3.236 \(\int x^2 \text{Erfi}(b x)^2 \, dx\)

Optimal. Leaf size=111 \[ -\frac{2 x^2 e^{b^2 x^2} \text{Erfi}(b x)}{3 \sqrt{\pi } b}+\frac{2 e^{b^2 x^2} \text{Erfi}(b x)}{3 \sqrt{\pi } b^3}-\frac{5 \text{Erfi}\left (\sqrt{2} b x\right )}{6 \sqrt{2 \pi } b^3}+\frac{x e^{2 b^2 x^2}}{3 \pi b^2}+\frac{1}{3} x^3 \text{Erfi}(b x)^2 \]

[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])

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Rubi [A]  time = 0.119623, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6366, 6387, 6384, 2204, 2212} \[ -\frac{2 x^2 e^{b^2 x^2} \text{Erfi}(b x)}{3 \sqrt{\pi } b}+\frac{2 e^{b^2 x^2} \text{Erfi}(b x)}{3 \sqrt{\pi } b^3}-\frac{5 \text{Erfi}\left (\sqrt{2} b x\right )}{6 \sqrt{2 \pi } b^3}+\frac{x e^{2 b^2 x^2}}{3 \pi b^2}+\frac{1}{3} x^3 \text{Erfi}(b x)^2 \]

Antiderivative was successfully verified.

[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 6366

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 6387

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]

Rule 6384

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 2204

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 2212

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])

Rubi steps

\begin{align*} \int x^2 \text{erfi}(b x)^2 \, dx &=\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]  time = 0.0301299, size = 87, normalized size = 0.78 \[ \frac{4 \pi b^3 x^3 \text{Erfi}(b x)^2-8 \sqrt{\pi } e^{b^2 x^2} \left (b^2 x^2-1\right ) \text{Erfi}(b x)+4 b x e^{2 b^2 x^2}-5 \sqrt{2 \pi } \text{Erfi}\left (\sqrt{2} b x\right )}{12 \pi b^3} \]

Antiderivative was successfully verified.

[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)

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Maple [F]  time = 0.042, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ({\it erfi} \left ( bx \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \operatorname{erfi}\left (b x\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2*erfi(b*x)^2, x)

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Fricas [A]  time = 2.47672, size = 232, normalized size = 2.09 \begin{align*} \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{erfi}\left (\sqrt{2} \sqrt{b^{2}} x\right )}{12 \, \pi b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)*erfi(sqrt(2)*sqrt(b^2)*x))/(pi*b^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \operatorname{erfi}^{2}{\left (b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*erfi(b*x)**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \operatorname{erfi}\left (b x\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2*erfi(b*x)^2, x)