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

Optimal. Leaf size=27 \[ \frac {b x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{\sqrt {\pi }} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {6378} \[ \frac {b x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{\sqrt {\pi }} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6378

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

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 27, normalized size = 1.00 \[ \frac {b x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{\sqrt {\pi }} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(erfi(b*x)*e^(-b^2*x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(erfi(b*x)*e^(-b^2*x^2), x)

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[ \int \erfi \left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(erfi(b*x)*e^(-b^2*x^2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int {\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfi}\left (b\,x\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: AttributeError

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