\(\int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^5} \, dx\) [172]

Optimal result

Integrand size = 19, antiderivative size = 134 \[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^5} \, dx=\frac {b e^c}{6 \sqrt {\pi } x^3}+\frac {b^3 e^c}{2 \sqrt {\pi } x}-\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{4 x^4}-\frac {b^2 e^{c+b^2 x^2} \text {erfc}(b x)}{4 x^2}+\frac {1}{4} b^4 e^c \operatorname {ExpIntegralEi}\left (b^2 x^2\right )-\frac {b^5 e^c x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};b^2 x^2\right )}{\sqrt {\pi }} \] Output:

1/6*b*exp(c)/Pi^(1/2)/x^3+1/2*b^3*exp(c)/Pi^(1/2)/x-1/4*exp(b^2*x^2+c)*erf 
c(b*x)/x^4-1/4*b^2*exp(b^2*x^2+c)*erfc(b*x)/x^2+1/4*b^4*exp(c)*Ei(b^2*x^2) 
-b^5*exp(c)*x*hypergeom([1/2, 1],[3/2, 3/2],b^2*x^2)/Pi^(1/2)
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.62 \[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^5} \, dx=-\frac {e^c \left (3 \sqrt {\pi } \left (e^{b^2 x^2} \left (1+b^2 x^2\right )-b^4 x^4 \operatorname {ExpIntegralEi}\left (b^2 x^2\right )\right )-8 b x \, _2F_2\left (-\frac {3}{2},1;-\frac {1}{2},\frac {3}{2};b^2 x^2\right )\right )}{12 \sqrt {\pi } x^4} \] Input:

Integrate[(E^(c + b^2*x^2)*Erfc[b*x])/x^5,x]
 

Output:

-1/12*(E^c*(3*Sqrt[Pi]*(E^(b^2*x^2)*(1 + b^2*x^2) - b^4*x^4*ExpIntegralEi[ 
b^2*x^2]) - 8*b*x*HypergeometricPFQ[{-3/2, 1}, {-1/2, 3/2}, b^2*x^2]))/(Sq 
rt[Pi]*x^4)
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00, 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, 15, 6943, 2639, 6942}

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.

\(\displaystyle \int \frac {e^{b^2 x^2+c} \text {erfc}(b x)}{x^5} \, dx\)

\(\Big \downarrow \) 6946

\(\displaystyle \frac {1}{2} b^2 \int \frac {e^{b^2 x^2+c} \text {erfc}(b x)}{x^3}dx-\frac {b \int \frac {e^c}{x^4}dx}{2 \sqrt {\pi }}-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{4 x^4}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {1}{2} b^2 \int \frac {e^{b^2 x^2+c} \text {erfc}(b x)}{x^3}dx-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{4 x^4}+\frac {b e^c}{6 \sqrt {\pi } x^3}\)

\(\Big \downarrow \) 6946

\(\displaystyle \frac {1}{2} b^2 \left (b^2 \int \frac {e^{b^2 x^2+c} \text {erfc}(b x)}{x}dx-\frac {b \int \frac {e^c}{x^2}dx}{\sqrt {\pi }}-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{2 x^2}\right )-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{4 x^4}+\frac {b e^c}{6 \sqrt {\pi } x^3}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {1}{2} b^2 \left (b^2 \int \frac {e^{b^2 x^2+c} \text {erfc}(b x)}{x}dx-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{2 x^2}+\frac {b e^c}{\sqrt {\pi } x}\right )-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{4 x^4}+\frac {b e^c}{6 \sqrt {\pi } x^3}\)

\(\Big \downarrow \) 6943

\(\displaystyle \frac {1}{2} b^2 \left (b^2 \left (\int \frac {e^{b^2 x^2+c}}{x}dx-\int \frac {e^{b^2 x^2+c} \text {erf}(b x)}{x}dx\right )-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{2 x^2}+\frac {b e^c}{\sqrt {\pi } x}\right )-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{4 x^4}+\frac {b e^c}{6 \sqrt {\pi } x^3}\)

\(\Big \downarrow \) 2639

\(\displaystyle \frac {1}{2} b^2 \left (b^2 \left (\frac {1}{2} e^c \operatorname {ExpIntegralEi}\left (b^2 x^2\right )-\int \frac {e^{b^2 x^2+c} \text {erf}(b x)}{x}dx\right )-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{2 x^2}+\frac {b e^c}{\sqrt {\pi } x}\right )-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{4 x^4}+\frac {b e^c}{6 \sqrt {\pi } x^3}\)

\(\Big \downarrow \) 6942

\(\displaystyle \frac {1}{2} b^2 \left (b^2 \left (\frac {1}{2} e^c \operatorname {ExpIntegralEi}\left (b^2 x^2\right )-\frac {2 b e^c x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};b^2 x^2\right )}{\sqrt {\pi }}\right )-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{2 x^2}+\frac {b e^c}{\sqrt {\pi } x}\right )-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{4 x^4}+\frac {b e^c}{6 \sqrt {\pi } x^3}\)

Input:

Int[(E^(c + b^2*x^2)*Erfc[b*x])/x^5,x]
 

Output:

(b*E^c)/(6*Sqrt[Pi]*x^3) - (E^(c + b^2*x^2)*Erfc[b*x])/(4*x^4) + (b^2*((b* 
E^c)/(Sqrt[Pi]*x) - (E^(c + b^2*x^2)*Erfc[b*x])/(2*x^2) + b^2*((E^c*ExpInt 
egralEi[b^2*x^2])/2 - (2*b*E^c*x*HypergeometricPFQ[{1/2, 1}, {3/2, 3/2}, b 
^2*x^2])/Sqrt[Pi])))/2
 

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 6942

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

rule 6943

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

\[\int \frac {{\mathrm e}^{b^{2} x^{2}+c} \operatorname {erfc}\left (b x \right )}{x^{5}}d x\]

Input:

int(exp(b^2*x^2+c)*erfc(b*x)/x^5,x)
 

Output:

int(exp(b^2*x^2+c)*erfc(b*x)/x^5,x)
 

Fricas [F]

\[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^5} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{5}} \,d x } \] Input:

integrate(exp(b^2*x^2+c)*erfc(b*x)/x^5,x, algorithm="fricas")
 

Output:

integral(-(erf(b*x) - 1)*e^(b^2*x^2 + c)/x^5, x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^5} \, dx=\text {Timed out} \] Input:

integrate(exp(b**2*x**2+c)*erfc(b*x)/x**5,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^5} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{5}} \,d x } \] Input:

integrate(exp(b^2*x^2+c)*erfc(b*x)/x^5,x, algorithm="maxima")
 

Output:

integrate(erfc(b*x)*e^(b^2*x^2 + c)/x^5, x)
 

Giac [F]

\[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^5} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{5}} \,d x } \] Input:

integrate(exp(b^2*x^2+c)*erfc(b*x)/x^5,x, algorithm="giac")
 

Output:

integrate(erfc(b*x)*e^(b^2*x^2 + c)/x^5, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^5} \, dx=\int \frac {{\mathrm {e}}^{b^2\,x^2+c}\,\mathrm {erfc}\left (b\,x\right )}{x^5} \,d x \] Input:

int((exp(c + b^2*x^2)*erfc(b*x))/x^5,x)
 

Output:

int((exp(c + b^2*x^2)*erfc(b*x))/x^5, x)
 

Reduce [F]

\[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^5} \, dx=\frac {e^{c} \left (3 \mathit {ei} \left (b^{2} x^{2}\right ) b^{4} \pi \,x^{4}+3 e^{b^{2} x^{2}} \mathrm {erf}\left (b x \right ) b^{2} \pi \,x^{2}+3 e^{b^{2} x^{2}} \mathrm {erf}\left (b x \right ) \pi -3 e^{b^{2} x^{2}} b^{2} \pi \,x^{2}-3 e^{b^{2} x^{2}} \pi +6 \sqrt {\pi }\, b^{3} x^{3}+2 \sqrt {\pi }\, b x -6 \left (\int \frac {e^{b^{2} x^{2}} \mathrm {erf}\left (b x \right )}{x}d x \right ) b^{4} \pi \,x^{4}\right )}{12 \pi \,x^{4}} \] Input:

int(exp(b^2*x^2+c)*erfc(b*x)/x^5,x)
 

Output:

(e**c*(3*ei(b**2*x**2)*b**4*pi*x**4 + 3*e**(b**2*x**2)*erf(b*x)*b**2*pi*x* 
*2 + 3*e**(b**2*x**2)*erf(b*x)*pi - 3*e**(b**2*x**2)*b**2*pi*x**2 - 3*e**( 
b**2*x**2)*pi + 6*sqrt(pi)*b**3*x**3 + 2*sqrt(pi)*b*x - 6*int((e**(b**2*x* 
*2)*erf(b*x))/x,x)*b**4*pi*x**4))/(12*pi*x**4)