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

Optimal result

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

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

Mathematica [A] (verified)

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

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

Output:

(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
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.12, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6919, 6946, 2639, 6928, 15}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 6919

\(\displaystyle -\frac {2 b \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^2}dx}{\sqrt {\pi }}-\frac {\text {erfc}(b x)^2}{2 x^2}\)

\(\Big \downarrow \) 6946

\(\displaystyle -\frac {2 b \left (-2 b^2 \int e^{-b^2 x^2} \text {erfc}(b x)dx-\frac {2 b \int \frac {e^{-2 b^2 x^2}}{x}dx}{\sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x}\right )}{\sqrt {\pi }}-\frac {\text {erfc}(b x)^2}{2 x^2}\)

\(\Big \downarrow \) 2639

\(\displaystyle -\frac {2 b \left (-2 b^2 \int e^{-b^2 x^2} \text {erfc}(b x)dx-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x}-\frac {b \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\sqrt {\pi }}\right )}{\sqrt {\pi }}-\frac {\text {erfc}(b x)^2}{2 x^2}\)

\(\Big \downarrow \) 6928

\(\displaystyle -\frac {2 b \left (\sqrt {\pi } b \int \text {erfc}(b x)d\text {erfc}(b x)-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x}-\frac {b \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\sqrt {\pi }}\right )}{\sqrt {\pi }}-\frac {\text {erfc}(b x)^2}{2 x^2}\)

\(\Big \downarrow \) 15

\(\displaystyle -\frac {2 b \left (-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x}-\frac {b \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\sqrt {\pi }}+\frac {1}{2} \sqrt {\pi } b \text {erfc}(b x)^2\right )}{\sqrt {\pi }}-\frac {\text {erfc}(b x)^2}{2 x^2}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 15

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

rule 2639

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] /; Free 
Q[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]
 

rule 6919

Int[Erfc[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(Erfc[b*x]^2 
/(m + 1)), x] + Simp[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 6928

Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(b_.)*(x_)]^(n_.), x_Symbol] :> Simp[(-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 6946

Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] 
:> Simp[x^(m + 1)*E^(c + d*x^2)*(Erfc[a + b*x]/(m + 1)), x] + (-Simp[2*(d/( 
m + 1))   Int[x^(m + 2)*E^(c + d*x^2)*Erfc[a + b*x], x], x] + Simp[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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.46 \[ \int \frac {\text {erfc}(b x)^2}{x^3} \, dx=-\frac {\pi - 4 \, \pi \sqrt {b^{2}} b x^{2} \operatorname {erf}\left (\sqrt {b^{2}} x\right ) - 4 \, b^{2} x^{2} {\rm Ei}\left (-2 \, b^{2} x^{2}\right ) + {\left (\pi + 2 \, \pi b^{2} x^{2}\right )} \operatorname {erf}\left (b x\right )^{2} + 4 \, \sqrt {\pi } {\left (b x \operatorname {erf}\left (b x\right ) - b x\right )} e^{\left (-b^{2} x^{2}\right )} - 2 \, \pi \operatorname {erf}\left (b x\right )}{2 \, \pi x^{2}} \] Input:

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

Output:

-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]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\text {erfc}(b x)^2}{x^3} \, dx=\frac {-e^{b^{2} x^{2}} \mathrm {erf}\left (b x \right )^{2} \pi +4 e^{b^{2} x^{2}} \mathrm {erf}\left (b x \right ) b^{2} \pi \,x^{2}+2 e^{b^{2} x^{2}} \mathrm {erf}\left (b x \right ) \pi +4 \sqrt {\pi }\, e^{b^{2} x^{2}} \left (\int \frac {\mathrm {erf}\left (b x \right )}{e^{b^{2} x^{2}} x^{2}}d x \right ) b \,x^{2}-e^{b^{2} x^{2}} \pi +4 \sqrt {\pi }\, b x}{2 e^{b^{2} x^{2}} \pi \,x^{2}} \] Input:

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

Output:

( - e**(b**2*x**2)*erf(b*x)**2*pi + 4*e**(b**2*x**2)*erf(b*x)*b**2*pi*x**2 
 + 2*e**(b**2*x**2)*erf(b*x)*pi + 4*sqrt(pi)*e**(b**2*x**2)*int(erf(b*x)/( 
e**(b**2*x**2)*x**2),x)*b*x**2 - e**(b**2*x**2)*pi + 4*sqrt(pi)*b*x)/(2*e* 
*(b**2*x**2)*pi*x**2)