3.116 \(\int \frac {\text {erfc}(b x)}{x^4} \, dx\)

Optimal. Leaf size=56 \[ \frac {b e^{-b^2 x^2}}{3 \sqrt {\pi } x^2}+\frac {b^3 \text {Ei}\left (-b^2 x^2\right )}{3 \sqrt {\pi }}-\frac {\text {erfc}(b x)}{3 x^3} \]

[Out]

-1/3*erfc(b*x)/x^3+1/3*b/exp(b^2*x^2)/x^2/Pi^(1/2)+1/3*b^3*Ei(-b^2*x^2)/Pi^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6362, 2214, 2210} \[ \frac {b^3 \text {Ei}\left (-b^2 x^2\right )}{3 \sqrt {\pi }}+\frac {b e^{-b^2 x^2}}{3 \sqrt {\pi } x^2}-\frac {\text {Erfc}(b x)}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[Erfc[b*x]/x^4,x]

[Out]

b/(3*E^(b^2*x^2)*Sqrt[Pi]*x^2) - Erfc[b*x]/(3*x^3) + (b^3*ExpIntegralEi[-(b^2*x^2)])/(3*Sqrt[Pi])

Rule 2210

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Simp[(F^a*ExpIntegralEi[
b*(c + d*x)^n*Log[F]])/(f*n), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]

Rule 2214

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*F^(a + b*(c + d*x)^n))/(d*(m + 1)), x] - Dist[(b*n*Log[F])/(m + 1), Int[(c + d*x)^(m + n)*F^(a + b*(c +
d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[-4, (m + 1)/n, 5] && IntegerQ[n
] && ((GtQ[n, 0] && LtQ[m, -1]) || (GtQ[-n, 0] && LeQ[-n, m + 1]))

Rule 6362

Int[Erfc[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*Erfc[a + b*x])/(
d*(m + 1)), x] + Dist[(2*b)/(Sqrt[Pi]*d*(m + 1)), Int[(c + d*x)^(m + 1)/E^(a + b*x)^2, x], x] /; FreeQ[{a, b,
c, d, m}, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\text {erfc}(b x)}{x^4} \, dx &=-\frac {\text {erfc}(b x)}{3 x^3}-\frac {(2 b) \int \frac {e^{-b^2 x^2}}{x^3} \, dx}{3 \sqrt {\pi }}\\ &=\frac {b e^{-b^2 x^2}}{3 \sqrt {\pi } x^2}-\frac {\text {erfc}(b x)}{3 x^3}+\frac {\left (2 b^3\right ) \int \frac {e^{-b^2 x^2}}{x} \, dx}{3 \sqrt {\pi }}\\ &=\frac {b e^{-b^2 x^2}}{3 \sqrt {\pi } x^2}-\frac {\text {erfc}(b x)}{3 x^3}+\frac {b^3 \text {Ei}\left (-b^2 x^2\right )}{3 \sqrt {\pi }}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 49, normalized size = 0.88 \[ \frac {1}{3} \left (\frac {b \left (b^2 \text {Ei}\left (-b^2 x^2\right )+\frac {e^{-b^2 x^2}}{x^2}\right )}{\sqrt {\pi }}-\frac {\text {erfc}(b x)}{x^3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Erfc[b*x]/x^4,x]

[Out]

(-(Erfc[b*x]/x^3) + (b*(1/(E^(b^2*x^2)*x^2) + b^2*ExpIntegralEi[-(b^2*x^2)]))/Sqrt[Pi])/3

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fricas [A]  time = 0.57, size = 51, normalized size = 0.91 \[ -\frac {\pi - \pi \operatorname {erf}\left (b x\right ) - \sqrt {\pi } {\left (b^{3} x^{3} {\rm Ei}\left (-b^{2} x^{2}\right ) + b x e^{\left (-b^{2} x^{2}\right )}\right )}}{3 \, \pi x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfc(b*x)/x^4,x, algorithm="fricas")

[Out]

-1/3*(pi - pi*erf(b*x) - sqrt(pi)*(b^3*x^3*Ei(-b^2*x^2) + b*x*e^(-b^2*x^2)))/(pi*x^3)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erfc}\left (b x\right )}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfc(b*x)/x^4,x, algorithm="giac")

[Out]

integrate(erfc(b*x)/x^4, x)

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maple [A]  time = 0.00, size = 53, normalized size = 0.95 \[ b^{3} \left (-\frac {\mathrm {erfc}\left (b x \right )}{3 b^{3} x^{3}}-\frac {2 \left (-\frac {{\mathrm e}^{-b^{2} x^{2}}}{2 b^{2} x^{2}}+\frac {\Ei \left (1, b^{2} x^{2}\right )}{2}\right )}{3 \sqrt {\pi }}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erfc(b*x)/x^4,x)

[Out]

b^3*(-1/3*erfc(b*x)/b^3/x^3-2/3/Pi^(1/2)*(-1/2/exp(b^2*x^2)/b^2/x^2+1/2*Ei(1,b^2*x^2)))

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maxima [A]  time = 1.96, size = 27, normalized size = 0.48 \[ \frac {b^{3} \Gamma \left (-1, b^{2} x^{2}\right )}{3 \, \sqrt {\pi }} - \frac {\operatorname {erfc}\left (b x\right )}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfc(b*x)/x^4,x, algorithm="maxima")

[Out]

1/3*b^3*gamma(-1, b^2*x^2)/sqrt(pi) - 1/3*erfc(b*x)/x^3

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mupad [B]  time = 0.18, size = 46, normalized size = 0.82 \[ \frac {b^3\,\mathrm {ei}\left (-b^2\,x^2\right )}{3\,\sqrt {\pi }}-\frac {\frac {\mathrm {erfc}\left (b\,x\right )}{3}-\frac {b\,x\,{\mathrm {e}}^{-b^2\,x^2}}{3\,\sqrt {\pi }}}{x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erfc(b*x)/x^4,x)

[Out]

(b^3*ei(-b^2*x^2))/(3*pi^(1/2)) - (erfc(b*x)/3 - (b*x*exp(-b^2*x^2))/(3*pi^(1/2)))/x^3

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sympy [A]  time = 1.75, size = 48, normalized size = 0.86 \[ - \frac {b^{3} \operatorname {E}_{1}\left (b^{2} x^{2}\right )}{3 \sqrt {\pi }} + \frac {b e^{- b^{2} x^{2}}}{3 \sqrt {\pi } x^{2}} - \frac {\operatorname {erfc}{\left (b x \right )}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfc(b*x)/x**4,x)

[Out]

-b**3*expint(1, b**2*x**2)/(3*sqrt(pi)) + b*exp(-b**2*x**2)/(3*sqrt(pi)*x**2) - erfc(b*x)/(3*x**3)

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