Integrand size = 145, antiderivative size = 30 \[ \int \frac {2205-13230 x+31080 x^2-36120 x^3+20900 x^4-4840 x^5+e^x \left (-105+430 x-645 x^2+430 x^3-110 x^4\right )}{441 x^2-2646 x^3+6657 x^4-8988 x^5+6868 x^6-2816 x^7+484 x^8+e^{2 x} \left (1-4 x+6 x^2-4 x^3+x^4\right )+e^x \left (-42 x+210 x^2-422 x^3+426 x^4-216 x^5+44 x^6\right )} \, dx=\frac {5}{-x+x^2+\frac {e^x}{21+\frac {x^2}{(1-x)^2}}} \]
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\[ \int \frac {2205-13230 x+31080 x^2-36120 x^3+20900 x^4-4840 x^5+e^x \left (-105+430 x-645 x^2+430 x^3-110 x^4\right )}{441 x^2-2646 x^3+6657 x^4-8988 x^5+6868 x^6-2816 x^7+484 x^8+e^{2 x} \left (1-4 x+6 x^2-4 x^3+x^4\right )+e^x \left (-42 x+210 x^2-422 x^3+426 x^4-216 x^5+44 x^6\right )} \, dx=\int \frac {2205-13230 x+31080 x^2-36120 x^3+20900 x^4-4840 x^5+e^x \left (-105+430 x-645 x^2+430 x^3-110 x^4\right )}{441 x^2-2646 x^3+6657 x^4-8988 x^5+6868 x^6-2816 x^7+484 x^8+e^{2 x} \left (1-4 x+6 x^2-4 x^3+x^4\right )+e^x \left (-42 x+210 x^2-422 x^3+426 x^4-216 x^5+44 x^6\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {5 \left (-\left ((-1+2 x) \left (21-42 x+22 x^2\right )^2\right )-e^x \left (21-86 x+129 x^2-86 x^3+22 x^4\right )\right )}{(1-x)^2 \left (e^x (-1+x)+x \left (21-42 x+22 x^2\right )\right )^2} \, dx \\ & = 5 \int \frac {-\left ((-1+2 x) \left (21-42 x+22 x^2\right )^2\right )-e^x \left (21-86 x+129 x^2-86 x^3+22 x^4\right )}{(1-x)^2 \left (e^x (-1+x)+x \left (21-42 x+22 x^2\right )\right )^2} \, dx \\ & = 5 \int \left (-\frac {-21+65 x-64 x^2+22 x^3}{(-1+x)^2 \left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )}+\frac {441-3087 x+8463 x^2-11760 x^3+8760 x^4-3300 x^5+484 x^6}{(-1+x)^2 \left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )^2}\right ) \, dx \\ & = -\left (5 \int \frac {-21+65 x-64 x^2+22 x^3}{(-1+x)^2 \left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )} \, dx\right )+5 \int \frac {441-3087 x+8463 x^2-11760 x^3+8760 x^4-3300 x^5+484 x^6}{(-1+x)^2 \left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )^2} \, dx \\ & = 5 \int \left (\frac {443}{\left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )^2}+\frac {1}{(-1+x)^2 \left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )^2}+\frac {3}{(-1+x) \left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )^2}-\frac {2204 x}{\left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )^2}+\frac {3612 x^2}{\left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )^2}-\frac {2332 x^3}{\left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )^2}+\frac {484 x^4}{\left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )^2}\right ) \, dx-5 \int \left (-\frac {20}{-e^x+21 x+e^x x-42 x^2+22 x^3}+\frac {2}{(-1+x)^2 \left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )}+\frac {3}{(-1+x) \left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )}+\frac {22 x}{-e^x+21 x+e^x x-42 x^2+22 x^3}\right ) \, dx \\ & = 5 \int \frac {1}{(-1+x)^2 \left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )^2} \, dx-10 \int \frac {1}{(-1+x)^2 \left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )} \, dx+15 \int \frac {1}{(-1+x) \left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )^2} \, dx-15 \int \frac {1}{(-1+x) \left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )} \, dx+100 \int \frac {1}{-e^x+21 x+e^x x-42 x^2+22 x^3} \, dx-110 \int \frac {x}{-e^x+21 x+e^x x-42 x^2+22 x^3} \, dx+2215 \int \frac {1}{\left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )^2} \, dx+2420 \int \frac {x^4}{\left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )^2} \, dx-11020 \int \frac {x}{\left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )^2} \, dx-11660 \int \frac {x^3}{\left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )^2} \, dx+18060 \int \frac {x^2}{\left (-e^x+21 x+e^x x-42 x^2+22 x^3\right )^2} \, dx \\ \end{align*}
Time = 4.52 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.30 \[ \int \frac {2205-13230 x+31080 x^2-36120 x^3+20900 x^4-4840 x^5+e^x \left (-105+430 x-645 x^2+430 x^3-110 x^4\right )}{441 x^2-2646 x^3+6657 x^4-8988 x^5+6868 x^6-2816 x^7+484 x^8+e^{2 x} \left (1-4 x+6 x^2-4 x^3+x^4\right )+e^x \left (-42 x+210 x^2-422 x^3+426 x^4-216 x^5+44 x^6\right )} \, dx=-\frac {5 \left (-21+42 x-22 x^2\right )}{(-1+x) \left (e^x (-1+x)+x \left (21-42 x+22 x^2\right )\right )} \]
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Time = 0.18 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40
method | result | size |
risch | \(\frac {110 x^{2}-210 x +105}{\left (-1+x \right ) \left (22 x^{3}-42 x^{2}+{\mathrm e}^{x} x +21 x -{\mathrm e}^{x}\right )}\) | \(42\) |
norman | \(\frac {110 x^{2}-210 x +105}{22 x^{4}+{\mathrm e}^{x} x^{2}-64 x^{3}-2 \,{\mathrm e}^{x} x +63 x^{2}+{\mathrm e}^{x}-21 x}\) | \(46\) |
parallelrisch | \(\frac {110 x^{2}-210 x +105}{22 x^{4}+{\mathrm e}^{x} x^{2}-64 x^{3}-2 \,{\mathrm e}^{x} x +63 x^{2}+{\mathrm e}^{x}-21 x}\) | \(46\) |
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Time = 0.26 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.47 \[ \int \frac {2205-13230 x+31080 x^2-36120 x^3+20900 x^4-4840 x^5+e^x \left (-105+430 x-645 x^2+430 x^3-110 x^4\right )}{441 x^2-2646 x^3+6657 x^4-8988 x^5+6868 x^6-2816 x^7+484 x^8+e^{2 x} \left (1-4 x+6 x^2-4 x^3+x^4\right )+e^x \left (-42 x+210 x^2-422 x^3+426 x^4-216 x^5+44 x^6\right )} \, dx=\frac {5 \, {\left (22 \, x^{2} - 42 \, x + 21\right )}}{22 \, x^{4} - 64 \, x^{3} + 63 \, x^{2} + {\left (x^{2} - 2 \, x + 1\right )} e^{x} - 21 \, x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (19) = 38\).
Time = 0.14 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.30 \[ \int \frac {2205-13230 x+31080 x^2-36120 x^3+20900 x^4-4840 x^5+e^x \left (-105+430 x-645 x^2+430 x^3-110 x^4\right )}{441 x^2-2646 x^3+6657 x^4-8988 x^5+6868 x^6-2816 x^7+484 x^8+e^{2 x} \left (1-4 x+6 x^2-4 x^3+x^4\right )+e^x \left (-42 x+210 x^2-422 x^3+426 x^4-216 x^5+44 x^6\right )} \, dx=\frac {110 x^{2} - 210 x + 105}{22 x^{4} - 64 x^{3} + 63 x^{2} - 21 x + \left (x^{2} - 2 x + 1\right ) e^{x}} \]
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Time = 0.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.47 \[ \int \frac {2205-13230 x+31080 x^2-36120 x^3+20900 x^4-4840 x^5+e^x \left (-105+430 x-645 x^2+430 x^3-110 x^4\right )}{441 x^2-2646 x^3+6657 x^4-8988 x^5+6868 x^6-2816 x^7+484 x^8+e^{2 x} \left (1-4 x+6 x^2-4 x^3+x^4\right )+e^x \left (-42 x+210 x^2-422 x^3+426 x^4-216 x^5+44 x^6\right )} \, dx=\frac {5 \, {\left (22 \, x^{2} - 42 \, x + 21\right )}}{22 \, x^{4} - 64 \, x^{3} + 63 \, x^{2} + {\left (x^{2} - 2 \, x + 1\right )} e^{x} - 21 \, x} \]
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Time = 0.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.53 \[ \int \frac {2205-13230 x+31080 x^2-36120 x^3+20900 x^4-4840 x^5+e^x \left (-105+430 x-645 x^2+430 x^3-110 x^4\right )}{441 x^2-2646 x^3+6657 x^4-8988 x^5+6868 x^6-2816 x^7+484 x^8+e^{2 x} \left (1-4 x+6 x^2-4 x^3+x^4\right )+e^x \left (-42 x+210 x^2-422 x^3+426 x^4-216 x^5+44 x^6\right )} \, dx=\frac {5 \, {\left (22 \, x^{2} - 42 \, x + 21\right )}}{22 \, x^{4} - 64 \, x^{3} + x^{2} e^{x} + 63 \, x^{2} - 2 \, x e^{x} - 21 \, x + e^{x}} \]
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Timed out. \[ \int \frac {2205-13230 x+31080 x^2-36120 x^3+20900 x^4-4840 x^5+e^x \left (-105+430 x-645 x^2+430 x^3-110 x^4\right )}{441 x^2-2646 x^3+6657 x^4-8988 x^5+6868 x^6-2816 x^7+484 x^8+e^{2 x} \left (1-4 x+6 x^2-4 x^3+x^4\right )+e^x \left (-42 x+210 x^2-422 x^3+426 x^4-216 x^5+44 x^6\right )} \, dx=\int -\frac {13230\,x+{\mathrm {e}}^x\,\left (110\,x^4-430\,x^3+645\,x^2-430\,x+105\right )-31080\,x^2+36120\,x^3-20900\,x^4+4840\,x^5-2205}{{\mathrm {e}}^{2\,x}\,\left (x^4-4\,x^3+6\,x^2-4\,x+1\right )-{\mathrm {e}}^x\,\left (-44\,x^6+216\,x^5-426\,x^4+422\,x^3-210\,x^2+42\,x\right )+441\,x^2-2646\,x^3+6657\,x^4-8988\,x^5+6868\,x^6-2816\,x^7+484\,x^8} \,d x \]
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