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

Optimal. Leaf size=62 \[ -\frac{e^{-b^2 x^2} \text{Erf}(b x)}{2 x^2}-\sqrt{2} b^2 \text{Erf}\left (\sqrt{2} b x\right )-\frac{b e^{-2 b^2 x^2}}{\sqrt{\pi } x} \]

[Out]

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

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Rubi [A]  time = 0.144876, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.075, Rules used = {6391, 2214, 2205} \[ -\frac{e^{-b^2 x^2} \text{Erf}(b x)}{2 x^2}-\sqrt{2} b^2 \text{Erf}\left (\sqrt{2} b x\right )-\frac{b e^{-2 b^2 x^2}}{\sqrt{\pi } x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6391

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[(x^(m + 1)*E^(c + d*x^2)*Erf
[a + b*x])/(m + 1), x] + (-Dist[(2*d)/(m + 1), Int[x^(m + 2)*E^(c + d*x^2)*Erf[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 2214

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*F^(a + b*(c + d*x)^n))/(d*(m + 1)), x] - Dist[(b*n*Log[F])/(m + 1), Int[(c + d*x)^(m + n)*F^(a + b*(c +
d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[-4, (m + 1)/n, 5] && IntegerQ[n
] && ((GtQ[n, 0] && LtQ[m, -1]) || (GtQ[-n, 0] && LeQ[-n, m + 1]))

Rule 2205

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erf[(c + d*x)*Rt[-(b*Log[F]),
2]])/(2*d*Rt[-(b*Log[F]), 2]), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rubi steps

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

Mathematica [A]  time = 0.108658, size = 62, normalized size = 1. \[ -\frac{e^{-b^2 x^2} \text{Erf}(b x)}{2 x^2}-\sqrt{2} b^2 \text{Erf}\left (\sqrt{2} b x\right )-\frac{b e^{-2 b^2 x^2}}{\sqrt{\pi } x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.304, size = 67, normalized size = 1.1 \begin{align*}{\frac{1}{b} \left ( -{\frac{{\it Erf} \left ( bx \right ) b}{2\,{{\rm e}^{{b}^{2}{x}^{2}}}{x}^{2}}}+{\frac{{b}^{3}}{\sqrt{\pi }} \left ( -{\frac{1}{ \left ({{\rm e}^{{b}^{2}{x}^{2}}} \right ) ^{2}bx}}-\sqrt{2}\sqrt{\pi }{\it Erf} \left ( bx\sqrt{2} \right ) \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-1/2*erf(b*x)*b/exp(b^2*x^2)/x^2+1/Pi^(1/2)*b^3*(-1/exp(b^2*x^2)^2/b/x-2^(1/2)*Pi^(1/2)*erf(b*x*2^(1/2))))/b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{-\frac{\sqrt{2} \sqrt{b^{2} x^{2}} b \Gamma \left (-\frac{1}{2}, 2 \, b^{2} x^{2}\right )}{2 \, x}}{\sqrt{\pi }} - \frac{\operatorname{erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*integrate(e^(-2*b^2*x^2)/x^2, x)/sqrt(pi) - 1/2*erf(b*x)*e^(-b^2*x^2)/x^2

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Fricas [A]  time = 2.59963, size = 176, normalized size = 2.84 \begin{align*} -\frac{2 \, \sqrt{2} \pi \sqrt{b^{2}} b x^{2} \operatorname{erf}\left (\sqrt{2} \sqrt{b^{2}} x\right ) + 2 \, \sqrt{\pi } b x e^{\left (-2 \, b^{2} x^{2}\right )} + \pi \operatorname{erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{2 \, \pi x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (b^{2} x^{2} + 1\right ) e^{- b^{2} x^{2}} \operatorname{erf}{\left (b x \right )}}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((b**2*x**2 + 1)*exp(-b**2*x**2)*erf(b*x)/x**3, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b^{2} \operatorname{erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x} + \frac{\operatorname{erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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