Integrand size = 204, antiderivative size = 32 \[ \int \frac {-x^6+e^6 \left (32 x^3-32 x^4\right )+e^3 \left (8 x^4-12 x^5\right )+e^{3+x} \left (-4 x^3+3 x^4-x^5+e^6 \left (-32 x+32 x^2-16 x^3\right )+e^3 \left (-24 x^2+20 x^3-8 x^4\right )\right )}{e^{15+3 x} \left (-16+24 x-12 x^2+2 x^3\right )+e^{12+2 x} \left (-48 x^2+72 x^3-36 x^4+6 x^5\right )+e^{9+x} \left (-48 x^4+72 x^5-36 x^6+6 x^7\right )+e^6 \left (-16 x^6+24 x^7-12 x^8+2 x^9\right )} \, dx=\frac {x^2 \left (-4-\frac {x}{e^3}\right )^2}{(4-2 x)^2 \left (e^{3+x}+x^2\right )^2} \]
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\[ \int \frac {-x^6+e^6 \left (32 x^3-32 x^4\right )+e^3 \left (8 x^4-12 x^5\right )+e^{3+x} \left (-4 x^3+3 x^4-x^5+e^6 \left (-32 x+32 x^2-16 x^3\right )+e^3 \left (-24 x^2+20 x^3-8 x^4\right )\right )}{e^{15+3 x} \left (-16+24 x-12 x^2+2 x^3\right )+e^{12+2 x} \left (-48 x^2+72 x^3-36 x^4+6 x^5\right )+e^{9+x} \left (-48 x^4+72 x^5-36 x^6+6 x^7\right )+e^6 \left (-16 x^6+24 x^7-12 x^8+2 x^9\right )} \, dx=\int \frac {-x^6+e^6 \left (32 x^3-32 x^4\right )+e^3 \left (8 x^4-12 x^5\right )+e^{3+x} \left (-4 x^3+3 x^4-x^5+e^6 \left (-32 x+32 x^2-16 x^3\right )+e^3 \left (-24 x^2+20 x^3-8 x^4\right )\right )}{e^{15+3 x} \left (-16+24 x-12 x^2+2 x^3\right )+e^{12+2 x} \left (-48 x^2+72 x^3-36 x^4+6 x^5\right )+e^{9+x} \left (-48 x^4+72 x^5-36 x^6+6 x^7\right )+e^6 \left (-16 x^6+24 x^7-12 x^8+2 x^9\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {x \left (4 e^3+x\right ) \left (8 e^3 (-1+x) x^2+x^4+e^{3+x} x \left (4-3 x+x^2\right )+4 e^{6+x} \left (2-2 x+x^2\right )\right )}{2 e^6 (2-x)^3 \left (e^{3+x}+x^2\right )^3} \, dx \\ & = \frac {\int \frac {x \left (4 e^3+x\right ) \left (8 e^3 (-1+x) x^2+x^4+e^{3+x} x \left (4-3 x+x^2\right )+4 e^{6+x} \left (2-2 x+x^2\right )\right )}{(2-x)^3 \left (e^{3+x}+x^2\right )^3} \, dx}{2 e^6} \\ & = \frac {\int \left (\frac {x^3 \left (4 e^3+x\right )^2}{(-2+x) \left (e^{3+x}+x^2\right )^3}+\frac {x \left (32 e^6+8 e^3 \left (3-4 e^3\right ) x+4 \left (1-5 e^3+4 e^6\right ) x^2-\left (3-8 e^3\right ) x^3+x^4\right )}{(2-x)^3 \left (e^{3+x}+x^2\right )^2}\right ) \, dx}{2 e^6} \\ & = \frac {\int \frac {x^3 \left (4 e^3+x\right )^2}{(-2+x) \left (e^{3+x}+x^2\right )^3} \, dx}{2 e^6}+\frac {\int \frac {x \left (32 e^6+8 e^3 \left (3-4 e^3\right ) x+4 \left (1-5 e^3+4 e^6\right ) x^2-\left (3-8 e^3\right ) x^3+x^4\right )}{(2-x)^3 \left (e^{3+x}+x^2\right )^2} \, dx}{2 e^6} \\ & = \frac {\int \left (\frac {16 \left (1+2 e^3\right )^2}{\left (e^{3+x}+x^2\right )^3}+\frac {32 \left (1+2 e^3\right )^2}{(-2+x) \left (e^{3+x}+x^2\right )^3}+\frac {8 \left (1+2 e^3\right )^2 x}{\left (e^{3+x}+x^2\right )^3}+\frac {4 \left (1+2 e^3\right )^2 x^2}{\left (e^{3+x}+x^2\right )^3}+\frac {2 \left (1+4 e^3\right ) x^3}{\left (e^{3+x}+x^2\right )^3}+\frac {x^4}{\left (e^{3+x}+x^2\right )^3}\right ) \, dx}{2 e^6}+\frac {\int \left (-\frac {2 \left (5+14 e^3+8 e^6\right )}{\left (e^{3+x}+x^2\right )^2}-\frac {16 \left (1+2 e^3\right )^2}{(-2+x)^3 \left (e^{3+x}+x^2\right )^2}-\frac {16 \left (2+7 e^3+6 e^6\right )}{(-2+x)^2 \left (e^{3+x}+x^2\right )^2}-\frac {32 \left (1+3 e^3+2 e^6\right )}{(-2+x) \left (e^{3+x}+x^2\right )^2}-\frac {\left (3+8 e^3\right ) x}{\left (e^{3+x}+x^2\right )^2}-\frac {x^2}{\left (e^{3+x}+x^2\right )^2}\right ) \, dx}{2 e^6} \\ & = \frac {\int \frac {x^4}{\left (e^{3+x}+x^2\right )^3} \, dx}{2 e^6}-\frac {\int \frac {x^2}{\left (e^{3+x}+x^2\right )^2} \, dx}{2 e^6}+\frac {\left (2 \left (1+2 e^3\right )^2\right ) \int \frac {x^2}{\left (e^{3+x}+x^2\right )^3} \, dx}{e^6}+\frac {\left (4 \left (1+2 e^3\right )^2\right ) \int \frac {x}{\left (e^{3+x}+x^2\right )^3} \, dx}{e^6}+\frac {\left (8 \left (1+2 e^3\right )^2\right ) \int \frac {1}{\left (e^{3+x}+x^2\right )^3} \, dx}{e^6}-\frac {\left (8 \left (1+2 e^3\right )^2\right ) \int \frac {1}{(-2+x)^3 \left (e^{3+x}+x^2\right )^2} \, dx}{e^6}+\frac {\left (16 \left (1+2 e^3\right )^2\right ) \int \frac {1}{(-2+x) \left (e^{3+x}+x^2\right )^3} \, dx}{e^6}+\frac {\left (1+4 e^3\right ) \int \frac {x^3}{\left (e^{3+x}+x^2\right )^3} \, dx}{e^6}-\frac {\left (3+8 e^3\right ) \int \frac {x}{\left (e^{3+x}+x^2\right )^2} \, dx}{2 e^6}-\frac {\left (16 \left (1+3 e^3+2 e^6\right )\right ) \int \frac {1}{(-2+x) \left (e^{3+x}+x^2\right )^2} \, dx}{e^6}-\frac {\left (8 \left (2+7 e^3+6 e^6\right )\right ) \int \frac {1}{(-2+x)^2 \left (e^{3+x}+x^2\right )^2} \, dx}{e^6}-\frac {\left (5+14 e^3+8 e^6\right ) \int \frac {1}{\left (e^{3+x}+x^2\right )^2} \, dx}{e^6} \\ \end{align*}
Time = 4.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09 \[ \int \frac {-x^6+e^6 \left (32 x^3-32 x^4\right )+e^3 \left (8 x^4-12 x^5\right )+e^{3+x} \left (-4 x^3+3 x^4-x^5+e^6 \left (-32 x+32 x^2-16 x^3\right )+e^3 \left (-24 x^2+20 x^3-8 x^4\right )\right )}{e^{15+3 x} \left (-16+24 x-12 x^2+2 x^3\right )+e^{12+2 x} \left (-48 x^2+72 x^3-36 x^4+6 x^5\right )+e^{9+x} \left (-48 x^4+72 x^5-36 x^6+6 x^7\right )+e^6 \left (-16 x^6+24 x^7-12 x^8+2 x^9\right )} \, dx=\frac {x^2 \left (4 e^3+x\right )^2}{4 e^6 (-2+x)^2 \left (e^{3+x}+x^2\right )^2} \]
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Time = 0.30 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.28
method | result | size |
risch | \(\frac {x^{2} \left (16 \,{\mathrm e}^{6}+8 x \,{\mathrm e}^{3}+x^{2}\right ) {\mathrm e}^{-6}}{4 \left (x^{2}-4 x +4\right ) \left (x^{2}+{\mathrm e}^{3+x}\right )^{2}}\) | \(41\) |
parallelrisch | \(\frac {\left (16 x^{2} {\mathrm e}^{6}+8 x^{3} {\mathrm e}^{3}+x^{4}\right ) {\mathrm e}^{-6}}{4 x^{6}-16 x^{5}+8 x^{4} {\mathrm e}^{3+x}+16 x^{4}-32 x^{3} {\mathrm e}^{3+x}+4 x^{2} {\mathrm e}^{2 x +6}+32 x^{2} {\mathrm e}^{3+x}-16 x \,{\mathrm e}^{2 x +6}+16 \,{\mathrm e}^{2 x +6}}\) | \(97\) |
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Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (28) = 56\).
Time = 0.25 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.44 \[ \int \frac {-x^6+e^6 \left (32 x^3-32 x^4\right )+e^3 \left (8 x^4-12 x^5\right )+e^{3+x} \left (-4 x^3+3 x^4-x^5+e^6 \left (-32 x+32 x^2-16 x^3\right )+e^3 \left (-24 x^2+20 x^3-8 x^4\right )\right )}{e^{15+3 x} \left (-16+24 x-12 x^2+2 x^3\right )+e^{12+2 x} \left (-48 x^2+72 x^3-36 x^4+6 x^5\right )+e^{9+x} \left (-48 x^4+72 x^5-36 x^6+6 x^7\right )+e^6 \left (-16 x^6+24 x^7-12 x^8+2 x^9\right )} \, dx=\frac {x^{4} e^{6} + 8 \, x^{3} e^{9} + 16 \, x^{2} e^{12}}{4 \, {\left ({\left (x^{6} - 4 \, x^{5} + 4 \, x^{4}\right )} e^{12} + {\left (x^{2} - 4 \, x + 4\right )} e^{\left (2 \, x + 18\right )} + 2 \, {\left (x^{4} - 4 \, x^{3} + 4 \, x^{2}\right )} e^{\left (x + 15\right )}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (29) = 58\).
Time = 0.14 (sec) , antiderivative size = 102, normalized size of antiderivative = 3.19 \[ \int \frac {-x^6+e^6 \left (32 x^3-32 x^4\right )+e^3 \left (8 x^4-12 x^5\right )+e^{3+x} \left (-4 x^3+3 x^4-x^5+e^6 \left (-32 x+32 x^2-16 x^3\right )+e^3 \left (-24 x^2+20 x^3-8 x^4\right )\right )}{e^{15+3 x} \left (-16+24 x-12 x^2+2 x^3\right )+e^{12+2 x} \left (-48 x^2+72 x^3-36 x^4+6 x^5\right )+e^{9+x} \left (-48 x^4+72 x^5-36 x^6+6 x^7\right )+e^6 \left (-16 x^6+24 x^7-12 x^8+2 x^9\right )} \, dx=\frac {x^{4} + 8 x^{3} e^{3} + 16 x^{2} e^{6}}{4 x^{6} e^{6} - 16 x^{5} e^{6} + 16 x^{4} e^{6} + \left (4 x^{2} e^{6} - 16 x e^{6} + 16 e^{6}\right ) e^{2 x + 6} + \left (8 x^{4} e^{6} - 32 x^{3} e^{6} + 32 x^{2} e^{6}\right ) e^{x + 3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (28) = 56\).
Time = 0.34 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.78 \[ \int \frac {-x^6+e^6 \left (32 x^3-32 x^4\right )+e^3 \left (8 x^4-12 x^5\right )+e^{3+x} \left (-4 x^3+3 x^4-x^5+e^6 \left (-32 x+32 x^2-16 x^3\right )+e^3 \left (-24 x^2+20 x^3-8 x^4\right )\right )}{e^{15+3 x} \left (-16+24 x-12 x^2+2 x^3\right )+e^{12+2 x} \left (-48 x^2+72 x^3-36 x^4+6 x^5\right )+e^{9+x} \left (-48 x^4+72 x^5-36 x^6+6 x^7\right )+e^6 \left (-16 x^6+24 x^7-12 x^8+2 x^9\right )} \, dx=\frac {x^{4} + 8 \, x^{3} e^{3} + 16 \, x^{2} e^{6}}{4 \, {\left (x^{6} e^{6} - 4 \, x^{5} e^{6} + 4 \, x^{4} e^{6} + {\left (x^{2} e^{12} - 4 \, x e^{12} + 4 \, e^{12}\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x^{4} e^{9} - 4 \, x^{3} e^{9} + 4 \, x^{2} e^{9}\right )} e^{x}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 226 vs. \(2 (28) = 56\).
Time = 0.36 (sec) , antiderivative size = 226, normalized size of antiderivative = 7.06 \[ \int \frac {-x^6+e^6 \left (32 x^3-32 x^4\right )+e^3 \left (8 x^4-12 x^5\right )+e^{3+x} \left (-4 x^3+3 x^4-x^5+e^6 \left (-32 x+32 x^2-16 x^3\right )+e^3 \left (-24 x^2+20 x^3-8 x^4\right )\right )}{e^{15+3 x} \left (-16+24 x-12 x^2+2 x^3\right )+e^{12+2 x} \left (-48 x^2+72 x^3-36 x^4+6 x^5\right )+e^{9+x} \left (-48 x^4+72 x^5-36 x^6+6 x^7\right )+e^6 \left (-16 x^6+24 x^7-12 x^8+2 x^9\right )} \, dx=\frac {{\left (x + 3\right )}^{4} e^{3} + 8 \, {\left (x + 3\right )}^{3} e^{6} - 12 \, {\left (x + 3\right )}^{3} e^{3} + 16 \, {\left (x + 3\right )}^{2} e^{9} - 72 \, {\left (x + 3\right )}^{2} e^{6} + 54 \, {\left (x + 3\right )}^{2} e^{3} - 96 \, {\left (x + 3\right )} e^{9} + 216 \, {\left (x + 3\right )} e^{6} - 108 \, {\left (x + 3\right )} e^{3} + 144 \, e^{9} - 216 \, e^{6} + 81 \, e^{3}}{4 \, {\left ({\left (x + 3\right )}^{6} e^{9} - 22 \, {\left (x + 3\right )}^{5} e^{9} + 199 \, {\left (x + 3\right )}^{4} e^{9} + 2 \, {\left (x + 3\right )}^{4} e^{\left (x + 12\right )} - 948 \, {\left (x + 3\right )}^{3} e^{9} - 32 \, {\left (x + 3\right )}^{3} e^{\left (x + 12\right )} + 2511 \, {\left (x + 3\right )}^{2} e^{9} + {\left (x + 3\right )}^{2} e^{\left (2 \, x + 15\right )} + 188 \, {\left (x + 3\right )}^{2} e^{\left (x + 12\right )} - 3510 \, {\left (x + 3\right )} e^{9} - 10 \, {\left (x + 3\right )} e^{\left (2 \, x + 15\right )} - 480 \, {\left (x + 3\right )} e^{\left (x + 12\right )} + 2025 \, e^{9} + 25 \, e^{\left (2 \, x + 15\right )} + 450 \, e^{\left (x + 12\right )}\right )}} \]
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Time = 12.67 (sec) , antiderivative size = 98, normalized size of antiderivative = 3.06 \[ \int \frac {-x^6+e^6 \left (32 x^3-32 x^4\right )+e^3 \left (8 x^4-12 x^5\right )+e^{3+x} \left (-4 x^3+3 x^4-x^5+e^6 \left (-32 x+32 x^2-16 x^3\right )+e^3 \left (-24 x^2+20 x^3-8 x^4\right )\right )}{e^{15+3 x} \left (-16+24 x-12 x^2+2 x^3\right )+e^{12+2 x} \left (-48 x^2+72 x^3-36 x^4+6 x^5\right )+e^{9+x} \left (-48 x^4+72 x^5-36 x^6+6 x^7\right )+e^6 \left (-16 x^6+24 x^7-12 x^8+2 x^9\right )} \, dx=-\frac {\frac {{\mathrm {e}}^{-6}\,x^7}{4}+\frac {{\mathrm {e}}^{-6}\,\left (8\,{\mathrm {e}}^3-4\right )\,x^6}{4}+\frac {{\mathrm {e}}^{-6}\,\left (16\,{\mathrm {e}}^6-32\,{\mathrm {e}}^3+4\right )\,x^5}{4}+\frac {{\mathrm {e}}^{-6}\,\left (32\,{\mathrm {e}}^3-64\,{\mathrm {e}}^6\right )\,x^4}{4}+16\,x^3}{\left (2\,x-x^2\right )\,{\left (x-2\right )}^3\,\left ({\mathrm {e}}^{2\,x+6}+2\,x^2\,{\mathrm {e}}^{x+3}+x^4\right )} \]
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