3.536 \(\int \frac{e^{-x/2}}{x^3} \, dx\)

Optimal. Leaf size=39 \[ \frac{1}{8} \text{ExpIntegralEi}\left (-\frac{x}{2}\right )-\frac{e^{-x/2}}{2 x^2}+\frac{e^{-x/2}}{4 x} \]

[Out]

-1/(2*E^(x/2)*x^2) + 1/(4*E^(x/2)*x) + ExpIntegralEi[-x/2]/8

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Rubi [A]  time = 0.0330314, antiderivative size = 39, normalized size of antiderivative = 1., 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]

Int[1/(E^(x/2)*x^3),x]

[Out]

-1/(2*E^(x/2)*x^2) + 1/(4*E^(x/2)*x) + ExpIntegralEi[-x/2]/8

Rule 2177

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*(b*F^(g*(e + f*x)))^n)/(d*(m + 1)), x] - Dist[(f*g*n*Log[F])/(d*(m + 1)), Int[(c + d*x)^(m + 1)*(b*F^(g*
(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] &&  !$UseGamma ===
True

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

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 = 0.0222686, size = 26, normalized size = 0.67 \[ \frac{1}{8} \left (\text{ExpIntegralEi}\left (-\frac{x}{2}\right )+\frac{2 e^{-x/2} (x-2)}{x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(E^(x/2)*x^3),x]

[Out]

((2*(-2 + x))/(E^(x/2)*x^2) + ExpIntegralEi[-x/2])/8

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Maple [A]  time = 0.001, size = 31, normalized size = 0.8 \begin{align*} -{\frac{1}{2\,{x}^{2}} \left ({{\rm e}^{{\frac{x}{2}}}} \right ) ^{-1}}+{\frac{1}{4\,x} \left ({{\rm e}^{{\frac{x}{2}}}} \right ) ^{-1}}-{\frac{1}{8}{\it Ei} \left ( 1,{\frac{x}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/exp(1/2*x)/x^3,x)

[Out]

-1/2/exp(1/2*x)/x^2+1/4/exp(1/2*x)/x-1/8*Ei(1,1/2*x)

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Maxima [A]  time = 1.02505, size = 9, normalized size = 0.23 \begin{align*} -\frac{1}{4} \, \Gamma \left (-2, \frac{1}{2} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/exp(1/2*x)/x^3,x, algorithm="maxima")

[Out]

-1/4*gamma(-2, 1/2*x)

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Fricas [A]  time = 1.77915, size = 66, normalized size = 1.69 \begin{align*} \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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/exp(1/2*x)/x^3,x, algorithm="fricas")

[Out]

1/8*(x^2*Ei(-1/2*x) + 2*(x - 2)*e^(-1/2*x))/x^2

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Sympy [C]  time = 1.44202, size = 32, normalized size = 0.82 \begin{align*} \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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/exp(1/2*x)/x**3,x)

[Out]

Ei(x*exp_polar(I*pi)/2)/8 + exp(-x/2)/(4*x) - exp(-x/2)/(2*x**2)

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Giac [A]  time = 1.11147, size = 36, normalized size = 0.92 \begin{align*} \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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/exp(1/2*x)/x^3,x, algorithm="giac")

[Out]

1/8*(x^2*Ei(-1/2*x) + 2*x*e^(-1/2*x) - 4*e^(-1/2*x))/x^2