3.218 \(\int \frac{\text{Erfi}(b x)}{x^2} \, dx\)

Optimal. Leaf size=25 \[ \frac{b \text{ExpIntegralEi}\left (b^2 x^2\right )}{\sqrt{\pi }}-\frac{\text{Erfi}(b x)}{x} \]

[Out]

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

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Rubi [A]  time = 0.0291507, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6363, 2210} \[ \frac{b \text{Ei}\left (b^2 x^2\right )}{\sqrt{\pi }}-\frac{\text{Erfi}(b x)}{x} \]

Antiderivative was successfully verified.

[In]

Int[Erfi[b*x]/x^2,x]

[Out]

-(Erfi[b*x]/x) + (b*ExpIntegralEi[b^2*x^2])/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 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^2} \, dx &=-\frac{\text{erfi}(b x)}{x}+\frac{(2 b) \int \frac{e^{b^2 x^2}}{x} \, dx}{\sqrt{\pi }}\\ &=-\frac{\text{erfi}(b x)}{x}+\frac{b \text{Ei}\left (b^2 x^2\right )}{\sqrt{\pi }}\\ \end{align*}

Mathematica [A]  time = 0.0122441, size = 25, normalized size = 1. \[ \frac{b \text{ExpIntegralEi}\left (b^2 x^2\right )}{\sqrt{\pi }}-\frac{\text{Erfi}(b x)}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[Erfi[b*x]/x^2,x]

[Out]

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

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Maple [A]  time = 0.043, size = 31, normalized size = 1.2 \begin{align*} b \left ( -{\frac{{\it erfi} \left ( bx \right ) }{bx}}-{\frac{{\it Ei} \left ( 1,-{b}^{2}{x}^{2} \right ) }{\sqrt{\pi }}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erfi(b*x)/x^2,x)

[Out]

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

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Maxima [A]  time = 1.29618, size = 31, normalized size = 1.24 \begin{align*} \frac{b{\rm Ei}\left (b^{2} x^{2}\right )}{\sqrt{\pi }} - \frac{\operatorname{erfi}\left (b x\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*Ei(b^2*x^2)/sqrt(pi) - erfi(b*x)/x

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Fricas [A]  time = 2.35493, size = 68, normalized size = 2.72 \begin{align*} \frac{\sqrt{\pi } b x{\rm Ei}\left (b^{2} x^{2}\right ) - \pi \operatorname{erfi}\left (b x\right )}{\pi x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 1.2032, size = 32, normalized size = 1.28 \begin{align*} - \frac{b \operatorname{E}_{1}\left (b^{2} x^{2} e^{i \pi }\right )}{\sqrt{\pi }} - \frac{i \operatorname{erfc}{\left (i b x \right )}}{x} + \frac{i}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x)/x**2,x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfi}\left (b x\right )}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(erfi(b*x)/x^2, x)