3.3.28 \(\int x^5 \text {erfi}(b x)^2 \, dx\) [228]

3.3.28.1 Optimal result

Integrand size = 10, antiderivative size = 175 \[ \int x^5 \text {erfi}(b x)^2 \, dx=\frac {11 e^{2 b^2 x^2}}{12 b^6 \pi }-\frac {7 e^{2 b^2 x^2} x^2}{12 b^4 \pi }+\frac {e^{2 b^2 x^2} x^4}{6 b^2 \pi }-\frac {5 e^{b^2 x^2} x \text {erfi}(b x)}{4 b^5 \sqrt {\pi }}+\frac {5 e^{b^2 x^2} x^3 \text {erfi}(b x)}{6 b^3 \sqrt {\pi }}-\frac {e^{b^2 x^2} x^5 \text {erfi}(b x)}{3 b \sqrt {\pi }}+\frac {5 \text {erfi}(b x)^2}{16 b^6}+\frac {1}{6} x^6 \text {erfi}(b x)^2 \]

output

11/12*exp(2*b^2*x^2)/b^6/Pi-7/12*exp(2*b^2*x^2)*x^2/b^4/Pi+1/6*exp(2*b^2*x 
^2)*x^4/b^2/Pi+5/16*erfi(b*x)^2/b^6+1/6*x^6*erfi(b*x)^2-5/4*exp(b^2*x^2)*x 
*erfi(b*x)/b^5/Pi^(1/2)+5/6*exp(b^2*x^2)*x^3*erfi(b*x)/b^3/Pi^(1/2)-1/3*ex 
p(b^2*x^2)*x^5*erfi(b*x)/b/Pi^(1/2)
 

3.3.28.2 Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.57 \[ \int x^5 \text {erfi}(b x)^2 \, dx=\frac {4 e^{2 b^2 x^2} \left (11-7 b^2 x^2+2 b^4 x^4\right )-4 b e^{b^2 x^2} \sqrt {\pi } x \left (15-10 b^2 x^2+4 b^4 x^4\right ) \text {erfi}(b x)+\pi \left (15+8 b^6 x^6\right ) \text {erfi}(b x)^2}{48 b^6 \pi } \]

input

Integrate[x^5*Erfi[b*x]^2,x]
 

output

(4*E^(2*b^2*x^2)*(11 - 7*b^2*x^2 + 2*b^4*x^4) - 4*b*E^(b^2*x^2)*Sqrt[Pi]*x 
*(15 - 10*b^2*x^2 + 4*b^4*x^4)*Erfi[b*x] + Pi*(15 + 8*b^6*x^6)*Erfi[b*x]^2 
)/(48*b^6*Pi)
 

3.3.28.3 Rubi [A] (verified)

Time = 1.24 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.54, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.200, Rules used = {6920, 6941, 2641, 2641, 2638, 6941, 2641, 2638, 6941, 2638, 6929, 15}

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.

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

\(\Big \downarrow \) 6920

\(\displaystyle \frac {1}{6} x^6 \text {erfi}(b x)^2-\frac {2 b \int e^{b^2 x^2} x^6 \text {erfi}(b x)dx}{3 \sqrt {\pi }}\)

\(\Big \downarrow \) 6941

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

\(\Big \downarrow \) 2641

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

\(\Big \downarrow \) 2641

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

\(\Big \downarrow \) 2638

\(\displaystyle \frac {1}{6} x^6 \text {erfi}(b x)^2-\frac {2 b \left (-\frac {5 \int e^{b^2 x^2} x^4 \text {erfi}(b x)dx}{2 b^2}+\frac {x^5 e^{b^2 x^2} \text {erfi}(b x)}{2 b^2}-\frac {\frac {x^4 e^{2 b^2 x^2}}{4 b^2}-\frac {\frac {x^2 e^{2 b^2 x^2}}{4 b^2}-\frac {e^{2 b^2 x^2}}{8 b^4}}{b^2}}{\sqrt {\pi } b}\right )}{3 \sqrt {\pi }}\)

\(\Big \downarrow \) 6941

\(\displaystyle \frac {1}{6} x^6 \text {erfi}(b x)^2-\frac {2 b \left (-\frac {5 \left (-\frac {3 \int e^{b^2 x^2} x^2 \text {erfi}(b x)dx}{2 b^2}-\frac {\int e^{2 b^2 x^2} x^3dx}{\sqrt {\pi } b}+\frac {x^3 e^{b^2 x^2} \text {erfi}(b x)}{2 b^2}\right )}{2 b^2}+\frac {x^5 e^{b^2 x^2} \text {erfi}(b x)}{2 b^2}-\frac {\frac {x^4 e^{2 b^2 x^2}}{4 b^2}-\frac {\frac {x^2 e^{2 b^2 x^2}}{4 b^2}-\frac {e^{2 b^2 x^2}}{8 b^4}}{b^2}}{\sqrt {\pi } b}\right )}{3 \sqrt {\pi }}\)

\(\Big \downarrow \) 2641

\(\displaystyle \frac {1}{6} x^6 \text {erfi}(b x)^2-\frac {2 b \left (-\frac {5 \left (-\frac {3 \int e^{b^2 x^2} x^2 \text {erfi}(b x)dx}{2 b^2}-\frac {\frac {x^2 e^{2 b^2 x^2}}{4 b^2}-\frac {\int e^{2 b^2 x^2} xdx}{2 b^2}}{\sqrt {\pi } b}+\frac {x^3 e^{b^2 x^2} \text {erfi}(b x)}{2 b^2}\right )}{2 b^2}+\frac {x^5 e^{b^2 x^2} \text {erfi}(b x)}{2 b^2}-\frac {\frac {x^4 e^{2 b^2 x^2}}{4 b^2}-\frac {\frac {x^2 e^{2 b^2 x^2}}{4 b^2}-\frac {e^{2 b^2 x^2}}{8 b^4}}{b^2}}{\sqrt {\pi } b}\right )}{3 \sqrt {\pi }}\)

\(\Big \downarrow \) 2638

\(\displaystyle \frac {1}{6} x^6 \text {erfi}(b x)^2-\frac {2 b \left (-\frac {5 \left (-\frac {3 \int e^{b^2 x^2} x^2 \text {erfi}(b x)dx}{2 b^2}+\frac {x^3 e^{b^2 x^2} \text {erfi}(b x)}{2 b^2}-\frac {\frac {x^2 e^{2 b^2 x^2}}{4 b^2}-\frac {e^{2 b^2 x^2}}{8 b^4}}{\sqrt {\pi } b}\right )}{2 b^2}+\frac {x^5 e^{b^2 x^2} \text {erfi}(b x)}{2 b^2}-\frac {\frac {x^4 e^{2 b^2 x^2}}{4 b^2}-\frac {\frac {x^2 e^{2 b^2 x^2}}{4 b^2}-\frac {e^{2 b^2 x^2}}{8 b^4}}{b^2}}{\sqrt {\pi } b}\right )}{3 \sqrt {\pi }}\)

\(\Big \downarrow \) 6941

\(\displaystyle \frac {1}{6} x^6 \text {erfi}(b x)^2-\frac {2 b \left (-\frac {5 \left (-\frac {3 \left (-\frac {\int e^{b^2 x^2} \text {erfi}(b x)dx}{2 b^2}-\frac {\int e^{2 b^2 x^2} xdx}{\sqrt {\pi } b}+\frac {x e^{b^2 x^2} \text {erfi}(b x)}{2 b^2}\right )}{2 b^2}+\frac {x^3 e^{b^2 x^2} \text {erfi}(b x)}{2 b^2}-\frac {\frac {x^2 e^{2 b^2 x^2}}{4 b^2}-\frac {e^{2 b^2 x^2}}{8 b^4}}{\sqrt {\pi } b}\right )}{2 b^2}+\frac {x^5 e^{b^2 x^2} \text {erfi}(b x)}{2 b^2}-\frac {\frac {x^4 e^{2 b^2 x^2}}{4 b^2}-\frac {\frac {x^2 e^{2 b^2 x^2}}{4 b^2}-\frac {e^{2 b^2 x^2}}{8 b^4}}{b^2}}{\sqrt {\pi } b}\right )}{3 \sqrt {\pi }}\)

\(\Big \downarrow \) 2638

\(\displaystyle \frac {1}{6} x^6 \text {erfi}(b x)^2-\frac {2 b \left (-\frac {5 \left (-\frac {3 \left (-\frac {\int e^{b^2 x^2} \text {erfi}(b x)dx}{2 b^2}+\frac {x e^{b^2 x^2} \text {erfi}(b x)}{2 b^2}-\frac {e^{2 b^2 x^2}}{4 \sqrt {\pi } b^3}\right )}{2 b^2}+\frac {x^3 e^{b^2 x^2} \text {erfi}(b x)}{2 b^2}-\frac {\frac {x^2 e^{2 b^2 x^2}}{4 b^2}-\frac {e^{2 b^2 x^2}}{8 b^4}}{\sqrt {\pi } b}\right )}{2 b^2}+\frac {x^5 e^{b^2 x^2} \text {erfi}(b x)}{2 b^2}-\frac {\frac {x^4 e^{2 b^2 x^2}}{4 b^2}-\frac {\frac {x^2 e^{2 b^2 x^2}}{4 b^2}-\frac {e^{2 b^2 x^2}}{8 b^4}}{b^2}}{\sqrt {\pi } b}\right )}{3 \sqrt {\pi }}\)

\(\Big \downarrow \) 6929

\(\displaystyle \frac {1}{6} x^6 \text {erfi}(b x)^2-\frac {2 b \left (-\frac {5 \left (-\frac {3 \left (-\frac {\sqrt {\pi } \int \text {erfi}(b x)d\text {erfi}(b x)}{4 b^3}+\frac {x e^{b^2 x^2} \text {erfi}(b x)}{2 b^2}-\frac {e^{2 b^2 x^2}}{4 \sqrt {\pi } b^3}\right )}{2 b^2}+\frac {x^3 e^{b^2 x^2} \text {erfi}(b x)}{2 b^2}-\frac {\frac {x^2 e^{2 b^2 x^2}}{4 b^2}-\frac {e^{2 b^2 x^2}}{8 b^4}}{\sqrt {\pi } b}\right )}{2 b^2}+\frac {x^5 e^{b^2 x^2} \text {erfi}(b x)}{2 b^2}-\frac {\frac {x^4 e^{2 b^2 x^2}}{4 b^2}-\frac {\frac {x^2 e^{2 b^2 x^2}}{4 b^2}-\frac {e^{2 b^2 x^2}}{8 b^4}}{b^2}}{\sqrt {\pi } b}\right )}{3 \sqrt {\pi }}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {1}{6} x^6 \text {erfi}(b x)^2-\frac {2 b \left (\frac {x^5 e^{b^2 x^2} \text {erfi}(b x)}{2 b^2}-\frac {\frac {x^4 e^{2 b^2 x^2}}{4 b^2}-\frac {\frac {x^2 e^{2 b^2 x^2}}{4 b^2}-\frac {e^{2 b^2 x^2}}{8 b^4}}{b^2}}{\sqrt {\pi } b}-\frac {5 \left (\frac {x^3 e^{b^2 x^2} \text {erfi}(b x)}{2 b^2}-\frac {\frac {x^2 e^{2 b^2 x^2}}{4 b^2}-\frac {e^{2 b^2 x^2}}{8 b^4}}{\sqrt {\pi } b}-\frac {3 \left (-\frac {\sqrt {\pi } \text {erfi}(b x)^2}{8 b^3}+\frac {x e^{b^2 x^2} \text {erfi}(b x)}{2 b^2}-\frac {e^{2 b^2 x^2}}{4 \sqrt {\pi } b^3}\right )}{2 b^2}\right )}{2 b^2}\right )}{3 \sqrt {\pi }}\)

input

Int[x^5*Erfi[b*x]^2,x]
 

output

(x^6*Erfi[b*x]^2)/6 - (2*b*(-(((E^(2*b^2*x^2)*x^4)/(4*b^2) - (-1/8*E^(2*b^ 
2*x^2)/b^4 + (E^(2*b^2*x^2)*x^2)/(4*b^2))/b^2)/(b*Sqrt[Pi])) + (E^(b^2*x^2 
)*x^5*Erfi[b*x])/(2*b^2) - (5*(-((-1/8*E^(2*b^2*x^2)/b^4 + (E^(2*b^2*x^2)* 
x^2)/(4*b^2))/(b*Sqrt[Pi])) + (E^(b^2*x^2)*x^3*Erfi[b*x])/(2*b^2) - (3*(-1 
/4*E^(2*b^2*x^2)/(b^3*Sqrt[Pi]) + (E^(b^2*x^2)*x*Erfi[b*x])/(2*b^2) - (Sqr 
t[Pi]*Erfi[b*x]^2)/(8*b^3)))/(2*b^2)))/(2*b^2)))/(3*Sqrt[Pi])
 

3.3.28.3.1 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 2638

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_ 
.), x_Symbol] :> Simp[(e + f*x)^n*(F^(a + b*(c + d*x)^n)/(b*f*n*(c + d*x)^n 
*Log[F])), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] && EqQ 
[d*e - c*f, 0]
 

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 6920

Int[Erfi[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(Erfi[b*x]^2 
/(m + 1)), x] - Simp[4*(b/(Sqrt[Pi]*(m + 1)))   Int[x^(m + 1)*E^(b^2*x^2)*E 
rfi[b*x], x], x] /; FreeQ[b, x] && (IGtQ[m, 0] || ILtQ[(m + 1)/2, 0])
 

rule 6929

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

3.3.28.4 Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.88

method	result	size
parallelrisch	\(\frac {8 \operatorname {erfi}\left (b x \right )^{2} x^{6} b^{6} \pi ^{\frac {3}{2}}-16 \,\operatorname {erfi}\left (b x \right ) {\mathrm e}^{b^{2} x^{2}} x^{5} b^{5} \pi +8 \,{\mathrm e}^{2 b^{2} x^{2}} x^{4} b^{4} \sqrt {\pi }+40 \,\operatorname {erfi}\left (b x \right ) {\mathrm e}^{b^{2} x^{2}} x^{3} b^{3} \pi -28 \,{\mathrm e}^{2 b^{2} x^{2}} x^{2} b^{2} \sqrt {\pi }-60 \,\operatorname {erfi}\left (b x \right ) x \,{\mathrm e}^{b^{2} x^{2}} b \pi +15 \operatorname {erfi}\left (b x \right )^{2} \pi ^{\frac {3}{2}}+44 \,{\mathrm e}^{2 b^{2} x^{2}} \sqrt {\pi }}{48 b^{6} \pi ^{\frac {3}{2}}}\)	\(154\)

input

int(x^5*erfi(b*x)^2,x,method=_RETURNVERBOSE)
 

output

1/48*(8*erfi(b*x)^2*x^6*b^6*Pi^(3/2)-16*erfi(b*x)*exp(b^2*x^2)*x^5*b^5*Pi+ 
8*exp(b^2*x^2)^2*x^4*b^4*Pi^(1/2)+40*erfi(b*x)*exp(b^2*x^2)*x^3*b^3*Pi-28* 
exp(b^2*x^2)^2*x^2*b^2*Pi^(1/2)-60*erfi(b*x)*x*exp(b^2*x^2)*b*Pi+15*erfi(b 
*x)^2*Pi^(3/2)+44*exp(b^2*x^2)^2*Pi^(1/2))/b^6/Pi^(3/2)
 

3.3.28.5 Fricas [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.55 \[ \int x^5 \text {erfi}(b x)^2 \, dx=-\frac {4 \, \sqrt {\pi } {\left (4 \, b^{5} x^{5} - 10 \, b^{3} x^{3} + 15 \, b x\right )} \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} - {\left (15 \, \pi + 8 \, \pi b^{6} x^{6}\right )} \operatorname {erfi}\left (b x\right )^{2} - 4 \, {\left (2 \, b^{4} x^{4} - 7 \, b^{2} x^{2} + 11\right )} e^{\left (2 \, b^{2} x^{2}\right )}}{48 \, \pi b^{6}} \]

input

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

output

-1/48*(4*sqrt(pi)*(4*b^5*x^5 - 10*b^3*x^3 + 15*b*x)*erfi(b*x)*e^(b^2*x^2) 
- (15*pi + 8*pi*b^6*x^6)*erfi(b*x)^2 - 4*(2*b^4*x^4 - 7*b^2*x^2 + 11)*e^(2 
*b^2*x^2))/(pi*b^6)
 

3.3.28.6 Sympy [A] (verification not implemented)

Time = 0.53 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.96 \[ \int x^5 \text {erfi}(b x)^2 \, dx=\begin {cases} \frac {x^{6} \operatorname {erfi}^{2}{\left (b x \right )}}{6} - \frac {x^{5} e^{b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}}{3 \sqrt {\pi } b} + \frac {x^{4} e^{2 b^{2} x^{2}}}{6 \pi b^{2}} + \frac {5 x^{3} e^{b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}}{6 \sqrt {\pi } b^{3}} - \frac {7 x^{2} e^{2 b^{2} x^{2}}}{12 \pi b^{4}} - \frac {5 x e^{b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}}{4 \sqrt {\pi } b^{5}} + \frac {11 e^{2 b^{2} x^{2}}}{12 \pi b^{6}} + \frac {5 \operatorname {erfi}^{2}{\left (b x \right )}}{16 b^{6}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]

input

integrate(x**5*erfi(b*x)**2,x)
 

output

Piecewise((x**6*erfi(b*x)**2/6 - x**5*exp(b**2*x**2)*erfi(b*x)/(3*sqrt(pi) 
*b) + x**4*exp(2*b**2*x**2)/(6*pi*b**2) + 5*x**3*exp(b**2*x**2)*erfi(b*x)/ 
(6*sqrt(pi)*b**3) - 7*x**2*exp(2*b**2*x**2)/(12*pi*b**4) - 5*x*exp(b**2*x* 
*2)*erfi(b*x)/(4*sqrt(pi)*b**5) + 11*exp(2*b**2*x**2)/(12*pi*b**6) + 5*erf 
i(b*x)**2/(16*b**6), Ne(b, 0)), (0, True))
 

3.3.28.7 Maxima [F]

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

input

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

output

integrate(x^5*erfi(b*x)^2, x)
 

3.3.28.8 Giac [F]

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

input

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

output

integrate(x^5*erfi(b*x)^2, x)
 

3.3.28.9 Mupad [B] (verification not implemented)

Time = 5.27 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.79 \[ \int x^5 \text {erfi}(b x)^2 \, dx=\frac {x^6\,{\mathrm {erfi}\left (b\,x\right )}^2}{6}+\frac {\frac {11\,{\mathrm {e}}^{2\,b^2\,x^2}}{12}+\frac {5\,\pi \,{\mathrm {erfi}\left (b\,x\right )}^2}{16}-\frac {7\,b^2\,x^2\,{\mathrm {e}}^{2\,b^2\,x^2}}{12}+\frac {b^4\,x^4\,{\mathrm {e}}^{2\,b^2\,x^2}}{6}+\frac {5\,b^3\,x^3\,\sqrt {\pi }\,{\mathrm {e}}^{b^2\,x^2}\,\mathrm {erfi}\left (b\,x\right )}{6}-\frac {b^5\,x^5\,\sqrt {\pi }\,{\mathrm {e}}^{b^2\,x^2}\,\mathrm {erfi}\left (b\,x\right )}{3}-\frac {5\,b\,x\,\sqrt {\pi }\,{\mathrm {e}}^{b^2\,x^2}\,\mathrm {erfi}\left (b\,x\right )}{4}}{b^6\,\pi } \]

input

int(x^5*erfi(b*x)^2,x)
 

output

(x^6*erfi(b*x)^2)/6 + ((11*exp(2*b^2*x^2))/12 + (5*pi*erfi(b*x)^2)/16 - (7 
*b^2*x^2*exp(2*b^2*x^2))/12 + (b^4*x^4*exp(2*b^2*x^2))/6 + (5*b^3*x^3*pi^( 
1/2)*exp(b^2*x^2)*erfi(b*x))/6 - (b^5*x^5*pi^(1/2)*exp(b^2*x^2)*erfi(b*x)) 
/3 - (5*b*x*pi^(1/2)*exp(b^2*x^2)*erfi(b*x))/4)/(b^6*pi)