3.232 \(\int \frac{\text{Erfi}(b x)^2}{x^3} \, dx\)

Optimal. Leaf size=65 \[ -\frac{2 b e^{b^2 x^2} \text{Erfi}(b x)}{\sqrt{\pi } x}+b^2 \text{Erfi}(b x)^2+\frac{2 b^2 \text{ExpIntegralEi}\left (2 b^2 x^2\right )}{\pi }-\frac{\text{Erfi}(b x)^2}{2 x^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0901547, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6366, 6393, 6375, 30, 2210} \[ -\frac{2 b e^{b^2 x^2} \text{Erfi}(b x)}{\sqrt{\pi } x}+b^2 \text{Erfi}(b x)^2+\frac{2 b^2 \text{Ei}\left (2 b^2 x^2\right )}{\pi }-\frac{\text{Erfi}(b x)^2}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[Erfi[b*x]^2/x^3,x]

[Out]

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

Rule 6366

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

Rule 6393

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[(x^(m + 1)*E^(c + d*x^2)*Er
fi[a + b*x])/(m + 1), x] + (-Dist[(2*d)/(m + 1), Int[x^(m + 2)*E^(c + d*x^2)*Erfi[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]

Rule 6375

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

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]

Rubi steps

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

Mathematica [A]  time = 0.0227362, size = 60, normalized size = 0.92 \[ -\frac{2 b e^{b^2 x^2} \text{Erfi}(b x)}{\sqrt{\pi } x}+\left (b^2-\frac{1}{2 x^2}\right ) \text{Erfi}(b x)^2+\frac{2 b^2 \text{ExpIntegralEi}\left (2 b^2 x^2\right )}{\pi } \]

Antiderivative was successfully verified.

[In]

Integrate[Erfi[b*x]^2/x^3,x]

[Out]

(-2*b*E^(b^2*x^2)*Erfi[b*x])/(Sqrt[Pi]*x) + (b^2 - 1/(2*x^2))*Erfi[b*x]^2 + (2*b^2*ExpIntegralEi[2*b^2*x^2])/P
i

________________________________________________________________________________________

Maple [F]  time = 0.042, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\it erfi} \left ( bx \right ) \right ) ^{2}}{{x}^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erfi(b*x)^2/x^3,x)

[Out]

int(erfi(b*x)^2/x^3,x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfi}\left (b x\right )^{2}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x)^2/x^3,x, algorithm="maxima")

[Out]

integrate(erfi(b*x)^2/x^3, x)

________________________________________________________________________________________

Fricas [A]  time = 2.33436, size = 153, normalized size = 2.35 \begin{align*} \frac{4 \, b^{2} x^{2}{\rm Ei}\left (2 \, b^{2} x^{2}\right ) - 4 \, \sqrt{\pi } b x \operatorname{erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} -{\left (\pi - 2 \, \pi b^{2} x^{2}\right )} \operatorname{erfi}\left (b x\right )^{2}}{2 \, \pi x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x)^2/x^3,x, algorithm="fricas")

[Out]

1/2*(4*b^2*x^2*Ei(2*b^2*x^2) - 4*sqrt(pi)*b*x*erfi(b*x)*e^(b^2*x^2) - (pi - 2*pi*b^2*x^2)*erfi(b*x)^2)/(pi*x^2
)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfi}^{2}{\left (b x \right )}}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x)**2/x**3,x)

[Out]

Integral(erfi(b*x)**2/x**3, x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfi}\left (b x\right )^{2}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x)^2/x^3,x, algorithm="giac")

[Out]

integrate(erfi(b*x)^2/x^3, x)