\(\int \frac {\text {erfc}(b x)}{x} \, dx\) [107]

Optimal result

Integrand size = 8, antiderivative size = 35 \[ \int \frac {\text {erfc}(b x)}{x} \, dx=-\frac {2 b x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};-b^2 x^2\right )}{\sqrt {\pi }}+\log (x) \]

[Out]

ln(x)-2*b*x*hypergeom([1/2, 1/2],[3/2, 3/2],-b^2*x^2)/Pi^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6494, 6493} \[ \int \frac {\text {erfc}(b x)}{x} \, dx=\log (x)-\frac {2 b x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};-b^2 x^2\right )}{\sqrt {\pi }} \]

[In]

Int[Erfc[b*x]/x,x]

[Out]

(-2*b*x*HypergeometricPFQ[{1/2, 1/2}, {3/2, 3/2}, -(b^2*x^2)])/Sqrt[Pi] + Log[x]

Rule 6493

Int[Erf[(b_.)*(x_)]/(x_), x_Symbol] :> Simp[2*b*(x/Sqrt[Pi])*HypergeometricPFQ[{1/2, 1/2}, {3/2, 3/2}, (-b^2)*
x^2], x] /; FreeQ[b, x]

Rule 6494

Int[Erfc[(b_.)*(x_)]/(x_), x_Symbol] :> Simp[Log[x], x] - Int[Erf[b*x]/x, x] /; FreeQ[b, x]

Rubi steps \begin{align*} \text {integral}& = \log (x)-\int \frac {\text {erf}(b x)}{x} \, dx \\ & = -\frac {2 b x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};-b^2 x^2\right )}{\sqrt {\pi }}+\log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.29 \[ \int \frac {\text {erfc}(b x)}{x} \, dx=-\frac {2 b x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};-b^2 x^2\right )}{\sqrt {\pi }}+(\text {erf}(b x)+\text {erfc}(b x)) \log (x) \]

[In]

Integrate[Erfc[b*x]/x,x]

[Out]

(-2*b*x*HypergeometricPFQ[{1/2, 1/2}, {3/2, 3/2}, -(b^2*x^2)])/Sqrt[Pi] + (Erf[b*x] + Erfc[b*x])*Log[x]

Maple [F]

\[\int \frac {\operatorname {erfc}\left (b x \right )}{x}d x\]

[In]

int(erfc(b*x)/x,x)

[Out]

int(erfc(b*x)/x,x)

Fricas [F]

\[ \int \frac {\text {erfc}(b x)}{x} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right )}{x} \,d x } \]

[In]

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

[Out]

integral(-(erf(b*x) - 1)/x, x)

Sympy [A] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.03 \[ \int \frac {\text {erfc}(b x)}{x} \, dx=- \frac {2 b x {{}_{2}F_{2}\left (\begin {matrix} \frac {1}{2}, \frac {1}{2} \\ \frac {3}{2}, \frac {3}{2} \end {matrix}\middle | {- b^{2} x^{2}} \right )}}{\sqrt {\pi }} + \frac {\log {\left (b^{2} x^{2} \right )}}{2} \]

[In]

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

[Out]

-2*b*x*hyper((1/2, 1/2), (3/2, 3/2), -b**2*x**2)/sqrt(pi) + log(b**2*x**2)/2

Maxima [F]

\[ \int \frac {\text {erfc}(b x)}{x} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right )}{x} \,d x } \]

[In]

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

[Out]

integrate(erfc(b*x)/x, x)

Giac [F]

\[ \int \frac {\text {erfc}(b x)}{x} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right )}{x} \,d x } \]

[In]

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

[Out]

integrate(erfc(b*x)/x, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {erfc}(b x)}{x} \, dx=\int \frac {\mathrm {erfc}\left (b\,x\right )}{x} \,d x \]

[In]

int(erfc(b*x)/x,x)

[Out]

int(erfc(b*x)/x, x)