3.27 \(\int \frac {\text {erf}(b x)^2}{x^5} \, dx\)

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

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

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Rubi [A]  time = 0.18, 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 = {6364, 6391, 6373, 30, 2210, 2214} \[ \frac {2 b^3 e^{-b^2 x^2} \text {Erf}(b x)}{3 \sqrt {\pi } x}-\frac {b e^{-b^2 x^2} \text {Erf}(b x)}{3 \sqrt {\pi } x^3}+\frac {1}{3} b^4 \text {Erf}(b x)^2-\frac {4 b^4 \text {Ei}\left (-2 b^2 x^2\right )}{3 \pi }-\frac {b^2 e^{-2 b^2 x^2}}{3 \pi x^2}-\frac {\text {Erf}(b x)^2}{4 x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-b^2/(3*E^(2*b^2*x^2)*Pi*x^2) - (b*Erf[b*x])/(3*E^(b^2*x^2)*Sqrt[Pi]*x^3) + (2*b^3*Erf[b*x])/(3*E^(b^2*x^2)*Sq
rt[Pi]*x) + (b^4*Erf[b*x]^2)/3 - Erf[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 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 6364

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

Rule 6373

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

Rule 6391

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[(x^(m + 1)*E^(c + d*x^2)*Erf
[a + b*x])/(m + 1), x] + (-Dist[(2*d)/(m + 1), Int[x^(m + 2)*E^(c + d*x^2)*Erf[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 {erf}(b x)^2}{x^5} \, dx &=-\frac {\text {erf}(b x)^2}{4 x^4}+\frac {b \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^4} \, dx}{\sqrt {\pi }}\\ &=-\frac {b e^{-b^2 x^2} \text {erf}(b x)}{3 \sqrt {\pi } x^3}-\frac {\text {erf}(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 {erf}(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 {erf}(b x)}{3 \sqrt {\pi } x^3}+\frac {2 b^3 e^{-b^2 x^2} \text {erf}(b x)}{3 \sqrt {\pi } x}-\frac {\text {erf}(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 {erf}(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 {erf}(b x)}{3 \sqrt {\pi } x^3}+\frac {2 b^3 e^{-b^2 x^2} \text {erf}(b x)}{3 \sqrt {\pi } x}-\frac {\text {erf}(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 ) \operatorname {Subst}(\int x \, dx,x,\text {erf}(b x))\\ &=-\frac {b^2 e^{-2 b^2 x^2}}{3 \pi x^2}-\frac {b e^{-b^2 x^2} \text {erf}(b x)}{3 \sqrt {\pi } x^3}+\frac {2 b^3 e^{-b^2 x^2} \text {erf}(b x)}{3 \sqrt {\pi } x}+\frac {1}{3} b^4 \text {erf}(b x)^2-\frac {\text {erf}(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 = 0.09, size = 97, normalized size = 0.78 \[ \frac {\left (4 b^4 x^4-3\right ) \text {erf}(b x)^2+\frac {4 b x e^{-b^2 x^2} \left (2 b^2 x^2-1\right ) \text {erf}(b x)}{\sqrt {\pi }}-\frac {4 b^2 x^2 \left (4 b^2 x^2 \text {Ei}\left (-2 b^2 x^2\right )+e^{-2 b^2 x^2}\right )}{\pi }}{12 x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*b*x*(-1 + 2*b^2*x^2)*Erf[b*x])/(E^(b^2*x^2)*Sqrt[Pi]) + (-3 + 4*b^4*x^4)*Erf[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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fricas [A]  time = 0.53, size = 94, normalized size = 0.75 \[ -\frac {16 \, b^{4} x^{4} {\rm Ei}\left (-2 \, b^{2} x^{2}\right ) + 4 \, b^{2} x^{2} e^{\left (-2 \, b^{2} x^{2}\right )} - 4 \, \sqrt {\pi } {\left (2 \, b^{3} x^{3} - b x\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} + {\left (3 \, \pi - 4 \, \pi b^{4} x^{4}\right )} \operatorname {erf}\left (b x\right )^{2}}{12 \, \pi x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erf}\left (b x\right )^{2}}{x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(b*x)^2/x^5,x, algorithm="giac")

[Out]

integrate(erf(b*x)^2/x^5, x)

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[ \int \frac {\erf \left (b x \right )^{2}}{x^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {b \int \frac {\operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{4}}\,{d x}}{\sqrt {\pi }} - \frac {\operatorname {erf}\left (b x\right )^{2}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*integrate(erf(b*x)*e^(-b^2*x^2)/x^4, x)/sqrt(pi) - 1/4*erf(b*x)^2/x^4

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {erf}\left (b\,x\right )}^2}{x^5} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(erf(b*x)^2/x^5, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erf}^{2}{\left (b x \right )}}{x^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(erf(b*x)**2/x**5, x)

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