Integrand size = 18, antiderivative size = 26 \[ \int \frac {(1-x)^2 x^4}{1+x^2} \, dx=x^2-\frac {x^4}{2}+\frac {x^5}{5}-\log \left (1+x^2\right ) \]
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Time = 0.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1643, 266} \[ \int \frac {(1-x)^2 x^4}{1+x^2} \, dx=\frac {x^5}{5}-\frac {x^4}{2}+x^2-\log \left (x^2+1\right ) \]
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Rule 266
Rule 1643
Rubi steps \begin{align*} \text {integral}& = \int \left (2 x-2 x^3+x^4-\frac {2 x}{1+x^2}\right ) \, dx \\ & = x^2-\frac {x^4}{2}+\frac {x^5}{5}-2 \int \frac {x}{1+x^2} \, dx \\ & = x^2-\frac {x^4}{2}+\frac {x^5}{5}-\log \left (1+x^2\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {(1-x)^2 x^4}{1+x^2} \, dx=x^2-\frac {x^4}{2}+\frac {x^5}{5}-\log \left (1+x^2\right ) \]
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Time = 0.22 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88
method | result | size |
default | \(x^{2}-\frac {x^{4}}{2}+\frac {x^{5}}{5}-\ln \left (x^{2}+1\right )\) | \(23\) |
norman | \(x^{2}-\frac {x^{4}}{2}+\frac {x^{5}}{5}-\ln \left (x^{2}+1\right )\) | \(23\) |
risch | \(x^{2}-\frac {x^{4}}{2}+\frac {x^{5}}{5}-\ln \left (x^{2}+1\right )\) | \(23\) |
parallelrisch | \(x^{2}-\frac {x^{4}}{2}+\frac {x^{5}}{5}-\ln \left (x^{2}+1\right )\) | \(23\) |
meijerg | \(-\frac {x \left (-5 x^{2}+15\right )}{15}+\frac {x^{2} \left (-3 x^{2}+6\right )}{6}-\ln \left (x^{2}+1\right )+\frac {x \left (21 x^{4}-35 x^{2}+105\right )}{105}\) | \(47\) |
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Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {(1-x)^2 x^4}{1+x^2} \, dx=\frac {1}{5} \, x^{5} - \frac {1}{2} \, x^{4} + x^{2} - \log \left (x^{2} + 1\right ) \]
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Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.73 \[ \int \frac {(1-x)^2 x^4}{1+x^2} \, dx=\frac {x^{5}}{5} - \frac {x^{4}}{2} + x^{2} - \log {\left (x^{2} + 1 \right )} \]
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Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {(1-x)^2 x^4}{1+x^2} \, dx=\frac {1}{5} \, x^{5} - \frac {1}{2} \, x^{4} + x^{2} - \log \left (x^{2} + 1\right ) \]
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none
Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {(1-x)^2 x^4}{1+x^2} \, dx=\frac {1}{5} \, x^{5} - \frac {1}{2} \, x^{4} + x^{2} - \log \left (x^{2} + 1\right ) \]
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Time = 0.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {(1-x)^2 x^4}{1+x^2} \, dx=x^2-\ln \left (x^2+1\right )-\frac {x^4}{2}+\frac {x^5}{5} \]
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