\(\int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^6} \, dx\) [282]

Optimal result

Integrand size = 18, antiderivative size = 144 \[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^6} \, dx=-\frac {b}{10 \sqrt {\pi } x^4}+\frac {2 b^3}{15 \sqrt {\pi } x^2}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{5 x^5}+\frac {2 b^2 e^{-b^2 x^2} \text {erfi}(b x)}{15 x^3}-\frac {4 b^4 e^{-b^2 x^2} \text {erfi}(b x)}{15 x}-\frac {8 b^7 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{15 \sqrt {\pi }}+\frac {8 b^5 \log (x)}{15 \sqrt {\pi }} \] Output:

-1/10*b/Pi^(1/2)/x^4+2/15*b^3/Pi^(1/2)/x^2-1/5*erfi(b*x)/exp(b^2*x^2)/x^5+ 
2/15*b^2*erfi(b*x)/exp(b^2*x^2)/x^3-4/15*b^4*erfi(b*x)/exp(b^2*x^2)/x-8/15 
*b^7*x^2*hypergeom([1, 1],[3/2, 2],-b^2*x^2)/Pi^(1/2)+8/15*b^5*ln(x)/Pi^(1 
/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.

Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.20 \[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^6} \, dx=-\frac {b G_{2,3}^{2,1}\left (b^2 x^2|\begin {array}{c} 0,3 \\ 0,2,-\frac {1}{2} \\\end {array}\right )}{2 x^4} \] Input:

Integrate[Erfi[b*x]/(E^(b^2*x^2)*x^6),x]
 

Output:

-1/2*(b*MeijerG[{{0}, {3}}, {{0, 2}, {-1/2}}, b^2*x^2])/x^4
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {6947, 15, 6947, 15, 6947, 14, 6932}

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.

\(\displaystyle \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^6} \, dx\)

\(\Big \downarrow \) 6947

\(\displaystyle -\frac {2}{5} b^2 \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^4}dx+\frac {2 b \int \frac {1}{x^5}dx}{5 \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{5 x^5}\)

\(\Big \downarrow \) 15

\(\displaystyle -\frac {2}{5} b^2 \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^4}dx-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{5 x^5}-\frac {b}{10 \sqrt {\pi } x^4}\)

\(\Big \downarrow \) 6947

\(\displaystyle -\frac {2}{5} b^2 \left (-\frac {2}{3} b^2 \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^2}dx+\frac {2 b \int \frac {1}{x^3}dx}{3 \sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{3 x^3}\right )-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{5 x^5}-\frac {b}{10 \sqrt {\pi } x^4}\)

\(\Big \downarrow \) 15

\(\displaystyle -\frac {2}{5} b^2 \left (-\frac {2}{3} b^2 \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^2}dx-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{3 x^3}-\frac {b}{3 \sqrt {\pi } x^2}\right )-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{5 x^5}-\frac {b}{10 \sqrt {\pi } x^4}\)

\(\Big \downarrow \) 6947

\(\displaystyle -\frac {2}{5} b^2 \left (-\frac {2}{3} b^2 \left (-2 b^2 \int e^{-b^2 x^2} \text {erfi}(b x)dx+\frac {2 b \int \frac {1}{x}dx}{\sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{x}\right )-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{3 x^3}-\frac {b}{3 \sqrt {\pi } x^2}\right )-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{5 x^5}-\frac {b}{10 \sqrt {\pi } x^4}\)

\(\Big \downarrow \) 14

\(\displaystyle -\frac {2}{5} b^2 \left (-\frac {2}{3} b^2 \left (-2 b^2 \int e^{-b^2 x^2} \text {erfi}(b x)dx-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{x}+\frac {2 b \log (x)}{\sqrt {\pi }}\right )-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{3 x^3}-\frac {b}{3 \sqrt {\pi } x^2}\right )-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{5 x^5}-\frac {b}{10 \sqrt {\pi } x^4}\)

\(\Big \downarrow \) 6932

\(\displaystyle -\frac {2}{5} b^2 \left (-\frac {2}{3} b^2 \left (-\frac {2 b^3 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{\sqrt {\pi }}-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{x}+\frac {2 b \log (x)}{\sqrt {\pi }}\right )-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{3 x^3}-\frac {b}{3 \sqrt {\pi } x^2}\right )-\frac {e^{-b^2 x^2} \text {erfi}(b x)}{5 x^5}-\frac {b}{10 \sqrt {\pi } x^4}\)

Input:

Int[Erfi[b*x]/(E^(b^2*x^2)*x^6),x]
 

Output:

-1/10*b/(Sqrt[Pi]*x^4) - Erfi[b*x]/(5*E^(b^2*x^2)*x^5) - (2*b^2*(-1/3*b/(S 
qrt[Pi]*x^2) - Erfi[b*x]/(3*E^(b^2*x^2)*x^3) - (2*b^2*(-(Erfi[b*x]/(E^(b^2 
*x^2)*x)) - (2*b^3*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, -(b^2*x^2)])/Sq 
rt[Pi] + (2*b*Log[x])/Sqrt[Pi]))/3))/5
 

Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[x], x] /; FreeQ[a, x]
 

rule 15

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

rule 6932

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

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]

\[\int \frac {\operatorname {erfi}\left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}}{x^{6}}d x\]

Input:

int(erfi(b*x)/exp(b^2*x^2)/x^6,x)
 

Output:

int(erfi(b*x)/exp(b^2*x^2)/x^6,x)
 

Fricas [F]

\[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^6} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{6}} \,d x } \] Input:

integrate(erfi(b*x)/exp(b^2*x^2)/x^6,x, algorithm="fricas")
 

Output:

integral(erfi(b*x)*e^(-b^2*x^2)/x^6, x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^6} \, dx=\text {Timed out} \] Input:

integrate(erfi(b*x)/exp(b**2*x**2)/x**6,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^6} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{6}} \,d x } \] Input:

integrate(erfi(b*x)/exp(b^2*x^2)/x^6,x, algorithm="maxima")
 

Output:

integrate(erfi(b*x)*e^(-b^2*x^2)/x^6, x)
 

Giac [F]

\[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^6} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{x^{6}} \,d x } \] Input:

integrate(erfi(b*x)/exp(b^2*x^2)/x^6,x, algorithm="giac")
 

Output:

integrate(erfi(b*x)*e^(-b^2*x^2)/x^6, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{-b^2 x^2} \text {erfi}(b x)}{x^6} \, dx=\int \frac {{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfi}\left (b\,x\right )}{x^6} \,d x \] Input:

int((exp(-b^2*x^2)*erfi(b*x))/x^6,x)
 

Output:

int((exp(-b^2*x^2)*erfi(b*x))/x^6, x)
 

Reduce [F]

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

int(erfi(b*x)/exp(b^2*x^2)/x^6,x)
 

Output:

 - int(erf(b*i*x)/(e**(b**2*x**2)*x**6),x)*i