Integrand size = 19, antiderivative size = 134 \[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^5} \, dx=\frac {b e^c}{6 \sqrt {\pi } x^3}+\frac {b^3 e^c}{2 \sqrt {\pi } x}-\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{4 x^4}-\frac {b^2 e^{c+b^2 x^2} \text {erfc}(b x)}{4 x^2}+\frac {1}{4} b^4 e^c \operatorname {ExpIntegralEi}\left (b^2 x^2\right )-\frac {b^5 e^c x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};b^2 x^2\right )}{\sqrt {\pi }} \]
1/4*b^4*exp(c)*Ei(b^2*x^2)-1/4*exp(b^2*x^2+c)*erfc(b*x)/x^4-1/4*b^2*exp(b^ 2*x^2+c)*erfc(b*x)/x^2+1/6*b*exp(c)/x^3/Pi^(1/2)+1/2*b^3*exp(c)/x/Pi^(1/2) -b^5*exp(c)*x*hypergeom([1/2, 1],[3/2, 3/2],b^2*x^2)/Pi^(1/2)
Time = 0.17 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.62 \[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^5} \, dx=-\frac {e^c \left (3 \sqrt {\pi } \left (e^{b^2 x^2} \left (1+b^2 x^2\right )-b^4 x^4 \operatorname {ExpIntegralEi}\left (b^2 x^2\right )\right )-8 b x \, _2F_2\left (-\frac {3}{2},1;-\frac {1}{2},\frac {3}{2};b^2 x^2\right )\right )}{12 \sqrt {\pi } x^4} \]
-1/12*(E^c*(3*Sqrt[Pi]*(E^(b^2*x^2)*(1 + b^2*x^2) - b^4*x^4*ExpIntegralEi[ b^2*x^2]) - 8*b*x*HypergeometricPFQ[{-3/2, 1}, {-1/2, 3/2}, b^2*x^2]))/(Sq rt[Pi]*x^4)
Time = 0.66 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {6946, 15, 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.
\(\displaystyle \int \frac {e^{b^2 x^2+c} \text {erfc}(b x)}{x^5} \, dx\) |
\(\Big \downarrow \) 6946 |
\(\displaystyle \frac {1}{2} b^2 \int \frac {e^{b^2 x^2+c} \text {erfc}(b x)}{x^3}dx-\frac {b \int \frac {e^c}{x^4}dx}{2 \sqrt {\pi }}-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{4 x^4}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {1}{2} b^2 \int \frac {e^{b^2 x^2+c} \text {erfc}(b x)}{x^3}dx-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{4 x^4}+\frac {b e^c}{6 \sqrt {\pi } x^3}\) |
\(\Big \downarrow \) 6946 |
\(\displaystyle \frac {1}{2} b^2 \left (b^2 \int \frac {e^{b^2 x^2+c} \text {erfc}(b x)}{x}dx-\frac {b \int \frac {e^c}{x^2}dx}{\sqrt {\pi }}-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{2 x^2}\right )-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{4 x^4}+\frac {b e^c}{6 \sqrt {\pi } x^3}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {1}{2} b^2 \left (b^2 \int \frac {e^{b^2 x^2+c} \text {erfc}(b x)}{x}dx-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{2 x^2}+\frac {b e^c}{\sqrt {\pi } x}\right )-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{4 x^4}+\frac {b e^c}{6 \sqrt {\pi } x^3}\) |
\(\Big \downarrow \) 6943 |
\(\displaystyle \frac {1}{2} b^2 \left (b^2 \left (\int \frac {e^{b^2 x^2+c}}{x}dx-\int \frac {e^{b^2 x^2+c} \text {erf}(b x)}{x}dx\right )-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{2 x^2}+\frac {b e^c}{\sqrt {\pi } x}\right )-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{4 x^4}+\frac {b e^c}{6 \sqrt {\pi } x^3}\) |
\(\Big \downarrow \) 2639 |
\(\displaystyle \frac {1}{2} b^2 \left (b^2 \left (\frac {1}{2} e^c \operatorname {ExpIntegralEi}\left (b^2 x^2\right )-\int \frac {e^{b^2 x^2+c} \text {erf}(b x)}{x}dx\right )-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{2 x^2}+\frac {b e^c}{\sqrt {\pi } x}\right )-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{4 x^4}+\frac {b e^c}{6 \sqrt {\pi } x^3}\) |
\(\Big \downarrow \) 6942 |
\(\displaystyle \frac {1}{2} b^2 \left (b^2 \left (\frac {1}{2} e^c \operatorname {ExpIntegralEi}\left (b^2 x^2\right )-\frac {2 b e^c x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};b^2 x^2\right )}{\sqrt {\pi }}\right )-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{2 x^2}+\frac {b e^c}{\sqrt {\pi } x}\right )-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{4 x^4}+\frac {b e^c}{6 \sqrt {\pi } x^3}\) |
(b*E^c)/(6*Sqrt[Pi]*x^3) - (E^(c + b^2*x^2)*Erfc[b*x])/(4*x^4) + (b^2*((b* E^c)/(Sqrt[Pi]*x) - (E^(c + b^2*x^2)*Erfc[b*x])/(2*x^2) + b^2*((E^c*ExpInt egralEi[b^2*x^2])/2 - (2*b*E^c*x*HypergeometricPFQ[{1/2, 1}, {3/2, 3/2}, b ^2*x^2])/Sqrt[Pi])))/2
3.2.72.3.1 Defintions of rubi rules used
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]
\[\int \frac {{\mathrm e}^{b^{2} x^{2}+c} \operatorname {erfc}\left (b x \right )}{x^{5}}d x\]
\[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^5} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{5}} \,d x } \]
Timed out. \[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^5} \, dx=\text {Timed out} \]
\[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^5} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{5}} \,d x } \]
\[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^5} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{5}} \,d x } \]
Timed out. \[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^5} \, dx=\int \frac {{\mathrm {e}}^{b^2\,x^2+c}\,\mathrm {erfc}\left (b\,x\right )}{x^5} \,d x \]