3.85 \(\int \frac{e^{-b^2 x^2} \text{Erf}(b x)}{x^4} \, dx\)

Optimal. Leaf size=108 \[ \frac{2 b^2 e^{-b^2 x^2} \text{Erf}(b x)}{3 x}-\frac{e^{-b^2 x^2} \text{Erf}(b x)}{3 x^3}+\frac{1}{3} \sqrt{\pi } b^3 \text{Erf}(b x)^2-\frac{4 b^3 \text{ExpIntegralEi}\left (-2 b^2 x^2\right )}{3 \sqrt{\pi }}-\frac{b e^{-2 b^2 x^2}}{3 \sqrt{\pi } x^2} \]

[Out]

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

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Rubi [A]  time = 0.147618, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {6391, 6373, 30, 2210, 2214} \[ \frac{2 b^2 e^{-b^2 x^2} \text{Erf}(b x)}{3 x}-\frac{e^{-b^2 x^2} \text{Erf}(b x)}{3 x^3}+\frac{1}{3} \sqrt{\pi } b^3 \text{Erf}(b x)^2-\frac{4 b^3 \text{Ei}\left (-2 b^2 x^2\right )}{3 \sqrt{\pi }}-\frac{b e^{-2 b^2 x^2}}{3 \sqrt{\pi } x^2} \]

Antiderivative was successfully verified.

[In]

Int[Erf[b*x]/(E^(b^2*x^2)*x^4),x]

[Out]

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

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

Mathematica [A]  time = 0.0656073, size = 85, normalized size = 0.79 \[ \frac{1}{3} \left (\frac{e^{-b^2 x^2} \left (2 b^2 x^2-1\right ) \text{Erf}(b x)}{x^3}+\sqrt{\pi } b^3 \text{Erf}(b x)^2+\frac{b \left (-4 b^2 \text{ExpIntegralEi}\left (-2 b^2 x^2\right )-\frac{e^{-2 b^2 x^2}}{x^2}\right )}{\sqrt{\pi }}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Erf[b*x]/(E^(b^2*x^2)*x^4),x]

[Out]

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

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Maple [F]  time = 0.412, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it Erf} \left ( bx \right ) }{{{\rm e}^{{b}^{2}{x}^{2}}}{x}^{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erf(b*x)/exp(b^2*x^2)/x^4,x)

[Out]

int(erf(b*x)/exp(b^2*x^2)/x^4,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(b*x)/exp(b^2*x^2)/x^4,x, algorithm="maxima")

[Out]

integrate(erf(b*x)*e^(-b^2*x^2)/x^4, x)

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Fricas [A]  time = 2.67961, size = 192, normalized size = 1.78 \begin{align*} -\frac{{\left (\pi - 2 \, \pi b^{2} x^{2}\right )} \operatorname{erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} - \sqrt{\pi }{\left (\pi b^{3} x^{3} \operatorname{erf}\left (b x\right )^{2} - 4 \, b^{3} x^{3}{\rm Ei}\left (-2 \, b^{2} x^{2}\right ) - b x e^{\left (-2 \, b^{2} x^{2}\right )}\right )}}{3 \, \pi x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(b*x)/exp(b^2*x^2)/x^4,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{- b^{2} x^{2}} \operatorname{erf}{\left (b x \right )}}{x^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(b*x)/exp(b**2*x**2)/x**4,x)

[Out]

Integral(exp(-b**2*x**2)*erf(b*x)/x**4, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(b*x)/exp(b^2*x^2)/x^4,x, algorithm="giac")

[Out]

integrate(erf(b*x)*e^(-b^2*x^2)/x^4, x)