3.209 \(\int x \text{Erfi}(b x) \, dx\)

Optimal. Leaf size=45 \[ \frac{\text{Erfi}(b x)}{4 b^2}-\frac{x e^{b^2 x^2}}{2 \sqrt{\pi } b}+\frac{1}{2} x^2 \text{Erfi}(b x) \]

[Out]

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

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Rubi [A]  time = 0.0331474, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6363, 2212, 2204} \[ \frac{\text{Erfi}(b x)}{4 b^2}-\frac{x e^{b^2 x^2}}{2 \sqrt{\pi } b}+\frac{1}{2} x^2 \text{Erfi}(b x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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]

Rubi steps

\begin{align*} \int x \text{erfi}(b x) \, dx &=\frac{1}{2} x^2 \text{erfi}(b x)-\frac{b \int e^{b^2 x^2} x^2 \, dx}{\sqrt{\pi }}\\ &=-\frac{e^{b^2 x^2} x}{2 b \sqrt{\pi }}+\frac{1}{2} x^2 \text{erfi}(b x)+\frac{\int e^{b^2 x^2} \, dx}{2 b \sqrt{\pi }}\\ &=-\frac{e^{b^2 x^2} x}{2 b \sqrt{\pi }}+\frac{\text{erfi}(b x)}{4 b^2}+\frac{1}{2} x^2 \text{erfi}(b x)\\ \end{align*}

Mathematica [A]  time = 0.0276535, size = 39, normalized size = 0.87 \[ \frac{1}{4} \left (\left (\frac{1}{b^2}+2 x^2\right ) \text{Erfi}(b x)-\frac{2 x e^{b^2 x^2}}{\sqrt{\pi } b}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.043, size = 45, normalized size = 1. \begin{align*}{\frac{1}{{b}^{2}} \left ({\frac{{b}^{2}{x}^{2}{\it erfi} \left ( bx \right ) }{2}}-{\frac{1}{\sqrt{\pi }} \left ({\frac{{{\rm e}^{{b}^{2}{x}^{2}}}bx}{2}}-{\frac{\sqrt{\pi }{\it erfi} \left ( bx \right ) }{4}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b^2*(1/2*b^2*x^2*erfi(b*x)-1/Pi^(1/2)*(1/2*exp(b^2*x^2)*b*x-1/4*Pi^(1/2)*erfi(b*x)))

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Maxima [C]  time = 1.03254, size = 59, normalized size = 1.31 \begin{align*} \frac{1}{2} \, x^{2} \operatorname{erfi}\left (b x\right ) - \frac{b{\left (\frac{2 \, x e^{\left (b^{2} x^{2}\right )}}{b^{2}} + \frac{i \, \sqrt{\pi } \operatorname{erf}\left (i \, b x\right )}{b^{3}}\right )}}{4 \, \sqrt{\pi }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.29701, size = 103, normalized size = 2.29 \begin{align*} -\frac{2 \, \sqrt{\pi } b x e^{\left (b^{2} x^{2}\right )} -{\left (\pi + 2 \, \pi b^{2} x^{2}\right )} \operatorname{erfi}\left (b x\right )}{4 \, \pi b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*sqrt(pi)*b*x*e^(b^2*x^2) - (pi + 2*pi*b^2*x^2)*erfi(b*x))/(pi*b^2)

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Sympy [A]  time = 0.23252, size = 39, normalized size = 0.87 \begin{align*} \begin{cases} \frac{x^{2} \operatorname{erfi}{\left (b x \right )}}{2} - \frac{x e^{b^{2} x^{2}}}{2 \sqrt{\pi } b} + \frac{\operatorname{erfi}{\left (b x \right )}}{4 b^{2}} & \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*erfi(b*x),x)

[Out]

Piecewise((x**2*erfi(b*x)/2 - x*exp(b**2*x**2)/(2*sqrt(pi)*b) + erfi(b*x)/(4*b**2), Ne(b, 0)), (0, True))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*erfi(b*x), x)