\(\int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^4} \, dx\) [177]

Optimal result

Integrand size = 19, antiderivative size = 134 \[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^4} \, dx=\frac {b e^c}{3 \sqrt {\pi } x^2}-\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{3 x^3}-\frac {2 b^2 e^{c+b^2 x^2} \text {erfc}(b x)}{3 x}+\frac {2}{3} b^3 e^c \sqrt {\pi } \text {erfi}(b x)-\frac {4 b^5 e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{3 \sqrt {\pi }}-\frac {4 b^3 e^c \log (x)}{3 \sqrt {\pi }} \] Output:

1/3*b*exp(c)/Pi^(1/2)/x^2-1/3*exp(b^2*x^2+c)*erfc(b*x)/x^3-2/3*b^2*exp(b^2 
*x^2+c)*erfc(b*x)/x+2/3*b^3*exp(c)*Pi^(1/2)*erfi(b*x)-4/3*b^5*exp(c)*x^2*h 
ypergeom([1, 1],[3/2, 2],b^2*x^2)/Pi^(1/2)-4/3*b^3*exp(c)*ln(x)/Pi^(1/2)
 

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.13 \[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^4} \, dx=\frac {e^c \left (-e^{b^2 x^2} \sqrt {\pi }+b x-2 b^2 e^{b^2 x^2} \sqrt {\pi } x^2+e^{b^2 x^2} \sqrt {\pi } \left (1+2 b^2 x^2\right ) \text {erf}(b x)+2 b^3 \pi x^3 \text {erfi}(b x)-2 b^3 \pi x^3 \text {erf}(b x) \text {erfi}(b x)+4 b^5 x^5 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )-4 b^3 x^3 \log (x)\right )}{3 \sqrt {\pi } x^3} \] Input:

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

Output:

(E^c*(-(E^(b^2*x^2)*Sqrt[Pi]) + b*x - 2*b^2*E^(b^2*x^2)*Sqrt[Pi]*x^2 + E^( 
b^2*x^2)*Sqrt[Pi]*(1 + 2*b^2*x^2)*Erf[b*x] + 2*b^3*Pi*x^3*Erfi[b*x] - 2*b^ 
3*Pi*x^3*Erf[b*x]*Erfi[b*x] + 4*b^5*x^5*HypergeometricPFQ[{1, 1}, {3/2, 2} 
, -(b^2*x^2)] - 4*b^3*x^3*Log[x]))/(3*Sqrt[Pi]*x^3)
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {6946, 15, 6946, 14, 6931, 2633, 6930}

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.

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

\(\Big \downarrow \) 6946

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 6946

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

\(\Big \downarrow \) 14

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

\(\Big \downarrow \) 6931

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

\(\Big \downarrow \) 2633

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

\(\Big \downarrow \) 6930

\(\displaystyle \frac {2}{3} b^2 \left (2 b^2 \left (\frac {\sqrt {\pi } e^c \text {erfi}(b x)}{2 b}-\frac {b e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{\sqrt {\pi }}\right )-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{x}-\frac {2 b e^c \log (x)}{\sqrt {\pi }}\right )-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{3 x^3}+\frac {b e^c}{3 \sqrt {\pi } x^2}\)

Input:

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

Output:

(b*E^c)/(3*Sqrt[Pi]*x^2) - (E^(c + b^2*x^2)*Erfc[b*x])/(3*x^3) + (2*b^2*(- 
((E^(c + b^2*x^2)*Erfc[b*x])/x) + 2*b^2*((E^c*Sqrt[Pi]*Erfi[b*x])/(2*b) - 
(b*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/Sqrt[Pi]) - (2*b* 
E^c*Log[x])/Sqrt[Pi]))/3
 

Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[x], x] /; FreeQ[a, x]
 

rule 15

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

rule 2633

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

rule 6930

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(b_.)*(x_)], x_Symbol] :> Simp[b*E^c*(x^2/ 
Sqrt[Pi])*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2], x] /; FreeQ[{b, c, 
d}, x] && EqQ[d, b^2]
 

rule 6931

Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(b_.)*(x_)], x_Symbol] :> Int[E^(c + d*x^ 
2), x] - Int[E^(c + d*x^2)*Erf[b*x], x] /; FreeQ[{b, c, d}, 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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(-(erf(b*x) - 1)*e^(b^2*x^2 + c)/x^4, x)
 

Sympy [A] (verification not implemented)

Time = 82.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.16 \[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^4} \, dx=- \frac {b^{3} {G_{3, 2}^{1, 3}\left (\begin {matrix} 2, \frac {5}{2}, 1 & \\2 & 0 \end {matrix} \middle | {\frac {1}{b^{2} x^{2}}} \right )} e^{c}}{2 \pi } \] Input:

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

Output:

-b**3*meijerg(((2, 5/2, 1), ()), ((2,), (0,)), 1/(b**2*x**2))*exp(c)/(2*pi 
)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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