\(\int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^2} \, dx\) [187]

Optimal result

Integrand size = 18, antiderivative size = 53 \[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^2} \, dx=-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x}+\frac {1}{2} b \sqrt {\pi } \text {erfc}(b x)^2-\frac {b \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\sqrt {\pi }} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6527, 6509, 30, 2241} \[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^2} \, dx=-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x}-\frac {b \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\sqrt {\pi }}+\frac {1}{2} \sqrt {\pi } b \text {erfc}(b x)^2 \]

[In]

Int[Erfc[b*x]/(E^(b^2*x^2)*x^2),x]

[Out]

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 6509

Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(b_.)*(x_)]^(n_.), x_Symbol] :> Dist[(-E^c)*(Sqrt[Pi]/(2*b)), Subst[Int[x^n,
x], x, Erfc[b*x]], x] /; FreeQ[{b, c, d, n}, x] && EqQ[d, -b^2]

Rule 6527

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

Rubi steps \begin{align*} \text {integral}& = -\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x}-\left (2 b^2\right ) \int e^{-b^2 x^2} \text {erfc}(b x) \, dx-\frac {(2 b) \int \frac {e^{-2 b^2 x^2}}{x} \, dx}{\sqrt {\pi }} \\ & = -\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x}-\frac {b \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\sqrt {\pi }}+\left (b \sqrt {\pi }\right ) \text {Subst}(\int x \, dx,x,\text {erfc}(b x)) \\ & = -\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x}+\frac {1}{2} b \sqrt {\pi } \text {erfc}(b x)^2-\frac {b \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\sqrt {\pi }} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^2} \, dx=-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x}+\frac {1}{2} b \sqrt {\pi } \text {erfc}(b x)^2-\frac {b \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\sqrt {\pi }} \]

[In]

Integrate[Erfc[b*x]/(E^(b^2*x^2)*x^2),x]

[Out]

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

Maple [F]

\[\int \frac {\operatorname {erfc}\left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}}{x^{2}}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.45 \[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^2} \, dx=-\frac {2 \, \pi ^{\frac {3}{2}} \sqrt {b^{2}} x \operatorname {erf}\left (\sqrt {b^{2}} x\right ) + 2 \, {\left (\pi - \pi \operatorname {erf}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )} - \sqrt {\pi } {\left (\pi b x \operatorname {erf}\left (b x\right )^{2} - 2 \, b x {\rm Ei}\left (-2 \, b^{2} x^{2}\right )\right )}}{2 \, \pi x} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^2} \, dx=\int \frac {e^{- b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}}{x^{2}}\, dx \]

[In]

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

[Out]

Integral(exp(-b**2*x**2)*erfc(b*x)/x**2, x)

Maxima [F]

\[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^2} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{2}} \,d x } \]

[In]

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

[Out]

integrate(erfc(b*x)*e^(-b^2*x^2)/x^2, x)

Giac [F]

\[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^2} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{2}} \,d x } \]

[In]

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

[Out]

integrate(erfc(b*x)*e^(-b^2*x^2)/x^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^2} \, dx=\int \frac {{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right )}{x^2} \,d x \]

[In]

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

[Out]

int((exp(-b^2*x^2)*erfc(b*x))/x^2, x)