3.26 \(\int \frac{\text{Erf}(b x)^2}{x^3} \, dx\)

Optimal. Leaf size=67 \[ -\frac{2 b e^{-b^2 x^2} \text{Erf}(b x)}{\sqrt{\pi } x}+b^2 \left (-\text{Erf}(b x)^2\right )+\frac{2 b^2 \text{ExpIntegralEi}\left (-2 b^2 x^2\right )}{\pi }-\frac{\text{Erf}(b x)^2}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0996888, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6364, 6391, 6373, 30, 2210} \[ -\frac{2 b e^{-b^2 x^2} \text{Erf}(b x)}{\sqrt{\pi } x}+b^2 \left (-\text{Erf}(b x)^2\right )+\frac{2 b^2 \text{Ei}\left (-2 b^2 x^2\right )}{\pi }-\frac{\text{Erf}(b x)^2}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[Erf[b*x]^2/x^3,x]

[Out]

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

Rule 6364

Int[Erf[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[(x^(m + 1)*Erf[b*x]^2)/(m + 1), x] - Dist[(4*b)/(Sqrt[Pi]*
(m + 1)), Int[(x^(m + 1)*Erf[b*x])/E^(b^2*x^2), x], x] /; FreeQ[b, x] && (IGtQ[m, 0] || ILtQ[(m + 1)/2, 0])

Rule 6391

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[(x^(m + 1)*E^(c + d*x^2)*Erf
[a + b*x])/(m + 1), x] + (-Dist[(2*d)/(m + 1), Int[x^(m + 2)*E^(c + d*x^2)*Erf[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 6373

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(b_.)*(x_)]^(n_.), x_Symbol] :> Dist[(E^c*Sqrt[Pi])/(2*b), Subst[Int[x^n, x],
x, Erf[b*x]], x] /; FreeQ[{b, c, d, n}, x] && EqQ[d, -b^2]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[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)^2}{x^3} \, dx &=-\frac{\text{erf}(b x)^2}{2 x^2}+\frac{(2 b) \int \frac{e^{-b^2 x^2} \text{erf}(b x)}{x^2} \, dx}{\sqrt{\pi }}\\ &=-\frac{2 b e^{-b^2 x^2} \text{erf}(b x)}{\sqrt{\pi } x}-\frac{\text{erf}(b x)^2}{2 x^2}+\frac{\left (4 b^2\right ) \int \frac{e^{-2 b^2 x^2}}{x} \, dx}{\pi }-\frac{\left (4 b^3\right ) \int e^{-b^2 x^2} \text{erf}(b x) \, dx}{\sqrt{\pi }}\\ &=-\frac{2 b e^{-b^2 x^2} \text{erf}(b x)}{\sqrt{\pi } x}-\frac{\text{erf}(b x)^2}{2 x^2}+\frac{2 b^2 \text{Ei}\left (-2 b^2 x^2\right )}{\pi }-\left (2 b^2\right ) \operatorname{Subst}(\int x \, dx,x,\text{erf}(b x))\\ &=-\frac{2 b e^{-b^2 x^2} \text{erf}(b x)}{\sqrt{\pi } x}-b^2 \text{erf}(b x)^2-\frac{\text{erf}(b x)^2}{2 x^2}+\frac{2 b^2 \text{Ei}\left (-2 b^2 x^2\right )}{\pi }\\ \end{align*}

Mathematica [A]  time = 0.0282474, size = 63, normalized size = 0.94 \[ -\frac{2 b e^{-b^2 x^2} \text{Erf}(b x)}{\sqrt{\pi } x}+\left (-b^2-\frac{1}{2 x^2}\right ) \text{Erf}(b x)^2+\frac{2 b^2 \text{ExpIntegralEi}\left (-2 b^2 x^2\right )}{\pi } \]

Antiderivative was successfully verified.

[In]

Integrate[Erf[b*x]^2/x^3,x]

[Out]

(-2*b*Erf[b*x])/(E^(b^2*x^2)*Sqrt[Pi]*x) + (-b^2 - 1/(2*x^2))*Erf[b*x]^2 + (2*b^2*ExpIntegralEi[-2*b^2*x^2])/P
i

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Maple [F]  time = 0.053, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\it Erf} \left ( bx \right ) \right ) ^{2}}{{x}^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erf(b*x)^2/x^3,x)

[Out]

int(erf(b*x)^2/x^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \, b \int \frac{\operatorname{erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{2}}\,{d x}}{\sqrt{\pi }} - \frac{\operatorname{erf}\left (b x\right )^{2}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(b*x)^2/x^3,x, algorithm="maxima")

[Out]

2*b*integrate(erf(b*x)*e^(-b^2*x^2)/x^2, x)/sqrt(pi) - 1/2*erf(b*x)^2/x^2

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Fricas [A]  time = 2.6121, size = 153, normalized size = 2.28 \begin{align*} \frac{4 \, b^{2} x^{2}{\rm Ei}\left (-2 \, b^{2} x^{2}\right ) - 4 \, \sqrt{\pi } b x \operatorname{erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} -{\left (\pi + 2 \, \pi b^{2} x^{2}\right )} \operatorname{erf}\left (b x\right )^{2}}{2 \, \pi x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(b*x)^2/x^3,x, algorithm="fricas")

[Out]

1/2*(4*b^2*x^2*Ei(-2*b^2*x^2) - 4*sqrt(pi)*b*x*erf(b*x)*e^(-b^2*x^2) - (pi + 2*pi*b^2*x^2)*erf(b*x)^2)/(pi*x^2
)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erf}^{2}{\left (b x \right )}}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(b*x)**2/x**3,x)

[Out]

Integral(erf(b*x)**2/x**3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(b*x)^2/x^3,x, algorithm="giac")

[Out]

integrate(erf(b*x)^2/x^3, x)