\(\int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x} \, dx\) [181]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x} \, dx=\text {Int}\left (\frac {e^{-b^2 x^2} \text {erfc}(b x)}{x},x\right ) \]

[Out]

Unintegrable(erfc(b*x)/exp(b^2*x^2)/x,x)

Rubi [N/A]

Not integrable

Time = 0.02 (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 = {} \[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x} \, dx=\int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.16 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x} \, dx=\int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.07 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.71 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x} \, dx=\int \frac {e^{- b^{2} x^{2}} \operatorname {erfc}{\left (b x \right )}}{x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

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

[In]

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

[Out]

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

Giac [N/A]

Not integrable

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

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 4.71 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {e^{-b^2 x^2} \text {erfc}(b x)}{x} \, dx=\int \frac {{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfc}\left (b\,x\right )}{x} \,d x \]

[In]

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

[Out]

int((exp(-b^2*x^2)*erfc(b*x))/x, x)