Integrand size = 118, antiderivative size = 31 \[ \int \frac {-364+406 x+189 x^2+e^{2 x} \left (405+162 x^2\right )+e^x \left (270+594 x-108 x^2+162 x^3\right )}{4800+19683 e^{4 x}-25920 x+54432 x^2-52488 x^3+19683 x^4+e^{3 x} (-52488+78732 x)+e^{2 x} \left (54432-157464 x+118098 x^2\right )+e^x \left (-25920+108864 x-157464 x^2+78732 x^3\right )} \, dx=\frac {-3-x-x^2}{3 \left (4+9 \left (2-3 \left (e^x+x\right )\right )^2\right )} \]
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\[ \int \frac {-364+406 x+189 x^2+e^{2 x} \left (405+162 x^2\right )+e^x \left (270+594 x-108 x^2+162 x^3\right )}{4800+19683 e^{4 x}-25920 x+54432 x^2-52488 x^3+19683 x^4+e^{3 x} (-52488+78732 x)+e^{2 x} \left (54432-157464 x+118098 x^2\right )+e^x \left (-25920+108864 x-157464 x^2+78732 x^3\right )} \, dx=\int \frac {-364+406 x+189 x^2+e^{2 x} \left (405+162 x^2\right )+e^x \left (270+594 x-108 x^2+162 x^3\right )}{4800+19683 e^{4 x}-25920 x+54432 x^2-52488 x^3+19683 x^4+e^{3 x} (-52488+78732 x)+e^{2 x} \left (54432-157464 x+118098 x^2\right )+e^x \left (-25920+108864 x-157464 x^2+78732 x^3\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {81 e^{2 x} \left (5+2 x^2\right )+7 \left (-52+58 x+27 x^2\right )+54 e^x \left (5+11 x-2 x^2+3 x^3\right )}{3 \left (40+81 e^{2 x}-108 x+81 x^2+54 e^x (-2+3 x)\right )^2} \, dx \\ & = \frac {1}{3} \int \frac {81 e^{2 x} \left (5+2 x^2\right )+7 \left (-52+58 x+27 x^2\right )+54 e^x \left (5+11 x-2 x^2+3 x^3\right )}{\left (40+81 e^{2 x}-108 x+81 x^2+54 e^x (-2+3 x)\right )^2} \, dx \\ & = \frac {1}{3} \int \left (-\frac {2 \left (3+x+x^2\right ) \left (94-135 e^x-189 x+81 e^x x+81 x^2\right )}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}+\frac {5+2 x^2}{40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2}\right ) \, dx \\ & = \frac {1}{3} \int \frac {5+2 x^2}{40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2} \, dx-\frac {2}{3} \int \frac {\left (3+x+x^2\right ) \left (94-135 e^x-189 x+81 e^x x+81 x^2\right )}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx \\ & = \frac {1}{3} \int \left (\frac {5}{40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2}+\frac {2 x^2}{40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2}\right ) \, dx-\frac {2}{3} \int \left (\frac {3 \left (94-135 e^x-189 x+81 e^x x+81 x^2\right )}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}+\frac {x \left (94-135 e^x-189 x+81 e^x x+81 x^2\right )}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}+\frac {x^2 \left (94-135 e^x-189 x+81 e^x x+81 x^2\right )}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}\right ) \, dx \\ & = -\left (\frac {2}{3} \int \frac {x \left (94-135 e^x-189 x+81 e^x x+81 x^2\right )}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx\right )-\frac {2}{3} \int \frac {x^2 \left (94-135 e^x-189 x+81 e^x x+81 x^2\right )}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx+\frac {2}{3} \int \frac {x^2}{40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2} \, dx+\frac {5}{3} \int \frac {1}{40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2} \, dx-2 \int \frac {94-135 e^x-189 x+81 e^x x+81 x^2}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx \\ & = \frac {2}{3} \int \frac {x^2}{40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2} \, dx-\frac {2}{3} \int \left (\frac {94 x}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}-\frac {135 e^x x}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}-\frac {189 x^2}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}+\frac {81 e^x x^2}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}+\frac {81 x^3}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}\right ) \, dx-\frac {2}{3} \int \left (\frac {94 x^2}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}-\frac {135 e^x x^2}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}-\frac {189 x^3}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}+\frac {81 e^x x^3}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}+\frac {81 x^4}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}\right ) \, dx+\frac {5}{3} \int \frac {1}{40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2} \, dx-2 \int \left (\frac {94}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}-\frac {135 e^x}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}-\frac {189 x}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}+\frac {81 e^x x}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}+\frac {81 x^2}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}\right ) \, dx \\ & = \frac {2}{3} \int \frac {x^2}{40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2} \, dx+\frac {5}{3} \int \frac {1}{40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2} \, dx-54 \int \frac {e^x x^2}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx-54 \int \frac {x^3}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx-54 \int \frac {e^x x^3}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx-54 \int \frac {x^4}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx-\frac {188}{3} \int \frac {x}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx-\frac {188}{3} \int \frac {x^2}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx+90 \int \frac {e^x x}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx+90 \int \frac {e^x x^2}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx+126 \int \frac {x^2}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx+126 \int \frac {x^3}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx-162 \int \frac {e^x x}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx-162 \int \frac {x^2}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx-188 \int \frac {1}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx+270 \int \frac {e^x}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx+378 \int \frac {x}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx \\ \end{align*}
Time = 3.53 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.26 \[ \int \frac {-364+406 x+189 x^2+e^{2 x} \left (405+162 x^2\right )+e^x \left (270+594 x-108 x^2+162 x^3\right )}{4800+19683 e^{4 x}-25920 x+54432 x^2-52488 x^3+19683 x^4+e^{3 x} (-52488+78732 x)+e^{2 x} \left (54432-157464 x+118098 x^2\right )+e^x \left (-25920+108864 x-157464 x^2+78732 x^3\right )} \, dx=-\frac {3+x+x^2}{3 \left (40+81 e^{2 x}-108 x+81 x^2+54 e^x (-2+3 x)\right )} \]
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Time = 0.10 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16
method | result | size |
risch | \(-\frac {x^{2}+x +3}{3 \left (81 \,{\mathrm e}^{2 x}+162 \,{\mathrm e}^{x} x +81 x^{2}-108 \,{\mathrm e}^{x}-108 x +40\right )}\) | \(36\) |
parallelrisch | \(\frac {-81 x^{2}-81 x -243}{19683 \,{\mathrm e}^{2 x}+39366 \,{\mathrm e}^{x} x +19683 x^{2}-26244 \,{\mathrm e}^{x}-26244 x +9720}\) | \(40\) |
norman | \(\frac {-\frac {7 x}{9}-\frac {4 \,{\mathrm e}^{x}}{9}+\frac {{\mathrm e}^{2 x}}{3}+\frac {2 \,{\mathrm e}^{x} x}{3}-\frac {203}{243}}{81 \,{\mathrm e}^{2 x}+162 \,{\mathrm e}^{x} x +81 x^{2}-108 \,{\mathrm e}^{x}-108 x +40}\) | \(49\) |
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Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13 \[ \int \frac {-364+406 x+189 x^2+e^{2 x} \left (405+162 x^2\right )+e^x \left (270+594 x-108 x^2+162 x^3\right )}{4800+19683 e^{4 x}-25920 x+54432 x^2-52488 x^3+19683 x^4+e^{3 x} (-52488+78732 x)+e^{2 x} \left (54432-157464 x+118098 x^2\right )+e^x \left (-25920+108864 x-157464 x^2+78732 x^3\right )} \, dx=-\frac {x^{2} + x + 3}{3 \, {\left (81 \, x^{2} + 54 \, {\left (3 \, x - 2\right )} e^{x} - 108 \, x + 81 \, e^{\left (2 \, x\right )} + 40\right )}} \]
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Time = 0.10 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {-364+406 x+189 x^2+e^{2 x} \left (405+162 x^2\right )+e^x \left (270+594 x-108 x^2+162 x^3\right )}{4800+19683 e^{4 x}-25920 x+54432 x^2-52488 x^3+19683 x^4+e^{3 x} (-52488+78732 x)+e^{2 x} \left (54432-157464 x+118098 x^2\right )+e^x \left (-25920+108864 x-157464 x^2+78732 x^3\right )} \, dx=\frac {- x^{2} - x - 3}{243 x^{2} - 324 x + \left (486 x - 324\right ) e^{x} + 243 e^{2 x} + 120} \]
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Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13 \[ \int \frac {-364+406 x+189 x^2+e^{2 x} \left (405+162 x^2\right )+e^x \left (270+594 x-108 x^2+162 x^3\right )}{4800+19683 e^{4 x}-25920 x+54432 x^2-52488 x^3+19683 x^4+e^{3 x} (-52488+78732 x)+e^{2 x} \left (54432-157464 x+118098 x^2\right )+e^x \left (-25920+108864 x-157464 x^2+78732 x^3\right )} \, dx=-\frac {x^{2} + x + 3}{3 \, {\left (81 \, x^{2} + 54 \, {\left (3 \, x - 2\right )} e^{x} - 108 \, x + 81 \, e^{\left (2 \, x\right )} + 40\right )}} \]
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Time = 0.39 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13 \[ \int \frac {-364+406 x+189 x^2+e^{2 x} \left (405+162 x^2\right )+e^x \left (270+594 x-108 x^2+162 x^3\right )}{4800+19683 e^{4 x}-25920 x+54432 x^2-52488 x^3+19683 x^4+e^{3 x} (-52488+78732 x)+e^{2 x} \left (54432-157464 x+118098 x^2\right )+e^x \left (-25920+108864 x-157464 x^2+78732 x^3\right )} \, dx=-\frac {x^{2} + x + 3}{3 \, {\left (81 \, x^{2} + 162 \, x e^{x} - 108 \, x + 81 \, e^{\left (2 \, x\right )} - 108 \, e^{x} + 40\right )}} \]
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Timed out. \[ \int \frac {-364+406 x+189 x^2+e^{2 x} \left (405+162 x^2\right )+e^x \left (270+594 x-108 x^2+162 x^3\right )}{4800+19683 e^{4 x}-25920 x+54432 x^2-52488 x^3+19683 x^4+e^{3 x} (-52488+78732 x)+e^{2 x} \left (54432-157464 x+118098 x^2\right )+e^x \left (-25920+108864 x-157464 x^2+78732 x^3\right )} \, dx=\int \frac {406\,x+{\mathrm {e}}^{2\,x}\,\left (162\,x^2+405\right )+189\,x^2+{\mathrm {e}}^x\,\left (162\,x^3-108\,x^2+594\,x+270\right )-364}{19683\,{\mathrm {e}}^{4\,x}-25920\,x+{\mathrm {e}}^{2\,x}\,\left (118098\,x^2-157464\,x+54432\right )+{\mathrm {e}}^{3\,x}\,\left (78732\,x-52488\right )+54432\,x^2-52488\,x^3+19683\,x^4+{\mathrm {e}}^x\,\left (78732\,x^3-157464\,x^2+108864\,x-25920\right )+4800} \,d x \]
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