3.166 \(\int \frac{e^{c+d x^2} \text{Erfc}(b x)}{x^4} \, dx\)

Optimal. Leaf size=154 \[ \frac{4}{3} d^2 \text{Unintegrable}\left (\text{Erfc}(b x) e^{c+d x^2},x\right )-\frac{2 b e^c d \text{ExpIntegralEi}\left (x^2 \left (-\left (b^2-d\right )\right )\right )}{3 \sqrt{\pi }}+\frac{b e^c \left (b^2-d\right ) \text{ExpIntegralEi}\left (x^2 \left (-\left (b^2-d\right )\right )\right )}{3 \sqrt{\pi }}+\frac{b e^{c-x^2 \left (b^2-d\right )}}{3 \sqrt{\pi } x^2}-\frac{2 d \text{Erfc}(b x) e^{c+d x^2}}{3 x}-\frac{\text{Erfc}(b x) e^{c+d x^2}}{3 x^3} \]

[Out]

(b*E^(c - (b^2 - d)*x^2))/(3*Sqrt[Pi]*x^2) - (E^(c + d*x^2)*Erfc[b*x])/(3*x^3) - (2*d*E^(c + d*x^2)*Erfc[b*x])
/(3*x) + (b*(b^2 - d)*E^c*ExpIntegralEi[-((b^2 - d)*x^2)])/(3*Sqrt[Pi]) - (2*b*d*E^c*ExpIntegralEi[-((b^2 - d)
*x^2)])/(3*Sqrt[Pi]) + (4*d^2*Unintegrable[E^(c + d*x^2)*Erfc[b*x], x])/3

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Rubi [A]  time = 0.268319, 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 = {} \[ \int \frac{e^{c+d x^2} \text{Erfc}(b x)}{x^4} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(E^(c + d*x^2)*Erfc[b*x])/x^4,x]

[Out]

(b*E^(c - (b^2 - d)*x^2))/(3*Sqrt[Pi]*x^2) - (E^(c + d*x^2)*Erfc[b*x])/(3*x^3) - (2*d*E^(c + d*x^2)*Erfc[b*x])
/(3*x) + (b*(b^2 - d)*E^c*ExpIntegralEi[-((b^2 - d)*x^2)])/(3*Sqrt[Pi]) - (2*b*d*E^c*ExpIntegralEi[-((b^2 - d)
*x^2)])/(3*Sqrt[Pi]) + (4*d^2*Defer[Int][E^(c + d*x^2)*Erfc[b*x], x])/3

Rubi steps

\begin{align*} \int \frac{e^{c+d x^2} \text{erfc}(b x)}{x^4} \, dx &=-\frac{e^{c+d x^2} \text{erfc}(b x)}{3 x^3}+\frac{1}{3} (2 d) \int \frac{e^{c+d x^2} \text{erfc}(b x)}{x^2} \, dx-\frac{(2 b) \int \frac{e^{c-\left (b^2-d\right ) x^2}}{x^3} \, dx}{3 \sqrt{\pi }}\\ &=\frac{b e^{c-\left (b^2-d\right ) x^2}}{3 \sqrt{\pi } x^2}-\frac{e^{c+d x^2} \text{erfc}(b x)}{3 x^3}-\frac{2 d e^{c+d x^2} \text{erfc}(b x)}{3 x}+\frac{1}{3} \left (4 d^2\right ) \int e^{c+d x^2} \text{erfc}(b x) \, dx+\frac{\left (2 b \left (b^2-d\right )\right ) \int \frac{e^{c+\left (-b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt{\pi }}-\frac{(4 b d) \int \frac{e^{c-\left (b^2-d\right ) x^2}}{x} \, dx}{3 \sqrt{\pi }}\\ &=\frac{b e^{c-\left (b^2-d\right ) x^2}}{3 \sqrt{\pi } x^2}-\frac{e^{c+d x^2} \text{erfc}(b x)}{3 x^3}-\frac{2 d e^{c+d x^2} \text{erfc}(b x)}{3 x}+\frac{b \left (b^2-d\right ) e^c \text{Ei}\left (-\left (b^2-d\right ) x^2\right )}{3 \sqrt{\pi }}-\frac{2 b d e^c \text{Ei}\left (-\left (b^2-d\right ) x^2\right )}{3 \sqrt{\pi }}+\frac{1}{3} \left (4 d^2\right ) \int e^{c+d x^2} \text{erfc}(b x) \, dx\\ \end{align*}

Mathematica [A]  time = 0.803698, size = 0, normalized size = 0. \[ \int \frac{e^{c+d x^2} \text{Erfc}(b x)}{x^4} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(E^(c + d*x^2)*Erfc[b*x])/x^4,x]

[Out]

Integrate[(E^(c + d*x^2)*Erfc[b*x])/x^4, x]

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Maple [A]  time = 0.309, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{d{x}^{2}+c}}{\it erfc} \left ( bx \right ) }{{x}^{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(d*x^2+c)*erfc(b*x)/x^4,x)

[Out]

int(exp(d*x^2+c)*erfc(b*x)/x^4,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfc}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(d*x^2+c)*erfc(b*x)/x^4,x, algorithm="maxima")

[Out]

integrate(erfc(b*x)*e^(d*x^2 + c)/x^4, x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (\operatorname{erf}\left (b x\right ) - 1\right )} e^{\left (d x^{2} + c\right )}}{x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(d*x^2+c)*erfc(b*x)/x^4,x, algorithm="fricas")

[Out]

integral(-(erf(b*x) - 1)*e^(d*x^2 + c)/x^4, x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} e^{c} \int \frac{e^{d x^{2}} \operatorname{erfc}{\left (b x \right )}}{x^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(d*x**2+c)*erfc(b*x)/x**4,x)

[Out]

exp(c)*Integral(exp(d*x**2)*erfc(b*x)/x**4, x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfc}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(d*x^2+c)*erfc(b*x)/x^4,x, algorithm="giac")

[Out]

integrate(erfc(b*x)*e^(d*x^2 + c)/x^4, x)