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

Optimal. Leaf size=30 \[ \frac {2 b x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};-b^2 x^2\right )}{\sqrt {\pi }} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 30, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.02, size = 30, normalized size = 1.00 \[ \frac {2 b x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};-b^2 x^2\right )}{\sqrt {\pi }} \]

Antiderivative was successfully verified.

[In]

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]

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fricas [F]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x}, x\right ) \]

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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x}\,{d x} \]

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)

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {\erfi \left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}}{x}\, dx \]

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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x}\,{d x} \]

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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfi}\left (b\,x\right )}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((exp(-b^2*x^2)*erfi(b*x))/x, x)

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: AttributeError} \]

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

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