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

Optimal. Leaf size=42 \[ b^2 (-\text{Erf}(b x))-\frac{b e^{-b^2 x^2}}{\sqrt{\pi } x}-\frac{\text{Erf}(b x)}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0386486, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6361, 2214, 2205} \[ b^2 (-\text{Erf}(b x))-\frac{b e^{-b^2 x^2}}{\sqrt{\pi } x}-\frac{\text{Erf}(b x)}{2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.036902, size = 42, normalized size = 1. \[ b^2 (-\text{Erf}(b x))-\frac{b e^{-b^2 x^2}}{\sqrt{\pi } x}-\frac{\text{Erf}(b x)}{2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.048, size = 50, normalized size = 1.2 \begin{align*}{b}^{2} \left ( -{\frac{{\it Erf} \left ( bx \right ) }{2\,{b}^{2}{x}^{2}}}+{\frac{1}{\sqrt{\pi }} \left ( -{\frac{1}{{{\rm e}^{{b}^{2}{x}^{2}}}bx}}-\sqrt{\pi }{\it Erf} \left ( bx \right ) \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.23835, size = 50, normalized size = 1.19 \begin{align*} -\frac{\sqrt{b^{2} x^{2}} b \Gamma \left (-\frac{1}{2}, b^{2} x^{2}\right )}{2 \, \sqrt{\pi } x} - \frac{\operatorname{erf}\left (b x\right )}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.51236, size = 103, normalized size = 2.45 \begin{align*} -\frac{2 \, \sqrt{\pi } b x e^{\left (-b^{2} x^{2}\right )} +{\left (\pi + 2 \, \pi b^{2} x^{2}\right )} \operatorname{erf}\left (b x\right )}{2 \, \pi x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.670192, size = 36, normalized size = 0.86 \begin{align*} - b^{2} \operatorname{erf}{\left (b x \right )} - \frac{b e^{- b^{2} x^{2}}}{\sqrt{\pi } x} - \frac{\operatorname{erf}{\left (b x \right )}}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b**2*erf(b*x) - b*exp(-b**2*x**2)/(sqrt(pi)*x) - erf(b*x)/(2*x**2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(erf(b*x)/x^3, x)