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

Optimal. Leaf size=105 \[ \frac{b^5 x \text{HypergeometricPFQ}\left (\left \{\frac{1}{2},1\right \},\left \{\frac{3}{2},\frac{3}{2}\right \},-b^2 x^2\right )}{\sqrt{\pi }}+\frac{b^2 e^{-b^2 x^2} \text{Erfi}(b x)}{4 x^2}-\frac{e^{-b^2 x^2} \text{Erfi}(b x)}{4 x^4}+\frac{b^3}{2 \sqrt{\pi } x}-\frac{b}{6 \sqrt{\pi } x^3} \]

[Out]

-b/(6*Sqrt[Pi]*x^3) + b^3/(2*Sqrt[Pi]*x) - Erfi[b*x]/(4*E^(b^2*x^2)*x^4) + (b^2*Erfi[b*x])/(4*E^(b^2*x^2)*x^2)
+ (b^5*x*HypergeometricPFQ[{1/2, 1}, {3/2, 3/2}, -(b^2*x^2)])/Sqrt[Pi]

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Rubi [A]  time = 0.113068, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6393, 6390, 30} \[ \frac{b^5 x \, _2F_2\left (\frac{1}{2},1;\frac{3}{2},\frac{3}{2};-b^2 x^2\right )}{\sqrt{\pi }}+\frac{b^2 e^{-b^2 x^2} \text{Erfi}(b x)}{4 x^2}-\frac{e^{-b^2 x^2} \text{Erfi}(b x)}{4 x^4}+\frac{b^3}{2 \sqrt{\pi } x}-\frac{b}{6 \sqrt{\pi } x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-b/(6*Sqrt[Pi]*x^3) + b^3/(2*Sqrt[Pi]*x) - Erfi[b*x]/(4*E^(b^2*x^2)*x^4) + (b^2*Erfi[b*x])/(4*E^(b^2*x^2)*x^2)
+ (b^5*x*HypergeometricPFQ[{1/2, 1}, {3/2, 3/2}, -(b^2*x^2)])/Sqrt[Pi]

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 \frac{e^{-b^2 x^2} \text{erfi}(b x)}{x^5} \, dx &=-\frac{e^{-b^2 x^2} \text{erfi}(b x)}{4 x^4}-\frac{1}{2} b^2 \int \frac{e^{-b^2 x^2} \text{erfi}(b x)}{x^3} \, dx+\frac{b \int \frac{1}{x^4} \, dx}{2 \sqrt{\pi }}\\ &=-\frac{b}{6 \sqrt{\pi } x^3}-\frac{e^{-b^2 x^2} \text{erfi}(b x)}{4 x^4}+\frac{b^2 e^{-b^2 x^2} \text{erfi}(b x)}{4 x^2}+\frac{1}{2} b^4 \int \frac{e^{-b^2 x^2} \text{erfi}(b x)}{x} \, dx-\frac{b^3 \int \frac{1}{x^2} \, dx}{2 \sqrt{\pi }}\\ &=-\frac{b}{6 \sqrt{\pi } x^3}+\frac{b^3}{2 \sqrt{\pi } x}-\frac{e^{-b^2 x^2} \text{erfi}(b x)}{4 x^4}+\frac{b^2 e^{-b^2 x^2} \text{erfi}(b x)}{4 x^2}+\frac{b^5 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 = 0.0164657, size = 34, normalized size = 0.32 \[ -\frac{2 b \text{HypergeometricPFQ}\left (\left \{-\frac{3}{2},1\right \},\left \{-\frac{1}{2},\frac{3}{2}\right \},-b^2 x^2\right )}{3 \sqrt{\pi } x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.639, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it erfi} \left ( bx \right ) }{{{\rm e}^{{b}^{2}{x}^{2}}}{x}^{5}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(erfi(b*x)/exp(b^2*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 ) e^{\left (-b^{2} x^{2}\right )}}{x^{5}}\,{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^5,x, algorithm="maxima")

[Out]

integrate(erfi(b*x)*e^(-b^2*x^2)/x^5, x)

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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^{5}}, 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^5,x, algorithm="fricas")

[Out]

integral(erfi(b*x)*e^(-b^2*x^2)/x^5, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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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^{5}}\,{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^5,x, algorithm="giac")

[Out]

integrate(erfi(b*x)*e^(-b^2*x^2)/x^5, x)