (336-210 x+42 x^2\right ) \log (3)+\left (72-45 x+9 x^2\right ) \log ^2(3)\right )}{-196 x+49 x^2+9 x^4 \log ^2(3)+e^{2 x} \left (-36 x+9 x^2\right ) \log ^2(3)+e^x \left (\left (-168 x+42 x^2\right ) \log (3)+\left (-36 x+9 x^2\right ) \log ^2(3)\right )} \, dx=-\int \frac {9\,x^3\,{\ln \left (3\right )}^2-49\,x+{\mathrm {e}}^x\,\left (\ln \left (3\right )\,\left (42\,x^2-210\,x+336\right )+{\ln \left (3\right )}^2\,\left (9\,x^2-45\,x+72\right )\right )+{\mathrm {e}}^{2\,x}\,{\ln \left (3\right )}^2\,\left (18\,x^2-81\,x+72\right )+392}{196\,x-9\,x^4\,{\ln \left (3\right )}^2+{\mathrm {e}}^x\,\left (\ln \left (3\right )\,\left (168\,x-42\,x^2\right )+{\ln \left (3\right )}^2\,\left (36\,x-9\,x^2\right )\right )-49\,x^2+{\mathrm {e}}^{2\,x}\,{\ln \left (3\right )}^2\,\left (36\,x-9\,x^2\right )} \,d x \]
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