\(\int e^{-b^2 x^2} x^4 \text {erfi}(b x) \, dx\) [277]

Optimal result

Integrand size = 18, antiderivative size = 109 \[ \int e^{-b^2 x^2} x^4 \text {erfi}(b x) \, dx=\frac {3 x^2}{4 b^3 \sqrt {\pi }}+\frac {x^4}{4 b \sqrt {\pi }}-\frac {3 e^{-b^2 x^2} x \text {erfi}(b x)}{4 b^4}-\frac {e^{-b^2 x^2} x^3 \text {erfi}(b x)}{2 b^2}+\frac {3 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{4 b^3 \sqrt {\pi }} \] Output:

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.39 \[ \int e^{-b^2 x^2} x^4 \text {erfi}(b x) \, dx=\frac {x^2 \left (1+b^2 x^2-\, _2F_2\left (1,1;-\frac {1}{2},2;-b^2 x^2\right )\right )}{4 b^3 \sqrt {\pi }} \] Input:

Integrate[(x^4*Erfi[b*x])/E^(b^2*x^2),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {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.

\(\displaystyle \int x^4 e^{-b^2 x^2} \text {erfi}(b x) \, dx\)

\(\Big \downarrow \) 6941

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 6941

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 6932

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

Input:

Int[(x^4*Erfi[b*x])/E^(b^2*x^2),x]
 

Output:

x^4/(4*b*Sqrt[Pi]) - (x^3*Erfi[b*x])/(2*b^2*E^(b^2*x^2)) + (3*(x^2/(2*b*Sq 
rt[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)
 

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]

\[\int x^{4} \operatorname {erfi}\left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}d x\]

Input:

int(x^4*erfi(b*x)/exp(b^2*x^2),x)
 

Output:

int(x^4*erfi(b*x)/exp(b^2*x^2),x)
 

Fricas [F]

\[ \int e^{-b^2 x^2} x^4 \text {erfi}(b x) \, dx=\int { x^{4} \operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} \,d x } \] Input:

integrate(x^4*erfi(b*x)/exp(b^2*x^2),x, algorithm="fricas")
 

Output:

integral(x^4*erfi(b*x)*e^(-b^2*x^2), x)
 

Sympy [F(-1)]

Timed out. \[ \int e^{-b^2 x^2} x^4 \text {erfi}(b x) \, dx=\text {Timed out} \] Input:

integrate(x**4*erfi(b*x)/exp(b**2*x**2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int e^{-b^2 x^2} x^4 \text {erfi}(b x) \, dx=\int { x^{4} \operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} \,d x } \] Input:

integrate(x^4*erfi(b*x)/exp(b^2*x^2),x, algorithm="maxima")
 

Output:

integrate(x^4*erfi(b*x)*e^(-b^2*x^2), x)
 

Giac [F]

\[ \int e^{-b^2 x^2} x^4 \text {erfi}(b x) \, dx=\int { x^{4} \operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} \,d x } \] Input:

integrate(x^4*erfi(b*x)/exp(b^2*x^2),x, algorithm="giac")
 

Output:

integrate(x^4*erfi(b*x)*e^(-b^2*x^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int e^{-b^2 x^2} x^4 \text {erfi}(b x) \, dx=\int x^4\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfi}\left (b\,x\right ) \,d x \] Input:

int(x^4*exp(-b^2*x^2)*erfi(b*x),x)
 

Output:

int(x^4*exp(-b^2*x^2)*erfi(b*x), x)
 

Reduce [F]

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

int(x^4*erfi(b*x)/exp(b^2*x^2),x)
 

Output:

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