3.7 \(\int \frac {\text {erf}(b x)}{x^7} \, dx\)

Optimal. Leaf size=96 \[ -\frac {4}{45} b^6 \text {erf}(b x)-\frac {b e^{-b^2 x^2}}{15 \sqrt {\pi } x^5}-\frac {4 b^5 e^{-b^2 x^2}}{45 \sqrt {\pi } x}+\frac {2 b^3 e^{-b^2 x^2}}{45 \sqrt {\pi } x^3}-\frac {\text {erf}(b x)}{6 x^6} \]

[Out]

-4/45*b^6*erf(b*x)-1/6*erf(b*x)/x^6-1/15*b/exp(b^2*x^2)/x^5/Pi^(1/2)+2/45*b^3/exp(b^2*x^2)/x^3/Pi^(1/2)-4/45*b
^5/exp(b^2*x^2)/x/Pi^(1/2)

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Rubi [A]  time = 0.09, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6361, 2214, 2205} \[ -\frac {4}{45} b^6 \text {Erf}(b x)-\frac {4 b^5 e^{-b^2 x^2}}{45 \sqrt {\pi } x}+\frac {2 b^3 e^{-b^2 x^2}}{45 \sqrt {\pi } x^3}-\frac {b e^{-b^2 x^2}}{15 \sqrt {\pi } x^5}-\frac {\text {Erf}(b x)}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Int[Erf[b*x]/x^7,x]

[Out]

-b/(15*E^(b^2*x^2)*Sqrt[Pi]*x^5) + (2*b^3)/(45*E^(b^2*x^2)*Sqrt[Pi]*x^3) - (4*b^5)/(45*E^(b^2*x^2)*Sqrt[Pi]*x)
- (4*b^6*Erf[b*x])/45 - Erf[b*x]/(6*x^6)

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]

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^7} \, dx &=-\frac {\text {erf}(b x)}{6 x^6}+\frac {b \int \frac {e^{-b^2 x^2}}{x^6} \, dx}{3 \sqrt {\pi }}\\ &=-\frac {b e^{-b^2 x^2}}{15 \sqrt {\pi } x^5}-\frac {\text {erf}(b x)}{6 x^6}-\frac {\left (2 b^3\right ) \int \frac {e^{-b^2 x^2}}{x^4} \, dx}{15 \sqrt {\pi }}\\ &=-\frac {b e^{-b^2 x^2}}{15 \sqrt {\pi } x^5}+\frac {2 b^3 e^{-b^2 x^2}}{45 \sqrt {\pi } x^3}-\frac {\text {erf}(b x)}{6 x^6}+\frac {\left (4 b^5\right ) \int \frac {e^{-b^2 x^2}}{x^2} \, dx}{45 \sqrt {\pi }}\\ &=-\frac {b e^{-b^2 x^2}}{15 \sqrt {\pi } x^5}+\frac {2 b^3 e^{-b^2 x^2}}{45 \sqrt {\pi } x^3}-\frac {4 b^5 e^{-b^2 x^2}}{45 \sqrt {\pi } x}-\frac {\text {erf}(b x)}{6 x^6}-\frac {\left (8 b^7\right ) \int e^{-b^2 x^2} \, dx}{45 \sqrt {\pi }}\\ &=-\frac {b e^{-b^2 x^2}}{15 \sqrt {\pi } x^5}+\frac {2 b^3 e^{-b^2 x^2}}{45 \sqrt {\pi } x^3}-\frac {4 b^5 e^{-b^2 x^2}}{45 \sqrt {\pi } x}-\frac {4}{45} b^6 \text {erf}(b x)-\frac {\text {erf}(b x)}{6 x^6}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 73, normalized size = 0.76 \[ \frac {e^{-b^2 x^2} \left (-8 b^5 x^5+4 b^3 x^3-\sqrt {\pi } e^{b^2 x^2} \left (8 b^6 x^6+15\right ) \text {erf}(b x)-6 b x\right )}{90 \sqrt {\pi } x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[Erf[b*x]/x^7,x]

[Out]

(-6*b*x + 4*b^3*x^3 - 8*b^5*x^5 - E^(b^2*x^2)*Sqrt[Pi]*(15 + 8*b^6*x^6)*Erf[b*x])/(90*E^(b^2*x^2)*Sqrt[Pi]*x^6
)

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fricas [A]  time = 0.40, size = 62, normalized size = 0.65 \[ -\frac {2 \, \sqrt {\pi } {\left (4 \, b^{5} x^{5} - 2 \, b^{3} x^{3} + 3 \, b x\right )} e^{\left (-b^{2} x^{2}\right )} + {\left (15 \, \pi + 8 \, \pi b^{6} x^{6}\right )} \operatorname {erf}\left (b x\right )}{90 \, \pi x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(b*x)/x^7,x, algorithm="fricas")

[Out]

-1/90*(2*sqrt(pi)*(4*b^5*x^5 - 2*b^3*x^3 + 3*b*x)*e^(-b^2*x^2) + (15*pi + 8*pi*b^6*x^6)*erf(b*x))/(pi*x^6)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erf}\left (b x\right )}{x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(b*x)/x^7,x, algorithm="giac")

[Out]

integrate(erf(b*x)/x^7, x)

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maple [A]  time = 0.01, size = 87, normalized size = 0.91 \[ b^{6} \left (-\frac {\erf \left (b x \right )}{6 b^{6} x^{6}}+\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}}}{5 b^{5} x^{5}}+\frac {2 \,{\mathrm e}^{-b^{2} x^{2}}}{15 b^{3} x^{3}}-\frac {4 \,{\mathrm e}^{-b^{2} x^{2}}}{15 b x}-\frac {4 \sqrt {\pi }\, \erf \left (b x \right )}{15}}{3 \sqrt {\pi }}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erf(b*x)/x^7,x)

[Out]

b^6*(-1/6*erf(b*x)/b^6/x^6+1/3/Pi^(1/2)*(-1/5/exp(b^2*x^2)/b^5/x^5+2/15/exp(b^2*x^2)/b^3/x^3-4/15/exp(b^2*x^2)
/b/x-4/15*Pi^(1/2)*erf(b*x)))

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maxima [A]  time = 0.59, size = 35, normalized size = 0.36 \[ -\frac {b^{6} {\left (x^{2}\right )}^{\frac {5}{2}} \Gamma \left (-\frac {5}{2}, b^{2} x^{2}\right )}{6 \, \sqrt {\pi } x^{5}} - \frac {\operatorname {erf}\left (b x\right )}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(b*x)/x^7,x, algorithm="maxima")

[Out]

-1/6*b^6*(x^2)^(5/2)*gamma(-5/2, b^2*x^2)/(sqrt(pi)*x^5) - 1/6*erf(b*x)/x^6

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mupad [B]  time = 0.16, size = 113, normalized size = 1.18 \[ -\frac {\mathrm {erf}\left (b\,x\right )}{6\,x^6}-\frac {3\,b\,{\mathrm {e}}^{-b^2\,x^2}-2\,b^3\,x^2\,{\mathrm {e}}^{-b^2\,x^2}+4\,b^5\,x^4\,{\mathrm {e}}^{-b^2\,x^2}+4\,b^5\,\sqrt {\pi }\,\sqrt {b^2}\,{\left (x^2\right )}^{5/2}-4\,b^5\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {b^2}\,\sqrt {x^2}\right )\,\sqrt {b^2}\,{\left (x^2\right )}^{5/2}}{45\,x^5\,\sqrt {\pi }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erf(b*x)/x^7,x)

[Out]

- erf(b*x)/(6*x^6) - (3*b*exp(-b^2*x^2) - 2*b^3*x^2*exp(-b^2*x^2) + 4*b^5*x^4*exp(-b^2*x^2) + 4*b^5*pi^(1/2)*(
b^2)^(1/2)*(x^2)^(5/2) - 4*b^5*pi^(1/2)*erfc((b^2)^(1/2)*(x^2)^(1/2))*(b^2)^(1/2)*(x^2)^(5/2))/(45*x^5*pi^(1/2
))

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sympy [A]  time = 3.02, size = 87, normalized size = 0.91 \[ - \frac {4 b^{6} \operatorname {erf}{\left (b x \right )}}{45} - \frac {4 b^{5} e^{- b^{2} x^{2}}}{45 \sqrt {\pi } x} + \frac {2 b^{3} e^{- b^{2} x^{2}}}{45 \sqrt {\pi } x^{3}} - \frac {b e^{- b^{2} x^{2}}}{15 \sqrt {\pi } x^{5}} - \frac {\operatorname {erf}{\left (b x \right )}}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(b*x)/x**7,x)

[Out]

-4*b**6*erf(b*x)/45 - 4*b**5*exp(-b**2*x**2)/(45*sqrt(pi)*x) + 2*b**3*exp(-b**2*x**2)/(45*sqrt(pi)*x**3) - b*e
xp(-b**2*x**2)/(15*sqrt(pi)*x**5) - erf(b*x)/(6*x**6)

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