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

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6501, 6522, 6519, 2235, 2243} \[ \int x^2 \text {erfi}(b x)^2 \, dx=-\frac {5 \text {erfi}\left (\sqrt {2} b x\right )}{6 \sqrt {2 \pi } b^3}-\frac {2 x^2 e^{b^2 x^2} \text {erfi}(b x)}{3 \sqrt {\pi } b}+\frac {x e^{2 b^2 x^2}}{3 \pi b^2}+\frac {2 e^{b^2 x^2} \text {erfi}(b x)}{3 \sqrt {\pi } b^3}+\frac {1}{3} x^3 \text {erfi}(b x)^2 \]

[In]

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

[Out]

(E^(2*b^2*x^2)*x)/(3*b^2*Pi) + (2*E^(b^2*x^2)*Erfi[b*x])/(3*b^3*Sqrt[Pi]) - (2*E^(b^2*x^2)*x^2*Erfi[b*x])/(3*b
*Sqrt[Pi]) + (x^3*Erfi[b*x]^2)/3 - (5*Erfi[Sqrt[2]*b*x])/(6*b^3*Sqrt[2*Pi])

Rule 2235

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 2243

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*Log[F])), x] - Dist[(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 6501

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

Rule 6519

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(a_.) + (b_.)*(x_)]*(x_), x_Symbol] :> Simp[E^(c + d*x^2)*(Erfi[a + b*x]/(2*
d)), x] - Dist[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 6522

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[x^(m - 1)*E^(c + d*x^2)*(Er
fi[a + b*x]/(2*d)), x] + (-Dist[(m - 1)/(2*d), Int[x^(m - 2)*E^(c + d*x^2)*Erfi[a + b*x], x], x] - Dist[b/(d*S
qrt[Pi]), Int[x^(m - 1)*E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2), x], x]) /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 1]

Rubi steps \begin{align*} \text {integral}& = \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 }} \\ & = -\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 {4 \int e^{2 b^2 x^2} x^2 \, dx}{3 \pi }+\frac {4 \int e^{b^2 x^2} x \text {erfi}(b x) \, dx}{3 b \sqrt {\pi }} \\ & = \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 {\int e^{2 b^2 x^2} \, dx}{3 b^2 \pi }-\frac {4 \int e^{2 b^2 x^2} \, dx}{3 b^2 \pi } \\ & = \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 {\sqrt {\frac {2}{\pi }} \text {erfi}\left (\sqrt {2} b x\right )}{3 b^3}-\frac {\text {erfi}\left (\sqrt {2} b x\right )}{6 b^3 \sqrt {2 \pi }} \\ \end{align*}

Mathematica [A] (verified)

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

[In]

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

[Out]

(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*P
i]*Erfi[Sqrt[2]*b*x])/(12*b^3*Pi)

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.86 \[ \int x^2 \text {erfi}(b x)^2 \, dx=\frac {4 \, \pi b^{4} x^{3} \operatorname {erfi}\left (b x\right )^{2} + 4 \, b^{2} x e^{\left (2 \, b^{2} x^{2}\right )} - 8 \, \sqrt {\pi } {\left (b^{3} x^{2} - b\right )} \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} + 5 \, \sqrt {2} \sqrt {\pi } \sqrt {-b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {-b^{2}} x\right )}{12 \, \pi b^{4}} \]

[In]

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

[Out]

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*sq
rt(2)*sqrt(pi)*sqrt(-b^2)*erf(sqrt(2)*sqrt(-b^2)*x))/(pi*b^4)

Sympy [F]

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

[In]

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

[Out]

Integral(x**2*erfi(b*x)**2, x)

Maxima [F]

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

[In]

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

[Out]

integrate(x^2*erfi(b*x)^2, x)

Giac [F]

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

[In]

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

[Out]

integrate(x^2*erfi(b*x)^2, x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(x^2*erfi(b*x)^2, x)