3.27 \(\int \frac{\text{Erf}(b x)^2}{x^5} \, dx\)

Optimal. Leaf size=125 \[ \frac{2 b^3 e^{-b^2 x^2} \text{Erf}(b x)}{3 \sqrt{\pi } x}-\frac{b e^{-b^2 x^2} \text{Erf}(b x)}{3 \sqrt{\pi } x^3}+\frac{1}{3} b^4 \text{Erf}(b x)^2-\frac{4 b^4 \text{ExpIntegralEi}\left (-2 b^2 x^2\right )}{3 \pi }-\frac{b^2 e^{-2 b^2 x^2}}{3 \pi x^2}-\frac{\text{Erf}(b x)^2}{4 x^4} \]

[Out]

-b^2/(3*E^(2*b^2*x^2)*Pi*x^2) - (b*Erf[b*x])/(3*E^(b^2*x^2)*Sqrt[Pi]*x^3) + (2*b^3*Erf[b*x])/(3*E^(b^2*x^2)*Sq
rt[Pi]*x) + (b^4*Erf[b*x]^2)/3 - Erf[b*x]^2/(4*x^4) - (4*b^4*ExpIntegralEi[-2*b^2*x^2])/(3*Pi)

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Rubi [A]  time = 0.18297, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6364, 6391, 6373, 30, 2210, 2214} \[ \frac{2 b^3 e^{-b^2 x^2} \text{Erf}(b x)}{3 \sqrt{\pi } x}-\frac{b e^{-b^2 x^2} \text{Erf}(b x)}{3 \sqrt{\pi } x^3}+\frac{1}{3} b^4 \text{Erf}(b x)^2-\frac{4 b^4 \text{Ei}\left (-2 b^2 x^2\right )}{3 \pi }-\frac{b^2 e^{-2 b^2 x^2}}{3 \pi x^2}-\frac{\text{Erf}(b x)^2}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[Erf[b*x]^2/x^5,x]

[Out]

-b^2/(3*E^(2*b^2*x^2)*Pi*x^2) - (b*Erf[b*x])/(3*E^(b^2*x^2)*Sqrt[Pi]*x^3) + (2*b^3*Erf[b*x])/(3*E^(b^2*x^2)*Sq
rt[Pi]*x) + (b^4*Erf[b*x]^2)/3 - Erf[b*x]^2/(4*x^4) - (4*b^4*ExpIntegralEi[-2*b^2*x^2])/(3*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]

Rule 2214

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*F^(a + b*(c + d*x)^n))/(d*(m + 1)), x] - Dist[(b*n*Log[F])/(m + 1), Int[(c + d*x)^(m + n)*F^(a + b*(c +
d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[-4, (m + 1)/n, 5] && IntegerQ[n
] && ((GtQ[n, 0] && LtQ[m, -1]) || (GtQ[-n, 0] && LeQ[-n, m + 1]))

Rubi steps

\begin{align*} \int \frac{\text{erf}(b x)^2}{x^5} \, dx &=-\frac{\text{erf}(b x)^2}{4 x^4}+\frac{b \int \frac{e^{-b^2 x^2} \text{erf}(b x)}{x^4} \, dx}{\sqrt{\pi }}\\ &=-\frac{b e^{-b^2 x^2} \text{erf}(b x)}{3 \sqrt{\pi } x^3}-\frac{\text{erf}(b x)^2}{4 x^4}+\frac{\left (2 b^2\right ) \int \frac{e^{-2 b^2 x^2}}{x^3} \, dx}{3 \pi }-\frac{\left (2 b^3\right ) \int \frac{e^{-b^2 x^2} \text{erf}(b x)}{x^2} \, dx}{3 \sqrt{\pi }}\\ &=-\frac{b^2 e^{-2 b^2 x^2}}{3 \pi x^2}-\frac{b e^{-b^2 x^2} \text{erf}(b x)}{3 \sqrt{\pi } x^3}+\frac{2 b^3 e^{-b^2 x^2} \text{erf}(b x)}{3 \sqrt{\pi } x}-\frac{\text{erf}(b x)^2}{4 x^4}-2 \frac{\left (4 b^4\right ) \int \frac{e^{-2 b^2 x^2}}{x} \, dx}{3 \pi }+\frac{\left (4 b^5\right ) \int e^{-b^2 x^2} \text{erf}(b x) \, dx}{3 \sqrt{\pi }}\\ &=-\frac{b^2 e^{-2 b^2 x^2}}{3 \pi x^2}-\frac{b e^{-b^2 x^2} \text{erf}(b x)}{3 \sqrt{\pi } x^3}+\frac{2 b^3 e^{-b^2 x^2} \text{erf}(b x)}{3 \sqrt{\pi } x}-\frac{\text{erf}(b x)^2}{4 x^4}-\frac{4 b^4 \text{Ei}\left (-2 b^2 x^2\right )}{3 \pi }+\frac{1}{3} \left (2 b^4\right ) \operatorname{Subst}(\int x \, dx,x,\text{erf}(b x))\\ &=-\frac{b^2 e^{-2 b^2 x^2}}{3 \pi x^2}-\frac{b e^{-b^2 x^2} \text{erf}(b x)}{3 \sqrt{\pi } x^3}+\frac{2 b^3 e^{-b^2 x^2} \text{erf}(b x)}{3 \sqrt{\pi } x}+\frac{1}{3} b^4 \text{erf}(b x)^2-\frac{\text{erf}(b x)^2}{4 x^4}-\frac{4 b^4 \text{Ei}\left (-2 b^2 x^2\right )}{3 \pi }\\ \end{align*}

Mathematica [A]  time = 0.0784899, size = 97, normalized size = 0.78 \[ \frac{\frac{4 b x e^{-b^2 x^2} \left (2 b^2 x^2-1\right ) \text{Erf}(b x)}{\sqrt{\pi }}+\left (4 b^4 x^4-3\right ) \text{Erf}(b x)^2-\frac{4 b^2 x^2 \left (4 b^2 x^2 \text{ExpIntegralEi}\left (-2 b^2 x^2\right )+e^{-2 b^2 x^2}\right )}{\pi }}{12 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[Erf[b*x]^2/x^5,x]

[Out]

((4*b*x*(-1 + 2*b^2*x^2)*Erf[b*x])/(E^(b^2*x^2)*Sqrt[Pi]) + (-3 + 4*b^4*x^4)*Erf[b*x]^2 - (4*b^2*x^2*(E^(-2*b^
2*x^2) + 4*b^2*x^2*ExpIntegralEi[-2*b^2*x^2]))/Pi)/(12*x^4)

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\it Erf} \left ( bx \right ) \right ) ^{2}}{{x}^{5}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erf(b*x)^2/x^5,x)

[Out]

int(erf(b*x)^2/x^5,x)

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Maxima [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(erf(b*x)^2/x^5,x, algorithm="maxima")

[Out]

Timed out

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Fricas [A]  time = 2.57718, size = 215, normalized size = 1.72 \begin{align*} -\frac{16 \, b^{4} x^{4}{\rm Ei}\left (-2 \, b^{2} x^{2}\right ) + 4 \, b^{2} x^{2} e^{\left (-2 \, b^{2} x^{2}\right )} - 4 \, \sqrt{\pi }{\left (2 \, b^{3} x^{3} - b x\right )} \operatorname{erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} +{\left (3 \, \pi - 4 \, \pi b^{4} x^{4}\right )} \operatorname{erf}\left (b x\right )^{2}}{12 \, \pi x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(b*x)^2/x^5,x, algorithm="fricas")

[Out]

-1/12*(16*b^4*x^4*Ei(-2*b^2*x^2) + 4*b^2*x^2*e^(-2*b^2*x^2) - 4*sqrt(pi)*(2*b^3*x^3 - b*x)*erf(b*x)*e^(-b^2*x^
2) + (3*pi - 4*pi*b^4*x^4)*erf(b*x)^2)/(pi*x^4)

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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^{5}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(b*x)**2/x**5,x)

[Out]

Integral(erf(b*x)**2/x**5, 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^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(b*x)^2/x^5,x, algorithm="giac")

[Out]

integrate(erf(b*x)^2/x^5, x)