\(\int e^{c+b^2 x^2} x^3 \text {erfi}(b x) \, dx\) [284]

Optimal result

Integrand size = 19, antiderivative size = 97 \[ \int e^{c+b^2 x^2} x^3 \text {erfi}(b x) \, dx=-\frac {e^{c+2 b^2 x^2} x}{4 b^3 \sqrt {\pi }}-\frac {e^{c+b^2 x^2} \text {erfi}(b x)}{2 b^4}+\frac {e^{c+b^2 x^2} x^2 \text {erfi}(b x)}{2 b^2}+\frac {5 e^c \text {erfi}\left (\sqrt {2} b x\right )}{8 \sqrt {2} b^4} \] Output:

-1/4*exp(2*b^2*x^2+c)*x/b^3/Pi^(1/2)-1/2*exp(b^2*x^2+c)*erfi(b*x)/b^4+1/2* 
exp(b^2*x^2+c)*x^2*erfi(b*x)/b^2+5/16*exp(c)*erfi(2^(1/2)*b*x)*2^(1/2)/b^4
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.79 \[ \int e^{c+b^2 x^2} x^3 \text {erfi}(b x) \, dx=\frac {e^c \left (-4 b e^{2 b^2 x^2} x+8 e^{b^2 x^2} \sqrt {\pi } \left (-1+b^2 x^2\right ) \text {erfi}(b x)+5 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} b x\right )\right )}{16 b^4 \sqrt {\pi }} \] Input:

Integrate[E^(c + b^2*x^2)*x^3*Erfi[b*x],x]
 

Output:

(E^c*(-4*b*E^(2*b^2*x^2)*x + 8*E^(b^2*x^2)*Sqrt[Pi]*(-1 + b^2*x^2)*Erfi[b* 
x] + 5*Sqrt[2*Pi]*Erfi[Sqrt[2]*b*x]))/(16*b^4*Sqrt[Pi])
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.41, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6941, 2641, 2633, 6938, 2633}

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.

\(\displaystyle \int x^3 e^{b^2 x^2+c} \text {erfi}(b x) \, dx\)

\(\Big \downarrow \) 6941

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

\(\Big \downarrow \) 2641

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

\(\Big \downarrow \) 2633

\(\displaystyle -\frac {\int e^{b^2 x^2+c} x \text {erfi}(b x)dx}{b^2}+\frac {x^2 e^{b^2 x^2+c} \text {erfi}(b x)}{2 b^2}-\frac {\frac {x e^{2 b^2 x^2+c}}{4 b^2}-\frac {\sqrt {\frac {\pi }{2}} e^c \text {erfi}\left (\sqrt {2} b x\right )}{8 b^3}}{\sqrt {\pi } b}\)

\(\Big \downarrow \) 6938

\(\displaystyle -\frac {\frac {e^{b^2 x^2+c} \text {erfi}(b x)}{2 b^2}-\frac {\int e^{2 b^2 x^2+c}dx}{\sqrt {\pi } b}}{b^2}+\frac {x^2 e^{b^2 x^2+c} \text {erfi}(b x)}{2 b^2}-\frac {\frac {x e^{2 b^2 x^2+c}}{4 b^2}-\frac {\sqrt {\frac {\pi }{2}} e^c \text {erfi}\left (\sqrt {2} b x\right )}{8 b^3}}{\sqrt {\pi } b}\)

\(\Big \downarrow \) 2633

\(\displaystyle \frac {x^2 e^{b^2 x^2+c} \text {erfi}(b x)}{2 b^2}-\frac {\frac {e^{b^2 x^2+c} \text {erfi}(b x)}{2 b^2}-\frac {e^c \text {erfi}\left (\sqrt {2} b x\right )}{2 \sqrt {2} b^2}}{b^2}-\frac {\frac {x e^{2 b^2 x^2+c}}{4 b^2}-\frac {\sqrt {\frac {\pi }{2}} e^c \text {erfi}\left (\sqrt {2} b x\right )}{8 b^3}}{\sqrt {\pi } b}\)

Input:

Int[E^(c + b^2*x^2)*x^3*Erfi[b*x],x]
 

Output:

(E^(c + b^2*x^2)*x^2*Erfi[b*x])/(2*b^2) - ((E^(c + b^2*x^2)*Erfi[b*x])/(2* 
b^2) - (E^c*Erfi[Sqrt[2]*b*x])/(2*Sqrt[2]*b^2))/b^2 - ((E^(c + 2*b^2*x^2)* 
x)/(4*b^2) - (E^c*Sqrt[Pi/2]*Erfi[Sqrt[2]*b*x])/(8*b^3))/(b*Sqrt[Pi])
 

Defintions of rubi rules used

rule 2633

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]
 

rule 2641

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_ 
.), x_Symbol] :> Simp[(c + d*x)^(m - n + 1)*(F^(a + b*(c + d*x)^n)/(b*d*n*L 
og[F])), x] - Simp[(m - n + 1)/(b*n*Log[F])   Int[(c + d*x)^(m - n)*F^(a + 
b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2*((m + 1)/ 
n)] && LtQ[0, (m + 1)/n, 5] && IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n 
, 0])
 

rule 6938

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(a_.) + (b_.)*(x_)]*(x_), x_Symbol] :> Si 
mp[E^(c + d*x^2)*(Erfi[a + b*x]/(2*d)), x] - Simp[b/(d*Sqrt[Pi])   Int[E^(a 
^2 + c + 2*a*b*x + (b^2 + d)*x^2), x], x] /; FreeQ[{a, b, c, d}, x]
 

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]

\[\int {\mathrm e}^{b^{2} x^{2}+c} x^{3} \operatorname {erfi}\left (b x \right )d x\]

Input:

int(exp(b^2*x^2+c)*x^3*erfi(b*x),x)
 

Output:

int(exp(b^2*x^2+c)*x^3*erfi(b*x),x)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.89 \[ \int e^{c+b^2 x^2} x^3 \text {erfi}(b x) \, dx=-\frac {4 \, \sqrt {\pi } b^{2} x e^{\left (2 \, b^{2} x^{2} + c\right )} + 5 \, \sqrt {2} \pi \sqrt {-b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {-b^{2}} x\right ) e^{c} - 8 \, {\left (\pi b^{3} x^{2} - \pi b\right )} \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{16 \, \pi b^{5}} \] Input:

integrate(exp(b^2*x^2+c)*x^3*erfi(b*x),x, algorithm="fricas")
 

Output:

-1/16*(4*sqrt(pi)*b^2*x*e^(2*b^2*x^2 + c) + 5*sqrt(2)*pi*sqrt(-b^2)*erf(sq 
rt(2)*sqrt(-b^2)*x)*e^c - 8*(pi*b^3*x^2 - pi*b)*erfi(b*x)*e^(b^2*x^2 + c)) 
/(pi*b^5)
 

Sympy [F]

\[ \int e^{c+b^2 x^2} x^3 \text {erfi}(b x) \, dx=e^{c} \int x^{3} e^{b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}\, dx \] Input:

integrate(exp(b**2*x**2+c)*x**3*erfi(b*x),x)
 

Output:

exp(c)*Integral(x**3*exp(b**2*x**2)*erfi(b*x), x)
 

Maxima [F]

\[ \int e^{c+b^2 x^2} x^3 \text {erfi}(b x) \, dx=\int { x^{3} \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} \,d x } \] Input:

integrate(exp(b^2*x^2+c)*x^3*erfi(b*x),x, algorithm="maxima")
 

Output:

integrate(x^3*erfi(b*x)*e^(b^2*x^2 + c), x)
 

Giac [F]

\[ \int e^{c+b^2 x^2} x^3 \text {erfi}(b x) \, dx=\int { x^{3} \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} \,d x } \] Input:

integrate(exp(b^2*x^2+c)*x^3*erfi(b*x),x, algorithm="giac")
 

Output:

integrate(x^3*erfi(b*x)*e^(b^2*x^2 + c), x)
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.21 \[ \int e^{c+b^2 x^2} x^3 \text {erfi}(b x) \, dx=\frac {\sqrt {2}\,{\mathrm {e}}^c\,\mathrm {erfi}\left (\sqrt {2}\,x\,\sqrt {b^2}\right )}{16\,b\,{\left (b^2\right )}^{3/2}}-\frac {x\,{\mathrm {e}}^{2\,b^2\,x^2+c}}{4\,b^3\,\sqrt {\pi }}-\mathrm {erfi}\left (b\,x\right )\,\left (\frac {{\mathrm {e}}^{b^2\,x^2+c}}{2\,b^4}-\frac {x^2\,{\mathrm {e}}^{b^2\,x^2+c}}{2\,b^2}\right )-\frac {\sqrt {2}\,\mathrm {erf}\left (\sqrt {2}\,x\,\sqrt {-b^2}\right )\,{\mathrm {e}}^c}{4\,b\,{\left (-b^2\right )}^{3/2}} \] Input:

int(x^3*exp(c + b^2*x^2)*erfi(b*x),x)
 

Output:

(2^(1/2)*exp(c)*erfi(2^(1/2)*x*(b^2)^(1/2)))/(16*b*(b^2)^(3/2)) - (x*exp(c 
 + 2*b^2*x^2))/(4*b^3*pi^(1/2)) - erfi(b*x)*(exp(c + b^2*x^2)/(2*b^4) - (x 
^2*exp(c + b^2*x^2))/(2*b^2)) - (2^(1/2)*erf(2^(1/2)*x*(-b^2)^(1/2))*exp(c 
))/(4*b*(-b^2)^(3/2))
 

Reduce [F]

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

int(exp(b^2*x^2+c)*x^3*erfi(b*x),x)
 

Output:

 - e**c*int(e**(b**2*x**2)*erf(b*i*x)*x**3,x)*i