\(\int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^5} \, dx\) [183]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^5} \, dx=\frac {b e^{-2 b^2 x^2}}{6 \sqrt {\pi } x^3}-\frac {7 b^3 e^{-2 b^2 x^2}}{6 \sqrt {\pi } x}-\frac {b^4 \text {erf}\left (\sqrt {2} b x\right )}{\sqrt {2}}-\frac {2}{3} \sqrt {2} b^4 \text {erf}\left (\sqrt {2} b x\right )-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{4 x^4}+\frac {b^2 e^{-b^2 x^2} \text {erfc}(b x)}{4 x^2}+\frac {1}{2} b^4 \text {Int}\left (\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x},x\right ) \]

[Out]

-1/4*erfc(b*x)/exp(b^2*x^2)/x^4+1/4*b^2*erfc(b*x)/exp(b^2*x^2)/x^2-7/6*b^4*erf(b*x*2^(1/2))*2^(1/2)+1/6*b/exp(
2*b^2*x^2)/x^3/Pi^(1/2)-7/6*b^3/exp(2*b^2*x^2)/x/Pi^(1/2)+1/2*b^4*Unintegrable(erfc(b*x)/exp(b^2*x^2)/x,x)

Rubi [N/A]

Not integrable

Time = 0.13 (sec) , antiderivative size = 18, 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 = {} \[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^5} \, dx=\int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^5} \, dx \]

[In]

Int[Erfc[b*x]/(E^(b^2*x^2)*x^5),x]

[Out]

b/(6*E^(2*b^2*x^2)*Sqrt[Pi]*x^3) - (7*b^3)/(6*E^(2*b^2*x^2)*Sqrt[Pi]*x) - (b^4*Erf[Sqrt[2]*b*x])/Sqrt[2] - (2*
Sqrt[2]*b^4*Erf[Sqrt[2]*b*x])/3 - Erfc[b*x]/(4*E^(b^2*x^2)*x^4) + (b^2*Erfc[b*x])/(4*E^(b^2*x^2)*x^2) + (b^4*D
efer[Int][Erfc[b*x]/(E^(b^2*x^2)*x), x])/2

Rubi steps \begin{align*} \text {integral}& = -\frac {e^{-b^2 x^2} \text {erfc}(b x)}{4 x^4}-\frac {1}{2} b^2 \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^3} \, dx-\frac {b \int \frac {e^{-2 b^2 x^2}}{x^4} \, dx}{2 \sqrt {\pi }} \\ & = \frac {b e^{-2 b^2 x^2}}{6 \sqrt {\pi } x^3}-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{4 x^4}+\frac {b^2 e^{-b^2 x^2} \text {erfc}(b x)}{4 x^2}+\frac {1}{2} b^4 \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x} \, dx+\frac {b^3 \int \frac {e^{-2 b^2 x^2}}{x^2} \, dx}{2 \sqrt {\pi }}+\frac {\left (2 b^3\right ) \int \frac {e^{-2 b^2 x^2}}{x^2} \, dx}{3 \sqrt {\pi }} \\ & = \frac {b e^{-2 b^2 x^2}}{6 \sqrt {\pi } x^3}-\frac {7 b^3 e^{-2 b^2 x^2}}{6 \sqrt {\pi } x}-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{4 x^4}+\frac {b^2 e^{-b^2 x^2} \text {erfc}(b x)}{4 x^2}+\frac {1}{2} b^4 \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x} \, dx-\frac {\left (2 b^5\right ) \int e^{-2 b^2 x^2} \, dx}{\sqrt {\pi }}-\frac {\left (8 b^5\right ) \int e^{-2 b^2 x^2} \, dx}{3 \sqrt {\pi }} \\ & = \frac {b e^{-2 b^2 x^2}}{6 \sqrt {\pi } x^3}-\frac {7 b^3 e^{-2 b^2 x^2}}{6 \sqrt {\pi } x}-\frac {b^4 \text {erf}\left (\sqrt {2} b x\right )}{\sqrt {2}}-\frac {2}{3} \sqrt {2} b^4 \text {erf}\left (\sqrt {2} b x\right )-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{4 x^4}+\frac {b^2 e^{-b^2 x^2} \text {erfc}(b x)}{4 x^2}+\frac {1}{2} b^4 \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.22 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^5} \, dx=\int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^5} \, dx \]

[In]

Integrate[Erfc[b*x]/(E^(b^2*x^2)*x^5),x]

[Out]

Integrate[Erfc[b*x]/(E^(b^2*x^2)*x^5), x]

Maple [N/A] (verified)

Not integrable

Time = 0.22 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

\[\int \frac {\operatorname {erfc}\left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}}{x^{5}}d x\]

[In]

int(erfc(b*x)/exp(b^2*x^2)/x^5,x)

[Out]

int(erfc(b*x)/exp(b^2*x^2)/x^5,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^5} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{5}} \,d x } \]

[In]

integrate(erfc(b*x)/exp(b^2*x^2)/x^5,x, algorithm="fricas")

[Out]

integral(-(erf(b*x) - 1)*e^(-b^2*x^2)/x^5, x)

Sympy [N/A]

Not integrable

Time = 16.09 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^5} \, dx=\int \frac {e^{- b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}}{x^{5}}\, dx \]

[In]

integrate(erfc(b*x)/exp(b**2*x**2)/x**5,x)

[Out]

Integral(exp(-b**2*x**2)*erfc(b*x)/x**5, x)

Maxima [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^5} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{5}} \,d x } \]

[In]

integrate(erfc(b*x)/exp(b^2*x^2)/x^5,x, algorithm="maxima")

[Out]

integrate(erfc(b*x)*e^(-b^2*x^2)/x^5, x)

Giac [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^5} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{5}} \,d x } \]

[In]

integrate(erfc(b*x)/exp(b^2*x^2)/x^5,x, algorithm="giac")

[Out]

integrate(erfc(b*x)*e^(-b^2*x^2)/x^5, x)

Mupad [N/A]

Not integrable

Time = 4.79 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x^5} \, dx=\int \frac {{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right )}{x^5} \,d x \]

[In]

int((exp(-b^2*x^2)*erfc(b*x))/x^5,x)

[Out]

int((exp(-b^2*x^2)*erfc(b*x))/x^5, x)