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

Optimal. Leaf size=63 \[ 2 d \text {Int}\left (\text {erfc}(b x) e^{c+d x^2},x\right )-\frac {b e^c \text {Ei}\left (-\left (\left (b^2-d\right ) x^2\right )\right )}{\sqrt {\pi }}-\frac {\text {erfc}(b x) e^{c+d x^2}}{x} \]

[Out]

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

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Rubi [A]  time = 0.11, 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 = {} \[ \int \frac {e^{c+d x^2} \text {Erfc}(b x)}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(E^(c + d*x^2)*Erfc[b*x])/x^2,x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.86, size = 0, normalized size = 0.00 \[ \int \frac {e^{c+d x^2} \text {erfc}(b x)}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (\operatorname {erf}\left (b x\right ) - 1\right )} e^{\left (d x^{2} + c\right )}}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{d \,x^{2}+c} \mathrm {erfc}\left (b x \right )}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {e}}^{d\,x^2+c}\,\mathrm {erfc}\left (b\,x\right )}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(c + d*x^2)*erfc(b*x))/x^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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