3.219 \(\int \frac{\text{Erfi}(b x)}{x^4} \, dx\)

Optimal. Leaf size=54 \[ \frac{b^3 \text{ExpIntegralEi}\left (b^2 x^2\right )}{3 \sqrt{\pi }}-\frac{b e^{b^2 x^2}}{3 \sqrt{\pi } x^2}-\frac{\text{Erfi}(b x)}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.050437, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6363, 2214, 2210} \[ \frac{b^3 \text{Ei}\left (b^2 x^2\right )}{3 \sqrt{\pi }}-\frac{b e^{b^2 x^2}}{3 \sqrt{\pi } x^2}-\frac{\text{Erfi}(b x)}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[Erfi[b*x]/x^4,x]

[Out]

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

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]

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 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]

Rubi steps

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

Mathematica [A]  time = 0.0214942, size = 50, normalized size = 0.93 \[ -\frac{-\frac{b^3 x^3 \text{ExpIntegralEi}\left (b^2 x^2\right )}{\sqrt{\pi }}+\frac{b x e^{b^2 x^2}}{\sqrt{\pi }}+\text{Erfi}(b x)}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Erfi[b*x]/x^4,x]

[Out]

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

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Maple [A]  time = 0.043, size = 52, normalized size = 1. \begin{align*}{b}^{3} \left ( -{\frac{{\it erfi} \left ( bx \right ) }{3\,{x}^{3}{b}^{3}}}+{\frac{2}{3\,\sqrt{\pi }} \left ( -{\frac{{{\rm e}^{{b}^{2}{x}^{2}}}}{2\,{b}^{2}{x}^{2}}}-{\frac{{\it Ei} \left ( 1,-{b}^{2}{x}^{2} \right ) }{2}} \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erfi(b*x)/x^4,x)

[Out]

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

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Maxima [A]  time = 1.28505, size = 38, normalized size = 0.7 \begin{align*} \frac{b^{3} \Gamma \left (-1, -b^{2} x^{2}\right )}{3 \, \sqrt{\pi }} - \frac{\operatorname{erfi}\left (b x\right )}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x)/x^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.28505, size = 109, normalized size = 2.02 \begin{align*} -\frac{\pi \operatorname{erfi}\left (b x\right ) - \sqrt{\pi }{\left (b^{3} x^{3}{\rm Ei}\left (b^{2} x^{2}\right ) - b x e^{\left (b^{2} x^{2}\right )}\right )}}{3 \, \pi x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x)/x^4,x, algorithm="fricas")

[Out]

-1/3*(pi*erfi(b*x) - sqrt(pi)*(b^3*x^3*Ei(b^2*x^2) - b*x*e^(b^2*x^2)))/(pi*x^3)

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Sympy [C]  time = 2.35422, size = 63, normalized size = 1.17 \begin{align*} - \frac{b^{3} \operatorname{E}_{1}\left (b^{2} x^{2} e^{i \pi }\right )}{3 \sqrt{\pi }} - \frac{b e^{b^{2} x^{2}}}{3 \sqrt{\pi } x^{2}} - \frac{i \operatorname{erfc}{\left (i b x \right )}}{3 x^{3}} + \frac{i}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x)/x**4,x)

[Out]

-b**3*expint(1, b**2*x**2*exp_polar(I*pi))/(3*sqrt(pi)) - b*exp(b**2*x**2)/(3*sqrt(pi)*x**2) - I*erfc(I*b*x)/(
3*x**3) + I/(3*x**3)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfi}\left (b x\right )}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x)/x^4,x, algorithm="giac")

[Out]

integrate(erfi(b*x)/x^4, x)