\(\int e^{-b^2 x^2} x^6 \text {erfi}(b x) \, dx\) [276]

Optimal result

Integrand size = 18, antiderivative size = 148 \[ \int e^{-b^2 x^2} x^6 \text {erfi}(b x) \, dx=\frac {15 x^2}{8 b^5 \sqrt {\pi }}+\frac {5 x^4}{8 b^3 \sqrt {\pi }}+\frac {x^6}{6 b \sqrt {\pi }}-\frac {15 e^{-b^2 x^2} x \text {erfi}(b x)}{8 b^6}-\frac {5 e^{-b^2 x^2} x^3 \text {erfi}(b x)}{4 b^4}-\frac {e^{-b^2 x^2} x^5 \text {erfi}(b x)}{2 b^2}+\frac {15 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{8 b^5 \sqrt {\pi }} \] Output:

15/8*x^2/b^5/Pi^(1/2)+5/8*x^4/b^3/Pi^(1/2)+1/6*x^6/b/Pi^(1/2)-15/8*x*erfi( 
b*x)/b^6/exp(b^2*x^2)-5/4*x^3*erfi(b*x)/b^4/exp(b^2*x^2)-1/2*x^5*erfi(b*x) 
/b^2/exp(b^2*x^2)+15/8*x^2*hypergeom([1, 1],[3/2, 2],-b^2*x^2)/b^5/Pi^(1/2 
)
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.35 \[ \int e^{-b^2 x^2} x^6 \text {erfi}(b x) \, dx=\frac {x^2 \left (9+3 b^2 x^2+4 b^4 x^4-9 \, _2F_2\left (1,1;-\frac {3}{2},2;-b^2 x^2\right )\right )}{24 b^5 \sqrt {\pi }} \] Input:

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

Output:

(x^2*(9 + 3*b^2*x^2 + 4*b^4*x^4 - 9*HypergeometricPFQ[{1, 1}, {-3/2, 2}, - 
(b^2*x^2)]))/(24*b^5*Sqrt[Pi])
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {6941, 15, 6941, 15, 6941, 15, 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.

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

\(\Big \downarrow \) 6941

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

\(\Big \downarrow \) 15

\(\displaystyle \frac {5 \int e^{-b^2 x^2} x^4 \text {erfi}(b x)dx}{2 b^2}-\frac {x^5 e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2}+\frac {x^6}{6 \sqrt {\pi } b}\)

\(\Big \downarrow \) 6941

\(\displaystyle \frac {5 \left (\frac {3 \int e^{-b^2 x^2} x^2 \text {erfi}(b x)dx}{2 b^2}+\frac {\int x^3dx}{\sqrt {\pi } b}-\frac {x^3 e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2}\right )}{2 b^2}-\frac {x^5 e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2}+\frac {x^6}{6 \sqrt {\pi } b}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {5 \left (\frac {3 \int e^{-b^2 x^2} x^2 \text {erfi}(b x)dx}{2 b^2}-\frac {x^3 e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2}+\frac {x^4}{4 \sqrt {\pi } b}\right )}{2 b^2}-\frac {x^5 e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2}+\frac {x^6}{6 \sqrt {\pi } b}\)

\(\Big \downarrow \) 6941

\(\displaystyle \frac {5 \left (\frac {3 \left (\frac {\int e^{-b^2 x^2} \text {erfi}(b x)dx}{2 b^2}+\frac {\int xdx}{\sqrt {\pi } b}-\frac {x e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2}\right )}{2 b^2}-\frac {x^3 e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2}+\frac {x^4}{4 \sqrt {\pi } b}\right )}{2 b^2}-\frac {x^5 e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2}+\frac {x^6}{6 \sqrt {\pi } b}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {5 \left (\frac {3 \left (\frac {\int e^{-b^2 x^2} \text {erfi}(b x)dx}{2 b^2}-\frac {x e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2}+\frac {x^2}{2 \sqrt {\pi } b}\right )}{2 b^2}-\frac {x^3 e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2}+\frac {x^4}{4 \sqrt {\pi } b}\right )}{2 b^2}-\frac {x^5 e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2}+\frac {x^6}{6 \sqrt {\pi } b}\)

\(\Big \downarrow \) 6932

\(\displaystyle \frac {5 \left (\frac {3 \left (\frac {x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{2 \sqrt {\pi } b}-\frac {x e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2}+\frac {x^2}{2 \sqrt {\pi } b}\right )}{2 b^2}-\frac {x^3 e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2}+\frac {x^4}{4 \sqrt {\pi } b}\right )}{2 b^2}-\frac {x^5 e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2}+\frac {x^6}{6 \sqrt {\pi } b}\)

Input:

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

Output:

x^6/(6*b*Sqrt[Pi]) - (x^5*Erfi[b*x])/(2*b^2*E^(b^2*x^2)) + (5*(x^4/(4*b*Sq 
rt[Pi]) - (x^3*Erfi[b*x])/(2*b^2*E^(b^2*x^2)) + (3*(x^2/(2*b*Sqrt[Pi]) - ( 
x*Erfi[b*x])/(2*b^2*E^(b^2*x^2)) + (x^2*HypergeometricPFQ[{1, 1}, {3/2, 2} 
, -(b^2*x^2)])/(2*b*Sqrt[Pi])))/(2*b^2)))/(2*b^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 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 6941

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]/(2*d)), x] + (-Simp[(m - 1)/ 
(2*d)   Int[x^(m - 2)*E^(c + d*x^2)*Erfi[a + b*x], x], x] - Simp[b/(d*Sqrt[ 
Pi])   Int[x^(m - 1)*E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2), x], x]) /; Free 
Q[{a, b, c, d}, x] && IGtQ[m, 1]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int e^{-b^2 x^2} x^6 \text {erfi}(b x) \, dx=\frac {12 \,\mathrm {erf}\left (b i x \right ) b^{4} i \pi \,x^{5}+30 \,\mathrm {erf}\left (b i x \right ) b^{2} i \pi \,x^{3}+45 \,\mathrm {erf}\left (b i x \right ) i \pi x +4 \sqrt {\pi }\, e^{b^{2} x^{2}} b^{5} x^{6}+15 \sqrt {\pi }\, e^{b^{2} x^{2}} b^{3} x^{4}+45 \sqrt {\pi }\, e^{b^{2} x^{2}} b \,x^{2}-45 e^{b^{2} x^{2}} \left (\int \frac {\mathrm {erf}\left (b i x \right )}{e^{b^{2} x^{2}}}d x \right ) i \pi }{24 e^{b^{2} x^{2}} b^{6} \pi } \] Input:

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

Output:

(12*erf(b*i*x)*b**4*i*pi*x**5 + 30*erf(b*i*x)*b**2*i*pi*x**3 + 45*erf(b*i* 
x)*i*pi*x + 4*sqrt(pi)*e**(b**2*x**2)*b**5*x**6 + 15*sqrt(pi)*e**(b**2*x** 
2)*b**3*x**4 + 45*sqrt(pi)*e**(b**2*x**2)*b*x**2 - 45*e**(b**2*x**2)*int(e 
rf(b*i*x)/e**(b**2*x**2),x)*i*pi)/(24*e**(b**2*x**2)*b**6*pi)