3.2.10 \(\int \frac {\text {Erfc}(b x)}{x^7} \, dx\) [110]

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

[Out]

4/45*b^6*erf(b*x)-1/6*erfc(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 = 0.06, 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 = {6497, 2245, 2236} \begin {gather*} \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 {Erfc}(b x)}{6 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Erfc[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 - Erfc[b*x]/(6*x^6)

Rule 2236

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 2245

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 6497

Int[Erfc[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Erfc[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 {erfc}(b x)}{x^7} \, dx &=-\frac {\text {erfc}(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 {erfc}(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 {erfc}(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 {erfc}(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 {erfc}(b x)}{6 x^6}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 62, normalized size = 0.65 \begin {gather*} \frac {1}{90} \left (\frac {2 b e^{-b^2 x^2} \left (3-2 b^2 x^2+4 b^4 x^4\right )}{\sqrt {\pi } x^5}+8 b^6 \text {Erf}(b x)-\frac {15 \text {Erfc}(b x)}{x^6}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.09, size = 87, normalized size = 0.91

method	result	size
derivativedivides	\(b^{6} \left (-\frac {\mathrm {erfc}\left (b x \right )}{6 b^{6} x^{6}}-\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}}}{5 b^{5} x^{5}}+\frac {2 \,{\mathrm e}^{-b^{2} x^{2}}}{15 b^{3} x^{3}}-\frac {4 \,{\mathrm e}^{-b^{2} x^{2}}}{15 x b}-\frac {4 \erf \left (b x \right ) \sqrt {\pi }}{15}}{3 \sqrt {\pi }}\right )\)	\(87\)
default	\(b^{6} \left (-\frac {\mathrm {erfc}\left (b x \right )}{6 b^{6} x^{6}}-\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}}}{5 b^{5} x^{5}}+\frac {2 \,{\mathrm e}^{-b^{2} x^{2}}}{15 b^{3} x^{3}}-\frac {4 \,{\mathrm e}^{-b^{2} x^{2}}}{15 x b}-\frac {4 \erf \left (b x \right ) \sqrt {\pi }}{15}}{3 \sqrt {\pi }}\right )\)	\(87\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erfc(b*x)/x^7,x,method=_RETURNVERBOSE)

[Out]

b^6*(-1/6/b^6/x^6*erfc(b*x)-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 = 0.30, size = 35, normalized size = 0.36 \begin {gather*} \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 {erfc}\left (b x\right )}{6 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfc(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*erfc(b*x)/x^6

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Fricas [A]
time = 0.37, size = 66, normalized size = 0.69 \begin {gather*} -\frac {15 \, \pi - 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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/90*(15*pi - 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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Sympy [A]
time = 0.67, size = 87, normalized size = 0.91 \begin {gather*} - \frac {4 b^{6} \operatorname {erfc}{\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 {erfc}{\left (b x \right )}}{6 x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*b**6*erfc(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*
exp(-b**2*x**2)/(15*sqrt(pi)*x**5) - erfc(b*x)/(6*x**6)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(erfc(b*x)/x^7, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {erfc}\left (b\,x\right )}{x^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(erfc(b*x)/x^7, x)

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