\(\int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^5} \, dx\) [80]

Optimal result

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

[Out]

-1/4*erf(b*x)/exp(b^2*x^2)/x^4+1/4*b^2*erf(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(erf(b*x)/exp(b^2*x^2)/x,x)

Rubi [N/A]

Not integrable

Time = 0.12 (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 = {} \[ \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^5} \, dx=\int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^5} \, dx \]

[In]

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

[Out]

-1/6*b/(E^(2*b^2*x^2)*Sqrt[Pi]*x^3) + (7*b^3)/(6*E^(2*b^2*x^2)*Sqrt[Pi]*x) - Erf[b*x]/(4*E^(b^2*x^2)*x^4) + (b
^2*Erf[b*x])/(4*E^(b^2*x^2)*x^2) + (b^4*Erf[Sqrt[2]*b*x])/Sqrt[2] + (2*Sqrt[2]*b^4*Erf[Sqrt[2]*b*x])/3 + (b^4*
Defer[Int][Erf[b*x]/(E^(b^2*x^2)*x), x])/2

Rubi steps \begin{align*} \text {integral}& = -\frac {e^{-b^2 x^2} \text {erf}(b x)}{4 x^4}-\frac {1}{2} b^2 \int \frac {e^{-b^2 x^2} \text {erf}(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 {erf}(b x)}{4 x^4}+\frac {b^2 e^{-b^2 x^2} \text {erf}(b x)}{4 x^2}+\frac {1}{2} b^4 \int \frac {e^{-b^2 x^2} \text {erf}(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 {erf}(b x)}{4 x^4}+\frac {b^2 e^{-b^2 x^2} \text {erf}(b x)}{4 x^2}+\frac {1}{2} b^4 \int \frac {e^{-b^2 x^2} \text {erf}(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 {e^{-b^2 x^2} \text {erf}(b x)}{4 x^4}+\frac {b^2 e^{-b^2 x^2} \text {erf}(b x)}{4 x^2}+\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 {1}{2} b^4 \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.11 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^5} \, dx=\int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^5} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.16 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^5} \, dx=\int { \frac {\operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{5}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 15.34 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^5} \, dx=\int \frac {e^{- b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{x^{5}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^5} \, dx=\int { \frac {\operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{5}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^5} \, dx=\int { \frac {\operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{5}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 5.97 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {e^{-b^2 x^2} \text {erf}(b x)}{x^5} \, dx=\int \frac {{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{x^5} \,d x \]

[In]

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

[Out]

int((exp(-b^2*x^2)*erf(b*x))/x^5, x)