Integrand size = 29, antiderivative size = 30 \[ \int \left (1-x+x^2-x^3+x^4\right ) \left (1+x+x^2+x^3+x^4\right ) \, dx=x+\frac {x^3}{3}+\frac {x^5}{5}+\frac {x^7}{7}+\frac {x^9}{9} \]
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Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {6820} \[ \int \left (1-x+x^2-x^3+x^4\right ) \left (1+x+x^2+x^3+x^4\right ) \, dx=\frac {x^9}{9}+\frac {x^7}{7}+\frac {x^5}{5}+\frac {x^3}{3}+x \]
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Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \left (1+x^2+x^4+x^6+x^8\right ) \, dx \\ & = x+\frac {x^3}{3}+\frac {x^5}{5}+\frac {x^7}{7}+\frac {x^9}{9} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \left (1-x+x^2-x^3+x^4\right ) \left (1+x+x^2+x^3+x^4\right ) \, dx=x+\frac {x^3}{3}+\frac {x^5}{5}+\frac {x^7}{7}+\frac {x^9}{9} \]
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Time = 0.08 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.77
method | result | size |
gosper | \(x +\frac {1}{3} x^{3}+\frac {1}{5} x^{5}+\frac {1}{7} x^{7}+\frac {1}{9} x^{9}\) | \(23\) |
default | \(x +\frac {1}{3} x^{3}+\frac {1}{5} x^{5}+\frac {1}{7} x^{7}+\frac {1}{9} x^{9}\) | \(23\) |
norman | \(x +\frac {1}{3} x^{3}+\frac {1}{5} x^{5}+\frac {1}{7} x^{7}+\frac {1}{9} x^{9}\) | \(23\) |
risch | \(x +\frac {1}{3} x^{3}+\frac {1}{5} x^{5}+\frac {1}{7} x^{7}+\frac {1}{9} x^{9}\) | \(23\) |
parallelrisch | \(x +\frac {1}{3} x^{3}+\frac {1}{5} x^{5}+\frac {1}{7} x^{7}+\frac {1}{9} x^{9}\) | \(23\) |
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none
Time = 0.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73 \[ \int \left (1-x+x^2-x^3+x^4\right ) \left (1+x+x^2+x^3+x^4\right ) \, dx=\frac {1}{9} \, x^{9} + \frac {1}{7} \, x^{7} + \frac {1}{5} \, x^{5} + \frac {1}{3} \, x^{3} + x \]
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Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.67 \[ \int \left (1-x+x^2-x^3+x^4\right ) \left (1+x+x^2+x^3+x^4\right ) \, dx=\frac {x^{9}}{9} + \frac {x^{7}}{7} + \frac {x^{5}}{5} + \frac {x^{3}}{3} + x \]
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none
Time = 0.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73 \[ \int \left (1-x+x^2-x^3+x^4\right ) \left (1+x+x^2+x^3+x^4\right ) \, dx=\frac {1}{9} \, x^{9} + \frac {1}{7} \, x^{7} + \frac {1}{5} \, x^{5} + \frac {1}{3} \, x^{3} + x \]
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none
Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73 \[ \int \left (1-x+x^2-x^3+x^4\right ) \left (1+x+x^2+x^3+x^4\right ) \, dx=\frac {1}{9} \, x^{9} + \frac {1}{7} \, x^{7} + \frac {1}{5} \, x^{5} + \frac {1}{3} \, x^{3} + x \]
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Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73 \[ \int \left (1-x+x^2-x^3+x^4\right ) \left (1+x+x^2+x^3+x^4\right ) \, dx=\frac {x^9}{9}+\frac {x^7}{7}+\frac {x^5}{5}+\frac {x^3}{3}+x \]
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