Integrand size = 19, antiderivative size = 80 \[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^2} \, dx=-\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x}+b e^c \sqrt {\pi } \text {erfi}(b x)-\frac {2 b^3 e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{\sqrt {\pi }}-\frac {2 b e^c \log (x)}{\sqrt {\pi }} \] Output:
-exp(b^2*x^2+c)*erfc(b*x)/x+b*exp(c)*Pi^(1/2)*erfi(b*x)-2*b^3*exp(c)*x^2*h ypergeom([1, 1],[3/2, 2],b^2*x^2)/Pi^(1/2)-2*b*exp(c)*ln(x)/Pi^(1/2)
Time = 0.20 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.24 \[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^2} \, dx=-\frac {e^c \left (e^{b^2 x^2} \sqrt {\pi }-b \pi x \text {erfi}(b x)+\text {erf}(b x) \left (-e^{b^2 x^2} \sqrt {\pi }+b \pi x \text {erfi}(b x)\right )-2 b^3 x^3 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )+2 b x \log (x)\right )}{\sqrt {\pi } x} \] Input:
Integrate[(E^(c + b^2*x^2)*Erfc[b*x])/x^2,x]
Output:
-((E^c*(E^(b^2*x^2)*Sqrt[Pi] - b*Pi*x*Erfi[b*x] + Erf[b*x]*(-(E^(b^2*x^2)* Sqrt[Pi]) + b*Pi*x*Erfi[b*x]) - 2*b^3*x^3*HypergeometricPFQ[{1, 1}, {3/2, 2}, -(b^2*x^2)] + 2*b*x*Log[x]))/(Sqrt[Pi]*x))
Time = 0.44 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.11, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6946, 14, 6931, 2633, 6930}
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.
\(\displaystyle \int \frac {e^{b^2 x^2+c} \text {erfc}(b x)}{x^2} \, dx\) |
\(\Big \downarrow \) 6946 |
\(\displaystyle 2 b^2 \int e^{b^2 x^2+c} \text {erfc}(b x)dx-\frac {2 b \int \frac {e^c}{x}dx}{\sqrt {\pi }}-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{x}\) |
\(\Big \downarrow \) 14 |
\(\displaystyle 2 b^2 \int e^{b^2 x^2+c} \text {erfc}(b x)dx-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{x}-\frac {2 b e^c \log (x)}{\sqrt {\pi }}\) |
\(\Big \downarrow \) 6931 |
\(\displaystyle 2 b^2 \left (\int e^{b^2 x^2+c}dx-\int e^{b^2 x^2+c} \text {erf}(b x)dx\right )-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{x}-\frac {2 b e^c \log (x)}{\sqrt {\pi }}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle 2 b^2 \left (\frac {\sqrt {\pi } e^c \text {erfi}(b x)}{2 b}-\int e^{b^2 x^2+c} \text {erf}(b x)dx\right )-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{x}-\frac {2 b e^c \log (x)}{\sqrt {\pi }}\) |
\(\Big \downarrow \) 6930 |
\(\displaystyle 2 b^2 \left (\frac {\sqrt {\pi } e^c \text {erfi}(b x)}{2 b}-\frac {b e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{\sqrt {\pi }}\right )-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{x}-\frac {2 b e^c \log (x)}{\sqrt {\pi }}\) |
Input:
Int[(E^(c + b^2*x^2)*Erfc[b*x])/x^2,x]
Output:
-((E^(c + b^2*x^2)*Erfc[b*x])/x) + 2*b^2*((E^c*Sqrt[Pi]*Erfi[b*x])/(2*b) - (b*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/Sqrt[Pi]) - (2*b *E^c*Log[x])/Sqrt[Pi]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ F, a, b, c, d}, x] && PosQ[b]
Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(b_.)*(x_)], x_Symbol] :> Simp[b*E^c*(x^2/ Sqrt[Pi])*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2], x] /; FreeQ[{b, c, d}, x] && EqQ[d, b^2]
Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(b_.)*(x_)], x_Symbol] :> Int[E^(c + d*x^ 2), x] - Int[E^(c + d*x^2)*Erf[b*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]
\[\int \frac {{\mathrm e}^{b^{2} x^{2}+c} \operatorname {erfc}\left (b x \right )}{x^{2}}d x\]
Input:
int(exp(b^2*x^2+c)*erfc(b*x)/x^2,x)
Output:
int(exp(b^2*x^2+c)*erfc(b*x)/x^2,x)
\[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^2} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{2}} \,d x } \] Input:
integrate(exp(b^2*x^2+c)*erfc(b*x)/x^2,x, algorithm="fricas")
Output:
integral(-(erf(b*x) - 1)*e^(b^2*x^2 + c)/x^2, x)
Result contains complex when optimal does not.
Time = 10.98 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00 \[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^2} \, dx=- \frac {2 b^{3} x^{2} e^{c} {{}_{2}F_{2}\left (\begin {matrix} 1, 1 \\ 2, \frac {5}{2} \end {matrix}\middle | {b^{2} x^{2}} \right )}}{3 \sqrt {\pi }} - \frac {b e^{c} \log {\left (b^{2} x^{2} \right )}}{\sqrt {\pi }} - i \sqrt {\pi } b e^{c} \operatorname {erf}{\left (i b x \right )} - \frac {e^{c} e^{b^{2} x^{2}}}{x} \] Input:
integrate(exp(b**2*x**2+c)*erfc(b*x)/x**2,x)
Output:
-2*b**3*x**2*exp(c)*hyper((1, 1), (2, 5/2), b**2*x**2)/(3*sqrt(pi)) - b*ex p(c)*log(b**2*x**2)/sqrt(pi) - I*sqrt(pi)*b*exp(c)*erf(I*b*x) - exp(c)*exp (b**2*x**2)/x
\[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^2} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{2}} \,d x } \] Input:
integrate(exp(b^2*x^2+c)*erfc(b*x)/x^2,x, algorithm="maxima")
Output:
integrate(erfc(b*x)*e^(b^2*x^2 + c)/x^2, x)
\[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^2} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{2}} \,d x } \] Input:
integrate(exp(b^2*x^2+c)*erfc(b*x)/x^2,x, algorithm="giac")
Output:
integrate(erfc(b*x)*e^(b^2*x^2 + c)/x^2, x)
Timed out. \[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^2} \, dx=\int \frac {{\mathrm {e}}^{b^2\,x^2+c}\,\mathrm {erfc}\left (b\,x\right )}{x^2} \,d x \] Input:
int((exp(c + b^2*x^2)*erfc(b*x))/x^2,x)
Output:
int((exp(c + b^2*x^2)*erfc(b*x))/x^2, x)
\[ \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^2} \, dx=e^{c} \left (\int \frac {e^{b^{2} x^{2}}}{x^{2}}d x -\left (\int \frac {e^{b^{2} x^{2}} \mathrm {erf}\left (b x \right )}{x^{2}}d x \right )\right ) \] Input:
int(exp(b^2*x^2+c)*erfc(b*x)/x^2,x)
Output:
e**c*(int(e**(b**2*x**2)/x**2,x) - int((e**(b**2*x**2)*erf(b*x))/x**2,x))