Integrand size = 37, antiderivative size = 26 \[ \int \frac {e^5+e^{\frac {3+x}{e}} (-e-x)+e \left (2-4 x+4 x^3\right )}{e} \, dx=2+x \left (2+e^4-e^{\frac {3+x}{e}}-2 x+x^3\right ) \]
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Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {12, 2207, 2225} \[ \int \frac {e^5+e^{\frac {3+x}{e}} (-e-x)+e \left (2-4 x+4 x^3\right )}{e} \, dx=x^4-2 x^2+e^4 x+2 x+e^{\frac {x+3}{e}+1}-e^{\frac {x+3}{e}} (x+e) \]
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Rule 12
Rule 2207
Rule 2225
Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (e^5+e^{\frac {3+x}{e}} (-e-x)+e \left (2-4 x+4 x^3\right )\right ) \, dx}{e} \\ & = e^4 x+\frac {\int e^{\frac {3+x}{e}} (-e-x) \, dx}{e}+\int \left (2-4 x+4 x^3\right ) \, dx \\ & = 2 x+e^4 x-2 x^2+x^4-e^{\frac {3+x}{e}} (e+x)+\int e^{\frac {3+x}{e}} \, dx \\ & = e^{1+\frac {3+x}{e}}+2 x+e^4 x-2 x^2+x^4-e^{\frac {3+x}{e}} (e+x) \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {e^5+e^{\frac {3+x}{e}} (-e-x)+e \left (2-4 x+4 x^3\right )}{e} \, dx=x \left (2+e^4-e^{\frac {3+x}{e}}-2 x+x^3\right ) \]
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Time = 0.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04
method | result | size |
risch | \(x \,{\mathrm e}^{4}+x^{4}-2 x^{2}+2 x -{\mathrm e}^{\left (3+x \right ) {\mathrm e}^{-1}} x\) | \(27\) |
norman | \(x^{4}+\left (2+{\mathrm e}^{4}\right ) x -2 x^{2}-{\mathrm e}^{\left (3+x \right ) {\mathrm e}^{-1}} x\) | \(28\) |
parallelrisch | \({\mathrm e}^{-1} \left (x^{4} {\mathrm e}-2 x^{2} {\mathrm e}-x \,{\mathrm e} \,{\mathrm e}^{\left (3+x \right ) {\mathrm e}^{-1}}+2 x \,{\mathrm e}+{\mathrm e} \,{\mathrm e}^{4} x \right )\) | \(45\) |
parts | \(x^{4}-2 x^{2}+2 x -{\mathrm e} \,{\mathrm e}^{{\mathrm e}^{-1} x +3 \,{\mathrm e}^{-1}}-{\mathrm e} \left ({\mathrm e}^{{\mathrm e}^{-1} x +3 \,{\mathrm e}^{-1}} \left ({\mathrm e}^{-1} x +3 \,{\mathrm e}^{-1}\right )-{\mathrm e}^{{\mathrm e}^{-1} x +3 \,{\mathrm e}^{-1}}\right )+3 \,{\mathrm e}^{{\mathrm e}^{-1} x +3 \,{\mathrm e}^{-1}}+x \,{\mathrm e}^{4}\) | \(100\) |
default | \({\mathrm e}^{-1} \left ({\mathrm e} \left (-{\mathrm e} \,{\mathrm e}^{{\mathrm e}^{-1} x +3 \,{\mathrm e}^{-1}}-{\mathrm e} \left ({\mathrm e}^{{\mathrm e}^{-1} x +3 \,{\mathrm e}^{-1}} \left ({\mathrm e}^{-1} x +3 \,{\mathrm e}^{-1}\right )-{\mathrm e}^{{\mathrm e}^{-1} x +3 \,{\mathrm e}^{-1}}\right )+3 \,{\mathrm e}^{{\mathrm e}^{-1} x +3 \,{\mathrm e}^{-1}}\right )+2 \,{\mathrm e} \left (\frac {1}{2} x^{4}-x^{2}+x \right )+{\mathrm e} \,{\mathrm e}^{4} x \right )\) | \(116\) |
derivativedivides | \(-{\mathrm e} \,{\mathrm e}^{{\mathrm e}^{-1} x +3 \,{\mathrm e}^{-1}}-{\mathrm e} \left ({\mathrm e}^{{\mathrm e}^{-1} x +3 \,{\mathrm e}^{-1}} \left ({\mathrm e}^{-1} x +3 \,{\mathrm e}^{-1}\right )-{\mathrm e}^{{\mathrm e}^{-1} x +3 \,{\mathrm e}^{-1}}\right )+3 \,{\mathrm e}^{{\mathrm e}^{-1} x +3 \,{\mathrm e}^{-1}}+2 \,{\mathrm e} \left (\frac {{\mathrm e}^{3} \left ({\mathrm e}^{-1} x +3 \,{\mathrm e}^{-1}\right )^{4}}{2}-6 \,{\mathrm e}^{2} \left ({\mathrm e}^{-1} x +3 \,{\mathrm e}^{-1}\right )^{3}+26 \left ({\mathrm e}^{-1} x +3 \,{\mathrm e}^{-1}\right )^{2} {\mathrm e}-47 \,{\mathrm e}^{-1} x -141 \,{\mathrm e}^{-1}\right )+{\mathrm e} \,{\mathrm e}^{4} \left ({\mathrm e}^{-1} x +3 \,{\mathrm e}^{-1}\right )\) | \(182\) |
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Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {e^5+e^{\frac {3+x}{e}} (-e-x)+e \left (2-4 x+4 x^3\right )}{e} \, dx=x^{4} - 2 \, x^{2} + x e^{4} - x e^{\left ({\left (x + 3\right )} e^{\left (-1\right )}\right )} + 2 \, x \]
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Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {e^5+e^{\frac {3+x}{e}} (-e-x)+e \left (2-4 x+4 x^3\right )}{e} \, dx=x^{4} - 2 x^{2} - x e^{\frac {x + 3}{e}} + x \left (2 + e^{4}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.42 \[ \int \frac {e^5+e^{\frac {3+x}{e}} (-e-x)+e \left (2-4 x+4 x^3\right )}{e} \, dx={\left (x e^{5} + {\left (x^{4} - 2 \, x^{2} + 2 \, x\right )} e - x e^{\left (x e^{\left (-1\right )} + 3 \, e^{\left (-1\right )} + 1\right )}\right )} e^{\left (-1\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (23) = 46\).
Time = 0.25 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.08 \[ \int \frac {e^5+e^{\frac {3+x}{e}} (-e-x)+e \left (2-4 x+4 x^3\right )}{e} \, dx={\left (x e^{5} + {\left (x^{4} - 2 \, x^{2} + 2 \, x\right )} e - {\left (x e - e^{2}\right )} e^{\left ({\left (x + 3\right )} e^{\left (-1\right )}\right )} - e^{\left ({\left (x + e + 3\right )} e^{\left (-1\right )} + 1\right )}\right )} e^{\left (-1\right )} \]
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Time = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {e^5+e^{\frac {3+x}{e}} (-e-x)+e \left (2-4 x+4 x^3\right )}{e} \, dx=2\,x+x\,{\mathrm {e}}^4-2\,x^2+x^4-x\,{\mathrm {e}}^{3\,{\mathrm {e}}^{-1}}\,{\mathrm {e}}^{x\,{\mathrm {e}}^{-1}} \]
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