3.303 \(\int (\frac{e^{-b^2 x^2} \text{Erfi}(b x)}{x^3}+\frac{b^2 e^{-b^2 x^2} \text{Erfi}(b x)}{x}) \, dx\)

Optimal. Leaf size=33 \[ -\frac{e^{-b^2 x^2} \text{Erfi}(b x)}{2 x^2}-\frac{b}{\sqrt{\pi } x} \]

[Out]

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

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Rubi [A]  time = 0.11913, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.075, Rules used = {6393, 6390, 30} \[ -\frac{e^{-b^2 x^2} \text{Erfi}(b x)}{2 x^2}-\frac{b}{\sqrt{\pi } x} \]

Antiderivative was successfully verified.

[In]

Int[Erfi[b*x]/(E^(b^2*x^2)*x^3) + (b^2*Erfi[b*x])/(E^(b^2*x^2)*x),x]

[Out]

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

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 6390

Int[(E^((c_.) + (d_.)*(x_)^2)*Erfi[(b_.)*(x_)])/(x_), x_Symbol] :> Simp[(2*b*E^c*x*HypergeometricPFQ[{1/2, 1},
{3/2, 3/2}, -(b^2*x^2)])/Sqrt[Pi], x] /; FreeQ[{b, c, d}, 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]

Rubi steps

\begin{align*} \int \left (\frac{e^{-b^2 x^2} \text{erfi}(b x)}{x^3}+\frac{b^2 e^{-b^2 x^2} \text{erfi}(b x)}{x}\right ) \, dx &=b^2 \int \frac{e^{-b^2 x^2} \text{erfi}(b x)}{x} \, dx+\int \frac{e^{-b^2 x^2} \text{erfi}(b x)}{x^3} \, dx\\ &=-\frac{e^{-b^2 x^2} \text{erfi}(b x)}{2 x^2}+\frac{2 b^3 x \, _2F_2\left (\frac{1}{2},1;\frac{3}{2},\frac{3}{2};-b^2 x^2\right )}{\sqrt{\pi }}-b^2 \int \frac{e^{-b^2 x^2} \text{erfi}(b x)}{x} \, dx+\frac{b \int \frac{1}{x^2} \, dx}{\sqrt{\pi }}\\ &=-\frac{b}{\sqrt{\pi } x}-\frac{e^{-b^2 x^2} \text{erfi}(b x)}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0577235, size = 33, normalized size = 1. \[ -\frac{e^{-b^2 x^2} \text{Erfi}(b x)}{2 x^2}-\frac{b}{\sqrt{\pi } x} \]

Antiderivative was successfully verified.

[In]

Integrate[Erfi[b*x]/(E^(b^2*x^2)*x^3) + (b^2*Erfi[b*x])/(E^(b^2*x^2)*x),x]

[Out]

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

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Maple [A]  time = 0.201, size = 41, normalized size = 1.2 \begin{align*}{\frac{-2\,{{\rm e}^{{b}^{2}{x}^{2}}}bx-\sqrt{\pi }{\it erfi} \left ( bx \right ) }{2\,\sqrt{\pi }{x}^{2}{{\rm e}^{{b}^{2}{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(b^2*erfi(b*x)*e^(-b^2*x^2)/x + erfi(b*x)*e^(-b^2*x^2)/x^3, x)

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Fricas [A]  time = 2.63857, size = 97, normalized size = 2.94 \begin{align*} -\frac{{\left (2 \, \sqrt{\pi } b x e^{\left (b^{2} x^{2}\right )} + \pi \operatorname{erfi}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )}}{2 \, \pi x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 137.103, size = 53, normalized size = 1.61 \begin{align*} \frac{2 b^{3} x{{}_{2}F_{2}\left (\begin{matrix} \frac{1}{2}, 1 \\ \frac{3}{2}, \frac{3}{2} \end{matrix}\middle |{- b^{2} x^{2}} \right )}}{\sqrt{\pi }} - \frac{2 b{{}_{2}F_{2}\left (\begin{matrix} - \frac{1}{2}, 1 \\ \frac{1}{2}, \frac{3}{2} \end{matrix}\middle |{- b^{2} x^{2}} \right )}}{\sqrt{\pi } x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b**3*x*hyper((1/2, 1), (3/2, 3/2), -b**2*x**2)/sqrt(pi) - 2*b*hyper((-1/2, 1), (1/2, 3/2), -b**2*x**2)/(sqrt
(pi)*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(b^2*erfi(b*x)*e^(-b^2*x^2)/x + erfi(b*x)*e^(-b^2*x^2)/x^3, x)