\(\int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x} \, dx\) [170]

Optimal result

Integrand size = 19, antiderivative size = 48 \[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x} \, dx=\frac {1}{2} e^c \operatorname {ExpIntegralEi}\left (b^2 x^2\right )-\frac {2 b e^c x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};b^2 x^2\right )}{\sqrt {\pi }} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {6524, 2241, 6523} \[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x} \, dx=\frac {1}{2} e^c \operatorname {ExpIntegralEi}\left (b^2 x^2\right )-\frac {2 b e^c x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};b^2 x^2\right )}{\sqrt {\pi }} \]

[In]

Int[(E^(c + b^2*x^2)*Erfc[b*x])/x,x]

[Out]

(E^c*ExpIntegralEi[b^2*x^2])/2 - (2*b*E^c*x*HypergeometricPFQ[{1/2, 1}, {3/2, 3/2}, b^2*x^2])/Sqrt[Pi]

Rule 2241

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Simp[F^a*(ExpIntegralEi[
b*(c + d*x)^n*Log[F]]/(f*n)), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]

Rule 6523

Int[(E^((c_.) + (d_.)*(x_)^2)*Erf[(b_.)*(x_)])/(x_), x_Symbol] :> Simp[2*b*E^c*(x/Sqrt[Pi])*HypergeometricPFQ[
{1/2, 1}, {3/2, 3/2}, b^2*x^2], x] /; FreeQ[{b, c, d}, x] && EqQ[d, b^2]

Rule 6524

Int[(E^((c_.) + (d_.)*(x_)^2)*Erfc[(b_.)*(x_)])/(x_), x_Symbol] :> Int[E^(c + d*x^2)/x, x] - Int[E^(c + d*x^2)
*(Erf[b*x]/x), x] /; FreeQ[{b, c, d}, x] && EqQ[d, b^2]

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.94 \[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x} \, dx=\frac {1}{2} e^c \left (\operatorname {ExpIntegralEi}\left (b^2 x^2\right )-\frac {4 b x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};b^2 x^2\right )}{\sqrt {\pi }}\right ) \]

[In]

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

[Out]

(E^c*(ExpIntegralEi[b^2*x^2] - (4*b*x*HypergeometricPFQ[{1/2, 1}, {3/2, 3/2}, b^2*x^2])/Sqrt[Pi]))/2

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 5.23 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.81 \[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x} \, dx=- \frac {2 b x e^{c} {{}_{2}F_{2}\left (\begin {matrix} \frac {1}{2}, 1 \\ \frac {3}{2}, \frac {3}{2} \end {matrix}\middle | {b^{2} x^{2}} \right )}}{\sqrt {\pi }} + \frac {e^{c} \operatorname {Ei}{\left (b^{2} x^{2} \right )}}{2} \]

[In]

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

[Out]

-2*b*x*exp(c)*hyper((1/2, 1), (3/2, 3/2), b**2*x**2)/sqrt(pi) + exp(c)*Ei(b**2*x**2)/2

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x} \, dx=\int \frac {{\mathrm {e}}^{b^2\,x^2+c}\,\mathrm {erfc}\left (b\,x\right )}{x} \,d x \]

[In]

int((exp(c + b^2*x^2)*erfc(b*x))/x,x)

[Out]

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