Integrand size = 193, antiderivative size = 23 \[ \int \frac {-20+60 x^2-80 x^3+60 x^4+e^{4 x} (-20+80 x)+e^{3 x} \left (80-240 x+240 x^2\right )+e^{2 x} \left (-120+240 x-280 x^2+240 x^3\right )+e^x \left (80-80 x+80 x^4\right )+\left (20-80 x+180 x^2-160 x^3+100 x^4+e^{4 x} (20+80 x)+e^{3 x} \left (-80-80 x+240 x^2\right )+e^{2 x} \left (120-160 x-40 x^2+240 x^3\right )+e^x \left (-80+240 x-320 x^2+160 x^3+80 x^4\right )\right ) \log (x)}{1+3 \log (x)+3 \log ^2(x)+\log ^3(x)} \, dx=5+\frac {20 x \left (x+\left (-1+e^x+x\right )^2\right )^2}{(1+\log (x))^2} \]
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\[ \int \frac {-20+60 x^2-80 x^3+60 x^4+e^{4 x} (-20+80 x)+e^{3 x} \left (80-240 x+240 x^2\right )+e^{2 x} \left (-120+240 x-280 x^2+240 x^3\right )+e^x \left (80-80 x+80 x^4\right )+\left (20-80 x+180 x^2-160 x^3+100 x^4+e^{4 x} (20+80 x)+e^{3 x} \left (-80-80 x+240 x^2\right )+e^{2 x} \left (120-160 x-40 x^2+240 x^3\right )+e^x \left (-80+240 x-320 x^2+160 x^3+80 x^4\right )\right ) \log (x)}{1+3 \log (x)+3 \log ^2(x)+\log ^3(x)} \, dx=\int \frac {-20+60 x^2-80 x^3+60 x^4+e^{4 x} (-20+80 x)+e^{3 x} \left (80-240 x+240 x^2\right )+e^{2 x} \left (-120+240 x-280 x^2+240 x^3\right )+e^x \left (80-80 x+80 x^4\right )+\left (20-80 x+180 x^2-160 x^3+100 x^4+e^{4 x} (20+80 x)+e^{3 x} \left (-80-80 x+240 x^2\right )+e^{2 x} \left (120-160 x-40 x^2+240 x^3\right )+e^x \left (-80+240 x-320 x^2+160 x^3+80 x^4\right )\right ) \log (x)}{1+3 \log (x)+3 \log ^2(x)+\log ^3(x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {20 \left (1+e^{2 x}+2 e^x (-1+x)-x+x^2\right ) \left (-1-x+3 x^2+e^{2 x} (-1+4 x)+e^x \left (2-2 x+4 x^2\right )+\left (1-3 x+5 x^2+e^{2 x} (1+4 x)+2 e^x \left (-1+x+2 x^2\right )\right ) \log (x)\right )}{(1+\log (x))^3} \, dx \\ & = 20 \int \frac {\left (1+e^{2 x}+2 e^x (-1+x)-x+x^2\right ) \left (-1-x+3 x^2+e^{2 x} (-1+4 x)+e^x \left (2-2 x+4 x^2\right )+\left (1-3 x+5 x^2+e^{2 x} (1+4 x)+2 e^x \left (-1+x+2 x^2\right )\right ) \log (x)\right )}{(1+\log (x))^3} \, dx \\ & = 20 \int \left (-\frac {1}{(1+\log (x))^3}+\frac {3 x^2}{(1+\log (x))^3}-\frac {4 x^3}{(1+\log (x))^3}+\frac {3 x^4}{(1+\log (x))^3}+\frac {\log (x)}{(1+\log (x))^3}-\frac {4 x \log (x)}{(1+\log (x))^3}+\frac {9 x^2 \log (x)}{(1+\log (x))^3}-\frac {8 x^3 \log (x)}{(1+\log (x))^3}+\frac {5 x^4 \log (x)}{(1+\log (x))^3}+\frac {e^{4 x} (-1+4 x+\log (x)+4 x \log (x))}{(1+\log (x))^3}+\frac {4 e^{3 x} \left (1-3 x+3 x^2-\log (x)-x \log (x)+3 x^2 \log (x)\right )}{(1+\log (x))^3}+\frac {2 e^{2 x} \left (-3+6 x-7 x^2+6 x^3+3 \log (x)-4 x \log (x)-x^2 \log (x)+6 x^3 \log (x)\right )}{(1+\log (x))^3}+\frac {4 e^x \left (1-x+x^4-\log (x)+3 x \log (x)-4 x^2 \log (x)+2 x^3 \log (x)+x^4 \log (x)\right )}{(1+\log (x))^3}\right ) \, dx \\ & = -\left (20 \int \frac {1}{(1+\log (x))^3} \, dx\right )+20 \int \frac {\log (x)}{(1+\log (x))^3} \, dx+20 \int \frac {e^{4 x} (-1+4 x+\log (x)+4 x \log (x))}{(1+\log (x))^3} \, dx+40 \int \frac {e^{2 x} \left (-3+6 x-7 x^2+6 x^3+3 \log (x)-4 x \log (x)-x^2 \log (x)+6 x^3 \log (x)\right )}{(1+\log (x))^3} \, dx+60 \int \frac {x^2}{(1+\log (x))^3} \, dx+60 \int \frac {x^4}{(1+\log (x))^3} \, dx-80 \int \frac {x^3}{(1+\log (x))^3} \, dx-80 \int \frac {x \log (x)}{(1+\log (x))^3} \, dx+80 \int \frac {e^{3 x} \left (1-3 x+3 x^2-\log (x)-x \log (x)+3 x^2 \log (x)\right )}{(1+\log (x))^3} \, dx+80 \int \frac {e^x \left (1-x+x^4-\log (x)+3 x \log (x)-4 x^2 \log (x)+2 x^3 \log (x)+x^4 \log (x)\right )}{(1+\log (x))^3} \, dx+100 \int \frac {x^4 \log (x)}{(1+\log (x))^3} \, dx-160 \int \frac {x^3 \log (x)}{(1+\log (x))^3} \, dx+180 \int \frac {x^2 \log (x)}{(1+\log (x))^3} \, dx \\ & = -\frac {160 \operatorname {ExpIntegralEi}(2 (1+\log (x))) \log (x)}{e^2}+\frac {810 \operatorname {ExpIntegralEi}(3 (1+\log (x))) \log (x)}{e^3}-\frac {1280 \operatorname {ExpIntegralEi}(4 (1+\log (x))) \log (x)}{e^4}+\frac {1250 \operatorname {ExpIntegralEi}(5 (1+\log (x))) \log (x)}{e^5}+\frac {10 x}{(1+\log (x))^2}-\frac {30 x^3}{(1+\log (x))^2}+\frac {40 x^4}{(1+\log (x))^2}-\frac {30 x^5}{(1+\log (x))^2}+\frac {40 x^2 \log (x)}{(1+\log (x))^2}-\frac {90 x^3 \log (x)}{(1+\log (x))^2}+\frac {80 x^4 \log (x)}{(1+\log (x))^2}-\frac {50 x^5 \log (x)}{(1+\log (x))^2}+\frac {80 x^2 \log (x)}{1+\log (x)}-\frac {270 x^3 \log (x)}{1+\log (x)}+\frac {320 x^4 \log (x)}{1+\log (x)}-\frac {250 x^5 \log (x)}{1+\log (x)}+\frac {20 e^{4 x} (x+x \log (x))}{(1+\log (x))^3}-10 \int \frac {1}{(1+\log (x))^2} \, dx+20 \int \left (-\frac {1}{(1+\log (x))^3}+\frac {1}{(1+\log (x))^2}\right ) \, dx+40 \int \left (-\frac {2 e^{2 x} \left (3-5 x+3 x^2\right )}{(1+\log (x))^3}+\frac {e^{2 x} \left (3-4 x-x^2+6 x^3\right )}{(1+\log (x))^2}\right ) \, dx+80 \int \left (-\frac {2 e^{3 x} (-1+x)}{(1+\log (x))^3}+\frac {e^{3 x} \left (-1-x+3 x^2\right )}{(1+\log (x))^2}\right ) \, dx+80 \int \left (-\frac {2 e^x \left (-1+2 x-2 x^2+x^3\right )}{(1+\log (x))^3}+\frac {e^x \left (-1+3 x-4 x^2+2 x^3+x^4\right )}{(1+\log (x))^2}\right ) \, dx+80 \int \left (\frac {2 \operatorname {ExpIntegralEi}(2 (1+\log (x)))}{e^2 x}-\frac {x (3+2 \log (x))}{2 (1+\log (x))^2}\right ) \, dx+90 \int \frac {x^2}{(1+\log (x))^2} \, dx-100 \int \left (\frac {25 \operatorname {ExpIntegralEi}(5 (1+\log (x)))}{2 e^5 x}-\frac {x^4 (6+5 \log (x))}{2 (1+\log (x))^2}\right ) \, dx+150 \int \frac {x^4}{(1+\log (x))^2} \, dx-160 \int \frac {x^3}{(1+\log (x))^2} \, dx+160 \int \left (\frac {8 \operatorname {ExpIntegralEi}(4 (1+\log (x)))}{e^4 x}-\frac {x^3 (5+4 \log (x))}{2 (1+\log (x))^2}\right ) \, dx-180 \int \left (\frac {9 \operatorname {ExpIntegralEi}(3 (1+\log (x)))}{2 e^3 x}-\frac {x^2 (4+3 \log (x))}{2 (1+\log (x))^2}\right ) \, dx \\ & = -\frac {160 \operatorname {ExpIntegralEi}(2 (1+\log (x))) \log (x)}{e^2}+\frac {810 \operatorname {ExpIntegralEi}(3 (1+\log (x))) \log (x)}{e^3}-\frac {1280 \operatorname {ExpIntegralEi}(4 (1+\log (x))) \log (x)}{e^4}+\frac {1250 \operatorname {ExpIntegralEi}(5 (1+\log (x))) \log (x)}{e^5}+\frac {10 x}{(1+\log (x))^2}-\frac {30 x^3}{(1+\log (x))^2}+\frac {40 x^4}{(1+\log (x))^2}-\frac {30 x^5}{(1+\log (x))^2}+\frac {40 x^2 \log (x)}{(1+\log (x))^2}-\frac {90 x^3 \log (x)}{(1+\log (x))^2}+\frac {80 x^4 \log (x)}{(1+\log (x))^2}-\frac {50 x^5 \log (x)}{(1+\log (x))^2}+\frac {10 x}{1+\log (x)}-\frac {90 x^3}{1+\log (x)}+\frac {160 x^4}{1+\log (x)}-\frac {150 x^5}{1+\log (x)}+\frac {80 x^2 \log (x)}{1+\log (x)}-\frac {270 x^3 \log (x)}{1+\log (x)}+\frac {320 x^4 \log (x)}{1+\log (x)}-\frac {250 x^5 \log (x)}{1+\log (x)}+\frac {20 e^{4 x} (x+x \log (x))}{(1+\log (x))^3}-10 \int \frac {1}{1+\log (x)} \, dx-20 \int \frac {1}{(1+\log (x))^3} \, dx+20 \int \frac {1}{(1+\log (x))^2} \, dx+40 \int \frac {e^{2 x} \left (3-4 x-x^2+6 x^3\right )}{(1+\log (x))^2} \, dx-40 \int \frac {x (3+2 \log (x))}{(1+\log (x))^2} \, dx+50 \int \frac {x^4 (6+5 \log (x))}{(1+\log (x))^2} \, dx-80 \int \frac {e^{2 x} \left (3-5 x+3 x^2\right )}{(1+\log (x))^3} \, dx+80 \int \frac {e^{3 x} \left (-1-x+3 x^2\right )}{(1+\log (x))^2} \, dx+80 \int \frac {e^x \left (-1+3 x-4 x^2+2 x^3+x^4\right )}{(1+\log (x))^2} \, dx-80 \int \frac {x^3 (5+4 \log (x))}{(1+\log (x))^2} \, dx+90 \int \frac {x^2 (4+3 \log (x))}{(1+\log (x))^2} \, dx-160 \int \frac {e^{3 x} (-1+x)}{(1+\log (x))^3} \, dx-160 \int \frac {e^x \left (-1+2 x-2 x^2+x^3\right )}{(1+\log (x))^3} \, dx+270 \int \frac {x^2}{1+\log (x)} \, dx-640 \int \frac {x^3}{1+\log (x)} \, dx+750 \int \frac {x^4}{1+\log (x)} \, dx-\frac {1250 \int \frac {\operatorname {ExpIntegralEi}(5 (1+\log (x)))}{x} \, dx}{e^5}+\frac {1280 \int \frac {\operatorname {ExpIntegralEi}(4 (1+\log (x)))}{x} \, dx}{e^4}-\frac {810 \int \frac {\operatorname {ExpIntegralEi}(3 (1+\log (x)))}{x} \, dx}{e^3}+\frac {160 \int \frac {\operatorname {ExpIntegralEi}(2 (1+\log (x)))}{x} \, dx}{e^2} \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.39 \[ \int \frac {-20+60 x^2-80 x^3+60 x^4+e^{4 x} (-20+80 x)+e^{3 x} \left (80-240 x+240 x^2\right )+e^{2 x} \left (-120+240 x-280 x^2+240 x^3\right )+e^x \left (80-80 x+80 x^4\right )+\left (20-80 x+180 x^2-160 x^3+100 x^4+e^{4 x} (20+80 x)+e^{3 x} \left (-80-80 x+240 x^2\right )+e^{2 x} \left (120-160 x-40 x^2+240 x^3\right )+e^x \left (-80+240 x-320 x^2+160 x^3+80 x^4\right )\right ) \log (x)}{1+3 \log (x)+3 \log ^2(x)+\log ^3(x)} \, dx=\frac {20 x \left (1+e^{2 x}+2 e^x (-1+x)-x+x^2\right )^2}{(1+\log (x))^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(89\) vs. \(2(22)=44\).
Time = 1.29 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.91
method | result | size |
risch | \(\frac {20 x \left (x^{4}+4 \,{\mathrm e}^{x} x^{3}+6 \,{\mathrm e}^{2 x} x^{2}+4 x \,{\mathrm e}^{3 x}+{\mathrm e}^{4 x}-2 x^{3}-8 \,{\mathrm e}^{x} x^{2}-10 x \,{\mathrm e}^{2 x}-4 \,{\mathrm e}^{3 x}+3 x^{2}+8 \,{\mathrm e}^{x} x +6 \,{\mathrm e}^{2 x}-2 x -4 \,{\mathrm e}^{x}+1\right )}{\left (\ln \left (x \right )+1\right )^{2}}\) | \(90\) |
parallelrisch | \(-\frac {-40 x^{5}-160 \,{\mathrm e}^{x} x^{4}-240 \,{\mathrm e}^{2 x} x^{3}-160 x^{2} {\mathrm e}^{3 x}-40 x \,{\mathrm e}^{4 x}+80 x^{4}+320 \,{\mathrm e}^{x} x^{3}+400 \,{\mathrm e}^{2 x} x^{2}+160 x \,{\mathrm e}^{3 x}-120 x^{3}-320 \,{\mathrm e}^{x} x^{2}-240 x \,{\mathrm e}^{2 x}+80 x^{2}+160 \,{\mathrm e}^{x} x -40 x}{2 \left (\ln \left (x \right )^{2}+2 \ln \left (x \right )+1\right )}\) | \(113\) |
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Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (22) = 44\).
Time = 0.26 (sec) , antiderivative size = 94, normalized size of antiderivative = 4.09 \[ \int \frac {-20+60 x^2-80 x^3+60 x^4+e^{4 x} (-20+80 x)+e^{3 x} \left (80-240 x+240 x^2\right )+e^{2 x} \left (-120+240 x-280 x^2+240 x^3\right )+e^x \left (80-80 x+80 x^4\right )+\left (20-80 x+180 x^2-160 x^3+100 x^4+e^{4 x} (20+80 x)+e^{3 x} \left (-80-80 x+240 x^2\right )+e^{2 x} \left (120-160 x-40 x^2+240 x^3\right )+e^x \left (-80+240 x-320 x^2+160 x^3+80 x^4\right )\right ) \log (x)}{1+3 \log (x)+3 \log ^2(x)+\log ^3(x)} \, dx=\frac {20 \, {\left (x^{5} - 2 \, x^{4} + 3 \, x^{3} - 2 \, x^{2} + x e^{\left (4 \, x\right )} + 4 \, {\left (x^{2} - x\right )} e^{\left (3 \, x\right )} + 2 \, {\left (3 \, x^{3} - 5 \, x^{2} + 3 \, x\right )} e^{\left (2 \, x\right )} + 4 \, {\left (x^{4} - 2 \, x^{3} + 2 \, x^{2} - x\right )} e^{x} + x\right )}}{\log \left (x\right )^{2} + 2 \, \log \left (x\right ) + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 704 vs. \(2 (22) = 44\).
Time = 0.29 (sec) , antiderivative size = 704, normalized size of antiderivative = 30.61 \[ \int \frac {-20+60 x^2-80 x^3+60 x^4+e^{4 x} (-20+80 x)+e^{3 x} \left (80-240 x+240 x^2\right )+e^{2 x} \left (-120+240 x-280 x^2+240 x^3\right )+e^x \left (80-80 x+80 x^4\right )+\left (20-80 x+180 x^2-160 x^3+100 x^4+e^{4 x} (20+80 x)+e^{3 x} \left (-80-80 x+240 x^2\right )+e^{2 x} \left (120-160 x-40 x^2+240 x^3\right )+e^x \left (-80+240 x-320 x^2+160 x^3+80 x^4\right )\right ) \log (x)}{1+3 \log (x)+3 \log ^2(x)+\log ^3(x)} \, dx=\frac {\left (20 x \log {\left (x \right )}^{6} + 120 x \log {\left (x \right )}^{5} + 300 x \log {\left (x \right )}^{4} + 400 x \log {\left (x \right )}^{3} + 300 x \log {\left (x \right )}^{2} + 120 x \log {\left (x \right )} + 20 x\right ) e^{4 x} + \left (80 x^{2} \log {\left (x \right )}^{6} + 480 x^{2} \log {\left (x \right )}^{5} + 1200 x^{2} \log {\left (x \right )}^{4} + 1600 x^{2} \log {\left (x \right )}^{3} + 1200 x^{2} \log {\left (x \right )}^{2} + 480 x^{2} \log {\left (x \right )} + 80 x^{2} - 80 x \log {\left (x \right )}^{6} - 480 x \log {\left (x \right )}^{5} - 1200 x \log {\left (x \right )}^{4} - 1600 x \log {\left (x \right )}^{3} - 1200 x \log {\left (x \right )}^{2} - 480 x \log {\left (x \right )} - 80 x\right ) e^{3 x} + \left (120 x^{3} \log {\left (x \right )}^{6} + 720 x^{3} \log {\left (x \right )}^{5} + 1800 x^{3} \log {\left (x \right )}^{4} + 2400 x^{3} \log {\left (x \right )}^{3} + 1800 x^{3} \log {\left (x \right )}^{2} + 720 x^{3} \log {\left (x \right )} + 120 x^{3} - 200 x^{2} \log {\left (x \right )}^{6} - 1200 x^{2} \log {\left (x \right )}^{5} - 3000 x^{2} \log {\left (x \right )}^{4} - 4000 x^{2} \log {\left (x \right )}^{3} - 3000 x^{2} \log {\left (x \right )}^{2} - 1200 x^{2} \log {\left (x \right )} - 200 x^{2} + 120 x \log {\left (x \right )}^{6} + 720 x \log {\left (x \right )}^{5} + 1800 x \log {\left (x \right )}^{4} + 2400 x \log {\left (x \right )}^{3} + 1800 x \log {\left (x \right )}^{2} + 720 x \log {\left (x \right )} + 120 x\right ) e^{2 x} + \left (80 x^{4} \log {\left (x \right )}^{6} + 480 x^{4} \log {\left (x \right )}^{5} + 1200 x^{4} \log {\left (x \right )}^{4} + 1600 x^{4} \log {\left (x \right )}^{3} + 1200 x^{4} \log {\left (x \right )}^{2} + 480 x^{4} \log {\left (x \right )} + 80 x^{4} - 160 x^{3} \log {\left (x \right )}^{6} - 960 x^{3} \log {\left (x \right )}^{5} - 2400 x^{3} \log {\left (x \right )}^{4} - 3200 x^{3} \log {\left (x \right )}^{3} - 2400 x^{3} \log {\left (x \right )}^{2} - 960 x^{3} \log {\left (x \right )} - 160 x^{3} + 160 x^{2} \log {\left (x \right )}^{6} + 960 x^{2} \log {\left (x \right )}^{5} + 2400 x^{2} \log {\left (x \right )}^{4} + 3200 x^{2} \log {\left (x \right )}^{3} + 2400 x^{2} \log {\left (x \right )}^{2} + 960 x^{2} \log {\left (x \right )} + 160 x^{2} - 80 x \log {\left (x \right )}^{6} - 480 x \log {\left (x \right )}^{5} - 1200 x \log {\left (x \right )}^{4} - 1600 x \log {\left (x \right )}^{3} - 1200 x \log {\left (x \right )}^{2} - 480 x \log {\left (x \right )} - 80 x\right ) e^{x}}{\log {\left (x \right )}^{8} + 8 \log {\left (x \right )}^{7} + 28 \log {\left (x \right )}^{6} + 56 \log {\left (x \right )}^{5} + 70 \log {\left (x \right )}^{4} + 56 \log {\left (x \right )}^{3} + 28 \log {\left (x \right )}^{2} + 8 \log {\left (x \right )} + 1} + \frac {20 x^{5} - 40 x^{4} + 60 x^{3} - 40 x^{2} + 20 x}{\log {\left (x \right )}^{2} + 2 \log {\left (x \right )} + 1} \]
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\[ \int \frac {-20+60 x^2-80 x^3+60 x^4+e^{4 x} (-20+80 x)+e^{3 x} \left (80-240 x+240 x^2\right )+e^{2 x} \left (-120+240 x-280 x^2+240 x^3\right )+e^x \left (80-80 x+80 x^4\right )+\left (20-80 x+180 x^2-160 x^3+100 x^4+e^{4 x} (20+80 x)+e^{3 x} \left (-80-80 x+240 x^2\right )+e^{2 x} \left (120-160 x-40 x^2+240 x^3\right )+e^x \left (-80+240 x-320 x^2+160 x^3+80 x^4\right )\right ) \log (x)}{1+3 \log (x)+3 \log ^2(x)+\log ^3(x)} \, dx=\int { \frac {20 \, {\left (3 \, x^{4} - 4 \, x^{3} + 3 \, x^{2} + {\left (4 \, x - 1\right )} e^{\left (4 \, x\right )} + 4 \, {\left (3 \, x^{2} - 3 \, x + 1\right )} e^{\left (3 \, x\right )} + 2 \, {\left (6 \, x^{3} - 7 \, x^{2} + 6 \, x - 3\right )} e^{\left (2 \, x\right )} + 4 \, {\left (x^{4} - x + 1\right )} e^{x} + {\left (5 \, x^{4} - 8 \, x^{3} + 9 \, x^{2} + {\left (4 \, x + 1\right )} e^{\left (4 \, x\right )} + 4 \, {\left (3 \, x^{2} - x - 1\right )} e^{\left (3 \, x\right )} + 2 \, {\left (6 \, x^{3} - x^{2} - 4 \, x + 3\right )} e^{\left (2 \, x\right )} + 4 \, {\left (x^{4} + 2 \, x^{3} - 4 \, x^{2} + 3 \, x - 1\right )} e^{x} - 4 \, x + 1\right )} \log \left (x\right ) - 1\right )}}{\log \left (x\right )^{3} + 3 \, \log \left (x\right )^{2} + 3 \, \log \left (x\right ) + 1} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (22) = 44\).
Time = 0.28 (sec) , antiderivative size = 107, normalized size of antiderivative = 4.65 \[ \int \frac {-20+60 x^2-80 x^3+60 x^4+e^{4 x} (-20+80 x)+e^{3 x} \left (80-240 x+240 x^2\right )+e^{2 x} \left (-120+240 x-280 x^2+240 x^3\right )+e^x \left (80-80 x+80 x^4\right )+\left (20-80 x+180 x^2-160 x^3+100 x^4+e^{4 x} (20+80 x)+e^{3 x} \left (-80-80 x+240 x^2\right )+e^{2 x} \left (120-160 x-40 x^2+240 x^3\right )+e^x \left (-80+240 x-320 x^2+160 x^3+80 x^4\right )\right ) \log (x)}{1+3 \log (x)+3 \log ^2(x)+\log ^3(x)} \, dx=\frac {20 \, {\left (x^{5} + 4 \, x^{4} e^{x} - 2 \, x^{4} + 6 \, x^{3} e^{\left (2 \, x\right )} - 8 \, x^{3} e^{x} + 3 \, x^{3} + 4 \, x^{2} e^{\left (3 \, x\right )} - 10 \, x^{2} e^{\left (2 \, x\right )} + 8 \, x^{2} e^{x} - 2 \, x^{2} + x e^{\left (4 \, x\right )} - 4 \, x e^{\left (3 \, x\right )} + 6 \, x e^{\left (2 \, x\right )} - 4 \, x e^{x} + x\right )}}{\log \left (x\right )^{2} + 2 \, \log \left (x\right ) + 1} \]
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Time = 9.90 (sec) , antiderivative size = 657, normalized size of antiderivative = 28.57 \[ \int \frac {-20+60 x^2-80 x^3+60 x^4+e^{4 x} (-20+80 x)+e^{3 x} \left (80-240 x+240 x^2\right )+e^{2 x} \left (-120+240 x-280 x^2+240 x^3\right )+e^x \left (80-80 x+80 x^4\right )+\left (20-80 x+180 x^2-160 x^3+100 x^4+e^{4 x} (20+80 x)+e^{3 x} \left (-80-80 x+240 x^2\right )+e^{2 x} \left (120-160 x-40 x^2+240 x^3\right )+e^x \left (-80+240 x-320 x^2+160 x^3+80 x^4\right )\right ) \log (x)}{1+3 \log (x)+3 \log ^2(x)+\log ^3(x)} \, dx=10\,x-\frac {10\,x\,\left (4\,{\mathrm {e}}^{3\,x}-6\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^x+12\,x\,{\mathrm {e}}^{2\,x}-12\,x\,{\mathrm {e}}^{3\,x}+4\,x\,{\mathrm {e}}^{4\,x}+4\,x^4\,{\mathrm {e}}^x-14\,x^2\,{\mathrm {e}}^{2\,x}+12\,x^2\,{\mathrm {e}}^{3\,x}+12\,x^3\,{\mathrm {e}}^{2\,x}-4\,x\,{\mathrm {e}}^x+3\,x^2-4\,x^3+3\,x^4-1\right )+10\,x\,\ln \left (x\right )\,\left (6\,{\mathrm {e}}^{2\,x}-4\,x-4\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^x-8\,x\,{\mathrm {e}}^{2\,x}-4\,x\,{\mathrm {e}}^{3\,x}+4\,x\,{\mathrm {e}}^{4\,x}-16\,x^2\,{\mathrm {e}}^x+8\,x^3\,{\mathrm {e}}^x+4\,x^4\,{\mathrm {e}}^x-2\,x^2\,{\mathrm {e}}^{2\,x}+12\,x^2\,{\mathrm {e}}^{3\,x}+12\,x^3\,{\mathrm {e}}^{2\,x}+12\,x\,{\mathrm {e}}^x+9\,x^2-8\,x^3+5\,x^4+1\right )}{{\ln \left (x\right )}^2+2\,\ln \left (x\right )+1}+{\mathrm {e}}^{4\,x}\,\left (160\,x^3+120\,x^2+10\,x\right )-\frac {20\,x\,\left (2\,x\,{\mathrm {e}}^{2\,x}-2\,x-8\,x\,{\mathrm {e}}^{3\,x}+4\,x\,{\mathrm {e}}^{4\,x}-10\,x^2\,{\mathrm {e}}^x+4\,x^3\,{\mathrm {e}}^x+12\,x^4\,{\mathrm {e}}^x+2\,x^5\,{\mathrm {e}}^x-10\,x^2\,{\mathrm {e}}^{2\,x}+6\,x^2\,{\mathrm {e}}^{3\,x}+16\,x^3\,{\mathrm {e}}^{2\,x}+8\,x^2\,{\mathrm {e}}^{4\,x}+18\,x^3\,{\mathrm {e}}^{3\,x}+12\,x^4\,{\mathrm {e}}^{2\,x}+4\,x\,{\mathrm {e}}^x+9\,x^2-12\,x^3+10\,x^4\right )+10\,x\,\ln \left (x\right )\,\left (6\,{\mathrm {e}}^{2\,x}-8\,x-4\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^x-4\,x\,{\mathrm {e}}^{2\,x}-20\,x\,{\mathrm {e}}^{3\,x}+12\,x\,{\mathrm {e}}^{4\,x}-36\,x^2\,{\mathrm {e}}^x+16\,x^3\,{\mathrm {e}}^x+28\,x^4\,{\mathrm {e}}^x+4\,x^5\,{\mathrm {e}}^x-22\,x^2\,{\mathrm {e}}^{2\,x}+24\,x^2\,{\mathrm {e}}^{3\,x}+44\,x^3\,{\mathrm {e}}^{2\,x}+16\,x^2\,{\mathrm {e}}^{4\,x}+36\,x^3\,{\mathrm {e}}^{3\,x}+24\,x^4\,{\mathrm {e}}^{2\,x}+20\,x\,{\mathrm {e}}^x+27\,x^2-32\,x^3+25\,x^4+1\right )}{\ln \left (x\right )+1}-{\mathrm {e}}^{3\,x}\,\left (-360\,x^4-240\,x^3+200\,x^2+40\,x\right )+{\mathrm {e}}^x\,\left (40\,x^6+280\,x^5+160\,x^4-360\,x^3+200\,x^2-40\,x\right )+{\mathrm {e}}^{2\,x}\,\left (240\,x^5+440\,x^4-220\,x^3-40\,x^2+60\,x\right )-80\,x^2+270\,x^3-320\,x^4+250\,x^5 \]
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