Integrand size = 42, antiderivative size = 25 \[ \int \frac {e^{40} \left (841 x+348 x^2-432 x^3-80 x^4+48 x^5\right )}{32-128 e^{20}+128 e^{40}} \, dx=\frac {x^2 \left (\frac {29}{4}+x-x^2\right )^2}{\left (-4+\frac {2}{e^{20}}\right )^2} \]
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Leaf count is larger than twice the leaf count of optimal. \(96\) vs. \(2(25)=50\).
Time = 0.01 (sec) , antiderivative size = 96, normalized size of antiderivative = 3.84, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {12} \[ \int \frac {e^{40} \left (841 x+348 x^2-432 x^3-80 x^4+48 x^5\right )}{32-128 e^{20}+128 e^{40}} \, dx=\frac {e^{40} x^6}{4 \left (1-2 e^{20}\right )^2}-\frac {e^{40} x^5}{2 \left (1-2 e^{20}\right )^2}-\frac {27 e^{40} x^4}{8 \left (1-2 e^{20}\right )^2}+\frac {29 e^{40} x^3}{8 \left (1-2 e^{20}\right )^2}+\frac {841 e^{40} x^2}{64 \left (1-2 e^{20}\right )^2} \]
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Rule 12
Rubi steps \begin{align*} \text {integral}& = \frac {e^{40} \int \left (841 x+348 x^2-432 x^3-80 x^4+48 x^5\right ) \, dx}{32 \left (1-2 e^{20}\right )^2} \\ & = \frac {841 e^{40} x^2}{64 \left (1-2 e^{20}\right )^2}+\frac {29 e^{40} x^3}{8 \left (1-2 e^{20}\right )^2}-\frac {27 e^{40} x^4}{8 \left (1-2 e^{20}\right )^2}-\frac {e^{40} x^5}{2 \left (1-2 e^{20}\right )^2}+\frac {e^{40} x^6}{4 \left (1-2 e^{20}\right )^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.76 \[ \int \frac {e^{40} \left (841 x+348 x^2-432 x^3-80 x^4+48 x^5\right )}{32-128 e^{20}+128 e^{40}} \, dx=\frac {e^{40} \left (\frac {841 x^2}{2}+116 x^3-108 x^4-16 x^5+8 x^6\right )}{32 \left (1-2 e^{20}\right )^2} \]
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Time = 0.09 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.76
method | result | size |
gosper | \(\frac {{\mathrm e}^{40} x^{2} \left (16 x^{4}-32 x^{3}-216 x^{2}+232 x +841\right )}{256 \,{\mathrm e}^{40}-256 \,{\mathrm e}^{20}+64}\) | \(44\) |
parallelrisch | \(\frac {{\mathrm e}^{40} \left (8 x^{6}-16 x^{5}-108 x^{4}+116 x^{3}+\frac {841}{2} x^{2}\right )}{128 \,{\mathrm e}^{40}-128 \,{\mathrm e}^{20}+32}\) | \(46\) |
default | \(\frac {{\mathrm e}^{40} \left (8 x^{6}-16 x^{5}-108 x^{4}+116 x^{3}+\frac {841}{2} x^{2}\right )}{128 \,{\mathrm e}^{40}-128 \,{\mathrm e}^{20}+32}\) | \(47\) |
norman | \(\frac {\frac {841 \,{\mathrm e}^{40} x^{2}}{64 \left (2 \,{\mathrm e}^{20}-1\right )}+\frac {29 \,{\mathrm e}^{40} x^{3}}{8 \left (2 \,{\mathrm e}^{20}-1\right )}-\frac {27 \,{\mathrm e}^{40} x^{4}}{8 \left (2 \,{\mathrm e}^{20}-1\right )}-\frac {{\mathrm e}^{40} x^{5}}{2 \left (2 \,{\mathrm e}^{20}-1\right )}+\frac {{\mathrm e}^{40} x^{6}}{8 \,{\mathrm e}^{20}-4}}{2 \,{\mathrm e}^{20}-1}\) | \(96\) |
risch | \(\frac {8 \,{\mathrm e}^{40} x^{6}}{128 \,{\mathrm e}^{40}-128 \,{\mathrm e}^{20}+32}-\frac {16 \,{\mathrm e}^{40} x^{5}}{128 \,{\mathrm e}^{40}-128 \,{\mathrm e}^{20}+32}-\frac {108 \,{\mathrm e}^{40} x^{4}}{128 \,{\mathrm e}^{40}-128 \,{\mathrm e}^{20}+32}+\frac {116 \,{\mathrm e}^{40} x^{3}}{128 \,{\mathrm e}^{40}-128 \,{\mathrm e}^{20}+32}+\frac {841 \,{\mathrm e}^{40} x^{2}}{2 \left (128 \,{\mathrm e}^{40}-128 \,{\mathrm e}^{20}+32\right )}\) | \(97\) |
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Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.68 \[ \int \frac {e^{40} \left (841 x+348 x^2-432 x^3-80 x^4+48 x^5\right )}{32-128 e^{20}+128 e^{40}} \, dx=\frac {{\left (16 \, x^{6} - 32 \, x^{5} - 216 \, x^{4} + 232 \, x^{3} + 841 \, x^{2}\right )} e^{40}}{64 \, {\left (4 \, e^{40} - 4 \, e^{20} + 1\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (20) = 40\).
Time = 0.03 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.88 \[ \int \frac {e^{40} \left (841 x+348 x^2-432 x^3-80 x^4+48 x^5\right )}{32-128 e^{20}+128 e^{40}} \, dx=\frac {x^{6} e^{40}}{- 16 e^{20} + 4 + 16 e^{40}} - \frac {x^{5} e^{40}}{- 8 e^{20} + 2 + 8 e^{40}} - \frac {27 x^{4} e^{40}}{- 32 e^{20} + 8 + 32 e^{40}} + \frac {29 x^{3} e^{40}}{- 32 e^{20} + 8 + 32 e^{40}} + \frac {841 x^{2} e^{40}}{- 256 e^{20} + 64 + 256 e^{40}} \]
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Time = 0.18 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.68 \[ \int \frac {e^{40} \left (841 x+348 x^2-432 x^3-80 x^4+48 x^5\right )}{32-128 e^{20}+128 e^{40}} \, dx=\frac {{\left (16 \, x^{6} - 32 \, x^{5} - 216 \, x^{4} + 232 \, x^{3} + 841 \, x^{2}\right )} e^{40}}{64 \, {\left (4 \, e^{40} - 4 \, e^{20} + 1\right )}} \]
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Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.68 \[ \int \frac {e^{40} \left (841 x+348 x^2-432 x^3-80 x^4+48 x^5\right )}{32-128 e^{20}+128 e^{40}} \, dx=\frac {{\left (16 \, x^{6} - 32 \, x^{5} - 216 \, x^{4} + 232 \, x^{3} + 841 \, x^{2}\right )} e^{40}}{64 \, {\left (4 \, e^{40} - 4 \, e^{20} + 1\right )}} \]
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Time = 8.23 (sec) , antiderivative size = 76, normalized size of antiderivative = 3.04 \[ \int \frac {e^{40} \left (841 x+348 x^2-432 x^3-80 x^4+48 x^5\right )}{32-128 e^{20}+128 e^{40}} \, dx=\frac {{\mathrm {e}}^{40}\,x^6}{4\,{\left (2\,{\mathrm {e}}^{20}-1\right )}^2}-\frac {{\mathrm {e}}^{40}\,x^5}{2\,{\left (2\,{\mathrm {e}}^{20}-1\right )}^2}-\frac {27\,{\mathrm {e}}^{40}\,x^4}{8\,{\left (2\,{\mathrm {e}}^{20}-1\right )}^2}+\frac {29\,{\mathrm {e}}^{40}\,x^3}{8\,{\left (2\,{\mathrm {e}}^{20}-1\right )}^2}+\frac {841\,{\mathrm {e}}^{40}\,x^2}{64\,{\left (2\,{\mathrm {e}}^{20}-1\right )}^2} \]
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