Optimal. Leaf size=39 \[ \frac {\text {Ei}\left (-\frac {x}{2}\right )}{8}-\frac {e^{-x/2}}{2 x^2}+\frac {e^{-x/2}}{4 x} \]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2177, 2178} \[ \frac {1}{8} \text {ExpIntegralEi}\left (-\frac {x}{2}\right )-\frac {e^{-x/2}}{2 x^2}+\frac {e^{-x/2}}{4 x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2177
Rule 2178
Rubi steps
\begin {align*} \int \frac {e^{-x/2}}{x^3} \, dx &=-\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 {\text {Ei}\left (-\frac {x}{2}\right )}{8}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 26, normalized size = 0.67 \[ \frac {1}{8} \left (\text {Ei}\left (-\frac {x}{2}\right )+\frac {2 e^{-x/2} (x-2)}{x^2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{-x/2}}{x^3} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.40, size = 23, normalized size = 0.59 \[ \frac {x^{2} {\rm Ei}\left (-\frac {1}{2} \, x\right ) + 2 \, {\left (x - 2\right )} e^{\left (-\frac {1}{2} \, x\right )}}{8 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.60, size = 27, normalized size = 0.69 \[ \frac {x^{2} {\rm Ei}\left (-\frac {1}{2} \, x\right ) + 2 \, x e^{\left (-\frac {1}{2} \, x\right )} - 4 \, e^{\left (-\frac {1}{2} \, x\right )}}{8 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 27, normalized size = 0.69
method | result | size |
risch | \(-\frac {{\mathrm e}^{-\frac {x}{2}}}{2 x^{2}}+\frac {{\mathrm e}^{-\frac {x}{2}}}{4 x}-\frac {\expIntegralEi \left (1, \frac {x}{2}\right )}{8}\) | \(27\) |
derivativedivides | \(-\frac {{\mathrm e}^{-\frac {x}{2}}}{2 x^{2}}+\frac {{\mathrm e}^{-\frac {x}{2}}}{4 x}-\frac {\expIntegralEi \left (1, \frac {x}{2}\right )}{8}\) | \(31\) |
default | \(-\frac {{\mathrm e}^{-\frac {x}{2}}}{2 x^{2}}+\frac {{\mathrm e}^{-\frac {x}{2}}}{4 x}-\frac {\expIntegralEi \left (1, \frac {x}{2}\right )}{8}\) | \(31\) |
meijerg | \(\frac {\frac {9}{4} x^{2}-6 x +6}{12 x^{2}}-\frac {\left (-\frac {3 x}{2}+3\right ) {\mathrm e}^{-\frac {x}{2}}}{6 x^{2}}-\frac {\ln \left (\frac {x}{2}\right )}{8}-\frac {\expIntegralEi \left (1, \frac {x}{2}\right )}{8}-\frac {3}{16}+\frac {\ln \relax (x )}{8}-\frac {\ln \relax (2)}{8}-\frac {1}{2 x^{2}}+\frac {1}{2 x}\) | \(63\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.58, size = 7, normalized size = 0.18 \[ -\frac {1}{4} \, \Gamma \left (-2, \frac {1}{2} \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.27, size = 22, normalized size = 0.56 \[ \frac {{\mathrm {e}}^{-\frac {x}{2}}\,\left (\frac {1}{x}-\frac {2}{x^2}\right )}{4}-\frac {\mathrm {expint}\left (\frac {x}{2}\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 1.34, size = 32, normalized size = 0.82 \[ \frac {\operatorname {Ei}{\left (\frac {x e^{i \pi }}{2} \right )}}{8} + \frac {e^{- \frac {x}{2}}}{4 x} - \frac {e^{- \frac {x}{2}}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________