Integrand size = 19, antiderivative size = 48 \[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x} \, dx=\frac {1}{2} e^c \operatorname {ExpIntegralEi}\left (b^2 x^2\right )-\frac {2 b e^c x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};b^2 x^2\right )}{\sqrt {\pi }} \] Output:
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)
Time = 0.11 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.94 \[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x} \, dx=\frac {1}{2} e^c \left (\operatorname {ExpIntegralEi}\left (b^2 x^2\right )-\frac {4 b x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};b^2 x^2\right )}{\sqrt {\pi }}\right ) \] Input:
Integrate[(E^(c + b^2*x^2)*Erfc[b*x])/x,x]
Output:
(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
Time = 0.38 (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 = {6943, 2639, 6942}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {e^{b^2 x^2+c} \text {erfc}(b x)}{x} \, dx\) |
\(\Big \downarrow \) 6943 |
\(\displaystyle \int \frac {e^{b^2 x^2+c}}{x}dx-\int \frac {e^{b^2 x^2+c} \text {erf}(b x)}{x}dx\) |
\(\Big \downarrow \) 2639 |
\(\displaystyle \frac {1}{2} e^c \operatorname {ExpIntegralEi}\left (b^2 x^2\right )-\int \frac {e^{b^2 x^2+c} \text {erf}(b x)}{x}dx\) |
\(\Big \downarrow \) 6942 |
\(\displaystyle \frac {1}{2} e^c \operatorname {ExpIntegralEi}\left (b^2 x^2\right )-\frac {2 b e^c x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};b^2 x^2\right )}{\sqrt {\pi }}\) |
Input:
Int[(E^(c + b^2*x^2)*Erfc[b*x])/x,x]
Output:
(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]
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] /; Free Q[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]
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] /; Fr eeQ[{b, c, d}, x] && EqQ[d, b^2]
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]
\[\int \frac {{\mathrm e}^{b^{2} x^{2}+c} \operatorname {erfc}\left (b x \right )}{x}d x\]
Input:
int(exp(b^2*x^2+c)*erfc(b*x)/x,x)
Output:
int(exp(b^2*x^2+c)*erfc(b*x)/x,x)
\[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x} \,d x } \] Input:
integrate(exp(b^2*x^2+c)*erfc(b*x)/x,x, algorithm="fricas")
Output:
integral(-(erf(b*x) - 1)*e^(b^2*x^2 + c)/x, x)
Time = 5.61 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.81 \[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x} \, dx=- \frac {2 b x e^{c} {{}_{2}F_{2}\left (\begin {matrix} \frac {1}{2}, 1 \\ \frac {3}{2}, \frac {3}{2} \end {matrix}\middle | {b^{2} x^{2}} \right )}}{\sqrt {\pi }} + \frac {e^{c} \operatorname {Ei}{\left (b^{2} x^{2} \right )}}{2} \] Input:
integrate(exp(b**2*x**2+c)*erfc(b*x)/x,x)
Output:
-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
\[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x} \,d x } \] Input:
integrate(exp(b^2*x^2+c)*erfc(b*x)/x,x, algorithm="maxima")
Output:
integrate(erfc(b*x)*e^(b^2*x^2 + c)/x, x)
\[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x} \,d x } \] Input:
integrate(exp(b^2*x^2+c)*erfc(b*x)/x,x, algorithm="giac")
Output:
integrate(erfc(b*x)*e^(b^2*x^2 + c)/x, x)
Timed out. \[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x} \, dx=\int \frac {{\mathrm {e}}^{b^2\,x^2+c}\,\mathrm {erfc}\left (b\,x\right )}{x} \,d x \] Input:
int((exp(c + b^2*x^2)*erfc(b*x))/x,x)
Output:
int((exp(c + b^2*x^2)*erfc(b*x))/x, x)
\[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x} \, dx=\frac {e^{c} \left (\mathit {ei} \left (b^{2} x^{2}\right )-2 \left (\int \frac {e^{b^{2} x^{2}} \mathrm {erf}\left (b x \right )}{x}d x \right )\right )}{2} \] Input:
int(exp(b^2*x^2+c)*erfc(b*x)/x,x)
Output:
(e**c*(ei(b**2*x**2) - 2*int((e**(b**2*x**2)*erf(b*x))/x,x)))/2