Integrand size = 248, antiderivative size = 25 \[ \int \frac {\left (-12+3 e^{\frac {x}{3+x^2+x^3}}+2 x\right ) \left (36+24 x^2+24 x^3+4 x^4+8 x^5+4 x^6+e^{\frac {x}{3+x^2+x^3}} \left (-9+3 x-6 x^2-7 x^3-3 x^4-2 x^5-x^6\right )\right )}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (432-180 x+306 x^2+168 x^3-60 x^4+88 x^5+10 x^6-16 x^7+2 x^8+e^{\frac {2 x}{3+x^2+x^3}} \left (27+18 x^2+18 x^3+3 x^4+6 x^5+3 x^6\right )+e^{\frac {x}{3+x^2+x^3}} \left (-216+45 x-144 x^2-114 x^3+6 x^4-43 x^5-14 x^6+5 x^7\right )\right )} \, dx=3-\frac {x}{-4+e^{\frac {x}{3+x^2 (1+x)}}+x} \]
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\[ \int \frac {\left (-12+3 e^{\frac {x}{3+x^2+x^3}}+2 x\right ) \left (36+24 x^2+24 x^3+4 x^4+8 x^5+4 x^6+e^{\frac {x}{3+x^2+x^3}} \left (-9+3 x-6 x^2-7 x^3-3 x^4-2 x^5-x^6\right )\right )}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (432-180 x+306 x^2+168 x^3-60 x^4+88 x^5+10 x^6-16 x^7+2 x^8+e^{\frac {2 x}{3+x^2+x^3}} \left (27+18 x^2+18 x^3+3 x^4+6 x^5+3 x^6\right )+e^{\frac {x}{3+x^2+x^3}} \left (-216+45 x-144 x^2-114 x^3+6 x^4-43 x^5-14 x^6+5 x^7\right )\right )} \, dx=\int \frac {\left (-12+3 e^{\frac {x}{3+x^2+x^3}}+2 x\right ) \left (36+24 x^2+24 x^3+4 x^4+8 x^5+4 x^6+e^{\frac {x}{3+x^2+x^3}} \left (-9+3 x-6 x^2-7 x^3-3 x^4-2 x^5-x^6\right )\right )}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (432-180 x+306 x^2+168 x^3-60 x^4+88 x^5+10 x^6-16 x^7+2 x^8+e^{\frac {2 x}{3+x^2+x^3}} \left (27+18 x^2+18 x^3+3 x^4+6 x^5+3 x^6\right )+e^{\frac {x}{3+x^2+x^3}} \left (-216+45 x-144 x^2-114 x^3+6 x^4-43 x^5-14 x^6+5 x^7\right )\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {4 \left (3+x^2+x^3\right )^2-e^{\frac {x}{3+x^2+x^3}} \left (9-3 x+6 x^2+7 x^3+3 x^4+2 x^5+x^6\right )}{\left (4-e^{\frac {x}{3+x^2+x^3}}-x\right )^2 \left (3+x^2+x^3\right )^2} \, dx \\ & = \int \left (\frac {x \left (21-3 x+2 x^2-x^3+3 x^4+2 x^5+x^6\right )}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2 \left (3+x^2+x^3\right )^2}-\frac {9-3 x+6 x^2+7 x^3+3 x^4+2 x^5+x^6}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (3+x^2+x^3\right )^2}\right ) \, dx \\ & = \int \frac {x \left (21-3 x+2 x^2-x^3+3 x^4+2 x^5+x^6\right )}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2 \left (3+x^2+x^3\right )^2} \, dx-\int \frac {9-3 x+6 x^2+7 x^3+3 x^4+2 x^5+x^6}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (3+x^2+x^3\right )^2} \, dx \\ & = -\int \left (\frac {1}{-4+e^{\frac {x}{3+x^2+x^3}}+x}+\frac {3-9 x+x^2}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (3+x^2+x^3\right )^2}+\frac {-1+2 x}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (3+x^2+x^3\right )}\right ) \, dx+\int \left (\frac {x}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2}-\frac {15-39 x+14 x^2}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2 \left (3+x^2+x^3\right )^2}+\frac {5-9 x+2 x^2}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2 \left (3+x^2+x^3\right )}\right ) \, dx \\ & = \int \frac {x}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2} \, dx-\int \frac {1}{-4+e^{\frac {x}{3+x^2+x^3}}+x} \, dx-\int \frac {3-9 x+x^2}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (3+x^2+x^3\right )^2} \, dx-\int \frac {15-39 x+14 x^2}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2 \left (3+x^2+x^3\right )^2} \, dx-\int \frac {-1+2 x}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (3+x^2+x^3\right )} \, dx+\int \frac {5-9 x+2 x^2}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2 \left (3+x^2+x^3\right )} \, dx \\ & = \int \frac {x}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2} \, dx-\int \frac {1}{-4+e^{\frac {x}{3+x^2+x^3}}+x} \, dx-\int \left (\frac {15}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2 \left (3+x^2+x^3\right )^2}-\frac {39 x}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2 \left (3+x^2+x^3\right )^2}+\frac {14 x^2}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2 \left (3+x^2+x^3\right )^2}\right ) \, dx-\int \left (\frac {3}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (3+x^2+x^3\right )^2}-\frac {9 x}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (3+x^2+x^3\right )^2}+\frac {x^2}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (3+x^2+x^3\right )^2}\right ) \, dx+\int \left (\frac {5}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2 \left (3+x^2+x^3\right )}-\frac {9 x}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2 \left (3+x^2+x^3\right )}+\frac {2 x^2}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2 \left (3+x^2+x^3\right )}\right ) \, dx-\int \left (-\frac {1}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (3+x^2+x^3\right )}+\frac {2 x}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (3+x^2+x^3\right )}\right ) \, dx \\ & = 2 \int \frac {x^2}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2 \left (3+x^2+x^3\right )} \, dx-2 \int \frac {x}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (3+x^2+x^3\right )} \, dx-3 \int \frac {1}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (3+x^2+x^3\right )^2} \, dx+5 \int \frac {1}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2 \left (3+x^2+x^3\right )} \, dx+9 \int \frac {x}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (3+x^2+x^3\right )^2} \, dx-9 \int \frac {x}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2 \left (3+x^2+x^3\right )} \, dx-14 \int \frac {x^2}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2 \left (3+x^2+x^3\right )^2} \, dx-15 \int \frac {1}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2 \left (3+x^2+x^3\right )^2} \, dx+39 \int \frac {x}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2 \left (3+x^2+x^3\right )^2} \, dx+\int \frac {x}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right )^2} \, dx-\int \frac {1}{-4+e^{\frac {x}{3+x^2+x^3}}+x} \, dx-\int \frac {x^2}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (3+x^2+x^3\right )^2} \, dx+\int \frac {1}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (3+x^2+x^3\right )} \, dx \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {\left (-12+3 e^{\frac {x}{3+x^2+x^3}}+2 x\right ) \left (36+24 x^2+24 x^3+4 x^4+8 x^5+4 x^6+e^{\frac {x}{3+x^2+x^3}} \left (-9+3 x-6 x^2-7 x^3-3 x^4-2 x^5-x^6\right )\right )}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (432-180 x+306 x^2+168 x^3-60 x^4+88 x^5+10 x^6-16 x^7+2 x^8+e^{\frac {2 x}{3+x^2+x^3}} \left (27+18 x^2+18 x^3+3 x^4+6 x^5+3 x^6\right )+e^{\frac {x}{3+x^2+x^3}} \left (-216+45 x-144 x^2-114 x^3+6 x^4-43 x^5-14 x^6+5 x^7\right )\right )} \, dx=-\frac {x}{-4+e^{\frac {x}{3+x^2+x^3}}+x} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.46 (sec) , antiderivative size = 197, normalized size of antiderivative = 7.88
\[-\frac {x \,{\mathrm e}^{-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (\frac {3 \,{\mathrm e}^{\frac {x}{x^{3}+x^{2}+3}}}{2}+x -6\right )}{{\mathrm e}^{\frac {x}{x^{3}+x^{2}+3}}+x -4}\right ) \left (-\operatorname {csgn}\left (\frac {i \left (\frac {3 \,{\mathrm e}^{\frac {x}{x^{3}+x^{2}+3}}}{2}+x -6\right )}{{\mathrm e}^{\frac {x}{x^{3}+x^{2}+3}}+x -4}\right )+\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{\frac {x}{x^{3}+x^{2}+3}}+x -4}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i \left (\frac {3 \,{\mathrm e}^{\frac {x}{x^{3}+x^{2}+3}}}{2}+x -6\right )}{{\mathrm e}^{\frac {x}{x^{3}+x^{2}+3}}+x -4}\right )+\operatorname {csgn}\left (i \left (\frac {3 \,{\mathrm e}^{\frac {x}{x^{3}+x^{2}+3}}}{2}+x -6\right )\right )\right )}{2}}}{{\mathrm e}^{\frac {x}{x^{3}+x^{2}+3}}+x -4}\]
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Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {\left (-12+3 e^{\frac {x}{3+x^2+x^3}}+2 x\right ) \left (36+24 x^2+24 x^3+4 x^4+8 x^5+4 x^6+e^{\frac {x}{3+x^2+x^3}} \left (-9+3 x-6 x^2-7 x^3-3 x^4-2 x^5-x^6\right )\right )}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (432-180 x+306 x^2+168 x^3-60 x^4+88 x^5+10 x^6-16 x^7+2 x^8+e^{\frac {2 x}{3+x^2+x^3}} \left (27+18 x^2+18 x^3+3 x^4+6 x^5+3 x^6\right )+e^{\frac {x}{3+x^2+x^3}} \left (-216+45 x-144 x^2-114 x^3+6 x^4-43 x^5-14 x^6+5 x^7\right )\right )} \, dx=-\frac {x}{x + e^{\left (\frac {x}{x^{3} + x^{2} + 3}\right )} - 4} \]
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Time = 0.18 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.68 \[ \int \frac {\left (-12+3 e^{\frac {x}{3+x^2+x^3}}+2 x\right ) \left (36+24 x^2+24 x^3+4 x^4+8 x^5+4 x^6+e^{\frac {x}{3+x^2+x^3}} \left (-9+3 x-6 x^2-7 x^3-3 x^4-2 x^5-x^6\right )\right )}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (432-180 x+306 x^2+168 x^3-60 x^4+88 x^5+10 x^6-16 x^7+2 x^8+e^{\frac {2 x}{3+x^2+x^3}} \left (27+18 x^2+18 x^3+3 x^4+6 x^5+3 x^6\right )+e^{\frac {x}{3+x^2+x^3}} \left (-216+45 x-144 x^2-114 x^3+6 x^4-43 x^5-14 x^6+5 x^7\right )\right )} \, dx=- \frac {x}{x + e^{\frac {x}{x^{3} + x^{2} + 3}} - 4} \]
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Time = 0.29 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {\left (-12+3 e^{\frac {x}{3+x^2+x^3}}+2 x\right ) \left (36+24 x^2+24 x^3+4 x^4+8 x^5+4 x^6+e^{\frac {x}{3+x^2+x^3}} \left (-9+3 x-6 x^2-7 x^3-3 x^4-2 x^5-x^6\right )\right )}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (432-180 x+306 x^2+168 x^3-60 x^4+88 x^5+10 x^6-16 x^7+2 x^8+e^{\frac {2 x}{3+x^2+x^3}} \left (27+18 x^2+18 x^3+3 x^4+6 x^5+3 x^6\right )+e^{\frac {x}{3+x^2+x^3}} \left (-216+45 x-144 x^2-114 x^3+6 x^4-43 x^5-14 x^6+5 x^7\right )\right )} \, dx=-\frac {x}{x + e^{\left (\frac {x}{x^{3} + x^{2} + 3}\right )} - 4} \]
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Time = 0.36 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {\left (-12+3 e^{\frac {x}{3+x^2+x^3}}+2 x\right ) \left (36+24 x^2+24 x^3+4 x^4+8 x^5+4 x^6+e^{\frac {x}{3+x^2+x^3}} \left (-9+3 x-6 x^2-7 x^3-3 x^4-2 x^5-x^6\right )\right )}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (432-180 x+306 x^2+168 x^3-60 x^4+88 x^5+10 x^6-16 x^7+2 x^8+e^{\frac {2 x}{3+x^2+x^3}} \left (27+18 x^2+18 x^3+3 x^4+6 x^5+3 x^6\right )+e^{\frac {x}{3+x^2+x^3}} \left (-216+45 x-144 x^2-114 x^3+6 x^4-43 x^5-14 x^6+5 x^7\right )\right )} \, dx=-\frac {x}{x + e^{\left (\frac {x}{x^{3} + x^{2} + 3}\right )} - 4} \]
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Time = 9.47 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {\left (-12+3 e^{\frac {x}{3+x^2+x^3}}+2 x\right ) \left (36+24 x^2+24 x^3+4 x^4+8 x^5+4 x^6+e^{\frac {x}{3+x^2+x^3}} \left (-9+3 x-6 x^2-7 x^3-3 x^4-2 x^5-x^6\right )\right )}{\left (-4+e^{\frac {x}{3+x^2+x^3}}+x\right ) \left (432-180 x+306 x^2+168 x^3-60 x^4+88 x^5+10 x^6-16 x^7+2 x^8+e^{\frac {2 x}{3+x^2+x^3}} \left (27+18 x^2+18 x^3+3 x^4+6 x^5+3 x^6\right )+e^{\frac {x}{3+x^2+x^3}} \left (-216+45 x-144 x^2-114 x^3+6 x^4-43 x^5-14 x^6+5 x^7\right )\right )} \, dx=-\frac {x}{x+{\mathrm {e}}^{\frac {x}{x^3+x^2+3}}-4} \]
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