3.281 \(\int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^4} \, dx\)

Optimal. Leaf size=105 \[ \frac {4 b^5 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{3 \sqrt {\pi }}-\frac {4 b^3 \log (x)}{3 \sqrt {\pi }}+\frac {2 b^2 e^{-b^2 x^2} \text {erfi}(b x)}{3 x}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{3 x^3}-\frac {b}{3 \sqrt {\pi } x^2} \]

[Out]

-1/3*erfi(b*x)/exp(b^2*x^2)/x^3+2/3*b^2*erfi(b*x)/exp(b^2*x^2)/x-1/3*b/x^2/Pi^(1/2)+4/3*b^5*x^2*Hypergeometric
PFQ([1, 1],[3/2, 2],-b^2*x^2)/Pi^(1/2)-4/3*b^3*ln(x)/Pi^(1/2)

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Rubi [A]  time = 0.10, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6393, 6378, 29, 30} \[ \frac {4 b^5 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{3 \sqrt {\pi }}+\frac {2 b^2 e^{-b^2 x^2} \text {Erfi}(b x)}{3 x}-\frac {e^{-b^2 x^2} \text {Erfi}(b x)}{3 x^3}-\frac {4 b^3 \log (x)}{3 \sqrt {\pi }}-\frac {b}{3 \sqrt {\pi } x^2} \]

Antiderivative was successfully verified.

[In]

Int[Erfi[b*x]/(E^(b^2*x^2)*x^4),x]

[Out]

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

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]

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 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]

Rubi steps

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

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Mathematica [C]  time = 0.02, size = 29, normalized size = 0.28 \[ -\frac {b G_{2,3}^{2,1}\left (b^2 x^2|\begin {array}{c} 0,2 \\ 0,1,-\frac {1}{2} \\\end {array}\right )}{2 x^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Erfi[b*x]/(E^(b^2*x^2)*x^4),x]

[Out]

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

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fricas [F]  time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x)/exp(b^2*x^2)/x^4,x, algorithm="fricas")

[Out]

integral(erfi(b*x)*e^(-b^2*x^2)/x^4, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x)/exp(b^2*x^2)/x^4,x, algorithm="giac")

[Out]

integrate(erfi(b*x)*e^(-b^2*x^2)/x^4, x)

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maple [F]  time = 0.30, size = 0, normalized size = 0.00 \[ \int \frac {\erfi \left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erfi(b*x)/exp(b^2*x^2)/x^4,x)

[Out]

int(erfi(b*x)/exp(b^2*x^2)/x^4,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x)/exp(b^2*x^2)/x^4,x, algorithm="maxima")

[Out]

integrate(erfi(b*x)*e^(-b^2*x^2)/x^4, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfi}\left (b\,x\right )}{x^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-b^2*x^2)*erfi(b*x))/x^4,x)

[Out]

int((exp(-b^2*x^2)*erfi(b*x))/x^4, 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**4,x)

[Out]

Exception raised: AttributeError

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