Integrand size = 201, antiderivative size = 37 \[ \int \frac {-60 x-21 x^2+5 x^3+135 x^4+135 x^5+99 x^6+59 x^7+18 x^8+2 x^9+e^{2 x} \left (135 x^2+135 x^3+99 x^4+59 x^5+18 x^6+2 x^7\right )+e^x \left (45+34 x-2 x^2-271 x^3-270 x^4-198 x^5-118 x^6-36 x^7-4 x^8\right )}{54 x^6+54 x^7+18 x^8+2 x^9+e^{2 x} \left (54 x^4+54 x^5+18 x^6+2 x^7\right )+e^x \left (-108 x^5-108 x^6-36 x^7-4 x^8\right )} \, dx=4+x+\frac {(5-x) \left (-x+\frac {1}{x (3+x)^2 \left (-e^x+x\right )}\right )}{2 x^2} \]
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\[ \int \frac {-60 x-21 x^2+5 x^3+135 x^4+135 x^5+99 x^6+59 x^7+18 x^8+2 x^9+e^{2 x} \left (135 x^2+135 x^3+99 x^4+59 x^5+18 x^6+2 x^7\right )+e^x \left (45+34 x-2 x^2-271 x^3-270 x^4-198 x^5-118 x^6-36 x^7-4 x^8\right )}{54 x^6+54 x^7+18 x^8+2 x^9+e^{2 x} \left (54 x^4+54 x^5+18 x^6+2 x^7\right )+e^x \left (-108 x^5-108 x^6-36 x^7-4 x^8\right )} \, dx=\int \frac {-60 x-21 x^2+5 x^3+135 x^4+135 x^5+99 x^6+59 x^7+18 x^8+2 x^9+e^{2 x} \left (135 x^2+135 x^3+99 x^4+59 x^5+18 x^6+2 x^7\right )+e^x \left (45+34 x-2 x^2-271 x^3-270 x^4-198 x^5-118 x^6-36 x^7-4 x^8\right )}{54 x^6+54 x^7+18 x^8+2 x^9+e^{2 x} \left (54 x^4+54 x^5+18 x^6+2 x^7\right )+e^x \left (-108 x^5-108 x^6-36 x^7-4 x^8\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{2 x} x^2 (3+x)^3 \left (5+2 x^2\right )+x \left (-60-21 x+5 x^2+135 x^3+135 x^4+99 x^5+59 x^6+18 x^7+2 x^8\right )-e^x \left (-45-34 x+2 x^2+271 x^3+270 x^4+198 x^5+118 x^6+36 x^7+4 x^8\right )}{2 \left (e^x-x\right )^2 x^4 (3+x)^3} \, dx \\ & = \frac {1}{2} \int \frac {e^{2 x} x^2 (3+x)^3 \left (5+2 x^2\right )+x \left (-60-21 x+5 x^2+135 x^3+135 x^4+99 x^5+59 x^6+18 x^7+2 x^8\right )-e^x \left (-45-34 x+2 x^2+271 x^3+270 x^4+198 x^5+118 x^6+36 x^7+4 x^8\right )}{\left (e^x-x\right )^2 x^4 (3+x)^3} \, dx \\ & = \frac {1}{2} \int \left (-\frac {5-6 x+x^2}{\left (e^x-x\right )^2 x^3 (3+x)^2}+\frac {5+2 x^2}{x^2}-\frac {-45-34 x+2 x^2+x^3}{\left (e^x-x\right ) x^4 (3+x)^3}\right ) \, dx \\ & = -\left (\frac {1}{2} \int \frac {5-6 x+x^2}{\left (e^x-x\right )^2 x^3 (3+x)^2} \, dx\right )+\frac {1}{2} \int \frac {5+2 x^2}{x^2} \, dx-\frac {1}{2} \int \frac {-45-34 x+2 x^2+x^3}{\left (e^x-x\right ) x^4 (3+x)^3} \, dx \\ & = \frac {1}{2} \int \left (2+\frac {5}{x^2}\right ) \, dx-\frac {1}{2} \int \left (\frac {5}{9 \left (e^x-x\right )^2 x^3}-\frac {28}{27 \left (e^x-x\right )^2 x^2}+\frac {20}{27 \left (e^x-x\right )^2 x}-\frac {32}{27 \left (e^x-x\right )^2 (3+x)^2}-\frac {20}{27 \left (e^x-x\right )^2 (3+x)}\right ) \, dx-\frac {1}{2} \int \left (-\frac {5}{3 \left (e^x-x\right ) x^4}+\frac {11}{27 \left (e^x-x\right ) x^3}+\frac {2}{9 \left (e^x-x\right ) x^2}-\frac {7}{27 \left (e^x-x\right ) x}+\frac {16}{27 \left (e^x-x\right ) (3+x)^3}+\frac {5}{9 \left (e^x-x\right ) (3+x)^2}+\frac {7}{27 \left (e^x-x\right ) (3+x)}\right ) \, dx \\ & = -\frac {5}{2 x}+x-\frac {1}{9} \int \frac {1}{\left (e^x-x\right ) x^2} \, dx+\frac {7}{54} \int \frac {1}{\left (e^x-x\right ) x} \, dx-\frac {7}{54} \int \frac {1}{\left (e^x-x\right ) (3+x)} \, dx-\frac {11}{54} \int \frac {1}{\left (e^x-x\right ) x^3} \, dx-\frac {5}{18} \int \frac {1}{\left (e^x-x\right )^2 x^3} \, dx-\frac {5}{18} \int \frac {1}{\left (e^x-x\right ) (3+x)^2} \, dx-\frac {8}{27} \int \frac {1}{\left (e^x-x\right ) (3+x)^3} \, dx-\frac {10}{27} \int \frac {1}{\left (e^x-x\right )^2 x} \, dx+\frac {10}{27} \int \frac {1}{\left (e^x-x\right )^2 (3+x)} \, dx+\frac {14}{27} \int \frac {1}{\left (e^x-x\right )^2 x^2} \, dx+\frac {16}{27} \int \frac {1}{\left (e^x-x\right )^2 (3+x)^2} \, dx+\frac {5}{6} \int \frac {1}{\left (e^x-x\right ) x^4} \, dx \\ \end{align*}
Time = 5.49 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.92 \[ \int \frac {-60 x-21 x^2+5 x^3+135 x^4+135 x^5+99 x^6+59 x^7+18 x^8+2 x^9+e^{2 x} \left (135 x^2+135 x^3+99 x^4+59 x^5+18 x^6+2 x^7\right )+e^x \left (45+34 x-2 x^2-271 x^3-270 x^4-198 x^5-118 x^6-36 x^7-4 x^8\right )}{54 x^6+54 x^7+18 x^8+2 x^9+e^{2 x} \left (54 x^4+54 x^5+18 x^6+2 x^7\right )+e^x \left (-108 x^5-108 x^6-36 x^7-4 x^8\right )} \, dx=\frac {1}{2} \left (-\frac {5}{x}+2 x+\frac {-5+x}{\left (e^x-x\right ) x^3 (3+x)^2}\right ) \]
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Time = 0.17 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.92
method | result | size |
risch | \(x -\frac {5}{2 x}-\frac {-5+x}{2 x^{3} \left (x^{2}+6 x +9\right ) \left (x -{\mathrm e}^{x}\right )}\) | \(34\) |
norman | \(\frac {\frac {5}{2}+x^{7}-69 x^{4}-\frac {59 x^{5}}{2}+69 \,{\mathrm e}^{x} x^{3}+\frac {59 \,{\mathrm e}^{x} x^{4}}{2}-\frac {x}{2}-\frac {45 x^{3}}{2}-x^{6} {\mathrm e}^{x}+\frac {45 \,{\mathrm e}^{x} x^{2}}{2}}{x^{3} \left (x -{\mathrm e}^{x}\right ) \left (3+x \right )^{2}}\) | \(69\) |
parallelrisch | \(\frac {2 x^{7}-2 x^{6} {\mathrm e}^{x}+5-59 x^{5}+59 \,{\mathrm e}^{x} x^{4}-138 x^{4}+138 \,{\mathrm e}^{x} x^{3}-45 x^{3}+45 \,{\mathrm e}^{x} x^{2}-x}{2 x^{3} \left (x^{3}-{\mathrm e}^{x} x^{2}+6 x^{2}-6 \,{\mathrm e}^{x} x +9 x -9 \,{\mathrm e}^{x}\right )}\) | \(89\) |
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Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (31) = 62\).
Time = 0.26 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.59 \[ \int \frac {-60 x-21 x^2+5 x^3+135 x^4+135 x^5+99 x^6+59 x^7+18 x^8+2 x^9+e^{2 x} \left (135 x^2+135 x^3+99 x^4+59 x^5+18 x^6+2 x^7\right )+e^x \left (45+34 x-2 x^2-271 x^3-270 x^4-198 x^5-118 x^6-36 x^7-4 x^8\right )}{54 x^6+54 x^7+18 x^8+2 x^9+e^{2 x} \left (54 x^4+54 x^5+18 x^6+2 x^7\right )+e^x \left (-108 x^5-108 x^6-36 x^7-4 x^8\right )} \, dx=\frac {2 \, x^{7} + 12 \, x^{6} + 13 \, x^{5} - 30 \, x^{4} - 45 \, x^{3} - {\left (2 \, x^{6} + 12 \, x^{5} + 13 \, x^{4} - 30 \, x^{3} - 45 \, x^{2}\right )} e^{x} - x + 5}{2 \, {\left (x^{6} + 6 \, x^{5} + 9 \, x^{4} - {\left (x^{5} + 6 \, x^{4} + 9 \, x^{3}\right )} e^{x}\right )}} \]
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Time = 0.10 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.14 \[ \int \frac {-60 x-21 x^2+5 x^3+135 x^4+135 x^5+99 x^6+59 x^7+18 x^8+2 x^9+e^{2 x} \left (135 x^2+135 x^3+99 x^4+59 x^5+18 x^6+2 x^7\right )+e^x \left (45+34 x-2 x^2-271 x^3-270 x^4-198 x^5-118 x^6-36 x^7-4 x^8\right )}{54 x^6+54 x^7+18 x^8+2 x^9+e^{2 x} \left (54 x^4+54 x^5+18 x^6+2 x^7\right )+e^x \left (-108 x^5-108 x^6-36 x^7-4 x^8\right )} \, dx=x + \frac {x - 5}{- 2 x^{6} - 12 x^{5} - 18 x^{4} + \left (2 x^{5} + 12 x^{4} + 18 x^{3}\right ) e^{x}} - \frac {5}{2 x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (31) = 62\).
Time = 0.24 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.59 \[ \int \frac {-60 x-21 x^2+5 x^3+135 x^4+135 x^5+99 x^6+59 x^7+18 x^8+2 x^9+e^{2 x} \left (135 x^2+135 x^3+99 x^4+59 x^5+18 x^6+2 x^7\right )+e^x \left (45+34 x-2 x^2-271 x^3-270 x^4-198 x^5-118 x^6-36 x^7-4 x^8\right )}{54 x^6+54 x^7+18 x^8+2 x^9+e^{2 x} \left (54 x^4+54 x^5+18 x^6+2 x^7\right )+e^x \left (-108 x^5-108 x^6-36 x^7-4 x^8\right )} \, dx=\frac {2 \, x^{7} + 12 \, x^{6} + 13 \, x^{5} - 30 \, x^{4} - 45 \, x^{3} - {\left (2 \, x^{6} + 12 \, x^{5} + 13 \, x^{4} - 30 \, x^{3} - 45 \, x^{2}\right )} e^{x} - x + 5}{2 \, {\left (x^{6} + 6 \, x^{5} + 9 \, x^{4} - {\left (x^{5} + 6 \, x^{4} + 9 \, x^{3}\right )} e^{x}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (31) = 62\).
Time = 0.27 (sec) , antiderivative size = 104, normalized size of antiderivative = 2.81 \[ \int \frac {-60 x-21 x^2+5 x^3+135 x^4+135 x^5+99 x^6+59 x^7+18 x^8+2 x^9+e^{2 x} \left (135 x^2+135 x^3+99 x^4+59 x^5+18 x^6+2 x^7\right )+e^x \left (45+34 x-2 x^2-271 x^3-270 x^4-198 x^5-118 x^6-36 x^7-4 x^8\right )}{54 x^6+54 x^7+18 x^8+2 x^9+e^{2 x} \left (54 x^4+54 x^5+18 x^6+2 x^7\right )+e^x \left (-108 x^5-108 x^6-36 x^7-4 x^8\right )} \, dx=\frac {2 \, x^{7} - 2 \, x^{6} e^{x} + 12 \, x^{6} - 12 \, x^{5} e^{x} + 13 \, x^{5} - 13 \, x^{4} e^{x} - 30 \, x^{4} + 30 \, x^{3} e^{x} - 45 \, x^{3} + 45 \, x^{2} e^{x} - x + 5}{2 \, {\left (x^{6} - x^{5} e^{x} + 6 \, x^{5} - 6 \, x^{4} e^{x} + 9 \, x^{4} - 9 \, x^{3} e^{x}\right )}} \]
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Time = 8.65 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.22 \[ \int \frac {-60 x-21 x^2+5 x^3+135 x^4+135 x^5+99 x^6+59 x^7+18 x^8+2 x^9+e^{2 x} \left (135 x^2+135 x^3+99 x^4+59 x^5+18 x^6+2 x^7\right )+e^x \left (45+34 x-2 x^2-271 x^3-270 x^4-198 x^5-118 x^6-36 x^7-4 x^8\right )}{54 x^6+54 x^7+18 x^8+2 x^9+e^{2 x} \left (54 x^4+54 x^5+18 x^6+2 x^7\right )+e^x \left (-108 x^5-108 x^6-36 x^7-4 x^8\right )} \, dx=x-\frac {5}{2\,x}+\frac {-x^3+3\,x^2+13\,x-15}{2\,x^3\,\left (x-{\mathrm {e}}^x\right )\,\left (x-1\right )\,{\left (x+3\right )}^3} \]
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