\(\int \frac {\text {erfc}(b x)}{x^4} \, dx\) [116]

Optimal result

Integrand size = 8, antiderivative size = 56 \[ \int \frac {\text {erfc}(b x)}{x^4} \, dx=\frac {b e^{-b^2 x^2}}{3 \sqrt {\pi } x^2}-\frac {\text {erfc}(b x)}{3 x^3}+\frac {b^3 \operatorname {ExpIntegralEi}\left (-b^2 x^2\right )}{3 \sqrt {\pi }} \]

[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)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6497, 2245, 2241} \[ \int \frac {\text {erfc}(b x)}{x^4} \, dx=\frac {b e^{-b^2 x^2}}{3 \sqrt {\pi } x^2}+\frac {b^3 \operatorname {ExpIntegralEi}\left (-b^2 x^2\right )}{3 \sqrt {\pi }}-\frac {\text {erfc}(b x)}{3 x^3} \]

[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 2241

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 2245

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 6497

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*} \text {integral}& = -\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 \operatorname {ExpIntegralEi}\left (-b^2 x^2\right )}{3 \sqrt {\pi }} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.88 \[ \int \frac {\text {erfc}(b x)}{x^4} \, dx=\frac {1}{3} \left (-\frac {\text {erfc}(b x)}{x^3}+\frac {b \left (\frac {e^{-b^2 x^2}}{x^2}+b^2 \operatorname {ExpIntegralEi}\left (-b^2 x^2\right )\right )}{\sqrt {\pi }}\right ) \]

[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

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.82

method	result	size
parts	\(-\frac {\operatorname {erfc}\left (b x \right )}{3 x^{3}}-\frac {2 b \left (-\frac {{\mathrm e}^{-b^{2} x^{2}}}{2 x^{2}}+\frac {b^{2} \operatorname {Ei}_{1}\left (b^{2} x^{2}\right )}{2}\right )}{3 \sqrt {\pi }}\)	\(46\)
derivativedivides	\(b^{3} \left (-\frac {\operatorname {erfc}\left (b x \right )}{3 b^{3} x^{3}}-\frac {2 \left (-\frac {{\mathrm e}^{-b^{2} x^{2}}}{2 x^{2} b^{2}}+\frac {\operatorname {Ei}_{1}\left (b^{2} x^{2}\right )}{2}\right )}{3 \sqrt {\pi }}\right )\)	\(53\)
default	\(b^{3} \left (-\frac {\operatorname {erfc}\left (b x \right )}{3 b^{3} x^{3}}-\frac {2 \left (-\frac {{\mathrm e}^{-b^{2} x^{2}}}{2 x^{2} b^{2}}+\frac {\operatorname {Ei}_{1}\left (b^{2} x^{2}\right )}{2}\right )}{3 \sqrt {\pi }}\right )\)	\(53\)

[In]

int(erfc(b*x)/x^4,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.91 \[ \int \frac {\text {erfc}(b x)}{x^4} \, dx=-\frac {\pi - \pi \operatorname {erf}\left (b x\right ) - \sqrt {\pi } {\left (b^{3} x^{3} {\rm Ei}\left (-b^{2} x^{2}\right ) + b x e^{\left (-b^{2} x^{2}\right )}\right )}}{3 \, \pi x^{3}} \]

[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)

Sympy [A] (verification not implemented)

Time = 1.12 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.86 \[ \int \frac {\text {erfc}(b x)}{x^4} \, dx=- \frac {b^{3} \operatorname {E}_{1}\left (b^{2} x^{2}\right )}{3 \sqrt {\pi }} + \frac {b e^{- b^{2} x^{2}}}{3 \sqrt {\pi } x^{2}} - \frac {\operatorname {erfc}{\left (b x \right )}}{3 x^{3}} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.48 \[ \int \frac {\text {erfc}(b x)}{x^4} \, dx=\frac {b^{3} \Gamma \left (-1, b^{2} x^{2}\right )}{3 \, \sqrt {\pi }} - \frac {\operatorname {erfc}\left (b x\right )}{3 \, x^{3}} \]

[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

Giac [F]

\[ \int \frac {\text {erfc}(b x)}{x^4} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right )}{x^{4}} \,d x } \]

[In]

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

[Out]

integrate(erfc(b*x)/x^4, x)

Mupad [B] (verification not implemented)

Time = 4.73 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.82 \[ \int \frac {\text {erfc}(b x)}{x^4} \, dx=\frac {b^3\,\mathrm {ei}\left (-b^2\,x^2\right )}{3\,\sqrt {\pi }}-\frac {\frac {\mathrm {erfc}\left (b\,x\right )}{3}-\frac {b\,x\,{\mathrm {e}}^{-b^2\,x^2}}{3\,\sqrt {\pi }}}{x^3} \]

[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