3.303 \(\int (\frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^3}+\frac {b^2 e^{-b^2 x^2} \text {erfi}(b x)}{x}) \, dx\)

Optimal. Leaf size=33 \[ -\frac {e^{-b^2 x^2} \text {erfi}(b x)}{2 x^2}-\frac {b}{\sqrt {\pi } x} \]

[Out]

-1/2*erfi(b*x)/exp(b^2*x^2)/x^2-b/x/Pi^(1/2)

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Rubi [A]  time = 0.12, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {6393, 6390, 30} \[ -\frac {e^{-b^2 x^2} \text {Erfi}(b x)}{2 x^2}-\frac {b}{\sqrt {\pi } x} \]

Antiderivative was successfully verified.

[In]

Int[Erfi[b*x]/(E^(b^2*x^2)*x^3) + (b^2*Erfi[b*x])/(E^(b^2*x^2)*x),x]

[Out]

-(b/(Sqrt[Pi]*x)) - Erfi[b*x]/(2*E^(b^2*x^2)*x^2)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[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 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 \left (\frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^3}+\frac {b^2 e^{-b^2 x^2} \text {erfi}(b x)}{x}\right ) \, dx &=b^2 \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x} \, dx+\int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^3} \, dx\\ &=-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{2 x^2}+\frac {2 b^3 x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};-b^2 x^2\right )}{\sqrt {\pi }}-b^2 \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x} \, dx+\frac {b \int \frac {1}{x^2} \, dx}{\sqrt {\pi }}\\ &=-\frac {b}{\sqrt {\pi } x}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{2 x^2}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 33, normalized size = 1.00 \[ -\frac {e^{-b^2 x^2} \text {erfi}(b x)}{2 x^2}-\frac {b}{\sqrt {\pi } x} \]

Antiderivative was successfully verified.

[In]

Integrate[Erfi[b*x]/(E^(b^2*x^2)*x^3) + (b^2*Erfi[b*x])/(E^(b^2*x^2)*x),x]

[Out]

-(b/(Sqrt[Pi]*x)) - Erfi[b*x]/(2*E^(b^2*x^2)*x^2)

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fricas [A]  time = 0.59, size = 39, normalized size = 1.18 \[ -\frac {{\left (2 \, \sqrt {\pi } b x e^{\left (b^{2} x^{2}\right )} + \pi \operatorname {erfi}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )}}{2 \, \pi x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x)/exp(b^2*x^2)/x^3+b^2*erfi(b*x)/exp(b^2*x^2)/x,x, algorithm="fricas")

[Out]

-1/2*(2*sqrt(pi)*b*x*e^(b^2*x^2) + pi*erfi(b*x))*e^(-b^2*x^2)/(pi*x^2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b^{2} \operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x} + \frac {\operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x)/exp(b^2*x^2)/x^3+b^2*erfi(b*x)/exp(b^2*x^2)/x,x, algorithm="giac")

[Out]

integrate(b^2*erfi(b*x)*e^(-b^2*x^2)/x + erfi(b*x)*e^(-b^2*x^2)/x^3, x)

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maple [A]  time = 0.18, size = 41, normalized size = 1.24 \[ \frac {\left (-2 \,{\mathrm e}^{b^{2} x^{2}} b x -\sqrt {\pi }\, \erfi \left (b x \right )\right ) {\mathrm e}^{-b^{2} x^{2}}}{2 \sqrt {\pi }\, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erfi(b*x)/exp(b^2*x^2)/x^3+b^2*erfi(b*x)/exp(b^2*x^2)/x,x)

[Out]

1/2*(-2*exp(b^2*x^2)*b*x-Pi^(1/2)*erfi(b*x))/Pi^(1/2)/x^2/exp(b^2*x^2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b^{2} \operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x} + \frac {\operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x)/exp(b^2*x^2)/x^3+b^2*erfi(b*x)/exp(b^2*x^2)/x,x, algorithm="maxima")

[Out]

integrate(b^2*erfi(b*x)*e^(-b^2*x^2)/x + erfi(b*x)*e^(-b^2*x^2)/x^3, x)

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mupad [B]  time = 0.17, size = 28, normalized size = 0.85 \[ -\frac {{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfi}\left (b\,x\right )}{2\,x^2}-\frac {b}{x\,\sqrt {\pi }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-b^2*x^2)*erfi(b*x))/x^3 + (b^2*exp(-b^2*x^2)*erfi(b*x))/x,x)

[Out]

- (exp(-b^2*x^2)*erfi(b*x))/(2*x^2) - b/(x*pi^(1/2))

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sympy [A]  time = 31.16, size = 53, normalized size = 1.61 \[ \frac {2 b^{3} x {{}_{2}F_{2}\left (\begin {matrix} \frac {1}{2}, 1 \\ \frac {3}{2}, \frac {3}{2} \end {matrix}\middle | {- b^{2} x^{2}} \right )}}{\sqrt {\pi }} - \frac {2 b {{}_{2}F_{2}\left (\begin {matrix} - \frac {1}{2}, 1 \\ \frac {1}{2}, \frac {3}{2} \end {matrix}\middle | {- b^{2} x^{2}} \right )}}{\sqrt {\pi } x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfi(b*x)/exp(b**2*x**2)/x**3+b**2*erfi(b*x)/exp(b**2*x**2)/x,x)

[Out]

2*b**3*x*hyper((1/2, 1), (3/2, 3/2), -b**2*x**2)/sqrt(pi) - 2*b*hyper((-1/2, 1), (1/2, 3/2), -b**2*x**2)/(sqrt
(pi)*x)

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