\(\int e^{c+d x^2} x^3 \text {erfi}(b x) \, dx\) [260]

Optimal result

Integrand size = 17, antiderivative size = 142 \[ \int e^{c+d x^2} x^3 \text {erfi}(b x) \, dx=-\frac {b e^{c+\left (b^2+d\right ) x^2} x}{2 d \left (b^2+d\right ) \sqrt {\pi }}-\frac {e^{c+d x^2} \text {erfi}(b x)}{2 d^2}+\frac {e^{c+d x^2} x^2 \text {erfi}(b x)}{2 d}+\frac {b e^c \text {erfi}\left (\sqrt {b^2+d} x\right )}{4 d \left (b^2+d\right )^{3/2}}+\frac {b e^c \text {erfi}\left (\sqrt {b^2+d} x\right )}{2 d^2 \sqrt {b^2+d}} \] Output:

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.64 \[ \int e^{c+d x^2} x^3 \text {erfi}(b x) \, dx=\frac {e^c \left (-\frac {2 b d e^{\left (b^2+d\right ) x^2} x}{\left (b^2+d\right ) \sqrt {\pi }}+2 e^{d x^2} \left (-1+d x^2\right ) \text {erfi}(b x)+\frac {\left (2 b^3+3 b d\right ) \text {erfi}\left (\sqrt {b^2+d} x\right )}{\left (b^2+d\right )^{3/2}}\right )}{4 d^2} \] Input:

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

Output:

(E^c*((-2*b*d*E^((b^2 + d)*x^2)*x)/((b^2 + d)*Sqrt[Pi]) + 2*E^(d*x^2)*(-1 
+ d*x^2)*Erfi[b*x] + ((2*b^3 + 3*b*d)*Erfi[Sqrt[b^2 + d]*x])/(b^2 + d)^(3/ 
2)))/(4*d^2)
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, 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.

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

\(\Big \downarrow \) 6941

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

\(\Big \downarrow \) 2641

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

\(\Big \downarrow \) 2633

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

\(\Big \downarrow \) 6938

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

\(\Big \downarrow \) 2633

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.06 \[ \int e^{c+d x^2} x^3 \text {erfi}(b x) \, dx=-\frac {\pi {\left (2 \, b^{3} + 3 \, b d\right )} \sqrt {-b^{2} - d} \operatorname {erf}\left (\sqrt {-b^{2} - d} x\right ) e^{c} + 2 \, \sqrt {\pi } {\left (b^{3} d + b d^{2}\right )} x e^{\left (b^{2} x^{2} + d x^{2} + c\right )} - 2 \, {\left (\pi {\left (b^{4} d + 2 \, b^{2} d^{2} + d^{3}\right )} x^{2} - \pi {\left (b^{4} + 2 \, b^{2} d + d^{2}\right )}\right )} \operatorname {erfi}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{4 \, \pi {\left (b^{4} d^{2} + 2 \, b^{2} d^{3} + d^{4}\right )}} \] Input:

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

Output:

-1/4*(pi*(2*b^3 + 3*b*d)*sqrt(-b^2 - d)*erf(sqrt(-b^2 - d)*x)*e^c + 2*sqrt 
(pi)*(b^3*d + b*d^2)*x*e^(b^2*x^2 + d*x^2 + c) - 2*(pi*(b^4*d + 2*b^2*d^2 
+ d^3)*x^2 - pi*(b^4 + 2*b^2*d + d^2))*erfi(b*x)*e^(d*x^2 + c))/(pi*(b^4*d 
^2 + 2*b^2*d^3 + d^4))
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.53 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.90 \[ \int e^{c+d x^2} x^3 \text {erfi}(b x) \, dx=\frac {b\,\mathrm {erfi}\left (x\,\sqrt {b^2+d}\right )\,{\mathrm {e}}^c}{4\,d\,{\left (b^2+d\right )}^{3/2}}-\mathrm {erfi}\left (b\,x\right )\,\left (\frac {{\mathrm {e}}^{d\,x^2+c}}{2\,d^2}-\frac {x^2\,{\mathrm {e}}^{d\,x^2+c}}{2\,d}\right )-\frac {b\,x\,{\mathrm {e}}^{b^2\,x^2+d\,x^2+c}}{2\,\sqrt {\pi }\,\left (b^2\,d+d^2\right )}+\frac {b\,{\mathrm {e}}^c\,\mathrm {erf}\left (x\,\sqrt {-b^2-d}\right )}{2\,d^2\,\sqrt {-b^2-d}} \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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