\(\int \frac {\text {erfc}(b x)}{x^5} \, dx\) [109]

Optimal result

Integrand size = 8, antiderivative size = 71 \[ \int \frac {\text {erfc}(b x)}{x^5} \, dx=\frac {b e^{-b^2 x^2}}{6 \sqrt {\pi } x^3}-\frac {b^3 e^{-b^2 x^2}}{3 \sqrt {\pi } x}-\frac {1}{3} b^4 \text {erf}(b x)-\frac {\text {erfc}(b x)}{4 x^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6497, 2245, 2236} \[ \int \frac {\text {erfc}(b x)}{x^5} \, dx=-\frac {1}{3} b^4 \text {erf}(b x)+\frac {b e^{-b^2 x^2}}{6 \sqrt {\pi } x^3}-\frac {b^3 e^{-b^2 x^2}}{3 \sqrt {\pi } x}-\frac {\text {erfc}(b x)}{4 x^4} \]

[In]

Int[Erfc[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 - Erfc[b*x]/(4*x^4)

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*} \text {integral}& = -\frac {\text {erfc}(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 {erfc}(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 {erfc}(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 {erfc}(b x)}{4 x^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.75 \[ \int \frac {\text {erfc}(b x)}{x^5} \, dx=\frac {1}{12} \left (\frac {2 e^{-b^2 x^2} \left (b-2 b^3 x^2\right )}{\sqrt {\pi } x^3}-4 b^4 \text {erf}(b x)-\frac {3 \text {erfc}(b x)}{x^4}\right ) \]

[In]

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

[Out]

((2*(b - 2*b^3*x^2))/(E^(b^2*x^2)*Sqrt[Pi]*x^3) - 4*b^4*Erf[b*x] - (3*Erfc[b*x])/x^4)/12

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.87

method	result	size
parts	\(-\frac {\operatorname {erfc}\left (b x \right )}{4 x^{4}}-\frac {b \left (-\frac {{\mathrm e}^{-b^{2} x^{2}}}{3 x^{3}}-\frac {2 b^{2} \left (-\frac {{\mathrm e}^{-b^{2} x^{2}}}{x}-b \sqrt {\pi }\, \operatorname {erf}\left (b x \right )\right )}{3}\right )}{2 \sqrt {\pi }}\)	\(62\)
parallelrisch	\(\frac {4 \,\operatorname {erfc}\left (b x \right ) x^{4} \sqrt {\pi }\, b^{4}-4 x^{3} {\mathrm e}^{-b^{2} x^{2}} b^{3}+2 \,{\mathrm e}^{-b^{2} x^{2}} b x -3 \,\operatorname {erfc}\left (b x \right ) \sqrt {\pi }}{12 \sqrt {\pi }\, x^{4}}\)	\(64\)
derivativedivides	\(b^{4} \left (-\frac {\operatorname {erfc}\left (b x \right )}{4 b^{4} x^{4}}-\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}}}{3 b^{3} x^{3}}+\frac {2 \,{\mathrm e}^{-b^{2} x^{2}}}{3 b x}+\frac {2 \,\operatorname {erf}\left (b x \right ) \sqrt {\pi }}{3}}{2 \sqrt {\pi }}\right )\)	\(69\)
default	\(b^{4} \left (-\frac {\operatorname {erfc}\left (b x \right )}{4 b^{4} x^{4}}-\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}}}{3 b^{3} x^{3}}+\frac {2 \,{\mathrm e}^{-b^{2} x^{2}}}{3 b x}+\frac {2 \,\operatorname {erf}\left (b x \right ) \sqrt {\pi }}{3}}{2 \sqrt {\pi }}\right )\)	\(69\)

[In]

int(erfc(b*x)/x^5,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.82 \[ \int \frac {\text {erfc}(b x)}{x^5} \, dx=-\frac {3 \, \pi + 2 \, \sqrt {\pi } {\left (2 \, b^{3} x^{3} - b x\right )} e^{\left (-b^{2} x^{2}\right )} - {\left (3 \, \pi - 4 \, \pi b^{4} x^{4}\right )} \operatorname {erf}\left (b x\right )}{12 \, \pi x^{4}} \]

[In]

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

[Out]

-1/12*(3*pi + 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)

Sympy [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.85 \[ \int \frac {\text {erfc}(b x)}{x^5} \, dx=\frac {b^{4} \operatorname {erfc}{\left (b x \right )}}{3} - \frac {b^{3} e^{- b^{2} x^{2}}}{3 \sqrt {\pi } x} + \frac {b e^{- b^{2} x^{2}}}{6 \sqrt {\pi } x^{3}} - \frac {\operatorname {erfc}{\left (b x \right )}}{4 x^{4}} \]

[In]

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

[Out]

b**4*erfc(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) - erfc(b*x)/(4*x*
*4)

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.49 \[ \int \frac {\text {erfc}(b x)}{x^5} \, dx=\frac {b^{4} {\left (x^{2}\right )}^{\frac {3}{2}} \Gamma \left (-\frac {3}{2}, b^{2} x^{2}\right )}{4 \, \sqrt {\pi } x^{3}} - \frac {\operatorname {erfc}\left (b x\right )}{4 \, x^{4}} \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\text {erfc}(b x)}{x^5} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right )}{x^{5}} \,d x } \]

[In]

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

[Out]

integrate(erfc(b*x)/x^5, x)

Mupad [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00 \[ \int \frac {\text {erfc}(b x)}{x^5} \, dx=-\frac {\frac {\mathrm {erfc}\left (b\,x\right )}{4}+\frac {b^3\,x^3\,{\mathrm {e}}^{-b^2\,x^2}}{3\,\sqrt {\pi }}-\frac {b\,x\,{\mathrm {e}}^{-b^2\,x^2}}{6\,\sqrt {\pi }}}{x^4}-\frac {b^5\,\mathrm {erfi}\left (x\,\sqrt {-b^2}\right )}{3\,\sqrt {-b^2}} \]

[In]

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

[Out]

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