\(\int \frac {\text {erfc}(b x)^2}{x^3} \, dx\) [129]

Optimal result

Integrand size = 10, antiderivative size = 67 \[ \int \frac {\text {erfc}(b x)^2}{x^3} \, dx=\frac {2 b e^{-b^2 x^2} \text {erfc}(b x)}{\sqrt {\pi } x}-b^2 \text {erfc}(b x)^2-\frac {\text {erfc}(b x)^2}{2 x^2}+\frac {2 b^2 \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\pi } \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6500, 6527, 6509, 30, 2241} \[ \int \frac {\text {erfc}(b x)^2}{x^3} \, dx=\frac {2 b e^{-b^2 x^2} \text {erfc}(b x)}{\sqrt {\pi } x}+b^2 \left (-\text {erfc}(b x)^2\right )+\frac {2 b^2 \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\pi }-\frac {\text {erfc}(b x)^2}{2 x^2} \]

[In]

Int[Erfc[b*x]^2/x^3,x]

[Out]

(2*b*Erfc[b*x])/(E^(b^2*x^2)*Sqrt[Pi]*x) - b^2*Erfc[b*x]^2 - Erfc[b*x]^2/(2*x^2) + (2*b^2*ExpIntegralEi[-2*b^2
*x^2])/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 6500

Int[Erfc[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(Erfc[b*x]^2/(m + 1)), x] + Dist[4*(b/(Sqrt[Pi]
*(m + 1))), Int[(x^(m + 1)*Erfc[b*x])/E^(b^2*x^2), x], x] /; FreeQ[b, x] && (IGtQ[m, 0] || ILtQ[(m + 1)/2, 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 {\text {erfc}(b x)^2}{2 x^2}-\frac {(2 b) \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^2} \, dx}{\sqrt {\pi }} \\ & = \frac {2 b e^{-b^2 x^2} \text {erfc}(b x)}{\sqrt {\pi } x}-\frac {\text {erfc}(b x)^2}{2 x^2}+\frac {\left (4 b^2\right ) \int \frac {e^{-2 b^2 x^2}}{x} \, dx}{\pi }+\frac {\left (4 b^3\right ) \int e^{-b^2 x^2} \text {erfc}(b x) \, dx}{\sqrt {\pi }} \\ & = \frac {2 b e^{-b^2 x^2} \text {erfc}(b x)}{\sqrt {\pi } x}-\frac {\text {erfc}(b x)^2}{2 x^2}+\frac {2 b^2 \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\pi }-\left (2 b^2\right ) \text {Subst}(\int x \, dx,x,\text {erfc}(b x)) \\ & = \frac {2 b e^{-b^2 x^2} \text {erfc}(b x)}{\sqrt {\pi } x}-b^2 \text {erfc}(b x)^2-\frac {\text {erfc}(b x)^2}{2 x^2}+\frac {2 b^2 \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\pi } \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.94 \[ \int \frac {\text {erfc}(b x)^2}{x^3} \, dx=\frac {2 b e^{-b^2 x^2} \text {erfc}(b x)}{\sqrt {\pi } x}+\left (-b^2-\frac {1}{2 x^2}\right ) \text {erfc}(b x)^2+\frac {2 b^2 \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\pi } \]

[In]

Integrate[Erfc[b*x]^2/x^3,x]

[Out]

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

Maple [F]

\[\int \frac {\operatorname {erfc}\left (b x \right )^{2}}{x^{3}}d x\]

[In]

int(erfc(b*x)^2/x^3,x)

[Out]

int(erfc(b*x)^2/x^3,x)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.46 \[ \int \frac {\text {erfc}(b x)^2}{x^3} \, dx=-\frac {\pi - 4 \, \pi \sqrt {b^{2}} b x^{2} \operatorname {erf}\left (\sqrt {b^{2}} x\right ) - 4 \, b^{2} x^{2} {\rm Ei}\left (-2 \, b^{2} x^{2}\right ) + {\left (\pi + 2 \, \pi b^{2} x^{2}\right )} \operatorname {erf}\left (b x\right )^{2} + 4 \, \sqrt {\pi } {\left (b x \operatorname {erf}\left (b x\right ) - b x\right )} e^{\left (-b^{2} x^{2}\right )} - 2 \, \pi \operatorname {erf}\left (b x\right )}{2 \, \pi x^{2}} \]

[In]

integrate(erfc(b*x)^2/x^3,x, algorithm="fricas")

[Out]

-1/2*(pi - 4*pi*sqrt(b^2)*b*x^2*erf(sqrt(b^2)*x) - 4*b^2*x^2*Ei(-2*b^2*x^2) + (pi + 2*pi*b^2*x^2)*erf(b*x)^2 +
4*sqrt(pi)*(b*x*erf(b*x) - b*x)*e^(-b^2*x^2) - 2*pi*erf(b*x))/(pi*x^2)

Sympy [F]

\[ \int \frac {\text {erfc}(b x)^2}{x^3} \, dx=\int \frac {\operatorname {erfc}^{2}{\left (b x \right )}}{x^{3}}\, dx \]

[In]

integrate(erfc(b*x)**2/x**3,x)

[Out]

Integral(erfc(b*x)**2/x**3, x)

Maxima [F]

\[ \int \frac {\text {erfc}(b x)^2}{x^3} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right )^{2}}{x^{3}} \,d x } \]

[In]

integrate(erfc(b*x)^2/x^3,x, algorithm="maxima")

[Out]

integrate(erfc(b*x)^2/x^3, x)

Giac [F]

\[ \int \frac {\text {erfc}(b x)^2}{x^3} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right )^{2}}{x^{3}} \,d x } \]

[In]

integrate(erfc(b*x)^2/x^3,x, algorithm="giac")

[Out]

integrate(erfc(b*x)^2/x^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {erfc}(b x)^2}{x^3} \, dx=\int \frac {{\mathrm {erfc}\left (b\,x\right )}^2}{x^3} \,d x \]

[In]

int(erfc(b*x)^2/x^3,x)

[Out]

int(erfc(b*x)^2/x^3, x)