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

Optimal. Leaf size=86 \[ -b^2 \text {Int}\left (\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x},x\right )+\sqrt {2} b^2 \text {erf}\left (\sqrt {2} b x\right )-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{2 x^2}+\frac {b e^{-2 b^2 x^2}}{\sqrt {\pi } x} \]

[Out]

-1/2*erfc(b*x)/exp(b^2*x^2)/x^2+b^2*erf(b*x*2^(1/2))*2^(1/2)+b/exp(2*b^2*x^2)/x/Pi^(1/2)-b^2*Unintegrable(erfc
(b*x)/exp(b^2*x^2)/x,x)

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Rubi [A]  time = 0.09, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {e^{-b^2 x^2} \text {Erfc}(b x)}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

b/(E^(2*b^2*x^2)*Sqrt[Pi]*x) + Sqrt[2]*b^2*Erf[Sqrt[2]*b*x] - Erfc[b*x]/(2*E^(b^2*x^2)*x^2) - b^2*Defer[Int][E
rfc[b*x]/(E^(b^2*x^2)*x), x]

Rubi steps

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

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Mathematica [A]  time = 0.30, size = 0, normalized size = 0.00 \[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (\operatorname {erf}\left (b x\right ) - 1\right )} e^{\left (-b^{2} x^{2}\right )}}{x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(erf(b*x) - 1)*e^(-b^2*x^2)/x^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.26, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {erfc}\left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right )}{x^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((exp(-b^2*x^2)*erfc(b*x))/x^3, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{- b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfc(b*x)/exp(b**2*x**2)/x**3,x)

[Out]

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

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