Optimal. Leaf size=39 \[ -\frac {e^{-x/2}}{2 x^2}+\frac {e^{-x/2}}{4 x}+\frac {\text {Ei}\left (-\frac {x}{2}\right )}{8} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, 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 = {2208, 2209}
\begin {gather*} \frac {1}{8} \text {ExpIntegralEi}\left (-\frac {x}{2}\right )-\frac {e^{-x/2}}{2 x^2}+\frac {e^{-x/2}}{4 x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2208
Rule 2209
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 \begin {gather*} \frac {1}{8} \left (\frac {2 e^{-x/2} (-2+x)}{x^2}+\text {Ei}\left (-\frac {x}{2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.04, size = 31, normalized size = 0.79
method | result | size |
risch | \(-\frac {{\mathrm e}^{-\frac {x}{2}}}{2 x^{2}}+\frac {{\mathrm e}^{-\frac {x}{2}}}{4 x}-\frac {\expIntegral \left (1, \frac {x}{2}\right )}{8}\) | \(27\) |
derivativedivides | \(-\frac {{\mathrm e}^{-\frac {x}{2}}}{2 x^{2}}+\frac {{\mathrm e}^{-\frac {x}{2}}}{4 x}-\frac {\expIntegral \left (1, \frac {x}{2}\right )}{8}\) | \(31\) |
default | \(-\frac {{\mathrm e}^{-\frac {x}{2}}}{2 x^{2}}+\frac {{\mathrm e}^{-\frac {x}{2}}}{4 x}-\frac {\expIntegral \left (1, \frac {x}{2}\right )}{8}\) | \(31\) |
meijerg | \(-\frac {1}{2 x^{2}}+\frac {1}{2 x}-\frac {3}{16}+\frac {\ln \left (x \right )}{8}-\frac {\ln \left (2\right )}{8}+\frac {\frac {9}{4} x^{2}-6 x +6}{12 x^{2}}-\frac {\left (-\frac {3 x}{2}+3\right ) {\mathrm e}^{-\frac {x}{2}}}{6 x^{2}}-\frac {\ln \left (\frac {x}{2}\right )}{8}-\frac {\expIntegral \left (1, \frac {x}{2}\right )}{8}\) | \(63\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 1.64, size = 7, normalized size = 0.18 \begin {gather*} -\frac {1}{4} \, \Gamma \left (-2, \frac {1}{2} \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.82, size = 23, normalized size = 0.59 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 0.64, size = 32, normalized size = 0.82 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.77, size = 27, normalized size = 0.69 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.27, size = 22, normalized size = 0.56 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________