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

Optimal. Leaf size=57 \[ -\frac{x^2 e^{b^2 x^2}}{3 \sqrt{\pi } b}+\frac{e^{b^2 x^2}}{3 \sqrt{\pi } b^3}+\frac{1}{3} x^3 \text{Erfi}(b x) \]

[Out]

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

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Rubi [A]  time = 0.0442204, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6363, 2212, 2209} \[ -\frac{x^2 e^{b^2 x^2}}{3 \sqrt{\pi } b}+\frac{e^{b^2 x^2}}{3 \sqrt{\pi } b^3}+\frac{1}{3} x^3 \text{Erfi}(b x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6363

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

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])

Rule 2209

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]

Rubi steps

\begin{align*} \int x^2 \text{erfi}(b x) \, dx &=\frac{1}{3} x^3 \text{erfi}(b x)-\frac{(2 b) \int e^{b^2 x^2} x^3 \, dx}{3 \sqrt{\pi }}\\ &=-\frac{e^{b^2 x^2} x^2}{3 b \sqrt{\pi }}+\frac{1}{3} x^3 \text{erfi}(b x)+\frac{2 \int e^{b^2 x^2} x \, dx}{3 b \sqrt{\pi }}\\ &=\frac{e^{b^2 x^2}}{3 b^3 \sqrt{\pi }}-\frac{e^{b^2 x^2} x^2}{3 b \sqrt{\pi }}+\frac{1}{3} x^3 \text{erfi}(b x)\\ \end{align*}

Mathematica [A]  time = 0.0249373, size = 41, normalized size = 0.72 \[ \frac{1}{3} \left (\frac{e^{b^2 x^2} \left (1-b^2 x^2\right )}{\sqrt{\pi } b^3}+x^3 \text{Erfi}(b x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((E^(b^2*x^2)*(1 - b^2*x^2))/(b^3*Sqrt[Pi]) + x^3*Erfi[b*x])/3

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Maple [A]  time = 0.042, size = 50, normalized size = 0.9 \begin{align*}{\frac{1}{{b}^{3}} \left ({\frac{{b}^{3}{x}^{3}{\it erfi} \left ( bx \right ) }{3}}-{\frac{2}{3\,\sqrt{\pi }} \left ({\frac{{b}^{2}{x}^{2}{{\rm e}^{{b}^{2}{x}^{2}}}}{2}}-{\frac{{{\rm e}^{{b}^{2}{x}^{2}}}}{2}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b^3*(1/3*b^3*x^3*erfi(b*x)-2/3/Pi^(1/2)*(1/2*b^2*x^2*exp(b^2*x^2)-1/2*exp(b^2*x^2)))

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Maxima [A]  time = 0.973776, size = 47, normalized size = 0.82 \begin{align*} \frac{1}{3} \, x^{3} \operatorname{erfi}\left (b x\right ) - \frac{{\left (b^{2} x^{2} - 1\right )} e^{\left (b^{2} x^{2}\right )}}{3 \, \sqrt{\pi } b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*erfi(b*x) - 1/3*(b^2*x^2 - 1)*e^(b^2*x^2)/(sqrt(pi)*b^3)

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Fricas [A]  time = 2.3121, size = 100, normalized size = 1.75 \begin{align*} \frac{\pi b^{3} x^{3} \operatorname{erfi}\left (b x\right ) - \sqrt{\pi }{\left (b^{2} x^{2} - 1\right )} e^{\left (b^{2} x^{2}\right )}}{3 \, \pi b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(pi*b^3*x^3*erfi(b*x) - sqrt(pi)*(b^2*x^2 - 1)*e^(b^2*x^2))/(pi*b^3)

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Sympy [A]  time = 0.494234, size = 49, normalized size = 0.86 \begin{align*} \begin{cases} \frac{x^{3} \operatorname{erfi}{\left (b x \right )}}{3} - \frac{x^{2} e^{b^{2} x^{2}}}{3 \sqrt{\pi } b} + \frac{e^{b^{2} x^{2}}}{3 \sqrt{\pi } b^{3}} & \text{for}\: b \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x**3*erfi(b*x)/3 - x**2*exp(b**2*x**2)/(3*sqrt(pi)*b) + exp(b**2*x**2)/(3*sqrt(pi)*b**3), Ne(b, 0))
, (0, True))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2*erfi(b*x), x)