Integrand size = 11, antiderivative size = 39 \[ \int \frac {e^{-x/2}}{x^3} \, dx=-\frac {e^{-x/2}}{2 x^2}+\frac {e^{-x/2}}{4 x}+\frac {\operatorname {ExpIntegralEi}\left (-\frac {x}{2}\right )}{8} \]
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Time = 0.03 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2208, 2209} \[ \int \frac {e^{-x/2}}{x^3} \, dx=\frac {\operatorname {ExpIntegralEi}\left (-\frac {x}{2}\right )}{8}-\frac {e^{-x/2}}{2 x^2}+\frac {e^{-x/2}}{4 x} \]
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Rule 2208
Rule 2209
Rubi steps \begin{align*} \text {integral}& = -\frac {e^{-x/2}}{2 x^2}-\frac {1}{4} \int \frac {e^{-x/2}}{x^2} \, dx \\ & = -\frac {e^{-x/2}}{2 x^2}+\frac {e^{-x/2}}{4 x}+\frac {1}{8} \int \frac {e^{-x/2}}{x} \, dx \\ & = -\frac {e^{-x/2}}{2 x^2}+\frac {e^{-x/2}}{4 x}+\frac {\operatorname {ExpIntegralEi}\left (-\frac {x}{2}\right )}{8} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.67 \[ \int \frac {e^{-x/2}}{x^3} \, dx=\frac {1}{8} \left (\frac {2 e^{-x/2} (-2+x)}{x^2}+\operatorname {ExpIntegralEi}\left (-\frac {x}{2}\right )\right ) \]
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Time = 0.07 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.69
method | result | size |
risch | \(-\frac {{\mathrm e}^{-\frac {x}{2}}}{2 x^{2}}+\frac {{\mathrm e}^{-\frac {x}{2}}}{4 x}-\frac {\operatorname {Ei}_{1}\left (\frac {x}{2}\right )}{8}\) | \(27\) |
derivativedivides | \(-\frac {{\mathrm e}^{-\frac {x}{2}}}{2 x^{2}}+\frac {{\mathrm e}^{-\frac {x}{2}}}{4 x}-\frac {\operatorname {Ei}_{1}\left (\frac {x}{2}\right )}{8}\) | \(31\) |
default | \(-\frac {{\mathrm e}^{-\frac {x}{2}}}{2 x^{2}}+\frac {{\mathrm e}^{-\frac {x}{2}}}{4 x}-\frac {\operatorname {Ei}_{1}\left (\frac {x}{2}\right )}{8}\) | \(31\) |
meijerg | \(-\frac {1}{2 x^{2}}+\frac {1}{2 x}-\frac {3}{16}+\frac {\ln \left (x \right )}{8}-\frac {\ln \left (2\right )}{8}+\frac {\frac {9}{4} x^{2}-6 x +6}{12 x^{2}}-\frac {\left (-\frac {3 x}{2}+3\right ) {\mathrm e}^{-\frac {x}{2}}}{6 x^{2}}-\frac {\ln \left (\frac {x}{2}\right )}{8}-\frac {\operatorname {Ei}_{1}\left (\frac {x}{2}\right )}{8}\) | \(63\) |
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none
Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.59 \[ \int \frac {e^{-x/2}}{x^3} \, dx=\frac {x^{2} {\rm Ei}\left (-\frac {1}{2} \, x\right ) + 2 \, {\left (x - 2\right )} e^{\left (-\frac {1}{2} \, x\right )}}{8 \, x^{2}} \]
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Result contains complex when optimal does not.
Time = 0.73 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.82 \[ \int \frac {e^{-x/2}}{x^3} \, dx=\frac {\operatorname {Ei}{\left (\frac {x e^{i \pi }}{2} \right )}}{8} + \frac {e^{- \frac {x}{2}}}{4 x} - \frac {e^{- \frac {x}{2}}}{2 x^{2}} \]
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none
Time = 0.22 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.18 \[ \int \frac {e^{-x/2}}{x^3} \, dx=-\frac {1}{4} \, \Gamma \left (-2, \frac {1}{2} \, x\right ) \]
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none
Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.69 \[ \int \frac {e^{-x/2}}{x^3} \, dx=\frac {x^{2} {\rm Ei}\left (-\frac {1}{2} \, x\right ) + 2 \, x e^{\left (-\frac {1}{2} \, x\right )} - 4 \, e^{\left (-\frac {1}{2} \, x\right )}}{8 \, x^{2}} \]
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Time = 0.36 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.56 \[ \int \frac {e^{-x/2}}{x^3} \, dx=\frac {{\mathrm {e}}^{-\frac {x}{2}}\,\left (\frac {1}{x}-\frac {2}{x^2}\right )}{4}-\frac {\mathrm {expint}\left (\frac {x}{2}\right )}{8} \]
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