\(\int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^5} \, dx\) [275]

Optimal result

Integrand size = 18, antiderivative size = 105 \[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^5} \, dx=-\frac {b}{6 \sqrt {\pi } x^3}+\frac {b^3}{2 \sqrt {\pi } x}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{4 x^4}+\frac {b^2 e^{-b^2 x^2} \text {erfi}(b x)}{4 x^2}+\frac {b^5 x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};-b^2 x^2\right )}{\sqrt {\pi }} \] Output:

-1/6*b/Pi^(1/2)/x^3+1/2*b^3/Pi^(1/2)/x-1/4*erfi(b*x)/exp(b^2*x^2)/x^4+1/4* 
b^2*erfi(b*x)/exp(b^2*x^2)/x^2+b^5*x*hypergeom([1/2, 1],[3/2, 3/2],-b^2*x^ 
2)/Pi^(1/2)
 

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.32 \[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^5} \, dx=-\frac {2 b \, _2F_2\left (-\frac {3}{2},1;-\frac {1}{2},\frac {3}{2};-b^2 x^2\right )}{3 \sqrt {\pi } x^3} \] Input:

Integrate[Erfi[b*x]/(E^(b^2*x^2)*x^5),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6947, 15, 6947, 15, 6944}

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.

\(\displaystyle \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^5} \, dx\)

\(\Big \downarrow \) 6947

\(\displaystyle -\frac {1}{2} b^2 \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^3}dx+\frac {b \int \frac {1}{x^4}dx}{2 \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{4 x^4}\)

\(\Big \downarrow \) 15

\(\displaystyle -\frac {1}{2} b^2 \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^3}dx-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{4 x^4}-\frac {b}{6 \sqrt {\pi } x^3}\)

\(\Big \downarrow \) 6947

\(\displaystyle -\frac {1}{2} b^2 \left (b^2 \left (-\int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x}dx\right )+\frac {b \int \frac {1}{x^2}dx}{\sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{2 x^2}\right )-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{4 x^4}-\frac {b}{6 \sqrt {\pi } x^3}\)

\(\Big \downarrow \) 15

\(\displaystyle -\frac {1}{2} b^2 \left (b^2 \left (-\int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x}dx\right )-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{2 x^2}-\frac {b}{\sqrt {\pi } x}\right )-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{4 x^4}-\frac {b}{6 \sqrt {\pi } x^3}\)

\(\Big \downarrow \) 6944

\(\displaystyle -\frac {1}{2} b^2 \left (-\frac {2 b^3 x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};-b^2 x^2\right )}{\sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{2 x^2}-\frac {b}{\sqrt {\pi } x}\right )-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{4 x^4}-\frac {b}{6 \sqrt {\pi } x^3}\)

Input:

Int[Erfi[b*x]/(E^(b^2*x^2)*x^5),x]
 

Output:

-1/6*b/(Sqrt[Pi]*x^3) - Erfi[b*x]/(4*E^(b^2*x^2)*x^4) - (b^2*(-(b/(Sqrt[Pi 
]*x)) - Erfi[b*x]/(2*E^(b^2*x^2)*x^2) - (2*b^3*x*HypergeometricPFQ[{1/2, 1 
}, {3/2, 3/2}, -(b^2*x^2)])/Sqrt[Pi]))/2
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 6944

Int[(E^((c_.) + (d_.)*(x_)^2)*Erfi[(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 6947

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] 
:> Simp[x^(m + 1)*E^(c + d*x^2)*(Erfi[a + b*x]/(m + 1)), x] + (-Simp[2*(d/( 
m + 1))   Int[x^(m + 2)*E^(c + d*x^2)*Erfi[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]
 

Maple [F]

\[\int \frac {\operatorname {erfi}\left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}}{x^{5}}d x\]

Input:

int(erfi(b*x)/exp(b^2*x^2)/x^5,x)
 

Output:

int(erfi(b*x)/exp(b^2*x^2)/x^5,x)
 

Fricas [F]

\[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^5} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{5}} \,d x } \] Input:

integrate(erfi(b*x)/exp(b^2*x^2)/x^5,x, algorithm="fricas")
 

Output:

integral(erfi(b*x)*e^(-b^2*x^2)/x^5, x)
 

Sympy [A] (verification not implemented)

Time = 63.45 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.30 \[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^5} \, dx=- \frac {2 b {{}_{2}F_{2}\left (\begin {matrix} - \frac {3}{2}, 1 \\ - \frac {1}{2}, \frac {3}{2} \end {matrix}\middle | {- b^{2} x^{2}} \right )}}{3 \sqrt {\pi } x^{3}} \] Input:

integrate(erfi(b*x)/exp(b**2*x**2)/x**5,x)
 

Output:

-2*b*hyper((-3/2, 1), (-1/2, 3/2), -b**2*x**2)/(3*sqrt(pi)*x**3)
 

Maxima [F]

\[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^5} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{5}} \,d x } \] Input:

integrate(erfi(b*x)/exp(b^2*x^2)/x^5,x, algorithm="maxima")
 

Output:

integrate(erfi(b*x)*e^(-b^2*x^2)/x^5, x)
 

Giac [F]

\[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^5} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{5}} \,d x } \] Input:

integrate(erfi(b*x)/exp(b^2*x^2)/x^5,x, algorithm="giac")
 

Output:

integrate(erfi(b*x)*e^(-b^2*x^2)/x^5, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^5} \, dx=\int \frac {{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfi}\left (b\,x\right )}{x^5} \,d x \] Input:

int((exp(-b^2*x^2)*erfi(b*x))/x^5,x)
 

Output:

int((exp(-b^2*x^2)*erfi(b*x))/x^5, x)
 

Reduce [F]

\[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^5} \, dx=\frac {-3 \,\mathrm {erf}\left (b i x \right ) b^{2} i \pi \,x^{2}+3 \,\mathrm {erf}\left (b i x \right ) i \pi +6 \sqrt {\pi }\, e^{b^{2} x^{2}} b^{3} x^{3}-2 \sqrt {\pi }\, e^{b^{2} x^{2}} b x -6 e^{b^{2} x^{2}} \left (\int \frac {\mathrm {erf}\left (b i x \right )}{e^{b^{2} x^{2}} x}d x \right ) b^{4} i \pi \,x^{4}}{12 e^{b^{2} x^{2}} \pi \,x^{4}} \] Input:

int(erfi(b*x)/exp(b^2*x^2)/x^5,x)
 

Output:

( - 3*erf(b*i*x)*b**2*i*pi*x**2 + 3*erf(b*i*x)*i*pi + 6*sqrt(pi)*e**(b**2* 
x**2)*b**3*x**3 - 2*sqrt(pi)*e**(b**2*x**2)*b*x - 6*e**(b**2*x**2)*int(erf 
(b*i*x)/(e**(b**2*x**2)*x),x)*b**4*i*pi*x**4)/(12*e**(b**2*x**2)*pi*x**4)