3.212 \(\int \frac {\text {erfi}(b x)}{x^5} \, dx\)

Optimal. Leaf size=69 \[ \frac {1}{3} b^4 \text {erfi}(b x)-\frac {b e^{b^2 x^2}}{6 \sqrt {\pi } x^3}-\frac {b^3 e^{b^2 x^2}}{3 \sqrt {\pi } x}-\frac {\text {erfi}(b x)}{4 x^4} \]

[Out]

1/3*b^4*erfi(b*x)-1/4*erfi(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)

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Rubi [A]  time = 0.06, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6363, 2214, 2204} \[ \frac {1}{3} b^4 \text {Erfi}(b x)-\frac {b^3 e^{b^2 x^2}}{3 \sqrt {\pi } x}-\frac {b e^{b^2 x^2}}{6 \sqrt {\pi } x^3}-\frac {\text {Erfi}(b x)}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[Erfi[b*x]/x^5,x]

[Out]

-(b*E^(b^2*x^2))/(6*Sqrt[Pi]*x^3) - (b^3*E^(b^2*x^2))/(3*Sqrt[Pi]*x) + (b^4*Erfi[b*x])/3 - Erfi[b*x]/(4*x^4)

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[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 6363

Int[Erfi[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*Erfi[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 {erfi}(b x)}{x^5} \, dx &=-\frac {\text {erfi}(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 {erfi}(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 {erfi}(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 {erfi}(b x)-\frac {\text {erfi}(b x)}{4 x^4}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 51, normalized size = 0.74 \[ \frac {\left (4 b^4 x^4-3\right ) \text {erfi}(b x)-\frac {2 b x e^{b^2 x^2} \left (2 b^2 x^2+1\right )}{\sqrt {\pi }}}{12 x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.55, size = 52, normalized size = 0.75 \[ -\frac {2 \, \sqrt {\pi } {\left (2 \, b^{3} x^{3} + b x\right )} e^{\left (b^{2} x^{2}\right )} + {\left (3 \, \pi - 4 \, \pi b^{4} x^{4}\right )} \operatorname {erfi}\left (b x\right )}{12 \, \pi x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(erfi(b*x)/x^5, x)

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maple [A]  time = 0.00, size = 65, normalized size = 0.94 \[ b^{4} \left (-\frac {\erfi \left (b x \right )}{4 b^{4} x^{4}}+\frac {-\frac {{\mathrm e}^{b^{2} x^{2}}}{3 b^{3} x^{3}}-\frac {2 \,{\mathrm e}^{b^{2} x^{2}}}{3 b x}+\frac {2 \sqrt {\pi }\, \erfi \left (b x \right )}{3}}{2 \sqrt {\pi }}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.38, size = 39, normalized size = 0.57 \[ -\frac {\left (-b^{2} x^{2}\right )^{\frac {3}{2}} b \Gamma \left (-\frac {3}{2}, -b^{2} x^{2}\right )}{4 \, \sqrt {\pi } x^{3}} - \frac {\operatorname {erfi}\left (b x\right )}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.08, size = 89, normalized size = 1.29 \[ \frac {b\,{\left (-b^2\,x^2\right )}^{3/2}}{3\,x^3}-\frac {\mathrm {erfi}\left (b\,x\right )}{4\,x^4}-\frac {b^3\,{\mathrm {e}}^{b^2\,x^2}}{3\,x\,\sqrt {\pi }}-\frac {b\,{\mathrm {e}}^{b^2\,x^2}}{6\,x^3\,\sqrt {\pi }}-\frac {b\,\mathrm {erfc}\left (\sqrt {-b^2\,x^2}\right )\,{\left (-b^2\,x^2\right )}^{3/2}}{3\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 1.10, size = 60, normalized size = 0.87 \[ \frac {b^{4} \operatorname {erfi}{\left (b x \right )}}{3} - \frac {b^{3} e^{b^{2} x^{2}}}{3 \sqrt {\pi } x} - \frac {b e^{b^{2} x^{2}}}{6 \sqrt {\pi } x^{3}} - \frac {\operatorname {erfi}{\left (b x \right )}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b**4*erfi(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) - erfi(b*x)/(4*x**4
)

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