Integrand size = 65, antiderivative size = 23 \[ \int \frac {1}{8} e^6 \left (64 x+e^2 \left (-576 x-48 x^3\right )+e^4 \left (1728 x+288 x^3+12 x^5\right )+e^6 \left (-1728 x-432 x^3-36 x^5-x^7\right )\right ) \, dx=1-4 e^{12} \left (3-\frac {1}{e^2}+\frac {x^2}{4}\right )^4 \]
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Leaf count is larger than twice the leaf count of optimal. \(91\) vs. \(2(23)=46\).
Time = 0.02 (sec) , antiderivative size = 91, normalized size of antiderivative = 3.96, number of steps used = 5, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.015, Rules used = {12} \[ \int \frac {1}{8} e^6 \left (64 x+e^2 \left (-576 x-48 x^3\right )+e^4 \left (1728 x+288 x^3+12 x^5\right )+e^6 \left (-1728 x-432 x^3-36 x^5-x^7\right )\right ) \, dx=-\frac {1}{64} e^{12} x^8-\frac {3 e^{12} x^6}{4}+\frac {e^{10} x^6}{4}-\frac {27 e^{12} x^4}{2}+9 e^{10} x^4-\frac {3 e^8 x^4}{2}-108 e^{12} x^2+108 e^{10} x^2-36 e^8 x^2+4 e^6 x^2 \]
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Rule 12
Rubi steps \begin{align*} \text {integral}& = \frac {1}{8} e^6 \int \left (64 x+e^2 \left (-576 x-48 x^3\right )+e^4 \left (1728 x+288 x^3+12 x^5\right )+e^6 \left (-1728 x-432 x^3-36 x^5-x^7\right )\right ) \, dx \\ & = 4 e^6 x^2+\frac {1}{8} e^8 \int \left (-576 x-48 x^3\right ) \, dx+\frac {1}{8} e^{10} \int \left (1728 x+288 x^3+12 x^5\right ) \, dx+\frac {1}{8} e^{12} \int \left (-1728 x-432 x^3-36 x^5-x^7\right ) \, dx \\ & = 4 e^6 x^2-36 e^8 x^2+108 e^{10} x^2-108 e^{12} x^2-\frac {3 e^8 x^4}{2}+9 e^{10} x^4-\frac {27 e^{12} x^4}{2}+\frac {e^{10} x^6}{4}-\frac {3 e^{12} x^6}{4}-\frac {e^{12} x^8}{64} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(64\) vs. \(2(23)=46\).
Time = 0.02 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.78 \[ \int \frac {1}{8} e^6 \left (64 x+e^2 \left (-576 x-48 x^3\right )+e^4 \left (1728 x+288 x^3+12 x^5\right )+e^6 \left (-1728 x-432 x^3-36 x^5-x^7\right )\right ) \, dx=-\frac {1}{8} e^6 \left (32 \left (-1+3 e^2\right )^3 x^2+12 e^2 \left (-1+3 e^2\right )^2 x^4+2 e^4 \left (-1+3 e^2\right ) x^6+\frac {e^6 x^8}{8}\right ) \]
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Time = 0.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09
method | result | size |
default | \(-\frac {{\mathrm e}^{12} {\mathrm e}^{-8} \left (x^{2} {\mathrm e}^{2}+12 \,{\mathrm e}^{2}-4\right )^{4}}{64}\) | \(25\) |
gosper | \(-\frac {{\mathrm e}^{12} \left ({\mathrm e}^{6} x^{6}+48 \,{\mathrm e}^{6} x^{4}-16 x^{4} {\mathrm e}^{4}+864 x^{2} {\mathrm e}^{6}-576 x^{2} {\mathrm e}^{4}+6912 \,{\mathrm e}^{6}+96 x^{2} {\mathrm e}^{2}-6912 \,{\mathrm e}^{4}+2304 \,{\mathrm e}^{2}-256\right ) x^{2} {\mathrm e}^{-6}}{64}\) | \(83\) |
risch | \(-\frac {x^{8} {\mathrm e}^{12}}{64}-\frac {3 \,{\mathrm e}^{12} x^{6}}{4}+\frac {{\mathrm e}^{6} x^{6} {\mathrm e}^{4}}{4}-\frac {27 x^{4} {\mathrm e}^{12}}{2}-\frac {3 \,{\mathrm e}^{6} x^{4} {\mathrm e}^{2}}{2}+9 \,{\mathrm e}^{6} x^{4} {\mathrm e}^{4}-108 \,{\mathrm e}^{12} x^{2}-36 \,{\mathrm e}^{6} x^{2} {\mathrm e}^{2}+108 \,{\mathrm e}^{6} x^{2} {\mathrm e}^{4}+4 x^{2} {\mathrm e}^{6}\) | \(88\) |
norman | \(\left (-\frac {3 \,{\mathrm e}^{12} \left (9 \,{\mathrm e}^{4}-6 \,{\mathrm e}^{2}+1\right ) x^{4}}{2}-\frac {{\mathrm e}^{12} {\mathrm e}^{4} x^{8}}{64}-4 \,{\mathrm e}^{12} \left (27 \,{\mathrm e}^{6}-27 \,{\mathrm e}^{4}+9 \,{\mathrm e}^{2}-1\right ) {\mathrm e}^{-2} x^{2}-\frac {{\mathrm e}^{12} {\mathrm e}^{2} \left (3 \,{\mathrm e}^{2}-1\right ) x^{6}}{4}\right ) {\mathrm e}^{-4}\) | \(89\) |
parallelrisch | \(\frac {{\mathrm e}^{12} {\mathrm e}^{-6} \left (-\frac {{\mathrm e}^{6} x^{8}}{8}-6 \,{\mathrm e}^{6} x^{6}+2 x^{6} {\mathrm e}^{4}-108 \,{\mathrm e}^{6} x^{4}+72 x^{4} {\mathrm e}^{4}-12 x^{4} {\mathrm e}^{2}-864 x^{2} {\mathrm e}^{6}+864 x^{2} {\mathrm e}^{4}-288 x^{2} {\mathrm e}^{2}+32 x^{2}\right )}{8}\) | \(94\) |
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Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (17) = 34\).
Time = 0.26 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.70 \[ \int \frac {1}{8} e^6 \left (64 x+e^2 \left (-576 x-48 x^3\right )+e^4 \left (1728 x+288 x^3+12 x^5\right )+e^6 \left (-1728 x-432 x^3-36 x^5-x^7\right )\right ) \, dx=4 \, x^{2} e^{6} - \frac {1}{64} \, {\left (x^{8} + 48 \, x^{6} + 864 \, x^{4} + 6912 \, x^{2}\right )} e^{12} + \frac {1}{4} \, {\left (x^{6} + 36 \, x^{4} + 432 \, x^{2}\right )} e^{10} - \frac {3}{2} \, {\left (x^{4} + 24 \, x^{2}\right )} e^{8} \]
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Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (19) = 38\).
Time = 0.04 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.96 \[ \int \frac {1}{8} e^6 \left (64 x+e^2 \left (-576 x-48 x^3\right )+e^4 \left (1728 x+288 x^3+12 x^5\right )+e^6 \left (-1728 x-432 x^3-36 x^5-x^7\right )\right ) \, dx=- \frac {x^{8} e^{12}}{64} + x^{6} \left (- \frac {3 e^{12}}{4} + \frac {e^{10}}{4}\right ) + x^{4} \left (- \frac {27 e^{12}}{2} - \frac {3 e^{8}}{2} + 9 e^{10}\right ) + x^{2} \left (- 108 e^{12} - 36 e^{8} + 4 e^{6} + 108 e^{10}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (17) = 34\).
Time = 0.17 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.78 \[ \int \frac {1}{8} e^6 \left (64 x+e^2 \left (-576 x-48 x^3\right )+e^4 \left (1728 x+288 x^3+12 x^5\right )+e^6 \left (-1728 x-432 x^3-36 x^5-x^7\right )\right ) \, dx=\frac {1}{64} \, {\left (256 \, x^{2} - {\left (x^{8} + 48 \, x^{6} + 864 \, x^{4} + 6912 \, x^{2}\right )} e^{6} + 16 \, {\left (x^{6} + 36 \, x^{4} + 432 \, x^{2}\right )} e^{4} - 96 \, {\left (x^{4} + 24 \, x^{2}\right )} e^{2}\right )} e^{6} \]
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Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (17) = 34\).
Time = 0.30 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.78 \[ \int \frac {1}{8} e^6 \left (64 x+e^2 \left (-576 x-48 x^3\right )+e^4 \left (1728 x+288 x^3+12 x^5\right )+e^6 \left (-1728 x-432 x^3-36 x^5-x^7\right )\right ) \, dx=\frac {1}{64} \, {\left (256 \, x^{2} - {\left (x^{8} + 48 \, x^{6} + 864 \, x^{4} + 6912 \, x^{2}\right )} e^{6} + 16 \, {\left (x^{6} + 36 \, x^{4} + 432 \, x^{2}\right )} e^{4} - 96 \, {\left (x^{4} + 24 \, x^{2}\right )} e^{2}\right )} e^{6} \]
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Time = 12.33 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.22 \[ \int \frac {1}{8} e^6 \left (64 x+e^2 \left (-576 x-48 x^3\right )+e^4 \left (1728 x+288 x^3+12 x^5\right )+e^6 \left (-1728 x-432 x^3-36 x^5-x^7\right )\right ) \, dx=-\frac {{\mathrm {e}}^{12}\,x^8}{64}-\frac {{\mathrm {e}}^{10}\,\left (3\,{\mathrm {e}}^2-1\right )\,x^6}{4}-\frac {3\,{\mathrm {e}}^8\,{\left (3\,{\mathrm {e}}^2-1\right )}^2\,x^4}{2}-4\,{\mathrm {e}}^6\,{\left (3\,{\mathrm {e}}^2-1\right )}^3\,x^2 \]
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