\(\int \frac {e^{c+d x^2} \text {erfc}(b x)}{x^4} \, dx\) [166]

Optimal result

Integrand size = 17, antiderivative size = 17 \[ \int \frac {e^{c+d x^2} \text {erfc}(b x)}{x^4} \, dx=\frac {b e^{c-\left (b^2-d\right ) x^2}}{3 \sqrt {\pi } x^2}-\frac {e^{c+d x^2} \text {erfc}(b x)}{3 x^3}-\frac {2 d e^{c+d x^2} \text {erfc}(b x)}{3 x}+\frac {b \left (b^2-d\right ) e^c \operatorname {ExpIntegralEi}\left (-\left (\left (b^2-d\right ) x^2\right )\right )}{3 \sqrt {\pi }}-\frac {2 b d e^c \operatorname {ExpIntegralEi}\left (-\left (\left (b^2-d\right ) x^2\right )\right )}{3 \sqrt {\pi }}+\frac {4}{3} d^2 \text {Int}\left (e^{c+d x^2} \text {erfc}(b x),x\right ) \]

[Out]

-1/3*exp(d*x^2+c)*erfc(b*x)/x^3-2/3*d*exp(d*x^2+c)*erfc(b*x)/x+1/3*b*exp(c-(b^2-d)*x^2)/x^2/Pi^(1/2)+1/3*b*(b^
2-d)*exp(c)*Ei(-(b^2-d)*x^2)/Pi^(1/2)-2/3*b*d*exp(c)*Ei(-(b^2-d)*x^2)/Pi^(1/2)+4/3*d^2*Unintegrable(exp(d*x^2+
c)*erfc(b*x),x)

Rubi [N/A]

Not integrable

Time = 0.19 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {e^{c+d x^2} \text {erfc}(b x)}{x^4} \, dx=\int \frac {e^{c+d x^2} \text {erfc}(b x)}{x^4} \, dx \]

[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*} \text {integral}& = -\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 \operatorname {ExpIntegralEi}\left (-\left (\left (b^2-d\right ) x^2\right )\right )}{3 \sqrt {\pi }}-\frac {2 b d e^c \operatorname {ExpIntegralEi}\left (-\left (\left (b^2-d\right ) x^2\right )\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 [N/A]

Not integrable

Time = 0.59 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {e^{c+d x^2} \text {erfc}(b x)}{x^4} \, dx=\int \frac {e^{c+d x^2} \text {erfc}(b x)}{x^4} \, dx \]

[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]

Maple [N/A] (verified)

Not integrable

Time = 0.19 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94

\[\int \frac {{\mathrm e}^{d \,x^{2}+c} \operatorname {erfc}\left (b x \right )}{x^{4}}d x\]

[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)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.24 \[ \int \frac {e^{c+d x^2} \text {erfc}(b x)}{x^4} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{x^{4}} \,d x } \]

[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)

Sympy [N/A]

Not integrable

Time = 13.98 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {e^{c+d x^2} \text {erfc}(b x)}{x^4} \, dx=e^{c} \int \frac {e^{d x^{2}} \operatorname {erfc}{\left (b x \right )}}{x^{4}}\, dx \]

[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)

Maxima [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \frac {e^{c+d x^2} \text {erfc}(b x)}{x^4} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{x^{4}} \,d x } \]

[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)

Giac [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \frac {e^{c+d x^2} \text {erfc}(b x)}{x^4} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{x^{4}} \,d x } \]

[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)

Mupad [N/A]

Not integrable

Time = 4.93 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \frac {e^{c+d x^2} \text {erfc}(b x)}{x^4} \, dx=\int \frac {{\mathrm {e}}^{d\,x^2+c}\,\mathrm {erfc}\left (b\,x\right )}{x^4} \,d x \]

[In]

int((exp(c + d*x^2)*erfc(b*x))/x^4,x)

[Out]

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