Integrand size = 43, antiderivative size = 21 \[ \int \frac {1}{20} \left (1-40 e^{-2 x}+40 e^{2 x}+e^x (-40-40 x)+e^{-x} (40-40 x)+40 x\right ) \, dx=-1+\frac {x}{20}+\left (e^{-x}-e^x+x\right )^2 \]
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Leaf count is larger than twice the leaf count of optimal. \(51\) vs. \(2(21)=42\).
Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.43, number of steps used = 8, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.070, Rules used = {12, 2225, 2207} \[ \int \frac {1}{20} \left (1-40 e^{-2 x}+40 e^{2 x}+e^x (-40-40 x)+e^{-x} (40-40 x)+40 x\right ) \, dx=x^2+\frac {x}{20}+e^{-2 x}+2 e^{-x}+2 e^x+e^{2 x}-2 e^{-x} (1-x)-2 e^x (x+1) \]
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Rule 12
Rule 2207
Rule 2225
Rubi steps \begin{align*} \text {integral}& = \frac {1}{20} \int \left (1-40 e^{-2 x}+40 e^{2 x}+e^x (-40-40 x)+e^{-x} (40-40 x)+40 x\right ) \, dx \\ & = \frac {x}{20}+x^2+\frac {1}{20} \int e^x (-40-40 x) \, dx+\frac {1}{20} \int e^{-x} (40-40 x) \, dx-2 \int e^{-2 x} \, dx+2 \int e^{2 x} \, dx \\ & = e^{-2 x}+e^{2 x}-2 e^{-x} (1-x)+\frac {x}{20}+x^2-2 e^x (1+x)-2 \int e^{-x} \, dx+2 \int e^x \, dx \\ & = e^{-2 x}+2 e^{-x}+2 e^x+e^{2 x}-2 e^{-x} (1-x)+\frac {x}{20}+x^2-2 e^x (1+x) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.57 \[ \int \frac {1}{20} \left (1-40 e^{-2 x}+40 e^{2 x}+e^x (-40-40 x)+e^{-x} (40-40 x)+40 x\right ) \, dx=e^{-2 x}+e^{2 x}+\frac {x}{20}+2 e^{-x} x-2 e^x x+x^2 \]
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Time = 0.04 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.33
method | result | size |
risch | \(\frac {x}{20}-2 \,{\mathrm e}^{x} x +2 x \,{\mathrm e}^{-x}+x^{2}+{\mathrm e}^{2 x}+{\mathrm e}^{-2 x}\) | \(28\) |
default | \(\frac {x}{20}-2 \,{\mathrm e}^{x} x +2 x \,{\mathrm e}^{-x}+x^{2}+{\mathrm e}^{2 x}+{\mathrm e}^{-2 x}\) | \(30\) |
parallelrisch | \(\frac {x}{20}-2 \,{\mathrm e}^{x} x +2 x \,{\mathrm e}^{-x}+x^{2}+{\mathrm e}^{2 x}+{\mathrm e}^{-2 x}\) | \(30\) |
parts | \(\frac {x}{20}-2 \,{\mathrm e}^{x} x +2 x \,{\mathrm e}^{-x}+x^{2}+{\mathrm e}^{2 x}+{\mathrm e}^{-2 x}\) | \(30\) |
norman | \(\left (1+{\mathrm e}^{4 x}+{\mathrm e}^{2 x} x^{2}+\frac {x \,{\mathrm e}^{2 x}}{20}-2 x \,{\mathrm e}^{3 x}+2 \,{\mathrm e}^{x} x \right ) {\mathrm e}^{-2 x}\) | \(39\) |
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Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (17) = 34\).
Time = 0.24 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.86 \[ \int \frac {1}{20} \left (1-40 e^{-2 x}+40 e^{2 x}+e^x (-40-40 x)+e^{-x} (40-40 x)+40 x\right ) \, dx=-\frac {1}{20} \, {\left (40 \, x e^{\left (3 \, x\right )} - {\left (20 \, x^{2} + x\right )} e^{\left (2 \, x\right )} - 40 \, x e^{x} - 20 \, e^{\left (4 \, x\right )} - 20\right )} e^{\left (-2 \, x\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (15) = 30\).
Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.48 \[ \int \frac {1}{20} \left (1-40 e^{-2 x}+40 e^{2 x}+e^x (-40-40 x)+e^{-x} (40-40 x)+40 x\right ) \, dx=x^{2} - 2 x e^{x} + \frac {x}{20} + 2 x e^{- x} + e^{2 x} + e^{- 2 x} \]
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Time = 0.21 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.29 \[ \int \frac {1}{20} \left (1-40 e^{-2 x}+40 e^{2 x}+e^x (-40-40 x)+e^{-x} (40-40 x)+40 x\right ) \, dx=x^{2} + 2 \, x e^{\left (-x\right )} - 2 \, x e^{x} + \frac {1}{20} \, x + e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )} \]
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Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.29 \[ \int \frac {1}{20} \left (1-40 e^{-2 x}+40 e^{2 x}+e^x (-40-40 x)+e^{-x} (40-40 x)+40 x\right ) \, dx=x^{2} + 2 \, x e^{\left (-x\right )} - 2 \, x e^{x} + \frac {1}{20} \, x + e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )} \]
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Time = 0.11 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.29 \[ \int \frac {1}{20} \left (1-40 e^{-2 x}+40 e^{2 x}+e^x (-40-40 x)+e^{-x} (40-40 x)+40 x\right ) \, dx=\frac {x}{20}+{\mathrm {e}}^{-2\,x}+{\mathrm {e}}^{2\,x}+2\,x\,{\mathrm {e}}^{-x}-2\,x\,{\mathrm {e}}^x+x^2 \]
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