Integrand size = 14, antiderivative size = 26 \[ \int \frac {x^3}{1-2 x+x^2} \, dx=\frac {1}{1-x}+2 x+\frac {x^2}{2}+3 \log (1-x) \]
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Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {27, 45} \[ \int \frac {x^3}{1-2 x+x^2} \, dx=\frac {x^2}{2}+2 x+\frac {1}{1-x}+3 \log (1-x) \]
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Rule 27
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^3}{(-1+x)^2} \, dx \\ & = \int \left (2+\frac {1}{(-1+x)^2}+\frac {3}{-1+x}+x\right ) \, dx \\ & = \frac {1}{1-x}+2 x+\frac {x^2}{2}+3 \log (1-x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {x^3}{1-2 x+x^2} \, dx=\frac {1}{2} \left (-5-\frac {2}{-1+x}+4 x+x^2+6 \log (-1+x)\right ) \]
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Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88
method | result | size |
default | \(2 x +\frac {x^{2}}{2}-\frac {1}{-1+x}+3 \ln \left (-1+x \right )\) | \(23\) |
risch | \(2 x +\frac {x^{2}}{2}-\frac {1}{-1+x}+3 \ln \left (-1+x \right )\) | \(23\) |
norman | \(\frac {\frac {3}{2} x^{2}+\frac {1}{2} x^{3}-3}{-1+x}+3 \ln \left (-1+x \right )\) | \(26\) |
meijerg | \(\frac {x \left (-2 x^{2}-6 x +12\right )}{4-4 x}+3 \ln \left (1-x \right )\) | \(30\) |
parallelrisch | \(\frac {x^{3}+6 \ln \left (-1+x \right ) x +3 x^{2}-6-6 \ln \left (-1+x \right )}{2 x -2}\) | \(31\) |
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Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {x^3}{1-2 x+x^2} \, dx=\frac {x^{3} + 3 \, x^{2} + 6 \, {\left (x - 1\right )} \log \left (x - 1\right ) - 4 \, x - 2}{2 \, {\left (x - 1\right )}} \]
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Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.73 \[ \int \frac {x^3}{1-2 x+x^2} \, dx=\frac {x^{2}}{2} + 2 x + 3 \log {\left (x - 1 \right )} - \frac {1}{x - 1} \]
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Time = 0.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {x^3}{1-2 x+x^2} \, dx=\frac {1}{2} \, x^{2} + 2 \, x - \frac {1}{x - 1} + 3 \, \log \left (x - 1\right ) \]
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Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {x^3}{1-2 x+x^2} \, dx=\frac {1}{2} \, x^{2} + 2 \, x - \frac {1}{x - 1} + 3 \, \log \left ({\left | x - 1 \right |}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {x^3}{1-2 x+x^2} \, dx=2\,x+3\,\ln \left (x-1\right )-\frac {1}{x-1}+\frac {x^2}{2} \]
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