3.233 \(\int \frac{\text{Erfi}(b x)^2}{x^5} \, dx\)

Optimal. Leaf size=123 \[ -\frac{2 b^3 e^{b^2 x^2} \text{Erfi}(b x)}{3 \sqrt{\pi } x}-\frac{b e^{b^2 x^2} \text{Erfi}(b x)}{3 \sqrt{\pi } x^3}+\frac{1}{3} b^4 \text{Erfi}(b x)^2+\frac{4 b^4 \text{ExpIntegralEi}\left (2 b^2 x^2\right )}{3 \pi }-\frac{b^2 e^{2 b^2 x^2}}{3 \pi x^2}-\frac{\text{Erfi}(b x)^2}{4 x^4} \]

[Out]

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

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Rubi [A]  time = 0.165433, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6366, 6393, 6375, 30, 2210, 2214} \[ -\frac{2 b^3 e^{b^2 x^2} \text{Erfi}(b x)}{3 \sqrt{\pi } x}-\frac{b e^{b^2 x^2} \text{Erfi}(b x)}{3 \sqrt{\pi } x^3}+\frac{1}{3} b^4 \text{Erfi}(b x)^2+\frac{4 b^4 \text{Ei}\left (2 b^2 x^2\right )}{3 \pi }-\frac{b^2 e^{2 b^2 x^2}}{3 \pi x^2}-\frac{\text{Erfi}(b x)^2}{4 x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]))

Rubi steps

\begin{align*} \int \frac{\text{erfi}(b x)^2}{x^5} \, dx &=-\frac{\text{erfi}(b x)^2}{4 x^4}+\frac{b \int \frac{e^{b^2 x^2} \text{erfi}(b x)}{x^4} \, dx}{\sqrt{\pi }}\\ &=-\frac{b e^{b^2 x^2} \text{erfi}(b x)}{3 \sqrt{\pi } x^3}-\frac{\text{erfi}(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{erfi}(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{erfi}(b x)}{3 \sqrt{\pi } x^3}-\frac{2 b^3 e^{b^2 x^2} \text{erfi}(b x)}{3 \sqrt{\pi } x}-\frac{\text{erfi}(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{erfi}(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{erfi}(b x)}{3 \sqrt{\pi } x^3}-\frac{2 b^3 e^{b^2 x^2} \text{erfi}(b x)}{3 \sqrt{\pi } x}-\frac{\text{erfi}(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{erfi}(b x))\\ &=-\frac{b^2 e^{2 b^2 x^2}}{3 \pi x^2}-\frac{b e^{b^2 x^2} \text{erfi}(b x)}{3 \sqrt{\pi } x^3}-\frac{2 b^3 e^{b^2 x^2} \text{erfi}(b x)}{3 \sqrt{\pi } x}+\frac{1}{3} b^4 \text{erfi}(b x)^2-\frac{\text{erfi}(b x)^2}{4 x^4}+\frac{4 b^4 \text{Ei}\left (2 b^2 x^2\right )}{3 \pi }\\ \end{align*}

Mathematica [A]  time = 0.0284174, size = 97, normalized size = 0.79 \[ \frac{-4 \sqrt{\pi } b x e^{b^2 x^2} \left (2 b^2 x^2+1\right ) \text{Erfi}(b x)+\pi \left (4 b^4 x^4-3\right ) \text{Erfi}(b x)^2-4 b^2 x^2 \left (e^{2 b^2 x^2}-4 b^2 x^2 \text{ExpIntegralEi}\left (2 b^2 x^2\right )\right )}{12 \pi x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.042, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\it erfi} \left ( bx \right ) \right ) ^{2}}{{x}^{5}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfi}\left (b x\right )^{2}}{x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(erfi(b*x)^2/x^5, x)

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Fricas [A]  time = 2.52266, size = 212, normalized size = 1.72 \begin{align*} \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{erfi}\left (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 )^{2}}{12 \, \pi x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(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)*erfi(b*x)*e^(b^2*x^2)
- (3*pi - 4*pi*b^4*x^4)*erfi(b*x)^2)/(pi*x^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfi}^{2}{\left (b x \right )}}{x^{5}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(erfi(b*x)**2/x**5, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfi}\left (b x\right )^{2}}{x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(erfi(b*x)^2/x^5, x)