\(\int x^2 \text {erfi}(b x)^2 \, dx\) [236]

Optimal result

Integrand size = 10, antiderivative size = 111 \[ \int x^2 \text {erfi}(b x)^2 \, dx=\frac {e^{2 b^2 x^2} x}{3 b^2 \pi }+\frac {2 e^{b^2 x^2} \text {erfi}(b x)}{3 b^3 \sqrt {\pi }}-\frac {2 e^{b^2 x^2} x^2 \text {erfi}(b x)}{3 b \sqrt {\pi }}+\frac {1}{3} x^3 \text {erfi}(b x)^2-\frac {5 \text {erfi}\left (\sqrt {2} b x\right )}{6 b^3 \sqrt {2 \pi }} \] Output:

1/3*exp(2*b^2*x^2)*x/b^2/Pi+2/3*exp(b^2*x^2)*erfi(b*x)/b^3/Pi^(1/2)-2/3*ex 
p(b^2*x^2)*x^2*erfi(b*x)/b/Pi^(1/2)+1/3*x^3*erfi(b*x)^2-5/12*erfi(2^(1/2)* 
b*x)/b^3*2^(1/2)/Pi^(1/2)
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.78 \[ \int x^2 \text {erfi}(b x)^2 \, dx=\frac {4 b e^{2 b^2 x^2} x-8 e^{b^2 x^2} \sqrt {\pi } \left (-1+b^2 x^2\right ) \text {erfi}(b x)+4 b^3 \pi x^3 \text {erfi}(b x)^2-5 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} b x\right )}{12 b^3 \pi } \] Input:

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

Output:

(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] + 4 
*b^3*Pi*x^3*Erfi[b*x]^2 - 5*Sqrt[2*Pi]*Erfi[Sqrt[2]*b*x])/(12*b^3*Pi)
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.34, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6920, 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.

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

\(\Big \downarrow \) 6920

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

\(\Big \downarrow \) 6941

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

\(\Big \downarrow \) 2641

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

\(\Big \downarrow \) 2633

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

\(\Big \downarrow \) 6938

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

\(\Big \downarrow \) 2633

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.86 \[ \int x^2 \text {erfi}(b x)^2 \, dx=\frac {4 \, \pi b^{4} x^{3} \operatorname {erfi}\left (b x\right )^{2} + 4 \, b^{2} x e^{\left (2 \, b^{2} x^{2}\right )} - 8 \, \sqrt {\pi } {\left (b^{3} x^{2} - b\right )} \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} + 5 \, \sqrt {2} \sqrt {\pi } \sqrt {-b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {-b^{2}} x\right )}{12 \, \pi b^{4}} \] Input:

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

Output:

1/12*(4*pi*b^4*x^3*erfi(b*x)^2 + 4*b^2*x*e^(2*b^2*x^2) - 8*sqrt(pi)*(b^3*x 
^2 - b)*erfi(b*x)*e^(b^2*x^2) + 5*sqrt(2)*sqrt(pi)*sqrt(-b^2)*erf(sqrt(2)* 
sqrt(-b^2)*x))/(pi*b^4)
 

Sympy [F]

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

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

Output:

Integral(x**2*erfi(b*x)**2, x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

 - int(erf(b*i*x)**2*x**2,x)