\(\int x^2 \text {erf}(b x) \, dx\) [10]

Optimal result

Integrand size = 8, antiderivative size = 59 \[ \int x^2 \text {erf}(b x) \, dx=\frac {e^{-b^2 x^2}}{3 b^3 \sqrt {\pi }}+\frac {e^{-b^2 x^2} x^2}{3 b \sqrt {\pi }}+\frac {1}{3} x^3 \text {erf}(b x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6496, 2243, 2240} \[ \int x^2 \text {erf}(b x) \, dx=\frac {x^2 e^{-b^2 x^2}}{3 \sqrt {\pi } b}+\frac {e^{-b^2 x^2}}{3 \sqrt {\pi } b^3}+\frac {1}{3} x^3 \text {erf}(b x) \]

[In]

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

[Out]

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

Rule 2240

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]

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}{3} x^3 \text {erf}(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 {erf}(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 {erf}(b x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.69 \[ \int x^2 \text {erf}(b x) \, dx=\frac {1}{3} \left (\frac {e^{-b^2 x^2} \left (1+b^2 x^2\right )}{b^3 \sqrt {\pi }}+x^3 \text {erf}(b x)\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.80

method	result	size
meijerg	\(\frac {-\frac {2}{3}+\frac {\left (2 b^{2} x^{2}+2\right ) {\mathrm e}^{-b^{2} x^{2}}}{3}+\frac {2 b^{3} x^{3} \operatorname {erf}\left (b x \right ) \sqrt {\pi }}{3}}{2 b^{3} \sqrt {\pi }}\)	\(47\)
parallelrisch	\(\frac {b^{3} x^{3} \operatorname {erf}\left (b x \right ) \sqrt {\pi }+x^{2} {\mathrm e}^{-b^{2} x^{2}} b^{2}+{\mathrm e}^{-b^{2} x^{2}}}{3 b^{3} \sqrt {\pi }}\)	\(49\)
parts	\(\frac {x^{3} \operatorname {erf}\left (b x \right )}{3}-\frac {2 b \left (-\frac {x^{2} {\mathrm e}^{-b^{2} x^{2}}}{2 b^{2}}-\frac {{\mathrm e}^{-b^{2} x^{2}}}{2 b^{4}}\right )}{3 \sqrt {\pi }}\)	\(49\)
derivativedivides	\(\frac {\frac {\operatorname {erf}\left (b x \right ) b^{3} x^{3}}{3}-\frac {2 \left (-\frac {x^{2} {\mathrm e}^{-b^{2} x^{2}} b^{2}}{2}-\frac {{\mathrm e}^{-b^{2} x^{2}}}{2}\right )}{3 \sqrt {\pi }}}{b^{3}}\)	\(54\)
default	\(\frac {\frac {\operatorname {erf}\left (b x \right ) b^{3} x^{3}}{3}-\frac {2 \left (-\frac {x^{2} {\mathrm e}^{-b^{2} x^{2}} b^{2}}{2}-\frac {{\mathrm e}^{-b^{2} x^{2}}}{2}\right )}{3 \sqrt {\pi }}}{b^{3}}\)	\(54\)

[In]

int(x^2*erf(b*x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.73 \[ \int x^2 \text {erf}(b x) \, dx=\frac {\pi b^{3} x^{3} \operatorname {erf}\left (b x\right ) + \sqrt {\pi } {\left (b^{2} x^{2} + 1\right )} e^{\left (-b^{2} x^{2}\right )}}{3 \, \pi b^{3}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.86 \[ \int x^2 \text {erf}(b x) \, dx=\begin {cases} \frac {x^{3} \operatorname {erf}{\left (b x \right )}}{3} + \frac {x^{2} e^{- b^{2} x^{2}}}{3 \sqrt {\pi } b} + \frac {e^{- b^{2} x^{2}}}{3 \sqrt {\pi } b^{3}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((x**3*erf(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))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.61 \[ \int x^2 \text {erf}(b x) \, dx=\frac {1}{3} \, x^{3} \operatorname {erf}\left (b x\right ) + \frac {{\left (b^{2} x^{2} + 1\right )} e^{\left (-b^{2} x^{2}\right )}}{3 \, \sqrt {\pi } b^{3}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.61 \[ \int x^2 \text {erf}(b x) \, dx=\frac {1}{3} \, x^{3} \operatorname {erf}\left (b x\right ) + \frac {{\left (b^{2} x^{2} + 1\right )} e^{\left (-b^{2} x^{2}\right )}}{3 \, \sqrt {\pi } b^{3}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.61 \[ \int x^2 \text {erf}(b x) \, dx=\frac {x^3\,\mathrm {erf}\left (b\,x\right )}{3}+\frac {{\mathrm {e}}^{-b^2\,x^2}\,\left (b^2\,x^2+1\right )}{3\,b^3\,\sqrt {\pi }} \]

[In]

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

[Out]

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