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

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

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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]

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*}

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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]

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

[Out]

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

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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]

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

[Out]

Could not integrate

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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]

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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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]

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

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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]

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

[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 = 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]

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

[Out]

-1/4*gamma(-2, 1/2*x)

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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]

int(exp(-x/2)/x^3,x)

[Out]

(exp(-x/2)*(1/x - 2/x^2))/4 - expint(x/2)/8

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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]

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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