\(\int x^3 \text {erf}(b x) \, dx\) [2]

Optimal result

Integrand size = 8, antiderivative size = 71 \[ \int x^3 \text {erf}(b x) \, dx=\frac {3 e^{-b^2 x^2} x}{8 b^3 \sqrt {\pi }}+\frac {e^{-b^2 x^2} x^3}{4 b \sqrt {\pi }}-\frac {3 \text {erf}(b x)}{16 b^4}+\frac {1}{4} x^4 \text {erf}(b x) \]

[Out]

-3/16*erf(b*x)/b^4+1/4*x^4*erf(b*x)+3/8*x/b^3/exp(b^2*x^2)/Pi^(1/2)+1/4*x^3/b/exp(b^2*x^2)/Pi^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6496, 2243, 2236} \[ \int x^3 \text {erf}(b x) \, dx=-\frac {3 \text {erf}(b x)}{16 b^4}+\frac {x^3 e^{-b^2 x^2}}{4 \sqrt {\pi } b}+\frac {3 x e^{-b^2 x^2}}{8 \sqrt {\pi } b^3}+\frac {1}{4} x^4 \text {erf}(b x) \]

[In]

Int[x^3*Erf[b*x],x]

[Out]

(3*x)/(8*b^3*E^(b^2*x^2)*Sqrt[Pi]) + x^3/(4*b*E^(b^2*x^2)*Sqrt[Pi]) - (3*Erf[b*x])/(16*b^4) + (x^4*Erf[b*x])/4

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}{4} x^4 \text {erf}(b x)-\frac {b \int e^{-b^2 x^2} x^4 \, dx}{2 \sqrt {\pi }} \\ & = \frac {e^{-b^2 x^2} x^3}{4 b \sqrt {\pi }}+\frac {1}{4} x^4 \text {erf}(b x)-\frac {3 \int e^{-b^2 x^2} x^2 \, dx}{4 b \sqrt {\pi }} \\ & = \frac {3 e^{-b^2 x^2} x}{8 b^3 \sqrt {\pi }}+\frac {e^{-b^2 x^2} x^3}{4 b \sqrt {\pi }}+\frac {1}{4} x^4 \text {erf}(b x)-\frac {3 \int e^{-b^2 x^2} \, dx}{8 b^3 \sqrt {\pi }} \\ & = \frac {3 e^{-b^2 x^2} x}{8 b^3 \sqrt {\pi }}+\frac {e^{-b^2 x^2} x^3}{4 b \sqrt {\pi }}-\frac {3 \text {erf}(b x)}{16 b^4}+\frac {1}{4} x^4 \text {erf}(b x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.89 \[ \int x^3 \text {erf}(b x) \, dx=e^{-b^2 x^2} \left (\frac {3 x}{8 b^3 \sqrt {\pi }}+\frac {x^3}{4 b \sqrt {\pi }}\right )-\frac {3 \text {erf}(b x)}{16 b^4}+\frac {1}{4} x^4 \text {erf}(b x) \]

[In]

Integrate[x^3*Erf[b*x],x]

[Out]

((3*x)/(8*b^3*Sqrt[Pi]) + x^3/(4*b*Sqrt[Pi]))/E^(b^2*x^2) - (3*Erf[b*x])/(16*b^4) + (x^4*Erf[b*x])/4

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.73

method	result	size
meijerg	\(\frac {\frac {b x \left (10 b^{2} x^{2}+15\right ) {\mathrm e}^{-b^{2} x^{2}}}{20}-\frac {\left (-20 b^{4} x^{4}+15\right ) \operatorname {erf}\left (b x \right ) \sqrt {\pi }}{40}}{2 b^{4} \sqrt {\pi }}\)	\(52\)
parallelrisch	\(\frac {4 \,\operatorname {erf}\left (b x \right ) x^{4} b^{4} \sqrt {\pi }+4 x^{3} {\mathrm e}^{-b^{2} x^{2}} b^{3}+6 \,{\mathrm e}^{-b^{2} x^{2}} b x -3 \,\operatorname {erf}\left (b x \right ) \sqrt {\pi }}{16 b^{4} \sqrt {\pi }}\)	\(64\)
derivativedivides	\(\frac {\frac {\operatorname {erf}\left (b x \right ) b^{4} x^{4}}{4}-\frac {-\frac {x^{3} {\mathrm e}^{-b^{2} x^{2}} b^{3}}{2}-\frac {3 \,{\mathrm e}^{-b^{2} x^{2}} b x}{4}+\frac {3 \,\operatorname {erf}\left (b x \right ) \sqrt {\pi }}{8}}{2 \sqrt {\pi }}}{b^{4}}\)	\(65\)
default	\(\frac {\frac {\operatorname {erf}\left (b x \right ) b^{4} x^{4}}{4}-\frac {-\frac {x^{3} {\mathrm e}^{-b^{2} x^{2}} b^{3}}{2}-\frac {3 \,{\mathrm e}^{-b^{2} x^{2}} b x}{4}+\frac {3 \,\operatorname {erf}\left (b x \right ) \sqrt {\pi }}{8}}{2 \sqrt {\pi }}}{b^{4}}\)	\(65\)
parts	\(\frac {x^{4} \operatorname {erf}\left (b x \right )}{4}-\frac {b \left (-\frac {x^{3} {\mathrm e}^{-b^{2} x^{2}}}{2 b^{2}}+\frac {-\frac {3 x \,{\mathrm e}^{-b^{2} x^{2}}}{4 b^{2}}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (b x \right )}{8 b^{3}}}{b^{2}}\right )}{2 \sqrt {\pi }}\)	\(68\)

[In]

int(x^3*erf(b*x),x,method=_RETURNVERBOSE)

[Out]

1/2/b^4/Pi^(1/2)*(1/20*b*x*(10*b^2*x^2+15)*exp(-b^2*x^2)-1/40*(-20*b^4*x^4+15)*erf(b*x)*Pi^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.77 \[ \int x^3 \text {erf}(b x) \, dx=\frac {2 \, \sqrt {\pi } {\left (2 \, b^{3} x^{3} + 3 \, b x\right )} e^{\left (-b^{2} x^{2}\right )} - {\left (3 \, \pi - 4 \, \pi b^{4} x^{4}\right )} \operatorname {erf}\left (b x\right )}{16 \, \pi b^{4}} \]

[In]

integrate(x^3*erf(b*x),x, algorithm="fricas")

[Out]

1/16*(2*sqrt(pi)*(2*b^3*x^3 + 3*b*x)*e^(-b^2*x^2) - (3*pi - 4*pi*b^4*x^4)*erf(b*x))/(pi*b^4)

Sympy [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.92 \[ \int x^3 \text {erf}(b x) \, dx=\begin {cases} \frac {x^{4} \operatorname {erf}{\left (b x \right )}}{4} + \frac {x^{3} e^{- b^{2} x^{2}}}{4 \sqrt {\pi } b} + \frac {3 x e^{- b^{2} x^{2}}}{8 \sqrt {\pi } b^{3}} - \frac {3 \operatorname {erf}{\left (b x \right )}}{16 b^{4}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

integrate(x**3*erf(b*x),x)

[Out]

Piecewise((x**4*erf(b*x)/4 + x**3*exp(-b**2*x**2)/(4*sqrt(pi)*b) + 3*x*exp(-b**2*x**2)/(8*sqrt(pi)*b**3) - 3*e
rf(b*x)/(16*b**4), Ne(b, 0)), (0, True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.77 \[ \int x^3 \text {erf}(b x) \, dx=\frac {1}{4} \, x^{4} \operatorname {erf}\left (b x\right ) + \frac {b {\left (\frac {2 \, {\left (2 \, b^{2} x^{3} + 3 \, x\right )} e^{\left (-b^{2} x^{2}\right )}}{b^{4}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (b x\right )}{b^{5}}\right )}}{16 \, \sqrt {\pi }} \]

[In]

integrate(x^3*erf(b*x),x, algorithm="maxima")

[Out]

1/4*x^4*erf(b*x) + 1/16*b*(2*(2*b^2*x^3 + 3*x)*e^(-b^2*x^2)/b^4 - 3*sqrt(pi)*erf(b*x)/b^5)/sqrt(pi)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.79 \[ \int x^3 \text {erf}(b x) \, dx=\frac {1}{4} \, x^{4} \operatorname {erf}\left (b x\right ) + \frac {b {\left (\frac {2 \, {\left (2 \, b^{2} x^{3} + 3 \, x\right )} e^{\left (-b^{2} x^{2}\right )}}{b^{4}} + \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-b x\right )}{b^{5}}\right )}}{16 \, \sqrt {\pi }} \]

[In]

integrate(x^3*erf(b*x),x, algorithm="giac")

[Out]

1/4*x^4*erf(b*x) + 1/16*b*(2*(2*b^2*x^3 + 3*x)*e^(-b^2*x^2)/b^4 + 3*sqrt(pi)*erf(-b*x)/b^5)/sqrt(pi)

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.24 \[ \int x^3 \text {erf}(b x) \, dx=\frac {x^4\,\mathrm {erf}\left (b\,x\right )}{4}-\frac {3\,b\,x^5}{16\,{\left (b^2\,x^2\right )}^{5/2}}+\frac {x^3\,{\mathrm {e}}^{-b^2\,x^2}}{4\,b\,\sqrt {\pi }}+\frac {3\,x\,{\mathrm {e}}^{-b^2\,x^2}}{8\,b^3\,\sqrt {\pi }}+\frac {3\,b\,x^5\,\mathrm {erfc}\left (\sqrt {b^2\,x^2}\right )}{16\,{\left (b^2\,x^2\right )}^{5/2}} \]

[In]

int(x^3*erf(b*x),x)

[Out]

(x^4*erf(b*x))/4 - (3*b*x^5)/(16*(b^2*x^2)^(5/2)) + (x^3*exp(-b^2*x^2))/(4*b*pi^(1/2)) + (3*x*exp(-b^2*x^2))/(
8*b^3*pi^(1/2)) + (3*b*x^5*erfc((b^2*x^2)^(1/2)))/(16*(b^2*x^2)^(5/2))