Integrand size = 39, antiderivative size = 28 \[ \int \frac {e^{-2/x} \left (32 x+e^{2/x} (-8192+48 x)+e^{\frac {1}{x}} (-512+512 x)\right )}{x^3} \, dx=\frac {16 (-3+x)}{x}+\frac {16 \left (16-e^{-1/x} x\right )^2}{x^2} \]
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Time = 0.17 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {6874, 2326, 2240, 37} \[ \int \frac {e^{-2/x} \left (32 x+e^{2/x} (-8192+48 x)+e^{\frac {1}{x}} (-512+512 x)\right )}{x^3} \, dx=\frac {(512-3 x)^2}{64 x^2}+16 e^{-2/x}-\frac {512 e^{-1/x}}{x} \]
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Rule 37
Rule 2240
Rule 2326
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {512 e^{-1/x} (-1+x)}{x^3}+\frac {32 e^{-2/x}}{x^2}+\frac {16 (-512+3 x)}{x^3}\right ) \, dx \\ & = 16 \int \frac {-512+3 x}{x^3} \, dx+32 \int \frac {e^{-2/x}}{x^2} \, dx+512 \int \frac {e^{-1/x} (-1+x)}{x^3} \, dx \\ & = 16 e^{-2/x}+\frac {(512-3 x)^2}{64 x^2}-\frac {512 e^{-1/x}}{x} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {e^{-2/x} \left (32 x+e^{2/x} (-8192+48 x)+e^{\frac {1}{x}} (-512+512 x)\right )}{x^3} \, dx=16 \left (e^{-2/x}+\frac {256}{x^2}-\frac {3}{x}-\frac {32 e^{-1/x}}{x}\right ) \]
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Time = 0.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07
method | result | size |
risch | \(\frac {-48 x +4096}{x^{2}}-\frac {512 \,{\mathrm e}^{-\frac {1}{x}}}{x}+16 \,{\mathrm e}^{-\frac {2}{x}}\) | \(30\) |
derivativedivides | \(\frac {4096}{x^{2}}-\frac {48}{x}+16 \,{\mathrm e}^{-\frac {2}{x}}-\frac {512 \,{\mathrm e}^{-\frac {1}{x}}}{x}\) | \(31\) |
default | \(\frac {4096}{x^{2}}-\frac {48}{x}+16 \,{\mathrm e}^{-\frac {2}{x}}-\frac {512 \,{\mathrm e}^{-\frac {1}{x}}}{x}\) | \(31\) |
parts | \(\frac {4096}{x^{2}}-\frac {48}{x}+16 \,{\mathrm e}^{-\frac {2}{x}}-\frac {512 \,{\mathrm e}^{-\frac {1}{x}}}{x}\) | \(31\) |
norman | \(\frac {\left (16 x^{2}+4096 \,{\mathrm e}^{\frac {2}{x}}-512 x \,{\mathrm e}^{\frac {1}{x}}-48 x \,{\mathrm e}^{\frac {2}{x}}\right ) {\mathrm e}^{-\frac {2}{x}}}{x^{2}}\) | \(41\) |
parallelrisch | \(-\frac {\left (48 \ln \left ({\mathrm e}^{\frac {1}{x}}\right ) x^{2} {\mathrm e}^{\frac {2}{x}}-16 x^{2}+512 x \,{\mathrm e}^{\frac {1}{x}}-4096 \,{\mathrm e}^{\frac {2}{x}}\right ) {\mathrm e}^{-\frac {2}{x}}}{x^{2}}\) | \(49\) |
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none
Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.25 \[ \int \frac {e^{-2/x} \left (32 x+e^{2/x} (-8192+48 x)+e^{\frac {1}{x}} (-512+512 x)\right )}{x^3} \, dx=\frac {16 \, {\left (x^{2} - {\left (3 \, x - 256\right )} e^{\frac {2}{x}} - 32 \, x e^{\frac {1}{x}}\right )} e^{\left (-\frac {2}{x}\right )}}{x^{2}} \]
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Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {e^{-2/x} \left (32 x+e^{2/x} (-8192+48 x)+e^{\frac {1}{x}} (-512+512 x)\right )}{x^3} \, dx=\frac {16 x e^{- \frac {2}{x}} - 512 e^{- \frac {1}{x}}}{x} + \frac {4096 - 48 x}{x^{2}} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.21 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {e^{-2/x} \left (32 x+e^{2/x} (-8192+48 x)+e^{\frac {1}{x}} (-512+512 x)\right )}{x^3} \, dx=-\frac {48}{x} + \frac {4096}{x^{2}} + 512 \, e^{\left (-\frac {1}{x}\right )} + 16 \, e^{\left (-\frac {2}{x}\right )} - 512 \, \Gamma \left (2, \frac {1}{x}\right ) \]
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none
Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {e^{-2/x} \left (32 x+e^{2/x} (-8192+48 x)+e^{\frac {1}{x}} (-512+512 x)\right )}{x^3} \, dx=-\frac {512 \, e^{\left (-\frac {1}{x}\right )}}{x} - \frac {48}{x} + \frac {4096}{x^{2}} + 16 \, e^{\left (-\frac {2}{x}\right )} \]
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Time = 10.58 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {e^{-2/x} \left (32 x+e^{2/x} (-8192+48 x)+e^{\frac {1}{x}} (-512+512 x)\right )}{x^3} \, dx=16\,{\mathrm {e}}^{-\frac {2}{x}}-\frac {48\,x-4096}{x^2}-\frac {512\,{\mathrm {e}}^{-\frac {1}{x}}}{x} \]
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