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

Optimal. Leaf size=87 \[ -b^2 \text{Unintegrable}\left (\frac{e^{-b^2 x^2} \text{Erf}(b x)}{x},x\right )-\frac{e^{-b^2 x^2} \text{Erf}(b x)}{2 x^2}-\sqrt{2} b^2 \text{Erf}\left (\sqrt{2} b x\right )-\frac{b e^{-2 b^2 x^2}}{\sqrt{\pi } x} \]

[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] - b^2*Unintegrab
le[Erf[b*x]/(E^(b^2*x^2)*x), x]

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Rubi [A]  time = 0.09838, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{e^{-b^2 x^2} \text{Erf}(b x)}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

Rubi steps

\begin{align*} \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}-b^2 \int \frac{e^{-b^2 x^2} \text{erf}(b x)}{x} \, dx+\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}-b^2 \int \frac{e^{-b^2 x^2} \text{erf}(b x)}{x} \, dx-\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 )-b^2 \int \frac{e^{-b^2 x^2} \text{erf}(b x)}{x} \, dx\\ \end{align*}

Mathematica [A]  time = 0.133205, size = 0, normalized size = 0. \[ \int \frac{e^{-b^2 x^2} \text{Erf}(b x)}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.315, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it Erf} \left ( bx \right ) }{{{\rm e}^{{b}^{2}{x}^{2}}}{x}^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \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,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{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,x)

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \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,x, algorithm="giac")

[Out]

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