3.115 \(\int \frac {\text {erfc}(b x)}{x^2} \, dx\)

Optimal. Leaf size=27 \[ -\frac {b \text {Ei}\left (-b^2 x^2\right )}{\sqrt {\pi }}-\frac {\text {erfc}(b x)}{x} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6362, 2210} \[ -\frac {b \text {Ei}\left (-b^2 x^2\right )}{\sqrt {\pi }}-\frac {\text {Erfc}(b x)}{x} \]

Antiderivative was successfully verified.

[In]

Int[Erfc[b*x]/x^2,x]

[Out]

-(Erfc[b*x]/x) - (b*ExpIntegralEi[-(b^2*x^2)])/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 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^2} \, dx &=-\frac {\text {erfc}(b x)}{x}-\frac {(2 b) \int \frac {e^{-b^2 x^2}}{x} \, dx}{\sqrt {\pi }}\\ &=-\frac {\text {erfc}(b x)}{x}-\frac {b \text {Ei}\left (-b^2 x^2\right )}{\sqrt {\pi }}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 27, normalized size = 1.00 \[ -\frac {b \text {Ei}\left (-b^2 x^2\right )}{\sqrt {\pi }}-\frac {\text {erfc}(b x)}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[Erfc[b*x]/x^2,x]

[Out]

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

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fricas [A]  time = 0.47, size = 32, normalized size = 1.19 \[ -\frac {\pi + \sqrt {\pi } b x {\rm Ei}\left (-b^{2} x^{2}\right ) - \pi \operatorname {erf}\left (b x\right )}{\pi x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfc(b*x)/x^2,x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erfc}\left (b x\right )}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfc(b*x)/x^2,x, algorithm="giac")

[Out]

integrate(erfc(b*x)/x^2, x)

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maple [A]  time = 0.00, size = 29, normalized size = 1.07 \[ b \left (-\frac {\mathrm {erfc}\left (b x \right )}{b x}+\frac {\Ei \left (1, b^{2} x^{2}\right )}{\sqrt {\pi }}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erfc(b*x)/x^2,x)

[Out]

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

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maxima [A]  time = 3.00, size = 25, normalized size = 0.93 \[ -\frac {b {\rm Ei}\left (-b^{2} x^{2}\right )}{\sqrt {\pi }} - \frac {\operatorname {erfc}\left (b x\right )}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfc(b*x)/x^2,x, algorithm="maxima")

[Out]

-b*Ei(-b^2*x^2)/sqrt(pi) - erfc(b*x)/x

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mupad [B]  time = 0.15, size = 25, normalized size = 0.93 \[ -\frac {\mathrm {erfc}\left (b\,x\right )}{x}-\frac {b\,\mathrm {ei}\left (-b^2\,x^2\right )}{\sqrt {\pi }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erfc(b*x)/x^2,x)

[Out]

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

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sympy [A]  time = 0.98, size = 20, normalized size = 0.74 \[ \frac {b \operatorname {E}_{1}\left (b^{2} x^{2}\right )}{\sqrt {\pi }} - \frac {\operatorname {erfc}{\left (b x \right )}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfc(b*x)/x**2,x)

[Out]

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

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