3.12 \(\int \frac{\text{Erf}(b x)}{x^2} \, dx\)

Optimal. Leaf size=26 \[ \frac{b \text{ExpIntegralEi}\left (-b^2 x^2\right )}{\sqrt{\pi }}-\frac{\text{Erf}(b x)}{x} \]

[Out]

-(Erf[b*x]/x) + (b*ExpIntegralEi[-(b^2*x^2)])/Sqrt[Pi]

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Rubi [A]  time = 0.0313195, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6361, 2210} \[ \frac{b \text{Ei}\left (-b^2 x^2\right )}{\sqrt{\pi }}-\frac{\text{Erf}(b x)}{x} \]

Antiderivative was successfully verified.

[In]

Int[Erf[b*x]/x^2,x]

[Out]

-(Erf[b*x]/x) + (b*ExpIntegralEi[-(b^2*x^2)])/Sqrt[Pi]

Rule 6361

Int[Erf[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*Erf[a + b*x])/(d*
(m + 1)), x] - Dist[(2*b)/(Sqrt[Pi]*d*(m + 1)), Int[(c + d*x)^(m + 1)/E^(a + b*x)^2, x], x] /; FreeQ[{a, b, c,
d, m}, x] && NeQ[m, -1]

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]

Rubi steps

\begin{align*} \int \frac{\text{erf}(b x)}{x^2} \, dx &=-\frac{\text{erf}(b x)}{x}+\frac{(2 b) \int \frac{e^{-b^2 x^2}}{x} \, dx}{\sqrt{\pi }}\\ &=-\frac{\text{erf}(b x)}{x}+\frac{b \text{Ei}\left (-b^2 x^2\right )}{\sqrt{\pi }}\\ \end{align*}

Mathematica [A]  time = 0.0391772, size = 26, normalized size = 1. \[ \frac{b \text{ExpIntegralEi}\left (-b^2 x^2\right )}{\sqrt{\pi }}-\frac{\text{Erf}(b x)}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[Erf[b*x]/x^2,x]

[Out]

-(Erf[b*x]/x) + (b*ExpIntegralEi[-(b^2*x^2)])/Sqrt[Pi]

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Maple [A]  time = 0.046, size = 30, normalized size = 1.2 \begin{align*} b \left ( -{\frac{{\it Erf} \left ( bx \right ) }{bx}}-{\frac{{\it Ei} \left ( 1,{b}^{2}{x}^{2} \right ) }{\sqrt{\pi }}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erf(b*x)/x^2,x)

[Out]

b*(-erf(b*x)/b/x-1/Pi^(1/2)*Ei(1,b^2*x^2))

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Maxima [A]  time = 1.14936, size = 32, normalized size = 1.23 \begin{align*} \frac{b{\rm Ei}\left (-b^{2} x^{2}\right )}{\sqrt{\pi }} - \frac{\operatorname{erf}\left (b x\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(b*x)/x^2,x, algorithm="maxima")

[Out]

b*Ei(-b^2*x^2)/sqrt(pi) - erf(b*x)/x

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Fricas [A]  time = 2.63074, size = 68, normalized size = 2.62 \begin{align*} \frac{\sqrt{\pi } b x{\rm Ei}\left (-b^{2} x^{2}\right ) - \pi \operatorname{erf}\left (b x\right )}{\pi x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(b*x)/x^2,x, algorithm="fricas")

[Out]

(sqrt(pi)*b*x*Ei(-b^2*x^2) - pi*erf(b*x))/(pi*x)

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Sympy [A]  time = 1.33391, size = 24, normalized size = 0.92 \begin{align*} - \frac{b \operatorname{E}_{1}\left (b^{2} x^{2}\right )}{\sqrt{\pi }} + \frac{\operatorname{erfc}{\left (b x \right )}}{x} - \frac{1}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(b*x)/x**2,x)

[Out]

-b*expint(1, b**2*x**2)/sqrt(pi) + erfc(b*x)/x - 1/x

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Giac [A]  time = 1.34871, size = 32, normalized size = 1.23 \begin{align*} \frac{b{\rm Ei}\left (-b^{2} x^{2}\right )}{\sqrt{\pi }} - \frac{\operatorname{erf}\left (b x\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(b*x)/x^2,x, algorithm="giac")

[Out]

b*Ei(-b^2*x^2)/sqrt(pi) - erf(b*x)/x