Integrand size = 71, antiderivative size = 18 \[ \int \frac {2 e^{6+2 x}+16 e^4 x^3+e^{10} (-10+2 x)+e^7 \left (-40 x+12 x^2\right )+e^x \left (e^8 (-8+2 x)+e^5 \left (8 x+4 x^2\right )\right )}{e^{10}} \, dx=\left (-5+e^{-2+x}+x+\frac {2 x^2}{e^3}\right )^2 \]
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Leaf count is larger than twice the leaf count of optimal. \(68\) vs. \(2(18)=36\).
Time = 0.07 (sec) , antiderivative size = 68, normalized size of antiderivative = 3.78, number of steps used = 15, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {12, 2225, 2227, 2207} \[ \int \frac {2 e^{6+2 x}+16 e^4 x^3+e^{10} (-10+2 x)+e^7 \left (-40 x+12 x^2\right )+e^x \left (e^8 (-8+2 x)+e^5 \left (8 x+4 x^2\right )\right )}{e^{10}} \, dx=\frac {4 x^4}{e^6}+\frac {4 x^3}{e^3}+4 e^{x-5} x^2-\frac {20 x^2}{e^3}-2 e^{x-2}+e^{2 x-4}+(5-x)^2-2 e^{x-2} (4-x) \]
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Rule 12
Rule 2207
Rule 2225
Rule 2227
Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (2 e^{6+2 x}+16 e^4 x^3+e^{10} (-10+2 x)+e^7 \left (-40 x+12 x^2\right )+e^x \left (e^8 (-8+2 x)+e^5 \left (8 x+4 x^2\right )\right )\right ) \, dx}{e^{10}} \\ & = (5-x)^2+\frac {4 x^4}{e^6}+\frac {\int e^x \left (e^8 (-8+2 x)+e^5 \left (8 x+4 x^2\right )\right ) \, dx}{e^{10}}+\frac {2 \int e^{6+2 x} \, dx}{e^{10}}+\frac {\int \left (-40 x+12 x^2\right ) \, dx}{e^3} \\ & = e^{-4+2 x}+(5-x)^2-\frac {20 x^2}{e^3}+\frac {4 x^3}{e^3}+\frac {4 x^4}{e^6}+\frac {\int \left (2 e^{8+x} (-4+x)+4 e^{5+x} x (2+x)\right ) \, dx}{e^{10}} \\ & = e^{-4+2 x}+(5-x)^2-\frac {20 x^2}{e^3}+\frac {4 x^3}{e^3}+\frac {4 x^4}{e^6}+\frac {2 \int e^{8+x} (-4+x) \, dx}{e^{10}}+\frac {4 \int e^{5+x} x (2+x) \, dx}{e^{10}} \\ & = e^{-4+2 x}-2 e^{-2+x} (4-x)+(5-x)^2-\frac {20 x^2}{e^3}+\frac {4 x^3}{e^3}+\frac {4 x^4}{e^6}-\frac {2 \int e^{8+x} \, dx}{e^{10}}+\frac {4 \int \left (2 e^{5+x} x+e^{5+x} x^2\right ) \, dx}{e^{10}} \\ & = -2 e^{-2+x}+e^{-4+2 x}-2 e^{-2+x} (4-x)+(5-x)^2-\frac {20 x^2}{e^3}+\frac {4 x^3}{e^3}+\frac {4 x^4}{e^6}+\frac {4 \int e^{5+x} x^2 \, dx}{e^{10}}+\frac {8 \int e^{5+x} x \, dx}{e^{10}} \\ & = -2 e^{-2+x}+e^{-4+2 x}-2 e^{-2+x} (4-x)+(5-x)^2+8 e^{-5+x} x-\frac {20 x^2}{e^3}+4 e^{-5+x} x^2+\frac {4 x^3}{e^3}+\frac {4 x^4}{e^6}-\frac {8 \int e^{5+x} \, dx}{e^{10}}-\frac {8 \int e^{5+x} x \, dx}{e^{10}} \\ & = -8 e^{-5+x}-2 e^{-2+x}+e^{-4+2 x}-2 e^{-2+x} (4-x)+(5-x)^2-\frac {20 x^2}{e^3}+4 e^{-5+x} x^2+\frac {4 x^3}{e^3}+\frac {4 x^4}{e^6}+\frac {8 \int e^{5+x} \, dx}{e^{10}} \\ & = -2 e^{-2+x}+e^{-4+2 x}-2 e^{-2+x} (4-x)+(5-x)^2-\frac {20 x^2}{e^3}+4 e^{-5+x} x^2+\frac {4 x^3}{e^3}+\frac {4 x^4}{e^6} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(79\) vs. \(2(18)=36\).
Time = 0.08 (sec) , antiderivative size = 79, normalized size of antiderivative = 4.39 \[ \int \frac {2 e^{6+2 x}+16 e^4 x^3+e^{10} (-10+2 x)+e^7 \left (-40 x+12 x^2\right )+e^x \left (e^8 (-8+2 x)+e^5 \left (8 x+4 x^2\right )\right )}{e^{10}} \, dx=\frac {2 \left (\frac {1}{2} e^{2+2 x}-5 e^6 x-10 e^3 x^2+\frac {e^6 x^2}{2}+2 e^{1+x} x^2+2 e^3 x^3+2 x^4+e^x \left (-5 e^4+e^4 x\right )\right )}{e^6} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(67\) vs. \(2(21)=42\).
Time = 0.21 (sec) , antiderivative size = 68, normalized size of antiderivative = 3.78
method | result | size |
parts | \({\mathrm e}^{-6} \left (x \,{\mathrm e}^{3}-5 \,{\mathrm e}^{3}+2 x^{2}\right )^{2}+{\mathrm e}^{-4} {\mathrm e}^{2 x}+2 \,{\mathrm e}^{-2} {\mathrm e}^{-3} \left ({\mathrm e}^{3} \left ({\mathrm e}^{x} x -{\mathrm e}^{x}\right )-4 \,{\mathrm e}^{x} {\mathrm e}^{3}+2 \,{\mathrm e}^{x} x^{2}\right )\) | \(68\) |
norman | \(\left ({\mathrm e}^{-2} {\mathrm e}^{3} {\mathrm e}^{2 x}+{\mathrm e}^{2} \left ({\mathrm e}^{3}-20\right ) x^{2}+4 x^{3} {\mathrm e}^{2}+4 \,{\mathrm e}^{x} x^{2}-10 \,{\mathrm e}^{x} {\mathrm e}^{3}-10 x \,{\mathrm e}^{2} {\mathrm e}^{3}+2 x \,{\mathrm e}^{3} {\mathrm e}^{x}+4 \,{\mathrm e}^{2} {\mathrm e}^{-3} x^{4}\right ) {\mathrm e}^{-2} {\mathrm e}^{-3}\) | \(77\) |
risch | \({\mathrm e}^{-6} {\mathrm e}^{6} x^{2}+4 \,{\mathrm e}^{-6} {\mathrm e}^{3} x^{3}+4 \,{\mathrm e}^{-6} x^{4}-10 \,{\mathrm e}^{-6} {\mathrm e}^{6} x -20 \,{\mathrm e}^{-6} {\mathrm e}^{3} x^{2}+25 \,{\mathrm e}^{-6} {\mathrm e}^{6}+{\mathrm e}^{2 x -4}+\left (2 x \,{\mathrm e}^{8}-10 \,{\mathrm e}^{8}+4 x^{2} {\mathrm e}^{5}\right ) {\mathrm e}^{x -10}\) | \(82\) |
parallelrisch | \({\mathrm e}^{-4} {\mathrm e}^{-6} \left ({\mathrm e}^{4} {\mathrm e}^{6} x^{2}+4 \,{\mathrm e}^{4} {\mathrm e}^{3} x^{3}+4 x^{4} {\mathrm e}^{4}-10 \,{\mathrm e}^{4} {\mathrm e}^{6} x -20 x^{2} {\mathrm e}^{3} {\mathrm e}^{4}+2 \,{\mathrm e}^{2} {\mathrm e}^{6} {\mathrm e}^{x} x +4 \,{\mathrm e}^{2} {\mathrm e}^{3} {\mathrm e}^{x} x^{2}-10 \,{\mathrm e}^{2} {\mathrm e}^{6} {\mathrm e}^{x}+{\mathrm e}^{6} {\mathrm e}^{2 x}\right )\) | \(106\) |
default | \({\mathrm e}^{-4} {\mathrm e}^{-6} \left (-8 \,{\mathrm e}^{2} {\mathrm e}^{6} {\mathrm e}^{x}+8 \,{\mathrm e}^{2} {\mathrm e}^{3} \left ({\mathrm e}^{x} x -{\mathrm e}^{x}\right )+4 \,{\mathrm e}^{2} {\mathrm e}^{3} \left ({\mathrm e}^{x} x^{2}-2 \,{\mathrm e}^{x} x +2 \,{\mathrm e}^{x}\right )+2 \,{\mathrm e}^{2} {\mathrm e}^{6} \left ({\mathrm e}^{x} x -{\mathrm e}^{x}\right )+2 \,{\mathrm e}^{4} {\mathrm e}^{6} \left (-5 x +\frac {1}{2} x^{2}\right )+4 \,{\mathrm e}^{4} {\mathrm e}^{3} \left (x^{3}-5 x^{2}\right )+4 x^{4} {\mathrm e}^{4}+{\mathrm e}^{6} {\mathrm e}^{2 x}\right )\) | \(129\) |
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Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (16) = 32\).
Time = 0.48 (sec) , antiderivative size = 56, normalized size of antiderivative = 3.11 \[ \int \frac {2 e^{6+2 x}+16 e^4 x^3+e^{10} (-10+2 x)+e^7 \left (-40 x+12 x^2\right )+e^x \left (e^8 (-8+2 x)+e^5 \left (8 x+4 x^2\right )\right )}{e^{10}} \, dx={\left (4 \, x^{4} + {\left (x^{2} - 10 \, x\right )} e^{6} + 4 \, {\left (x^{3} - 5 \, x^{2}\right )} e^{3} + 2 \, {\left (2 \, x^{2} e + {\left (x - 5\right )} e^{4}\right )} e^{x} + e^{\left (2 \, x + 2\right )}\right )} e^{\left (-6\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (19) = 38\).
Time = 0.12 (sec) , antiderivative size = 66, normalized size of antiderivative = 3.67 \[ \int \frac {2 e^{6+2 x}+16 e^4 x^3+e^{10} (-10+2 x)+e^7 \left (-40 x+12 x^2\right )+e^x \left (e^8 (-8+2 x)+e^5 \left (8 x+4 x^2\right )\right )}{e^{10}} \, dx=\frac {4 x^{4}}{e^{6}} + \frac {4 x^{3}}{e^{3}} + \frac {x^{2} \left (-20 + e^{3}\right )}{e^{3}} - 10 x + \frac {\left (4 x^{2} e^{4} + 2 x e^{7} - 10 e^{7}\right ) e^{x} + e^{5} e^{2 x}}{e^{9}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (16) = 32\).
Time = 0.22 (sec) , antiderivative size = 60, normalized size of antiderivative = 3.33 \[ \int \frac {2 e^{6+2 x}+16 e^4 x^3+e^{10} (-10+2 x)+e^7 \left (-40 x+12 x^2\right )+e^x \left (e^8 (-8+2 x)+e^5 \left (8 x+4 x^2\right )\right )}{e^{10}} \, dx={\left (4 \, x^{4} e^{4} + {\left (x^{2} - 10 \, x\right )} e^{10} + 4 \, {\left (x^{3} - 5 \, x^{2}\right )} e^{7} + 2 \, {\left (2 \, x^{2} e^{5} + x e^{8} - 5 \, e^{8}\right )} e^{x} + e^{\left (2 \, x + 6\right )}\right )} e^{\left (-10\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (16) = 32\).
Time = 0.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 3.22 \[ \int \frac {2 e^{6+2 x}+16 e^4 x^3+e^{10} (-10+2 x)+e^7 \left (-40 x+12 x^2\right )+e^x \left (e^8 (-8+2 x)+e^5 \left (8 x+4 x^2\right )\right )}{e^{10}} \, dx={\left (4 \, x^{4} e^{4} + 4 \, x^{2} e^{\left (x + 5\right )} + {\left (x^{2} - 10 \, x\right )} e^{10} + 4 \, {\left (x^{3} - 5 \, x^{2}\right )} e^{7} + 2 \, {\left (x - 5\right )} e^{\left (x + 8\right )} + e^{\left (2 \, x + 6\right )}\right )} e^{\left (-10\right )} \]
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Time = 0.13 (sec) , antiderivative size = 57, normalized size of antiderivative = 3.17 \[ \int \frac {2 e^{6+2 x}+16 e^4 x^3+e^{10} (-10+2 x)+e^7 \left (-40 x+12 x^2\right )+e^x \left (e^8 (-8+2 x)+e^5 \left (8 x+4 x^2\right )\right )}{e^{10}} \, dx={\mathrm {e}}^{2\,x-4}-10\,{\mathrm {e}}^{x-2}-10\,x+2\,x\,{\mathrm {e}}^{x-2}-x^2\,\left (20\,{\mathrm {e}}^{-3}-1\right )+4\,x^2\,{\mathrm {e}}^{x-5}+4\,x^3\,{\mathrm {e}}^{-3}+4\,x^4\,{\mathrm {e}}^{-6} \]
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