Integrand size = 19, antiderivative size = 88 \[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^3} \, dx=\frac {b e^c}{\sqrt {\pi } x}-\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{2 x^2}+\frac {1}{2} b^2 e^c \operatorname {ExpIntegralEi}\left (b^2 x^2\right )-\frac {2 b^3 e^c x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};b^2 x^2\right )}{\sqrt {\pi }} \] Output:
b*exp(c)/Pi^(1/2)/x-1/2*exp(b^2*x^2+c)*erfc(b*x)/x^2+1/2*b^2*exp(c)*Ei(b^2 *x^2)-2*b^3*exp(c)*x*hypergeom([1/2, 1],[3/2, 3/2],b^2*x^2)/Pi^(1/2)
Time = 0.15 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.74 \[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^3} \, dx=-\frac {e^c \left (e^{b^2 x^2}-b^2 x^2 \operatorname {ExpIntegralEi}\left (b^2 x^2\right )-\frac {4 b x \, _2F_2\left (-\frac {1}{2},1;\frac {1}{2},\frac {3}{2};b^2 x^2\right )}{\sqrt {\pi }}\right )}{2 x^2} \] Input:
Integrate[(E^(c + b^2*x^2)*Erfc[b*x])/x^3,x]
Output:
-1/2*(E^c*(E^(b^2*x^2) - b^2*x^2*ExpIntegralEi[b^2*x^2] - (4*b*x*Hypergeom etricPFQ[{-1/2, 1}, {1/2, 3/2}, b^2*x^2])/Sqrt[Pi]))/x^2
Time = 0.52 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6946, 15, 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.
\(\displaystyle \int \frac {e^{b^2 x^2+c} \text {erfc}(b x)}{x^3} \, dx\) |
\(\Big \downarrow \) 6946 |
\(\displaystyle b^2 \int \frac {e^{b^2 x^2+c} \text {erfc}(b x)}{x}dx-\frac {b \int \frac {e^c}{x^2}dx}{\sqrt {\pi }}-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{2 x^2}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle b^2 \int \frac {e^{b^2 x^2+c} \text {erfc}(b x)}{x}dx-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{2 x^2}+\frac {b e^c}{\sqrt {\pi } x}\) |
\(\Big \downarrow \) 6943 |
\(\displaystyle b^2 \left (\int \frac {e^{b^2 x^2+c}}{x}dx-\int \frac {e^{b^2 x^2+c} \text {erf}(b x)}{x}dx\right )-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{2 x^2}+\frac {b e^c}{\sqrt {\pi } x}\) |
\(\Big \downarrow \) 2639 |
\(\displaystyle b^2 \left (\frac {1}{2} e^c \operatorname {ExpIntegralEi}\left (b^2 x^2\right )-\int \frac {e^{b^2 x^2+c} \text {erf}(b x)}{x}dx\right )-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{2 x^2}+\frac {b e^c}{\sqrt {\pi } x}\) |
\(\Big \downarrow \) 6942 |
\(\displaystyle b^2 \left (\frac {1}{2} e^c \operatorname {ExpIntegralEi}\left (b^2 x^2\right )-\frac {2 b e^c x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};b^2 x^2\right )}{\sqrt {\pi }}\right )-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{2 x^2}+\frac {b e^c}{\sqrt {\pi } x}\) |
Input:
Int[(E^(c + b^2*x^2)*Erfc[b*x])/x^3,x]
Output:
(b*E^c)/(Sqrt[Pi]*x) - (E^(c + b^2*x^2)*Erfc[b*x])/(2*x^2) + b^2*((E^c*Exp IntegralEi[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[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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[E^((c_.) + (d_.)*(x_)^2)*Erfc[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[x^(m + 1)*E^(c + d*x^2)*(Erfc[a + b*x]/(m + 1)), x] + (-Simp[2*(d/( m + 1)) Int[x^(m + 2)*E^(c + d*x^2)*Erfc[a + b*x], x], x] + Simp[2*(b/((m + 1)*Sqrt[Pi])) Int[x^(m + 1)*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2), x] , x]) /; FreeQ[{a, b, c, d}, x] && ILtQ[m, -1]
\[\int \frac {{\mathrm e}^{b^{2} x^{2}+c} \operatorname {erfc}\left (b x \right )}{x^{3}}d x\]
Input:
int(exp(b^2*x^2+c)*erfc(b*x)/x^3,x)
Output:
int(exp(b^2*x^2+c)*erfc(b*x)/x^3,x)
\[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^3} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{3}} \,d x } \] Input:
integrate(exp(b^2*x^2+c)*erfc(b*x)/x^3,x, algorithm="fricas")
Output:
integral(-(erf(b*x) - 1)*e^(b^2*x^2 + c)/x^3, x)
Time = 30.80 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.69 \[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^3} \, dx=\frac {b^{2} e^{c} \operatorname {Ei}{\left (b^{2} x^{2} \right )}}{2} + \frac {2 b e^{c} {{}_{2}F_{2}\left (\begin {matrix} - \frac {1}{2}, 1 \\ \frac {1}{2}, \frac {3}{2} \end {matrix}\middle | {b^{2} x^{2}} \right )}}{\sqrt {\pi } x} - \frac {e^{c} e^{b^{2} x^{2}}}{2 x^{2}} \] Input:
integrate(exp(b**2*x**2+c)*erfc(b*x)/x**3,x)
Output:
b**2*exp(c)*Ei(b**2*x**2)/2 + 2*b*exp(c)*hyper((-1/2, 1), (1/2, 3/2), b**2 *x**2)/(sqrt(pi)*x) - exp(c)*exp(b**2*x**2)/(2*x**2)
\[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^3} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{3}} \,d x } \] Input:
integrate(exp(b^2*x^2+c)*erfc(b*x)/x^3,x, algorithm="maxima")
Output:
integrate(erfc(b*x)*e^(b^2*x^2 + c)/x^3, x)
\[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^3} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{3}} \,d x } \] Input:
integrate(exp(b^2*x^2+c)*erfc(b*x)/x^3,x, algorithm="giac")
Output:
integrate(erfc(b*x)*e^(b^2*x^2 + c)/x^3, x)
Timed out. \[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^3} \, dx=\int \frac {{\mathrm {e}}^{b^2\,x^2+c}\,\mathrm {erfc}\left (b\,x\right )}{x^3} \,d x \] Input:
int((exp(c + b^2*x^2)*erfc(b*x))/x^3,x)
Output:
int((exp(c + b^2*x^2)*erfc(b*x))/x^3, x)
\[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^3} \, dx=\frac {e^{c} \left (\mathit {ei} \left (b^{2} x^{2}\right ) b^{2} x^{2}-e^{b^{2} x^{2}}-2 \left (\int \frac {e^{b^{2} x^{2}} \mathrm {erf}\left (b x \right )}{x^{3}}d x \right ) x^{2}\right )}{2 x^{2}} \] Input:
int(exp(b^2*x^2+c)*erfc(b*x)/x^3,x)
Output:
(e**c*(ei(b**2*x**2)*b**2*x**2 - e**(b**2*x**2) - 2*int((e**(b**2*x**2)*er f(b*x))/x**3,x)*x**2))/(2*x**2)