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

Optimal. Leaf size=30 \[ \frac{2 b x \text{HypergeometricPFQ}\left (\left \{\frac{1}{2},1\right \},\left \{\frac{3}{2},\frac{3}{2}\right \},-b^2 x^2\right )}{\sqrt{\pi }} \]

[Out]

(2*b*x*HypergeometricPFQ[{1/2, 1}, {3/2, 3/2}, -(b^2*x^2)])/Sqrt[Pi]

________________________________________________________________________________________

Rubi [A]  time = 0.0364644, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {6390} \[ \frac{2 b x \, _2F_2\left (\frac{1}{2},1;\frac{3}{2},\frac{3}{2};-b^2 x^2\right )}{\sqrt{\pi }} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b*x*HypergeometricPFQ[{1/2, 1}, {3/2, 3/2}, -(b^2*x^2)])/Sqrt[Pi]

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]

Rubi steps

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

Mathematica [A]  time = 0.0153554, size = 30, normalized size = 1. \[ \frac{2 b x \text{HypergeometricPFQ}\left (\left \{\frac{1}{2},1\right \},\left \{\frac{3}{2},\frac{3}{2}\right \},-b^2 x^2\right )}{\sqrt{\pi }} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b*x*HypergeometricPFQ[{1/2, 1}, {3/2, 3/2}, -(b^2*x^2)])/Sqrt[Pi]

________________________________________________________________________________________

Maple [F]  time = 0.094, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it erfi} \left ( bx \right ) }{{{\rm e}^{{b}^{2}{x}^{2}}}x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(erfi(b*x)*e^(-b^2*x^2)/x, x)

________________________________________________________________________________________

Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: AttributeError

________________________________________________________________________________________

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

[Out]

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