\(\int \frac {\text {erfc}(b x)}{x^3} \, dx\) [108]

Optimal result

Integrand size = 8, antiderivative size = 40 \[ \int \frac {\text {erfc}(b x)}{x^3} \, dx=\frac {b e^{-b^2 x^2}}{\sqrt {\pi } x}+b^2 \text {erf}(b x)-\frac {\text {erfc}(b x)}{2 x^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 40, 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, 2236} \[ \int \frac {\text {erfc}(b x)}{x^3} \, dx=b^2 \text {erf}(b x)+\frac {b e^{-b^2 x^2}}{\sqrt {\pi } x}-\frac {\text {erfc}(b x)}{2 x^2} \]

[In]

Int[Erfc[b*x]/x^3,x]

[Out]

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

Rule 2236

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F],
2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int \frac {\text {erfc}(b x)}{x^3} \, dx=\frac {b e^{-b^2 x^2}}{\sqrt {\pi } x}+b^2 \text {erf}(b x)-\frac {\text {erfc}(b x)}{2 x^2} \]

[In]

Integrate[Erfc[b*x]/x^3,x]

[Out]

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

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.05

method	result	size
parts	\(-\frac {\operatorname {erfc}\left (b x \right )}{2 x^{2}}-\frac {b \left (-\frac {{\mathrm e}^{-b^{2} x^{2}}}{x}-b \sqrt {\pi }\, \operatorname {erf}\left (b x \right )\right )}{\sqrt {\pi }}\)	\(42\)
parallelrisch	\(-\frac {2 x^{2} \operatorname {erfc}\left (b x \right ) \sqrt {\pi }\, b^{2}-2 \,{\mathrm e}^{-b^{2} x^{2}} b x +\operatorname {erfc}\left (b x \right ) \sqrt {\pi }}{2 \sqrt {\pi }\, x^{2}}\)	\(46\)
derivativedivides	\(b^{2} \left (-\frac {\operatorname {erfc}\left (b x \right )}{2 b^{2} x^{2}}-\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}}}{b x}-\operatorname {erf}\left (b x \right ) \sqrt {\pi }}{\sqrt {\pi }}\right )\)	\(51\)
default	\(b^{2} \left (-\frac {\operatorname {erfc}\left (b x \right )}{2 b^{2} x^{2}}-\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}}}{b x}-\operatorname {erf}\left (b x \right ) \sqrt {\pi }}{\sqrt {\pi }}\right )\)	\(51\)

[In]

int(erfc(b*x)/x^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.08 \[ \int \frac {\text {erfc}(b x)}{x^3} \, dx=-\frac {\pi - 2 \, \sqrt {\pi } b x e^{\left (-b^{2} x^{2}\right )} - {\left (\pi + 2 \, \pi b^{2} x^{2}\right )} \operatorname {erf}\left (b x\right )}{2 \, \pi x^{2}} \]

[In]

integrate(erfc(b*x)/x^3,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.85 \[ \int \frac {\text {erfc}(b x)}{x^3} \, dx=- b^{2} \operatorname {erfc}{\left (b x \right )} + \frac {b e^{- b^{2} x^{2}}}{\sqrt {\pi } x} - \frac {\operatorname {erfc}{\left (b x \right )}}{2 x^{2}} \]

[In]

integrate(erfc(b*x)/x**3,x)

[Out]

-b**2*erfc(b*x) + b*exp(-b**2*x**2)/(sqrt(pi)*x) - erfc(b*x)/(2*x**2)

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.88 \[ \int \frac {\text {erfc}(b x)}{x^3} \, dx=\frac {b^{2} \sqrt {x^{2}} \Gamma \left (-\frac {1}{2}, b^{2} x^{2}\right )}{2 \, \sqrt {\pi } x} - \frac {\operatorname {erfc}\left (b x\right )}{2 \, x^{2}} \]

[In]

integrate(erfc(b*x)/x^3,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {\text {erfc}(b x)}{x^3} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right )}{x^{3}} \,d x } \]

[In]

integrate(erfc(b*x)/x^3,x, algorithm="giac")

[Out]

integrate(erfc(b*x)/x^3, x)

Mupad [B] (verification not implemented)

Time = 4.91 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.95 \[ \int \frac {\text {erfc}(b x)}{x^3} \, dx=-b^2\,\mathrm {erfc}\left (b\,x\right )-\frac {\frac {\mathrm {erfc}\left (b\,x\right )}{2}-\frac {b\,x\,{\mathrm {e}}^{-b^2\,x^2}}{\sqrt {\pi }}}{x^2} \]

[In]

int(erfc(b*x)/x^3,x)

[Out]

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