3.2.30 \(\int \frac {\text {Erfc}(b x)^2}{x^5} \, dx\) [130]

Optimal. Leaf size=125 \[ -\frac {b^2 e^{-2 b^2 x^2}}{3 \pi x^2}+\frac {b e^{-b^2 x^2} \text {Erfc}(b x)}{3 \sqrt {\pi } x^3}-\frac {2 b^3 e^{-b^2 x^2} \text {Erfc}(b x)}{3 \sqrt {\pi } x}+\frac {1}{3} b^4 \text {Erfc}(b x)^2-\frac {\text {Erfc}(b x)^2}{4 x^4}-\frac {4 b^4 \text {Ei}\left (-2 b^2 x^2\right )}{3 \pi } \]

[Out]

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

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Rubi [A]
time = 0.12, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6500, 6527, 6509, 30, 2241, 2245} \begin {gather*} \frac {1}{3} b^4 \text {Erfc}(b x)^2+\frac {b e^{-b^2 x^2} \text {Erfc}(b x)}{3 \sqrt {\pi } x^3}-\frac {b^2 e^{-2 b^2 x^2}}{3 \pi x^2}-\frac {4 b^4 \text {Ei}\left (-2 b^2 x^2\right )}{3 \pi }-\frac {2 b^3 e^{-b^2 x^2} \text {Erfc}(b x)}{3 \sqrt {\pi } x}-\frac {\text {Erfc}(b x)^2}{4 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Erfc[b*x]^2/x^5,x]

[Out]

-1/3*b^2/(E^(2*b^2*x^2)*Pi*x^2) + (b*Erfc[b*x])/(3*E^(b^2*x^2)*Sqrt[Pi]*x^3) - (2*b^3*Erfc[b*x])/(3*E^(b^2*x^2
)*Sqrt[Pi]*x) + (b^4*Erfc[b*x]^2)/3 - Erfc[b*x]^2/(4*x^4) - (4*b^4*ExpIntegralEi[-2*b^2*x^2])/(3*Pi)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2241

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 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 6500

Int[Erfc[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(Erfc[b*x]^2/(m + 1)), x] + Dist[4*(b/(Sqrt[Pi]
*(m + 1))), Int[(x^(m + 1)*Erfc[b*x])/E^(b^2*x^2), x], x] /; FreeQ[b, x] && (IGtQ[m, 0] || ILtQ[(m + 1)/2, 0])

Rule 6509

Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(b_.)*(x_)]^(n_.), x_Symbol] :> Dist[(-E^c)*(Sqrt[Pi]/(2*b)), Subst[Int[x^n,
x], x, Erfc[b*x]], x] /; FreeQ[{b, c, d, n}, x] && EqQ[d, -b^2]

Rule 6527

Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[x^(m + 1)*E^(c + d*x^2)*(Er
fc[a + b*x]/(m + 1)), x] + (-Dist[2*(d/(m + 1)), Int[x^(m + 2)*E^(c + d*x^2)*Erfc[a + b*x], x], x] + Dist[2*(b
/((m + 1)*Sqrt[Pi])), Int[x^(m + 1)*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2), x], x]) /; FreeQ[{a, b, c, d}, x]
&& ILtQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\text {erfc}(b x)^2}{x^5} \, dx &=-\frac {\text {erfc}(b x)^2}{4 x^4}-\frac {b \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^4} \, dx}{\sqrt {\pi }}\\ &=\frac {b e^{-b^2 x^2} \text {erfc}(b x)}{3 \sqrt {\pi } x^3}-\frac {\text {erfc}(b x)^2}{4 x^4}+\frac {\left (2 b^2\right ) \int \frac {e^{-2 b^2 x^2}}{x^3} \, dx}{3 \pi }+\frac {\left (2 b^3\right ) \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^2} \, dx}{3 \sqrt {\pi }}\\ &=-\frac {b^2 e^{-2 b^2 x^2}}{3 \pi x^2}+\frac {b e^{-b^2 x^2} \text {erfc}(b x)}{3 \sqrt {\pi } x^3}-\frac {2 b^3 e^{-b^2 x^2} \text {erfc}(b x)}{3 \sqrt {\pi } x}-\frac {\text {erfc}(b x)^2}{4 x^4}-2 \frac {\left (4 b^4\right ) \int \frac {e^{-2 b^2 x^2}}{x} \, dx}{3 \pi }-\frac {\left (4 b^5\right ) \int e^{-b^2 x^2} \text {erfc}(b x) \, dx}{3 \sqrt {\pi }}\\ &=-\frac {b^2 e^{-2 b^2 x^2}}{3 \pi x^2}+\frac {b e^{-b^2 x^2} \text {erfc}(b x)}{3 \sqrt {\pi } x^3}-\frac {2 b^3 e^{-b^2 x^2} \text {erfc}(b x)}{3 \sqrt {\pi } x}-\frac {\text {erfc}(b x)^2}{4 x^4}-\frac {4 b^4 \text {Ei}\left (-2 b^2 x^2\right )}{3 \pi }+\frac {1}{3} \left (2 b^4\right ) \text {Subst}(\int x \, dx,x,\text {erfc}(b x))\\ &=-\frac {b^2 e^{-2 b^2 x^2}}{3 \pi x^2}+\frac {b e^{-b^2 x^2} \text {erfc}(b x)}{3 \sqrt {\pi } x^3}-\frac {2 b^3 e^{-b^2 x^2} \text {erfc}(b x)}{3 \sqrt {\pi } x}+\frac {1}{3} b^4 \text {erfc}(b x)^2-\frac {\text {erfc}(b x)^2}{4 x^4}-\frac {4 b^4 \text {Ei}\left (-2 b^2 x^2\right )}{3 \pi }\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 97, normalized size = 0.78 \begin {gather*} \frac {-\frac {4 b e^{-b^2 x^2} x \left (-1+2 b^2 x^2\right ) \text {Erfc}(b x)}{\sqrt {\pi }}+\left (-3+4 b^4 x^4\right ) \text {Erfc}(b x)^2-\frac {4 b^2 x^2 \left (e^{-2 b^2 x^2}+4 b^2 x^2 \text {Ei}\left (-2 b^2 x^2\right )\right )}{\pi }}{12 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Erfc[b*x]^2/x^5,x]

[Out]

((-4*b*x*(-1 + 2*b^2*x^2)*Erfc[b*x])/(E^(b^2*x^2)*Sqrt[Pi]) + (-3 + 4*b^4*x^4)*Erfc[b*x]^2 - (4*b^2*x^2*(E^(-2
*b^2*x^2) + 4*b^2*x^2*ExpIntegralEi[-2*b^2*x^2]))/Pi)/(12*x^4)

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\mathrm {erfc}\left (b x \right )^{2}}{x^{5}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erfc(b*x)^2/x^5,x)

[Out]

int(erfc(b*x)^2/x^5,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfc(b*x)^2/x^5,x, algorithm="maxima")

[Out]

integrate(erfc(b*x)^2/x^5, x)

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Fricas [A]
time = 0.36, size = 141, normalized size = 1.13 \begin {gather*} -\frac {3 \, \pi + 8 \, \pi \sqrt {b^{2}} b^{3} x^{4} \operatorname {erf}\left (\sqrt {b^{2}} x\right ) + 16 \, b^{4} x^{4} {\rm Ei}\left (-2 \, b^{2} x^{2}\right ) + 4 \, b^{2} x^{2} e^{\left (-2 \, b^{2} x^{2}\right )} + {\left (3 \, \pi - 4 \, \pi b^{4} x^{4}\right )} \operatorname {erf}\left (b x\right )^{2} + 4 \, \sqrt {\pi } {\left (2 \, b^{3} x^{3} - b x - {\left (2 \, b^{3} x^{3} - b x\right )} \operatorname {erf}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )} - 6 \, \pi \operatorname {erf}\left (b x\right )}{12 \, \pi x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfc(b*x)^2/x^5,x, algorithm="fricas")

[Out]

-1/12*(3*pi + 8*pi*sqrt(b^2)*b^3*x^4*erf(sqrt(b^2)*x) + 16*b^4*x^4*Ei(-2*b^2*x^2) + 4*b^2*x^2*e^(-2*b^2*x^2) +
(3*pi - 4*pi*b^4*x^4)*erf(b*x)^2 + 4*sqrt(pi)*(2*b^3*x^3 - b*x - (2*b^3*x^3 - b*x)*erf(b*x))*e^(-b^2*x^2) - 6
*pi*erf(b*x))/(pi*x^4)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {erfc}^{2}{\left (b x \right )}}{x^{5}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfc(b*x)**2/x**5,x)

[Out]

Integral(erfc(b*x)**2/x**5, x)

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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)^2/x^5,x, algorithm="giac")

[Out]

integrate(erfc(b*x)^2/x^5, 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 )}^2}{x^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erfc(b*x)^2/x^5,x)

[Out]

int(erfc(b*x)^2/x^5, x)

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