Integrand size = 24, antiderivative size = 28 \[ \int \frac {1}{8} \left (-16+3 x^2+e^{2/x} (-8+8 x)\right ) \, dx=-2 x+\frac {1}{2} \left (3+\frac {1}{4} x^2 \left (4 e^{2/x}+x\right )\right ) \]
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Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {12, 2258, 2237, 2241, 2245} \[ \int \frac {1}{8} \left (-16+3 x^2+e^{2/x} (-8+8 x)\right ) \, dx=\frac {x^3}{8}+\frac {1}{2} e^{2/x} x^2-2 x \]
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Rule 12
Rule 2237
Rule 2241
Rule 2245
Rule 2258
Rubi steps \begin{align*} \text {integral}& = \frac {1}{8} \int \left (-16+3 x^2+e^{2/x} (-8+8 x)\right ) \, dx \\ & = -2 x+\frac {x^3}{8}+\frac {1}{8} \int e^{2/x} (-8+8 x) \, dx \\ & = -2 x+\frac {x^3}{8}+\frac {1}{8} \int \left (-8 e^{2/x}+8 e^{2/x} x\right ) \, dx \\ & = -2 x+\frac {x^3}{8}-\int e^{2/x} \, dx+\int e^{2/x} x \, dx \\ & = -2 x-e^{2/x} x+\frac {1}{2} e^{2/x} x^2+\frac {x^3}{8}-2 \int \frac {e^{2/x}}{x} \, dx+\int e^{2/x} \, dx \\ & = -2 x+\frac {1}{2} e^{2/x} x^2+\frac {x^3}{8}+2 \operatorname {ExpIntegralEi}\left (\frac {2}{x}\right )+2 \int \frac {e^{2/x}}{x} \, dx \\ & = -2 x+\frac {1}{2} e^{2/x} x^2+\frac {x^3}{8} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {1}{8} \left (-16+3 x^2+e^{2/x} (-8+8 x)\right ) \, dx=-2 x+\frac {1}{2} e^{2/x} x^2+\frac {x^3}{8} \]
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Time = 0.14 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75
method | result | size |
derivativedivides | \(\frac {x^{3}}{8}-2 x +\frac {x^{2} {\mathrm e}^{\frac {2}{x}}}{2}\) | \(21\) |
default | \(\frac {x^{3}}{8}-2 x +\frac {x^{2} {\mathrm e}^{\frac {2}{x}}}{2}\) | \(21\) |
norman | \(\frac {x^{3}}{8}-2 x +\frac {x^{2} {\mathrm e}^{\frac {2}{x}}}{2}\) | \(21\) |
risch | \(\frac {x^{3}}{8}-2 x +\frac {x^{2} {\mathrm e}^{\frac {2}{x}}}{2}\) | \(21\) |
parallelrisch | \(\frac {x^{3}}{8}-2 x +\frac {x^{2} {\mathrm e}^{\frac {2}{x}}}{2}\) | \(21\) |
parts | \(\frac {x^{3}}{8}-2 x +\frac {x^{2} {\mathrm e}^{\frac {2}{x}}}{2}\) | \(21\) |
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Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {1}{8} \left (-16+3 x^2+e^{2/x} (-8+8 x)\right ) \, dx=\frac {1}{8} \, x^{3} + \frac {1}{2} \, x^{2} e^{\frac {2}{x}} - 2 \, x \]
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Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.61 \[ \int \frac {1}{8} \left (-16+3 x^2+e^{2/x} (-8+8 x)\right ) \, dx=\frac {x^{3}}{8} + \frac {x^{2} e^{\frac {2}{x}}}{2} - 2 x \]
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Time = 0.19 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {1}{8} \left (-16+3 x^2+e^{2/x} (-8+8 x)\right ) \, dx=\frac {1}{8} \, x^{3} + \frac {1}{2} \, x^{2} e^{\frac {2}{x}} - 2 \, x \]
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Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {1}{8} \left (-16+3 x^2+e^{2/x} (-8+8 x)\right ) \, dx=\frac {1}{8} \, x^{3} + \frac {1}{2} \, x^{2} e^{\frac {2}{x}} - 2 \, x \]
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Time = 9.18 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.61 \[ \int \frac {1}{8} \left (-16+3 x^2+e^{2/x} (-8+8 x)\right ) \, dx=\frac {x\,\left (4\,x\,{\mathrm {e}}^{2/x}+x^2-16\right )}{8} \]
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