3.171 \(\int \frac{e^{c+b^2 x^2} \text{Erfc}(b x)}{x^3} \, dx\)

Optimal. Leaf size=88 \[ -\frac{2 b^3 e^c x \text{HypergeometricPFQ}\left (\left \{\frac{1}{2},1\right \},\left \{\frac{3}{2},\frac{3}{2}\right \},b^2 x^2\right )}{\sqrt{\pi }}-\frac{e^{b^2 x^2+c} \text{Erfc}(b x)}{2 x^2}+\frac{1}{2} b^2 e^c \text{ExpIntegralEi}\left (b^2 x^2\right )+\frac{b e^c}{\sqrt{\pi } x} \]

[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]

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Rubi [A]  time = 0.165603, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {6392, 6389, 2210, 6388, 12, 30} \[ -\frac{2 b^3 e^c x \, _2F_2\left (\frac{1}{2},1;\frac{3}{2},\frac{3}{2};b^2 x^2\right )}{\sqrt{\pi }}-\frac{e^{b^2 x^2+c} \text{Erfc}(b x)}{2 x^2}+\frac{1}{2} b^2 e^c \text{Ei}\left (b^2 x^2\right )+\frac{b e^c}{\sqrt{\pi } x} \]

Antiderivative was successfully verified.

[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 6392

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]

Rule 6389

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 2210

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 6388

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

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]

Rubi steps

\begin{align*} \int \frac{e^{c+b^2 x^2} \text{erfc}(b x)}{x^3} \, dx &=-\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 \text{Ei}\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]  time = 0.207471, size = 65, normalized size = 0.74 \[ -\frac{e^c \left (-\frac{4 b x \text{HypergeometricPFQ}\left (\left \{-\frac{1}{2},1\right \},\left \{\frac{1}{2},\frac{3}{2}\right \},b^2 x^2\right )}{\sqrt{\pi }}-b^2 x^2 \text{ExpIntegralEi}\left (b^2 x^2\right )+e^{b^2 x^2}\right )}{2 x^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(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]))/(2*x^2)

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Maple [F]  time = 0.277, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{{b}^{2}{x}^{2}+c}}{\it erfc} \left ( bx \right ) }{{x}^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (\operatorname{erf}\left (b x\right ) - 1\right )} e^{\left (b^{2} x^{2} + c\right )}}{x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)