3.160 \(\int \frac {e^{c+d x^2} \text {erfc}(b x)}{x^3} \, dx\)

Optimal. Leaf size=99 \[ d \text {Int}\left (\frac {\text {erfc}(b x) e^{c+d x^2}}{x},x\right )+b e^c \sqrt {b^2-d} \text {erf}\left (x \sqrt {b^2-d}\right )+\frac {b e^{c-x^2 \left (b^2-d\right )}}{\sqrt {\pi } x}-\frac {\text {erfc}(b x) e^{c+d x^2}}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.15, 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 = {} \[ \int \frac {e^{c+d x^2} \text {Erfc}(b x)}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(E^(c + d*x^2)*Erfc[b*x])/x^3,x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.78, size = 0, normalized size = 0.00 \[ \int \frac {e^{c+d x^2} \text {erfc}(b x)}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (\operatorname {erf}\left (b x\right ) - 1\right )} e^{\left (d x^{2} + c\right )}}{x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(erfc(b*x)*e^(d*x^2 + c)/x^3, x)

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maple [A]  time = 0.27, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{d \,x^{2}+c} \mathrm {erfc}\left (b x \right )}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(erfc(b*x)*e^(d*x^2 + c)/x^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(c + d*x^2)*erfc(b*x))/x^3,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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