\(\int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^4} \, dx\) [188]

Optimal result

Integrand size = 18, antiderivative size = 108 \[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^4} \, dx=\frac {b e^{-2 b^2 x^2}}{3 \sqrt {\pi } x^2}-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{3 x^3}+\frac {2 b^2 e^{-b^2 x^2} \text {erfc}(b x)}{3 x}-\frac {1}{3} b^3 \sqrt {\pi } \text {erfc}(b x)^2+\frac {4 b^3 \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{3 \sqrt {\pi }} \] Output:

1/3*b/exp(2*b^2*x^2)/Pi^(1/2)/x^2-1/3*erfc(b*x)/exp(b^2*x^2)/x^3+2/3*b^2*e 
rfc(b*x)/exp(b^2*x^2)/x-1/3*b^3*Pi^(1/2)*erfc(b*x)^2+4/3*b^3*Ei(-2*b^2*x^2 
)/Pi^(1/2)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.79 \[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^4} \, dx=\frac {1}{3} \left (\frac {e^{-b^2 x^2} \left (-1+2 b^2 x^2\right ) \text {erfc}(b x)}{x^3}-b^3 \sqrt {\pi } \text {erfc}(b x)^2+\frac {b \left (\frac {e^{-2 b^2 x^2}}{x^2}+4 b^2 \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )\right )}{\sqrt {\pi }}\right ) \] Input:

Integrate[Erfc[b*x]/(E^(b^2*x^2)*x^4),x]
 

Output:

(((-1 + 2*b^2*x^2)*Erfc[b*x])/(E^(b^2*x^2)*x^3) - b^3*Sqrt[Pi]*Erfc[b*x]^2 
 + (b*(1/(E^(2*b^2*x^2)*x^2) + 4*b^2*ExpIntegralEi[-2*b^2*x^2]))/Sqrt[Pi]) 
/3
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.15, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {6946, 2643, 2639, 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.

\(\displaystyle \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^4} \, dx\)

\(\Big \downarrow \) 6946

\(\displaystyle -\frac {2}{3} b^2 \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^2}dx-\frac {2 b \int \frac {e^{-2 b^2 x^2}}{x^3}dx}{3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{3 x^3}\)

\(\Big \downarrow \) 2643

\(\displaystyle -\frac {2}{3} b^2 \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^2}dx-\frac {2 b \left (-2 b^2 \int \frac {e^{-2 b^2 x^2}}{x}dx-\frac {e^{-2 b^2 x^2}}{2 x^2}\right )}{3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{3 x^3}\)

\(\Big \downarrow \) 2639

\(\displaystyle -\frac {2}{3} b^2 \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^2}dx-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{3 x^3}-\frac {2 b \left (b^2 \left (-\operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )\right )-\frac {e^{-2 b^2 x^2}}{2 x^2}\right )}{3 \sqrt {\pi }}\)

\(\Big \downarrow \) 6946

\(\displaystyle -\frac {2}{3} b^2 \left (-2 b^2 \int e^{-b^2 x^2} \text {erfc}(b x)dx-\frac {2 b \int \frac {e^{-2 b^2 x^2}}{x}dx}{\sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x}\right )-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{3 x^3}-\frac {2 b \left (b^2 \left (-\operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )\right )-\frac {e^{-2 b^2 x^2}}{2 x^2}\right )}{3 \sqrt {\pi }}\)

\(\Big \downarrow \) 2639

\(\displaystyle -\frac {2}{3} b^2 \left (-2 b^2 \int e^{-b^2 x^2} \text {erfc}(b x)dx-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x}-\frac {b \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\sqrt {\pi }}\right )-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{3 x^3}-\frac {2 b \left (b^2 \left (-\operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )\right )-\frac {e^{-2 b^2 x^2}}{2 x^2}\right )}{3 \sqrt {\pi }}\)

\(\Big \downarrow \) 6928

\(\displaystyle -\frac {2}{3} b^2 \left (\sqrt {\pi } b \int \text {erfc}(b x)d\text {erfc}(b x)-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x}-\frac {b \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\sqrt {\pi }}\right )-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{3 x^3}-\frac {2 b \left (b^2 \left (-\operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )\right )-\frac {e^{-2 b^2 x^2}}{2 x^2}\right )}{3 \sqrt {\pi }}\)

\(\Big \downarrow \) 15

\(\displaystyle -\frac {2}{3} b^2 \left (-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x}-\frac {b \operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )}{\sqrt {\pi }}+\frac {1}{2} \sqrt {\pi } b \text {erfc}(b x)^2\right )-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{3 x^3}-\frac {2 b \left (b^2 \left (-\operatorname {ExpIntegralEi}\left (-2 b^2 x^2\right )\right )-\frac {e^{-2 b^2 x^2}}{2 x^2}\right )}{3 \sqrt {\pi }}\)

Input:

Int[Erfc[b*x]/(E^(b^2*x^2)*x^4),x]
 

Output:

-1/3*Erfc[b*x]/(E^(b^2*x^2)*x^3) - (2*b*(-1/2*1/(E^(2*b^2*x^2)*x^2) - b^2* 
ExpIntegralEi[-2*b^2*x^2]))/(3*Sqrt[Pi]) - (2*b^2*(-(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 
]))/3
 

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 2643

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] - Simp[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 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]

\[\int \frac {\operatorname {erfc}\left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}}{x^{4}}d x\]

Input:

int(erfc(b*x)/exp(b^2*x^2)/x^4,x)
 

Output:

int(erfc(b*x)/exp(b^2*x^2)/x^4,x)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.13 \[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^4} \, dx=\frac {2 \, \pi ^{\frac {3}{2}} \sqrt {b^{2}} b^{2} x^{3} \operatorname {erf}\left (\sqrt {b^{2}} x\right ) - {\left (\pi - 2 \, \pi b^{2} x^{2} - {\left (\pi - 2 \, \pi b^{2} x^{2}\right )} \operatorname {erf}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )} - \sqrt {\pi } {\left (\pi b^{3} x^{3} \operatorname {erf}\left (b x\right )^{2} - 4 \, b^{3} x^{3} {\rm Ei}\left (-2 \, b^{2} x^{2}\right ) - b x e^{\left (-2 \, b^{2} x^{2}\right )}\right )}}{3 \, \pi x^{3}} \] Input:

integrate(erfc(b*x)/exp(b^2*x^2)/x^4,x, algorithm="fricas")
 

Output:

1/3*(2*pi^(3/2)*sqrt(b^2)*b^2*x^3*erf(sqrt(b^2)*x) - (pi - 2*pi*b^2*x^2 - 
(pi - 2*pi*b^2*x^2)*erf(b*x))*e^(-b^2*x^2) - sqrt(pi)*(pi*b^3*x^3*erf(b*x) 
^2 - 4*b^3*x^3*Ei(-2*b^2*x^2) - b*x*e^(-2*b^2*x^2)))/(pi*x^3)
 

Sympy [F]

\[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^4} \, dx=\int \frac {e^{- b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}}{x^{4}}\, dx \] Input:

integrate(erfc(b*x)/exp(b**2*x**2)/x**4,x)
 

Output:

Integral(exp(-b**2*x**2)*erfc(b*x)/x**4, x)
 

Maxima [F]

\[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^4} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{4}} \,d x } \] Input:

integrate(erfc(b*x)/exp(b^2*x^2)/x^4,x, algorithm="maxima")
 

Output:

integrate(erfc(b*x)*e^(-b^2*x^2)/x^4, x)
 

Giac [F]

\[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^4} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{4}} \,d x } \] Input:

integrate(erfc(b*x)/exp(b^2*x^2)/x^4,x, algorithm="giac")
 

Output:

integrate(erfc(b*x)*e^(-b^2*x^2)/x^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^4} \, dx=\int \frac {{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right )}{x^4} \,d x \] Input:

int((exp(-b^2*x^2)*erfc(b*x))/x^4,x)
 

Output:

int((exp(-b^2*x^2)*erfc(b*x))/x^4, x)
 

Reduce [F]

\[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^4} \, dx=\frac {-3 e^{b^{2} x^{2}} \left (\int \frac {\mathrm {erf}\left (b x \right )}{e^{b^{2} x^{2}} x^{4}}d x \right ) x^{3}-2 e^{b^{2} x^{2}} \left (\int \frac {1}{e^{b^{2} x^{2}} x^{2}}d x \right ) b^{2} x^{3}-1}{3 e^{b^{2} x^{2}} x^{3}} \] Input:

int(erfc(b*x)/exp(b^2*x^2)/x^4,x)
 

Output:

( - 3*e**(b**2*x**2)*int(erf(b*x)/(e**(b**2*x**2)*x**4),x)*x**3 - 2*e**(b* 
*2*x**2)*int(1/(e**(b**2*x**2)*x**2),x)*b**2*x**3 - 1)/(3*e**(b**2*x**2)*x 
**3)