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

Optimal. Leaf size=71 \[ \frac{1}{3} b^4 \text{Erf}(b x)+\frac{b^3 e^{-b^2 x^2}}{3 \sqrt{\pi } x}-\frac{b e^{-b^2 x^2}}{6 \sqrt{\pi } x^3}-\frac{\text{Erf}(b x)}{4 x^4} \]

[Out]

-b/(6*E^(b^2*x^2)*Sqrt[Pi]*x^3) + b^3/(3*E^(b^2*x^2)*Sqrt[Pi]*x) + (b^4*Erf[b*x])/3 - Erf[b*x]/(4*x^4)

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Rubi [A]  time = 0.0650876, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6361, 2214, 2205} \[ \frac{1}{3} b^4 \text{Erf}(b x)+\frac{b^3 e^{-b^2 x^2}}{3 \sqrt{\pi } x}-\frac{b e^{-b^2 x^2}}{6 \sqrt{\pi } x^3}-\frac{\text{Erf}(b x)}{4 x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-b/(6*E^(b^2*x^2)*Sqrt[Pi]*x^3) + b^3/(3*E^(b^2*x^2)*Sqrt[Pi]*x) + (b^4*Erf[b*x])/3 - Erf[b*x]/(4*x^4)

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 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]))

Rule 2205

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erf[(c + d*x)*Rt[-(b*Log[F]),
2]])/(2*d*Rt[-(b*Log[F]), 2]), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{\text{erf}(b x)}{x^5} \, dx &=-\frac{\text{erf}(b x)}{4 x^4}+\frac{b \int \frac{e^{-b^2 x^2}}{x^4} \, dx}{2 \sqrt{\pi }}\\ &=-\frac{b e^{-b^2 x^2}}{6 \sqrt{\pi } x^3}-\frac{\text{erf}(b x)}{4 x^4}-\frac{b^3 \int \frac{e^{-b^2 x^2}}{x^2} \, dx}{3 \sqrt{\pi }}\\ &=-\frac{b e^{-b^2 x^2}}{6 \sqrt{\pi } x^3}+\frac{b^3 e^{-b^2 x^2}}{3 \sqrt{\pi } x}-\frac{\text{erf}(b x)}{4 x^4}+\frac{\left (2 b^5\right ) \int e^{-b^2 x^2} \, dx}{3 \sqrt{\pi }}\\ &=-\frac{b e^{-b^2 x^2}}{6 \sqrt{\pi } x^3}+\frac{b^3 e^{-b^2 x^2}}{3 \sqrt{\pi } x}+\frac{1}{3} b^4 \text{erf}(b x)-\frac{\text{erf}(b x)}{4 x^4}\\ \end{align*}

Mathematica [A]  time = 0.0174827, size = 63, normalized size = 0.89 \[ \frac{1}{3} b^4 \text{Erf}(b x)+e^{-b^2 x^2} \left (\frac{b^3}{3 \sqrt{\pi } x}-\frac{b}{6 \sqrt{\pi } x^3}\right )-\frac{\text{Erf}(b x)}{4 x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-b/(6*Sqrt[Pi]*x^3) + b^3/(3*Sqrt[Pi]*x))/E^(b^2*x^2) + (b^4*Erf[b*x])/3 - Erf[b*x]/(4*x^4)

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Maple [A]  time = 0.046, size = 69, normalized size = 1. \begin{align*}{b}^{4} \left ( -{\frac{{\it Erf} \left ( bx \right ) }{4\,{b}^{4}{x}^{4}}}+{\frac{1}{2\,\sqrt{\pi }} \left ( -{\frac{1}{3\,{{\rm e}^{{b}^{2}{x}^{2}}}{b}^{3}{x}^{3}}}+{\frac{2}{3\,{{\rm e}^{{b}^{2}{x}^{2}}}bx}}+{\frac{2\,\sqrt{\pi }{\it Erf} \left ( bx \right ) }{3}} \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^4*(-1/4*erf(b*x)/b^4/x^4+1/2/Pi^(1/2)*(-1/3/exp(b^2*x^2)/b^3/x^3+2/3/exp(b^2*x^2)/b/x+2/3*Pi^(1/2)*erf(b*x))
)

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Maxima [A]  time = 1.13255, size = 50, normalized size = 0.7 \begin{align*} -\frac{\left (b^{2} x^{2}\right )^{\frac{3}{2}} b \Gamma \left (-\frac{3}{2}, b^{2} x^{2}\right )}{4 \, \sqrt{\pi } x^{3}} - \frac{\operatorname{erf}\left (b x\right )}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(b^2*x^2)^(3/2)*b*gamma(-3/2, b^2*x^2)/(sqrt(pi)*x^3) - 1/4*erf(b*x)/x^4

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Fricas [A]  time = 2.5562, size = 124, normalized size = 1.75 \begin{align*} \frac{2 \, \sqrt{\pi }{\left (2 \, b^{3} x^{3} - 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 )}{12 \, \pi x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.66474, size = 60, normalized size = 0.85 \begin{align*} \frac{b^{4} \operatorname{erf}{\left (b x \right )}}{3} + \frac{b^{3} e^{- b^{2} x^{2}}}{3 \sqrt{\pi } x} - \frac{b e^{- b^{2} x^{2}}}{6 \sqrt{\pi } x^{3}} - \frac{\operatorname{erf}{\left (b x \right )}}{4 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b**4*erf(b*x)/3 + b**3*exp(-b**2*x**2)/(3*sqrt(pi)*x) - b*exp(-b**2*x**2)/(6*sqrt(pi)*x**3) - erf(b*x)/(4*x**4
)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erf}\left (b x\right )}{x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(erf(b*x)/x^5, x)