\(\int e^{c+b^2 x^2} x^5 \text {erfi}(b x) \, dx\) [283]

Optimal result

Integrand size = 19, antiderivative size = 144 \[ \int e^{c+b^2 x^2} x^5 \text {erfi}(b x) \, dx=\frac {11 e^{c+2 b^2 x^2} x}{16 b^5 \sqrt {\pi }}-\frac {e^{c+2 b^2 x^2} x^3}{4 b^3 \sqrt {\pi }}+\frac {e^{c+b^2 x^2} \text {erfi}(b x)}{b^6}-\frac {e^{c+b^2 x^2} x^2 \text {erfi}(b x)}{b^4}+\frac {e^{c+b^2 x^2} x^4 \text {erfi}(b x)}{2 b^2}-\frac {43 e^c \text {erfi}\left (\sqrt {2} b x\right )}{32 \sqrt {2} b^6} \] Output:

11/16*exp(2*b^2*x^2+c)*x/b^5/Pi^(1/2)-1/4*exp(2*b^2*x^2+c)*x^3/b^3/Pi^(1/2 
)+exp(b^2*x^2+c)*erfi(b*x)/b^6-exp(b^2*x^2+c)*x^2*erfi(b*x)/b^4+1/2*exp(b^ 
2*x^2+c)*x^4*erfi(b*x)/b^2-43/64*exp(c)*erfi(2^(1/2)*b*x)*2^(1/2)/b^6
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.66 \[ \int e^{c+b^2 x^2} x^5 \text {erfi}(b x) \, dx=\frac {e^c \left (-4 b e^{2 b^2 x^2} x \left (-11+4 b^2 x^2\right )+32 e^{b^2 x^2} \sqrt {\pi } \left (2-2 b^2 x^2+b^4 x^4\right ) \text {erfi}(b x)-43 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} b x\right )\right )}{64 b^6 \sqrt {\pi }} \] Input:

Integrate[E^(c + b^2*x^2)*x^5*Erfi[b*x],x]
 

Output:

(E^c*(-4*b*E^(2*b^2*x^2)*x*(-11 + 4*b^2*x^2) + 32*E^(b^2*x^2)*Sqrt[Pi]*(2 
- 2*b^2*x^2 + b^4*x^4)*Erfi[b*x] - 43*Sqrt[2*Pi]*Erfi[Sqrt[2]*b*x]))/(64*b 
^6*Sqrt[Pi])
 

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.78, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {6941, 2641, 2641, 2633, 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.

\(\displaystyle \int x^5 e^{b^2 x^2+c} \text {erfi}(b x) \, dx\)

\(\Big \downarrow \) 6941

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

\(\Big \downarrow \) 2641

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

\(\Big \downarrow \) 2641

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

\(\Big \downarrow \) 2633

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

\(\Big \downarrow \) 6941

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

\(\Big \downarrow \) 2641

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

\(\Big \downarrow \) 2633

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

\(\Big \downarrow \) 6938

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

\(\Big \downarrow \) 2633

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

Input:

Int[E^(c + b^2*x^2)*x^5*Erfi[b*x],x]
 

Output:

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

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]

\[\int {\mathrm e}^{b^{2} x^{2}+c} x^{5} \operatorname {erfi}\left (b x \right )d x\]

Input:

int(exp(b^2*x^2+c)*x^5*erfi(b*x),x)
 

Output:

int(exp(b^2*x^2+c)*x^5*erfi(b*x),x)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.74 \[ \int e^{c+b^2 x^2} x^5 \text {erfi}(b x) \, dx=\frac {43 \, \sqrt {2} \pi \sqrt {-b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {-b^{2}} x\right ) e^{c} + 32 \, {\left (\pi b^{5} x^{4} - 2 \, \pi b^{3} x^{2} + 2 \, \pi b\right )} \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} - 4 \, \sqrt {\pi } {\left (4 \, b^{4} x^{3} - 11 \, b^{2} x\right )} e^{\left (2 \, b^{2} x^{2} + c\right )}}{64 \, \pi b^{7}} \] Input:

integrate(exp(b^2*x^2+c)*x^5*erfi(b*x),x, algorithm="fricas")
 

Output:

1/64*(43*sqrt(2)*pi*sqrt(-b^2)*erf(sqrt(2)*sqrt(-b^2)*x)*e^c + 32*(pi*b^5* 
x^4 - 2*pi*b^3*x^2 + 2*pi*b)*erfi(b*x)*e^(b^2*x^2 + c) - 4*sqrt(pi)*(4*b^4 
*x^3 - 11*b^2*x)*e^(2*b^2*x^2 + c))/(pi*b^7)
 

Sympy [F]

\[ \int e^{c+b^2 x^2} x^5 \text {erfi}(b x) \, dx=e^{c} \int x^{5} e^{b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}\, dx \] Input:

integrate(exp(b**2*x**2+c)*x**5*erfi(b*x),x)
 

Output:

exp(c)*Integral(x**5*exp(b**2*x**2)*erfi(b*x), x)
 

Maxima [F]

\[ \int e^{c+b^2 x^2} x^5 \text {erfi}(b x) \, dx=\int { x^{5} \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} \,d x } \] Input:

integrate(exp(b^2*x^2+c)*x^5*erfi(b*x),x, algorithm="maxima")
 

Output:

integrate(x^5*erfi(b*x)*e^(b^2*x^2 + c), x)
 

Giac [F]

\[ \int e^{c+b^2 x^2} x^5 \text {erfi}(b x) \, dx=\int { x^{5} \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} \,d x } \] Input:

integrate(exp(b^2*x^2+c)*x^5*erfi(b*x),x, algorithm="giac")
 

Output:

integrate(x^5*erfi(b*x)*e^(b^2*x^2 + c), x)
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 3.82 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.43 \[ \int e^{c+b^2 x^2} x^5 \text {erfi}(b x) \, dx=\mathrm {erfi}\left (b\,x\right )\,\left (\frac {{\mathrm {e}}^{b^2\,x^2+c}}{b^6}+\frac {x^4\,{\mathrm {e}}^{b^2\,x^2+c}}{2\,b^2}-\frac {x^2\,{\mathrm {e}}^{b^2\,x^2+c}}{b^4}\right )-\frac {3\,x^5\,{\mathrm {e}}^c}{8\,b\,{\left (-2\,b^2\,x^2\right )}^{5/2}}+\frac {11\,x\,{\mathrm {e}}^{2\,b^2\,x^2+c}}{16\,b^5\,\sqrt {\pi }}-\frac {x^3\,{\mathrm {e}}^{2\,b^2\,x^2+c}}{4\,b^3\,\sqrt {\pi }}-\frac {\sqrt {2}\,{\mathrm {e}}^c\,\mathrm {erfi}\left (\sqrt {2}\,x\,\sqrt {b^2}\right )}{8\,b^3\,{\left (b^2\right )}^{3/2}}+\frac {3\,x^5\,{\mathrm {e}}^c\,\mathrm {erfc}\left (\sqrt {-2\,b^2\,x^2}\right )}{8\,b\,{\left (-2\,b^2\,x^2\right )}^{5/2}}-\frac {\sqrt {2}\,\mathrm {erf}\left (\sqrt {2}\,x\,\sqrt {-b^2}\right )\,{\mathrm {e}}^c}{2\,b\,{\left (-b^2\right )}^{5/2}} \] Input:

int(x^5*exp(c + b^2*x^2)*erfi(b*x),x)
 

Output:

erfi(b*x)*(exp(c + b^2*x^2)/b^6 + (x^4*exp(c + b^2*x^2))/(2*b^2) - (x^2*ex 
p(c + b^2*x^2))/b^4) - (3*x^5*exp(c))/(8*b*(-2*b^2*x^2)^(5/2)) + (11*x*exp 
(c + 2*b^2*x^2))/(16*b^5*pi^(1/2)) - (x^3*exp(c + 2*b^2*x^2))/(4*b^3*pi^(1 
/2)) - (2^(1/2)*exp(c)*erfi(2^(1/2)*x*(b^2)^(1/2)))/(8*b^3*(b^2)^(3/2)) + 
(3*x^5*exp(c)*erfc((-2*b^2*x^2)^(1/2)))/(8*b*(-2*b^2*x^2)^(5/2)) - (2^(1/2 
)*erf(2^(1/2)*x*(-b^2)^(1/2))*exp(c))/(2*b*(-b^2)^(5/2))
 

Reduce [F]

\[ \int e^{c+b^2 x^2} x^5 \text {erfi}(b x) \, dx=-e^{c} \left (\int e^{b^{2} x^{2}} \mathrm {erf}\left (b i x \right ) x^{5}d x \right ) i \] Input:

int(exp(b^2*x^2+c)*x^5*erfi(b*x),x)
 

Output:

 - e**c*int(e**(b**2*x**2)*erf(b*i*x)*x**5,x)*i