3.13 \(\int \frac {\text {erf}(b x)}{x^4} \, dx\)

Optimal. Leaf size=56 \[ -\frac {b e^{-b^2 x^2}}{3 \sqrt {\pi } x^2}-\frac {b^3 \text {Ei}\left (-b^2 x^2\right )}{3 \sqrt {\pi }}-\frac {\text {erf}(b x)}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 56, normalized size of antiderivative = 1.00, 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, 2210} \[ -\frac {b^3 \text {Ei}\left (-b^2 x^2\right )}{3 \sqrt {\pi }}-\frac {b e^{-b^2 x^2}}{3 \sqrt {\pi } x^2}-\frac {\text {Erf}(b x)}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[Erf[b*x]/x^4,x]

[Out]

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

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]))

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]

Rubi steps

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

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Mathematica [A]  time = 0.05, size = 47, normalized size = 0.84 \[ -\frac {\frac {b x \left (b^2 x^2 \text {Ei}\left (-b^2 x^2\right )+e^{-b^2 x^2}\right )}{\sqrt {\pi }}+\text {erf}(b x)}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Erf[b*x]/x^4,x]

[Out]

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

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fricas [A]  time = 0.42, size = 48, normalized size = 0.86 \[ -\frac {\pi \operatorname {erf}\left (b x\right ) + \sqrt {\pi } {\left (b^{3} x^{3} {\rm Ei}\left (-b^{2} x^{2}\right ) + b x e^{\left (-b^{2} x^{2}\right )}\right )}}{3 \, \pi x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.24, size = 51, normalized size = 0.91 \[ -\frac {\operatorname {erf}\left (b x\right )}{3 \, x^{3}} - \frac {b^{6} x^{2} {\rm Ei}\left (-b^{2} x^{2}\right ) + b^{4} e^{\left (-b^{2} x^{2}\right )}}{3 \, \sqrt {\pi } b^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*erf(b*x)/x^3 - 1/3*(b^6*x^2*Ei(-b^2*x^2) + b^4*e^(-b^2*x^2))/(sqrt(pi)*b^3*x^2)

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maple [A]  time = 0.01, size = 53, normalized size = 0.95 \[ b^{3} \left (-\frac {\erf \left (b x \right )}{3 b^{3} x^{3}}+\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}}}{3 b^{2} x^{2}}+\frac {\Ei \left (1, b^{2} x^{2}\right )}{3}}{\sqrt {\pi }}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.53, size = 27, normalized size = 0.48 \[ -\frac {b^{3} \Gamma \left (-1, b^{2} x^{2}\right )}{3 \, \sqrt {\pi }} - \frac {\operatorname {erf}\left (b x\right )}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.18, size = 45, normalized size = 0.80 \[ -\frac {\mathrm {erf}\left (b\,x\right )}{3\,x^3}-\frac {b^3\,\mathrm {ei}\left (-b^2\,x^2\right )}{3\,\sqrt {\pi }}-\frac {b\,{\mathrm {e}}^{-b^2\,x^2}}{3\,x^2\,\sqrt {\pi }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- erf(b*x)/(3*x^3) - (b^3*ei(-b^2*x^2))/(3*pi^(1/2)) - (b*exp(-b^2*x^2))/(3*x^2*pi^(1/2))

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sympy [A]  time = 2.43, size = 54, normalized size = 0.96 \[ \frac {b^{3} \operatorname {E}_{1}\left (b^{2} x^{2}\right )}{3 \sqrt {\pi }} - \frac {b e^{- b^{2} x^{2}}}{3 \sqrt {\pi } x^{2}} + \frac {\operatorname {erfc}{\left (b x \right )}}{3 x^{3}} - \frac {1}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b**3*expint(1, b**2*x**2)/(3*sqrt(pi)) - b*exp(-b**2*x**2)/(3*sqrt(pi)*x**2) + erfc(b*x)/(3*x**3) - 1/(3*x**3)

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