Integrand size = 49, antiderivative size = 30 \[ \int \frac {8-240 x^2+7800 x^4-64000 x^6+390625 x^9+1600000 x^{10}-4875000 x^{12}+3750000 x^{14}-3125000 x^{16}}{390625 x^9} \, dx=-5-\frac {e}{3}+e^3-\left (1+\left (\frac {1}{5 x}-x\right )^4\right )^2+x \]
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Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.87, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.041, Rules used = {12, 14} \[ \int \frac {8-240 x^2+7800 x^4-64000 x^6+390625 x^9+1600000 x^{10}-4875000 x^{12}+3750000 x^{14}-3125000 x^{16}}{390625 x^9} \, dx=-x^8-\frac {1}{390625 x^8}+\frac {8 x^6}{5}+\frac {8}{78125 x^6}-\frac {78 x^4}{25}-\frac {78}{15625 x^4}+\frac {256 x^2}{125}+\frac {256}{3125 x^2}+x \]
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Rule 12
Rule 14
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {8-240 x^2+7800 x^4-64000 x^6+390625 x^9+1600000 x^{10}-4875000 x^{12}+3750000 x^{14}-3125000 x^{16}}{x^9} \, dx}{390625} \\ & = \frac {\int \left (390625+\frac {8}{x^9}-\frac {240}{x^7}+\frac {7800}{x^5}-\frac {64000}{x^3}+1600000 x-4875000 x^3+3750000 x^5-3125000 x^7\right ) \, dx}{390625} \\ & = -\frac {1}{390625 x^8}+\frac {8}{78125 x^6}-\frac {78}{15625 x^4}+\frac {256}{3125 x^2}+x+\frac {256 x^2}{125}-\frac {78 x^4}{25}+\frac {8 x^6}{5}-x^8 \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.87 \[ \int \frac {8-240 x^2+7800 x^4-64000 x^6+390625 x^9+1600000 x^{10}-4875000 x^{12}+3750000 x^{14}-3125000 x^{16}}{390625 x^9} \, dx=-\frac {1}{390625 x^8}+\frac {8}{78125 x^6}-\frac {78}{15625 x^4}+\frac {256}{3125 x^2}+x+\frac {256 x^2}{125}-\frac {78 x^4}{25}+\frac {8 x^6}{5}-x^8 \]
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Time = 0.35 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.43
method | result | size |
default | \(-x^{8}+\frac {8 x^{6}}{5}-\frac {78 x^{4}}{25}+\frac {256 x^{2}}{125}+x -\frac {78}{15625 x^{4}}-\frac {1}{390625 x^{8}}+\frac {8}{78125 x^{6}}+\frac {256}{3125 x^{2}}\) | \(43\) |
norman | \(\frac {-\frac {1}{390625}+x^{9}+\frac {8}{78125} x^{2}-\frac {78}{15625} x^{4}+\frac {256}{3125} x^{6}+\frac {256}{125} x^{10}-\frac {78}{25} x^{12}+\frac {8}{5} x^{14}-x^{16}}{x^{8}}\) | \(45\) |
risch | \(-x^{8}+\frac {8 x^{6}}{5}-\frac {78 x^{4}}{25}+\frac {256 x^{2}}{125}+x +\frac {32000 x^{6}-1950 x^{4}+40 x^{2}-1}{390625 x^{8}}\) | \(45\) |
gosper | \(-\frac {390625 x^{16}-625000 x^{14}+1218750 x^{12}-800000 x^{10}-390625 x^{9}-32000 x^{6}+1950 x^{4}-40 x^{2}+1}{390625 x^{8}}\) | \(48\) |
parallelrisch | \(-\frac {390625 x^{16}-625000 x^{14}+1218750 x^{12}-800000 x^{10}-390625 x^{9}-32000 x^{6}+1950 x^{4}-40 x^{2}+1}{390625 x^{8}}\) | \(48\) |
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Time = 0.23 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.57 \[ \int \frac {8-240 x^2+7800 x^4-64000 x^6+390625 x^9+1600000 x^{10}-4875000 x^{12}+3750000 x^{14}-3125000 x^{16}}{390625 x^9} \, dx=-\frac {390625 \, x^{16} - 625000 \, x^{14} + 1218750 \, x^{12} - 800000 \, x^{10} - 390625 \, x^{9} - 32000 \, x^{6} + 1950 \, x^{4} - 40 \, x^{2} + 1}{390625 \, x^{8}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (22) = 44\).
Time = 0.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.53 \[ \int \frac {8-240 x^2+7800 x^4-64000 x^6+390625 x^9+1600000 x^{10}-4875000 x^{12}+3750000 x^{14}-3125000 x^{16}}{390625 x^9} \, dx=- x^{8} + \frac {8 x^{6}}{5} - \frac {78 x^{4}}{25} + \frac {256 x^{2}}{125} + x - \frac {- 32000 x^{6} + 1950 x^{4} - 40 x^{2} + 1}{390625 x^{8}} \]
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Time = 0.17 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.47 \[ \int \frac {8-240 x^2+7800 x^4-64000 x^6+390625 x^9+1600000 x^{10}-4875000 x^{12}+3750000 x^{14}-3125000 x^{16}}{390625 x^9} \, dx=-x^{8} + \frac {8}{5} \, x^{6} - \frac {78}{25} \, x^{4} + \frac {256}{125} \, x^{2} + x + \frac {32000 \, x^{6} - 1950 \, x^{4} + 40 \, x^{2} - 1}{390625 \, x^{8}} \]
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Time = 0.26 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.47 \[ \int \frac {8-240 x^2+7800 x^4-64000 x^6+390625 x^9+1600000 x^{10}-4875000 x^{12}+3750000 x^{14}-3125000 x^{16}}{390625 x^9} \, dx=-x^{8} + \frac {8}{5} \, x^{6} - \frac {78}{25} \, x^{4} + \frac {256}{125} \, x^{2} + x + \frac {32000 \, x^{6} - 1950 \, x^{4} + 40 \, x^{2} - 1}{390625 \, x^{8}} \]
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Time = 9.32 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.43 \[ \int \frac {8-240 x^2+7800 x^4-64000 x^6+390625 x^9+1600000 x^{10}-4875000 x^{12}+3750000 x^{14}-3125000 x^{16}}{390625 x^9} \, dx=x+\frac {\frac {256\,x^6}{3125}-\frac {78\,x^4}{15625}+\frac {8\,x^2}{78125}-\frac {1}{390625}}{x^8}+\frac {256\,x^2}{125}-\frac {78\,x^4}{25}+\frac {8\,x^6}{5}-x^8 \]
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