\(\int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^3} \, dx\) [171]

Optimal result

Integrand size = 19, antiderivative size = 88 \[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^3} \, dx=\frac {b e^c}{\sqrt {\pi } x}-\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{2 x^2}+\frac {1}{2} b^2 e^c \operatorname {ExpIntegralEi}\left (b^2 x^2\right )-\frac {2 b^3 e^c x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};b^2 x^2\right )}{\sqrt {\pi }} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6527, 6524, 2241, 6523, 12, 30} \[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^3} \, dx=-\frac {2 b^3 e^c x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};b^2 x^2\right )}{\sqrt {\pi }}-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{2 x^2}+\frac {1}{2} b^2 e^c \operatorname {ExpIntegralEi}\left (b^2 x^2\right )+\frac {b e^c}{\sqrt {\pi } x} \]

[In]

Int[(E^(c + b^2*x^2)*Erfc[b*x])/x^3,x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 6523

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] /; FreeQ[{b, c, d}, x] && EqQ[d, b^2]

Rule 6524

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 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 {e^{c+b^2 x^2} \text {erfc}(b x)}{2 x^2}+b^2 \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x} \, dx-\frac {b \int \frac {e^c}{x^2} \, dx}{\sqrt {\pi }} \\ & = -\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{2 x^2}+b^2 \int \frac {e^{c+b^2 x^2}}{x} \, dx-b^2 \int \frac {e^{c+b^2 x^2} \text {erf}(b x)}{x} \, dx-\frac {\left (b e^c\right ) \int \frac {1}{x^2} \, dx}{\sqrt {\pi }} \\ & = \frac {b e^c}{\sqrt {\pi } x}-\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{2 x^2}+\frac {1}{2} b^2 e^c \operatorname {ExpIntegralEi}\left (b^2 x^2\right )-\frac {2 b^3 e^c x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};b^2 x^2\right )}{\sqrt {\pi }} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.74 \[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^3} \, dx=-\frac {e^c \left (e^{b^2 x^2}-b^2 x^2 \operatorname {ExpIntegralEi}\left (b^2 x^2\right )-\frac {4 b x \, _2F_2\left (-\frac {1}{2},1;\frac {1}{2},\frac {3}{2};b^2 x^2\right )}{\sqrt {\pi }}\right )}{2 x^2} \]

[In]

Integrate[(E^(c + b^2*x^2)*Erfc[b*x])/x^3,x]

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(-(erf(b*x) - 1)*e^(b^2*x^2 + c)/x^3, x)

Sympy [A] (verification not implemented)

Time = 28.84 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.69 \[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^3} \, dx=\frac {b^{2} e^{c} \operatorname {Ei}{\left (b^{2} x^{2} \right )}}{2} + \frac {2 b e^{c} {{}_{2}F_{2}\left (\begin {matrix} - \frac {1}{2}, 1 \\ \frac {1}{2}, \frac {3}{2} \end {matrix}\middle | {b^{2} x^{2}} \right )}}{\sqrt {\pi } x} - \frac {e^{c} e^{b^{2} x^{2}}}{2 x^{2}} \]

[In]

integrate(exp(b**2*x**2+c)*erfc(b*x)/x**3,x)

[Out]

b**2*exp(c)*Ei(b**2*x**2)/2 + 2*b*exp(c)*hyper((-1/2, 1), (1/2, 3/2), b**2*x**2)/(sqrt(pi)*x) - exp(c)*exp(b**
2*x**2)/(2*x**2)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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