3.282 \(\int \frac{e^{-b^2 x^2} \text{Erfi}(b x)}{x^6} \, dx\)

Optimal. Leaf size=144 \[ -\frac{8 b^7 x^2 \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},-b^2 x^2\right )}{15 \sqrt{\pi }}-\frac{4 b^4 e^{-b^2 x^2} \text{Erfi}(b x)}{15 x}+\frac{2 b^2 e^{-b^2 x^2} \text{Erfi}(b x)}{15 x^3}-\frac{e^{-b^2 x^2} \text{Erfi}(b x)}{5 x^5}+\frac{2 b^3}{15 \sqrt{\pi } x^2}+\frac{8 b^5 \log (x)}{15 \sqrt{\pi }}-\frac{b}{10 \sqrt{\pi } x^4} \]

[Out]

-b/(10*Sqrt[Pi]*x^4) + (2*b^3)/(15*Sqrt[Pi]*x^2) - Erfi[b*x]/(5*E^(b^2*x^2)*x^5) + (2*b^2*Erfi[b*x])/(15*E^(b^
2*x^2)*x^3) - (4*b^4*Erfi[b*x])/(15*E^(b^2*x^2)*x) - (8*b^7*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, -(b^2*x^2)
])/(15*Sqrt[Pi]) + (8*b^5*Log[x])/(15*Sqrt[Pi])

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Rubi [A]  time = 0.136732, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6393, 6378, 29, 30} \[ -\frac{8 b^7 x^2 \, _2F_2\left (1,1;\frac{3}{2},2;-b^2 x^2\right )}{15 \sqrt{\pi }}-\frac{4 b^4 e^{-b^2 x^2} \text{Erfi}(b x)}{15 x}+\frac{2 b^2 e^{-b^2 x^2} \text{Erfi}(b x)}{15 x^3}-\frac{e^{-b^2 x^2} \text{Erfi}(b x)}{5 x^5}+\frac{2 b^3}{15 \sqrt{\pi } x^2}+\frac{8 b^5 \log (x)}{15 \sqrt{\pi }}-\frac{b}{10 \sqrt{\pi } x^4} \]

Antiderivative was successfully verified.

[In]

Int[Erfi[b*x]/(E^(b^2*x^2)*x^6),x]

[Out]

-b/(10*Sqrt[Pi]*x^4) + (2*b^3)/(15*Sqrt[Pi]*x^2) - Erfi[b*x]/(5*E^(b^2*x^2)*x^5) + (2*b^2*Erfi[b*x])/(15*E^(b^
2*x^2)*x^3) - (4*b^4*Erfi[b*x])/(15*E^(b^2*x^2)*x) - (8*b^7*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, -(b^2*x^2)
])/(15*Sqrt[Pi]) + (8*b^5*Log[x])/(15*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 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]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

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^6} \, dx &=-\frac{e^{-b^2 x^2} \text{erfi}(b x)}{5 x^5}-\frac{1}{5} \left (2 b^2\right ) \int \frac{e^{-b^2 x^2} \text{erfi}(b x)}{x^4} \, dx+\frac{(2 b) \int \frac{1}{x^5} \, dx}{5 \sqrt{\pi }}\\ &=-\frac{b}{10 \sqrt{\pi } x^4}-\frac{e^{-b^2 x^2} \text{erfi}(b x)}{5 x^5}+\frac{2 b^2 e^{-b^2 x^2} \text{erfi}(b x)}{15 x^3}+\frac{1}{15} \left (4 b^4\right ) \int \frac{e^{-b^2 x^2} \text{erfi}(b x)}{x^2} \, dx-\frac{\left (4 b^3\right ) \int \frac{1}{x^3} \, dx}{15 \sqrt{\pi }}\\ &=-\frac{b}{10 \sqrt{\pi } x^4}+\frac{2 b^3}{15 \sqrt{\pi } x^2}-\frac{e^{-b^2 x^2} \text{erfi}(b x)}{5 x^5}+\frac{2 b^2 e^{-b^2 x^2} \text{erfi}(b x)}{15 x^3}-\frac{4 b^4 e^{-b^2 x^2} \text{erfi}(b x)}{15 x}-\frac{1}{15} \left (8 b^6\right ) \int e^{-b^2 x^2} \text{erfi}(b x) \, dx+\frac{\left (8 b^5\right ) \int \frac{1}{x} \, dx}{15 \sqrt{\pi }}\\ &=-\frac{b}{10 \sqrt{\pi } x^4}+\frac{2 b^3}{15 \sqrt{\pi } x^2}-\frac{e^{-b^2 x^2} \text{erfi}(b x)}{5 x^5}+\frac{2 b^2 e^{-b^2 x^2} \text{erfi}(b x)}{15 x^3}-\frac{4 b^4 e^{-b^2 x^2} \text{erfi}(b x)}{15 x}-\frac{8 b^7 x^2 \, _2F_2\left (1,1;\frac{3}{2},2;-b^2 x^2\right )}{15 \sqrt{\pi }}+\frac{8 b^5 \log (x)}{15 \sqrt{\pi }}\\ \end{align*}

Mathematica [C]  time = 0.0182751, size = 29, normalized size = 0.2 \[ -\frac{b G_{2,3}^{2,1}\left (b^2 x^2|\begin{array}{c} 0,3 \\ 0,2,-\frac{1}{2} \\\end{array}\right )}{2 x^4} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Erfi[b*x]/(E^(b^2*x^2)*x^6),x]

[Out]

-(b*MeijerG[{{0}, {3}}, {{0, 2}, {-1/2}}, b^2*x^2])/(2*x^4)

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Maple [F]  time = 0.645, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it erfi} \left ( bx \right ) }{{{\rm e}^{{b}^{2}{x}^{2}}}{x}^{6}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erfi(b*x)/exp(b^2*x^2)/x^6,x)

[Out]

int(erfi(b*x)/exp(b^2*x^2)/x^6,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^{6}}\,{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^6,x, algorithm="maxima")

[Out]

integrate(erfi(b*x)*e^(-b^2*x^2)/x^6, 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^{6}}, 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^6,x, algorithm="fricas")

[Out]

integral(erfi(b*x)*e^(-b^2*x^2)/x^6, 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**6,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^{6}}\,{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^6,x, algorithm="giac")

[Out]

integrate(erfi(b*x)*e^(-b^2*x^2)/x^6, x)