Integrand size = 91, antiderivative size = 22 \[ \int \left (2 x-6 x^2+4 x^3+e^x \left (8 x-8 x^2-4 x^3\right )+e^{3 x} \left (12 x^2+12 x^3\right )+e^{2 x} \left (8 x+14 x^2-4 x^3-4 x^4\right )+e^{4 x} \left (4 x^3+4 x^4\right )\right ) \, dx=\left (-x+x^2-e^x x \left (2+e^x x\right )\right )^2 \]
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Leaf count is larger than twice the leaf count of optimal. \(77\) vs. \(2(22)=44\).
Time = 0.26 (sec) , antiderivative size = 77, normalized size of antiderivative = 3.50, number of steps used = 51, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.055, Rules used = {1608, 2227, 2207, 2225, 1607} \[ \int \left (2 x-6 x^2+4 x^3+e^x \left (8 x-8 x^2-4 x^3\right )+e^{3 x} \left (12 x^2+12 x^3\right )+e^{2 x} \left (8 x+14 x^2-4 x^3-4 x^4\right )+e^{4 x} \left (4 x^3+4 x^4\right )\right ) \, dx=-2 e^{2 x} x^4+e^{4 x} x^4+x^4-4 e^x x^3+2 e^{2 x} x^3+4 e^{3 x} x^3-2 x^3+4 e^x x^2+4 e^{2 x} x^2+x^2 \]
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Rule 1607
Rule 1608
Rule 2207
Rule 2225
Rule 2227
Rubi steps \begin{align*} \text {integral}& = x^2-2 x^3+x^4+\int e^x \left (8 x-8 x^2-4 x^3\right ) \, dx+\int e^{3 x} \left (12 x^2+12 x^3\right ) \, dx+\int e^{2 x} \left (8 x+14 x^2-4 x^3-4 x^4\right ) \, dx+\int e^{4 x} \left (4 x^3+4 x^4\right ) \, dx \\ & = x^2-2 x^3+x^4+\int e^{4 x} x^3 (4+4 x) \, dx+\int e^{3 x} x^2 (12+12 x) \, dx+\int e^x x \left (8-8 x-4 x^2\right ) \, dx+\int \left (8 e^{2 x} x+14 e^{2 x} x^2-4 e^{2 x} x^3-4 e^{2 x} x^4\right ) \, dx \\ & = x^2-2 x^3+x^4-4 \int e^{2 x} x^3 \, dx-4 \int e^{2 x} x^4 \, dx+8 \int e^{2 x} x \, dx+14 \int e^{2 x} x^2 \, dx+\int \left (8 e^x x-8 e^x x^2-4 e^x x^3\right ) \, dx+\int \left (12 e^{3 x} x^2+12 e^{3 x} x^3\right ) \, dx+\int \left (4 e^{4 x} x^3+4 e^{4 x} x^4\right ) \, dx \\ & = 4 e^{2 x} x+x^2+7 e^{2 x} x^2-2 x^3-2 e^{2 x} x^3+x^4-2 e^{2 x} x^4-4 \int e^{2 x} \, dx-4 \int e^x x^3 \, dx+4 \int e^{4 x} x^3 \, dx+4 \int e^{4 x} x^4 \, dx+6 \int e^{2 x} x^2 \, dx+8 \int e^x x \, dx-8 \int e^x x^2 \, dx+8 \int e^{2 x} x^3 \, dx+12 \int e^{3 x} x^2 \, dx+12 \int e^{3 x} x^3 \, dx-14 \int e^{2 x} x \, dx \\ & = -2 e^{2 x}+8 e^x x-3 e^{2 x} x+x^2-8 e^x x^2+10 e^{2 x} x^2+4 e^{3 x} x^2-2 x^3-4 e^x x^3+2 e^{2 x} x^3+4 e^{3 x} x^3+e^{4 x} x^3+x^4-2 e^{2 x} x^4+e^{4 x} x^4-3 \int e^{4 x} x^2 \, dx-4 \int e^{4 x} x^3 \, dx-6 \int e^{2 x} x \, dx+7 \int e^{2 x} \, dx-8 \int e^x \, dx-8 \int e^{3 x} x \, dx+12 \int e^x x^2 \, dx-12 \int e^{2 x} x^2 \, dx-12 \int e^{3 x} x^2 \, dx+16 \int e^x x \, dx \\ & = -8 e^x+\frac {3 e^{2 x}}{2}+24 e^x x-6 e^{2 x} x-\frac {8}{3} e^{3 x} x+x^2+4 e^x x^2+4 e^{2 x} x^2-\frac {3}{4} e^{4 x} x^2-2 x^3-4 e^x x^3+2 e^{2 x} x^3+4 e^{3 x} x^3+x^4-2 e^{2 x} x^4+e^{4 x} x^4+\frac {3}{2} \int e^{4 x} x \, dx+\frac {8}{3} \int e^{3 x} \, dx+3 \int e^{2 x} \, dx+3 \int e^{4 x} x^2 \, dx+8 \int e^{3 x} x \, dx+12 \int e^{2 x} x \, dx-16 \int e^x \, dx-24 \int e^x x \, dx \\ & = -24 e^x+3 e^{2 x}+\frac {8 e^{3 x}}{9}+\frac {3}{8} e^{4 x} x+x^2+4 e^x x^2+4 e^{2 x} x^2-2 x^3-4 e^x x^3+2 e^{2 x} x^3+4 e^{3 x} x^3+x^4-2 e^{2 x} x^4+e^{4 x} x^4-\frac {3}{8} \int e^{4 x} \, dx-\frac {3}{2} \int e^{4 x} x \, dx-\frac {8}{3} \int e^{3 x} \, dx-6 \int e^{2 x} \, dx+24 \int e^x \, dx \\ & = -\frac {3 e^{4 x}}{32}+x^2+4 e^x x^2+4 e^{2 x} x^2-2 x^3-4 e^x x^3+2 e^{2 x} x^3+4 e^{3 x} x^3+x^4-2 e^{2 x} x^4+e^{4 x} x^4+\frac {3}{8} \int e^{4 x} \, dx \\ & = x^2+4 e^x x^2+4 e^{2 x} x^2-2 x^3-4 e^x x^3+2 e^{2 x} x^3+4 e^{3 x} x^3+x^4-2 e^{2 x} x^4+e^{4 x} x^4 \\ \end{align*}
Time = 2.84 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \left (2 x-6 x^2+4 x^3+e^x \left (8 x-8 x^2-4 x^3\right )+e^{3 x} \left (12 x^2+12 x^3\right )+e^{2 x} \left (8 x+14 x^2-4 x^3-4 x^4\right )+e^{4 x} \left (4 x^3+4 x^4\right )\right ) \, dx=x^2 \left (1+2 e^x-x+e^{2 x} x\right )^2 \]
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Leaf count of result is larger than twice the leaf count of optimal. \(64\) vs. \(2(20)=40\).
Time = 0.13 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.95
method | result | size |
risch | \(4 x^{3} {\mathrm e}^{3 x}+{\mathrm e}^{4 x} x^{4}+\left (-2 x^{4}+2 x^{3}+4 x^{2}\right ) {\mathrm e}^{2 x}+\left (-4 x^{3}+4 x^{2}\right ) {\mathrm e}^{x}+x^{4}-2 x^{3}+x^{2}\) | \(65\) |
default | \(-2 \,{\mathrm e}^{2 x} x^{4}+{\mathrm e}^{4 x} x^{4}-4 \,{\mathrm e}^{x} x^{3}+2 \,{\mathrm e}^{2 x} x^{3}+4 x^{3} {\mathrm e}^{3 x}+x^{4}+4 \,{\mathrm e}^{x} x^{2}+4 \,{\mathrm e}^{2 x} x^{2}-2 x^{3}+x^{2}\) | \(71\) |
norman | \(-2 \,{\mathrm e}^{2 x} x^{4}+{\mathrm e}^{4 x} x^{4}-4 \,{\mathrm e}^{x} x^{3}+2 \,{\mathrm e}^{2 x} x^{3}+4 x^{3} {\mathrm e}^{3 x}+x^{4}+4 \,{\mathrm e}^{x} x^{2}+4 \,{\mathrm e}^{2 x} x^{2}-2 x^{3}+x^{2}\) | \(71\) |
parallelrisch | \(-2 \,{\mathrm e}^{2 x} x^{4}+{\mathrm e}^{4 x} x^{4}-4 \,{\mathrm e}^{x} x^{3}+2 \,{\mathrm e}^{2 x} x^{3}+4 x^{3} {\mathrm e}^{3 x}+x^{4}+4 \,{\mathrm e}^{x} x^{2}+4 \,{\mathrm e}^{2 x} x^{2}-2 x^{3}+x^{2}\) | \(71\) |
parts | \(-2 \,{\mathrm e}^{2 x} x^{4}+{\mathrm e}^{4 x} x^{4}-4 \,{\mathrm e}^{x} x^{3}+2 \,{\mathrm e}^{2 x} x^{3}+4 x^{3} {\mathrm e}^{3 x}+x^{4}+4 \,{\mathrm e}^{x} x^{2}+4 \,{\mathrm e}^{2 x} x^{2}-2 x^{3}+x^{2}\) | \(71\) |
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Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (19) = 38\).
Time = 0.26 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.82 \[ \int \left (2 x-6 x^2+4 x^3+e^x \left (8 x-8 x^2-4 x^3\right )+e^{3 x} \left (12 x^2+12 x^3\right )+e^{2 x} \left (8 x+14 x^2-4 x^3-4 x^4\right )+e^{4 x} \left (4 x^3+4 x^4\right )\right ) \, dx=x^{4} e^{\left (4 \, x\right )} + x^{4} + 4 \, x^{3} e^{\left (3 \, x\right )} - 2 \, x^{3} + x^{2} - 2 \, {\left (x^{4} - x^{3} - 2 \, x^{2}\right )} e^{\left (2 \, x\right )} - 4 \, {\left (x^{3} - x^{2}\right )} e^{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (17) = 34\).
Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.86 \[ \int \left (2 x-6 x^2+4 x^3+e^x \left (8 x-8 x^2-4 x^3\right )+e^{3 x} \left (12 x^2+12 x^3\right )+e^{2 x} \left (8 x+14 x^2-4 x^3-4 x^4\right )+e^{4 x} \left (4 x^3+4 x^4\right )\right ) \, dx=x^{4} e^{4 x} + x^{4} + 4 x^{3} e^{3 x} - 2 x^{3} + x^{2} + \left (- 4 x^{3} + 4 x^{2}\right ) e^{x} + \left (- 2 x^{4} + 2 x^{3} + 4 x^{2}\right ) e^{2 x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (19) = 38\).
Time = 0.18 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.82 \[ \int \left (2 x-6 x^2+4 x^3+e^x \left (8 x-8 x^2-4 x^3\right )+e^{3 x} \left (12 x^2+12 x^3\right )+e^{2 x} \left (8 x+14 x^2-4 x^3-4 x^4\right )+e^{4 x} \left (4 x^3+4 x^4\right )\right ) \, dx=x^{4} e^{\left (4 \, x\right )} + x^{4} + 4 \, x^{3} e^{\left (3 \, x\right )} - 2 \, x^{3} + x^{2} - 2 \, {\left (x^{4} - x^{3} - 2 \, x^{2}\right )} e^{\left (2 \, x\right )} - 4 \, {\left (x^{3} - x^{2}\right )} e^{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (19) = 38\).
Time = 0.27 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.82 \[ \int \left (2 x-6 x^2+4 x^3+e^x \left (8 x-8 x^2-4 x^3\right )+e^{3 x} \left (12 x^2+12 x^3\right )+e^{2 x} \left (8 x+14 x^2-4 x^3-4 x^4\right )+e^{4 x} \left (4 x^3+4 x^4\right )\right ) \, dx=x^{4} e^{\left (4 \, x\right )} + x^{4} + 4 \, x^{3} e^{\left (3 \, x\right )} - 2 \, x^{3} + x^{2} - 2 \, {\left (x^{4} - x^{3} - 2 \, x^{2}\right )} e^{\left (2 \, x\right )} - 4 \, {\left (x^{3} - x^{2}\right )} e^{x} \]
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Time = 12.78 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \left (2 x-6 x^2+4 x^3+e^x \left (8 x-8 x^2-4 x^3\right )+e^{3 x} \left (12 x^2+12 x^3\right )+e^{2 x} \left (8 x+14 x^2-4 x^3-4 x^4\right )+e^{4 x} \left (4 x^3+4 x^4\right )\right ) \, dx=x^2\,{\left (2\,{\mathrm {e}}^x-x+x\,{\mathrm {e}}^{2\,x}+1\right )}^2 \]
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