\(\int x \text {erf}(b x) \, dx\) [3]

Optimal result

Integrand size = 6, antiderivative size = 46 \[ \int x \text {erf}(b x) \, dx=\frac {e^{-b^2 x^2} x}{2 b \sqrt {\pi }}-\frac {\text {erf}(b x)}{4 b^2}+\frac {1}{2} x^2 \text {erf}(b x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6496, 2243, 2236} \[ \int x \text {erf}(b x) \, dx=-\frac {\text {erf}(b x)}{4 b^2}+\frac {x e^{-b^2 x^2}}{2 \sqrt {\pi } b}+\frac {1}{2} x^2 \text {erf}(b x) \]

[In]

Int[x*Erf[b*x],x]

[Out]

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

Rule 2236

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F],
2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[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 6496

Int[Erf[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Erf[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]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \text {erf}(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 {erf}(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 {erf}(b x)}{4 b^2}+\frac {1}{2} x^2 \text {erf}(b x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.91 \[ \int x \text {erf}(b x) \, dx=\frac {1}{4} \left (\frac {2 e^{-b^2 x^2} x}{b \sqrt {\pi }}+\left (-\frac {1}{b^2}+2 x^2\right ) \text {erf}(b x)\right ) \]

[In]

Integrate[x*Erf[b*x],x]

[Out]

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

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.89

method	result	size
meijerg	\(\frac {{\mathrm e}^{-b^{2} x^{2}} b x -\frac {\left (-6 b^{2} x^{2}+3\right ) \operatorname {erf}\left (b x \right ) \sqrt {\pi }}{6}}{2 b^{2} \sqrt {\pi }}\)	\(41\)
parts	\(\frac {x^{2} \operatorname {erf}\left (b x \right )}{2}-\frac {b \left (-\frac {x \,{\mathrm e}^{-b^{2} x^{2}}}{2 b^{2}}+\frac {\sqrt {\pi }\, \operatorname {erf}\left (b x \right )}{4 b^{3}}\right )}{\sqrt {\pi }}\)	\(45\)
derivativedivides	\(\frac {\frac {\operatorname {erf}\left (b x \right ) b^{2} x^{2}}{2}-\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}} b x}{2}+\frac {\operatorname {erf}\left (b x \right ) \sqrt {\pi }}{4}}{\sqrt {\pi }}}{b^{2}}\)	\(47\)
default	\(\frac {\frac {\operatorname {erf}\left (b x \right ) b^{2} x^{2}}{2}-\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}} b x}{2}+\frac {\operatorname {erf}\left (b x \right ) \sqrt {\pi }}{4}}{\sqrt {\pi }}}{b^{2}}\)	\(47\)
parallelrisch	\(\frac {2 x^{2} \operatorname {erf}\left (b x \right ) \sqrt {\pi }\, b^{2}+2 \,{\mathrm e}^{-b^{2} x^{2}} b x -\operatorname {erf}\left (b x \right ) \sqrt {\pi }}{4 \sqrt {\pi }\, b^{2}}\)	\(47\)

[In]

int(x*erf(b*x),x,method=_RETURNVERBOSE)

[Out]

1/2/b^2/Pi^(1/2)*(exp(-b^2*x^2)*b*x-1/6*(-6*b^2*x^2+3)*erf(b*x)*Pi^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.91 \[ \int x \text {erf}(b x) \, dx=\frac {2 \, \sqrt {\pi } b x e^{\left (-b^{2} x^{2}\right )} - {\left (\pi - 2 \, \pi b^{2} x^{2}\right )} \operatorname {erf}\left (b x\right )}{4 \, \pi b^{2}} \]

[In]

integrate(x*erf(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)*erf(b*x))/(pi*b^2)

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.85 \[ \int x \text {erf}(b x) \, dx=\begin {cases} \frac {x^{2} \operatorname {erf}{\left (b x \right )}}{2} + \frac {x e^{- b^{2} x^{2}}}{2 \sqrt {\pi } b} - \frac {\operatorname {erf}{\left (b x \right )}}{4 b^{2}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

integrate(x*erf(b*x),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.96 \[ \int x \text {erf}(b x) \, dx=\frac {1}{2} \, x^{2} \operatorname {erf}\left (b x\right ) + \frac {b {\left (\frac {2 \, x e^{\left (-b^{2} x^{2}\right )}}{b^{2}} - \frac {\sqrt {\pi } \operatorname {erf}\left (b x\right )}{b^{3}}\right )}}{4 \, \sqrt {\pi }} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.96 \[ \int x \text {erf}(b x) \, dx=\frac {1}{2} \, x^{2} \operatorname {erf}\left (b x\right ) + \frac {b {\left (\frac {2 \, x e^{\left (-b^{2} x^{2}\right )}}{b^{2}} + \frac {\sqrt {\pi } \operatorname {erf}\left (-b x\right )}{b^{3}}\right )}}{4 \, \sqrt {\pi }} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.04 \[ \int x \text {erf}(b x) \, dx=\frac {x^2\,\mathrm {erf}\left (b\,x\right )}{2}+\frac {b\,\mathrm {erfi}\left (x\,\sqrt {-b^2}\right )}{4\,{\left (-b^2\right )}^{3/2}}+\frac {x\,{\mathrm {e}}^{-b^2\,x^2}}{2\,b\,\sqrt {\pi }} \]

[In]

int(x*erf(b*x),x)

[Out]

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