Integrand size = 9, antiderivative size = 45 \[ \int \frac {x^3}{(-1+x)^{12}} \, dx=\frac {1}{11 (1-x)^{11}}-\frac {3}{10 (1-x)^{10}}+\frac {1}{3 (1-x)^9}-\frac {1}{8 (1-x)^8} \]
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Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {45} \[ \int \frac {x^3}{(-1+x)^{12}} \, dx=-\frac {1}{8 (1-x)^8}+\frac {1}{3 (1-x)^9}-\frac {3}{10 (1-x)^{10}}+\frac {1}{11 (1-x)^{11}} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{(-1+x)^{12}}+\frac {3}{(-1+x)^{11}}+\frac {3}{(-1+x)^{10}}+\frac {1}{(-1+x)^9}\right ) \, dx \\ & = \frac {1}{11 (1-x)^{11}}-\frac {3}{10 (1-x)^{10}}+\frac {1}{3 (1-x)^9}-\frac {1}{8 (1-x)^8} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.53 \[ \int \frac {x^3}{(-1+x)^{12}} \, dx=\frac {1-11 x+55 x^2-165 x^3}{1320 (-1+x)^{11}} \]
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Time = 0.20 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.49
method | result | size |
norman | \(\frac {-\frac {1}{8} x^{3}+\frac {1}{24} x^{2}-\frac {1}{120} x +\frac {1}{1320}}{\left (-1+x \right )^{11}}\) | \(22\) |
risch | \(\frac {-\frac {1}{8} x^{3}+\frac {1}{24} x^{2}-\frac {1}{120} x +\frac {1}{1320}}{\left (-1+x \right )^{11}}\) | \(22\) |
gosper | \(-\frac {165 x^{3}-55 x^{2}+11 x -1}{1320 \left (-1+x \right )^{11}}\) | \(23\) |
parallelrisch | \(\frac {-165 x^{3}+55 x^{2}-11 x +1}{1320 \left (-1+x \right )^{11}}\) | \(23\) |
default | \(-\frac {1}{3 \left (-1+x \right )^{9}}-\frac {1}{11 \left (-1+x \right )^{11}}-\frac {1}{8 \left (-1+x \right )^{8}}-\frac {3}{10 \left (-1+x \right )^{10}}\) | \(30\) |
meijerg | \(\frac {x^{4} \left (-x^{7}+11 x^{6}-55 x^{5}+165 x^{4}-330 x^{3}+462 x^{2}-462 x +330\right )}{1320 \left (1-x \right )^{11}}\) | \(48\) |
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Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (29) = 58\).
Time = 0.23 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.60 \[ \int \frac {x^3}{(-1+x)^{12}} \, dx=-\frac {165 \, x^{3} - 55 \, x^{2} + 11 \, x - 1}{1320 \, {\left (x^{11} - 11 \, x^{10} + 55 \, x^{9} - 165 \, x^{8} + 330 \, x^{7} - 462 \, x^{6} + 462 \, x^{5} - 330 \, x^{4} + 165 \, x^{3} - 55 \, x^{2} + 11 \, x - 1\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (32) = 64\).
Time = 0.06 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.56 \[ \int \frac {x^3}{(-1+x)^{12}} \, dx=\frac {- 165 x^{3} + 55 x^{2} - 11 x + 1}{1320 x^{11} - 14520 x^{10} + 72600 x^{9} - 217800 x^{8} + 435600 x^{7} - 609840 x^{6} + 609840 x^{5} - 435600 x^{4} + 217800 x^{3} - 72600 x^{2} + 14520 x - 1320} \]
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Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (29) = 58\).
Time = 0.19 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.60 \[ \int \frac {x^3}{(-1+x)^{12}} \, dx=-\frac {165 \, x^{3} - 55 \, x^{2} + 11 \, x - 1}{1320 \, {\left (x^{11} - 11 \, x^{10} + 55 \, x^{9} - 165 \, x^{8} + 330 \, x^{7} - 462 \, x^{6} + 462 \, x^{5} - 330 \, x^{4} + 165 \, x^{3} - 55 \, x^{2} + 11 \, x - 1\right )}} \]
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none
Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.49 \[ \int \frac {x^3}{(-1+x)^{12}} \, dx=-\frac {165 \, x^{3} - 55 \, x^{2} + 11 \, x - 1}{1320 \, {\left (x - 1\right )}^{11}} \]
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Time = 0.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.64 \[ \int \frac {x^3}{(-1+x)^{12}} \, dx=-\frac {1}{8\,{\left (x-1\right )}^8}-\frac {1}{3\,{\left (x-1\right )}^9}-\frac {3}{10\,{\left (x-1\right )}^{10}}-\frac {1}{11\,{\left (x-1\right )}^{11}} \]
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