\(\int x^4 \text {erf}(b x)^2 \, dx\) [29]

Optimal result

Integrand size = 10, antiderivative size = 165 \[ \int x^4 \text {erf}(b x)^2 \, dx=\frac {11 e^{-2 b^2 x^2} x}{20 b^4 \pi }+\frac {e^{-2 b^2 x^2} x^3}{5 b^2 \pi }+\frac {4 e^{-b^2 x^2} \text {erf}(b x)}{5 b^5 \sqrt {\pi }}+\frac {4 e^{-b^2 x^2} x^2 \text {erf}(b x)}{5 b^3 \sqrt {\pi }}+\frac {2 e^{-b^2 x^2} x^4 \text {erf}(b x)}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erf}(b x)^2-\frac {43 \text {erf}\left (\sqrt {2} b x\right )}{40 b^5 \sqrt {2 \pi }} \]

[Out]

11/20*x/b^4/exp(2*b^2*x^2)/Pi+1/5*x^3/b^2/exp(2*b^2*x^2)/Pi+1/5*x^5*erf(b*x)^2+4/5*erf(b*x)/b^5/exp(b^2*x^2)/P
i^(1/2)+4/5*x^2*erf(b*x)/b^3/exp(b^2*x^2)/Pi^(1/2)+2/5*x^4*erf(b*x)/b/exp(b^2*x^2)/Pi^(1/2)-43/80*erf(b*x*2^(1
/2))/b^5*2^(1/2)/Pi^(1/2)

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6499, 6520, 6517, 2236, 2243} \[ \int x^4 \text {erf}(b x)^2 \, dx=-\frac {43 \text {erf}\left (\sqrt {2} b x\right )}{40 \sqrt {2 \pi } b^5}+\frac {2 x^4 e^{-b^2 x^2} \text {erf}(b x)}{5 \sqrt {\pi } b}+\frac {x^3 e^{-2 b^2 x^2}}{5 \pi b^2}+\frac {4 e^{-b^2 x^2} \text {erf}(b x)}{5 \sqrt {\pi } b^5}+\frac {11 x e^{-2 b^2 x^2}}{20 \pi b^4}+\frac {4 x^2 e^{-b^2 x^2} \text {erf}(b x)}{5 \sqrt {\pi } b^3}+\frac {1}{5} x^5 \text {erf}(b x)^2 \]

[In]

Int[x^4*Erf[b*x]^2,x]

[Out]

(11*x)/(20*b^4*E^(2*b^2*x^2)*Pi) + x^3/(5*b^2*E^(2*b^2*x^2)*Pi) + (4*Erf[b*x])/(5*b^5*E^(b^2*x^2)*Sqrt[Pi]) +
(4*x^2*Erf[b*x])/(5*b^3*E^(b^2*x^2)*Sqrt[Pi]) + (2*x^4*Erf[b*x])/(5*b*E^(b^2*x^2)*Sqrt[Pi]) + (x^5*Erf[b*x]^2)
/5 - (43*Erf[Sqrt[2]*b*x])/(40*b^5*Sqrt[2*Pi])

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 6499

Int[Erf[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(Erf[b*x]^2/(m + 1)), x] - Dist[4*(b/(Sqrt[Pi]*(
m + 1))), Int[(x^(m + 1)*Erf[b*x])/E^(b^2*x^2), x], x] /; FreeQ[b, x] && (IGtQ[m, 0] || ILtQ[(m + 1)/2, 0])

Rule 6517

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

Rule 6520

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} x^5 \text {erf}(b x)^2-\frac {(4 b) \int e^{-b^2 x^2} x^5 \text {erf}(b x) \, dx}{5 \sqrt {\pi }} \\ & = \frac {2 e^{-b^2 x^2} x^4 \text {erf}(b x)}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erf}(b x)^2-\frac {4 \int e^{-2 b^2 x^2} x^4 \, dx}{5 \pi }-\frac {8 \int e^{-b^2 x^2} x^3 \text {erf}(b x) \, dx}{5 b \sqrt {\pi }} \\ & = \frac {e^{-2 b^2 x^2} x^3}{5 b^2 \pi }+\frac {4 e^{-b^2 x^2} x^2 \text {erf}(b x)}{5 b^3 \sqrt {\pi }}+\frac {2 e^{-b^2 x^2} x^4 \text {erf}(b x)}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erf}(b x)^2-\frac {3 \int e^{-2 b^2 x^2} x^2 \, dx}{5 b^2 \pi }-\frac {8 \int e^{-2 b^2 x^2} x^2 \, dx}{5 b^2 \pi }-\frac {8 \int e^{-b^2 x^2} x \text {erf}(b x) \, dx}{5 b^3 \sqrt {\pi }} \\ & = \frac {11 e^{-2 b^2 x^2} x}{20 b^4 \pi }+\frac {e^{-2 b^2 x^2} x^3}{5 b^2 \pi }+\frac {4 e^{-b^2 x^2} \text {erf}(b x)}{5 b^5 \sqrt {\pi }}+\frac {4 e^{-b^2 x^2} x^2 \text {erf}(b x)}{5 b^3 \sqrt {\pi }}+\frac {2 e^{-b^2 x^2} x^4 \text {erf}(b x)}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erf}(b x)^2-\frac {3 \int e^{-2 b^2 x^2} \, dx}{20 b^4 \pi }-\frac {2 \int e^{-2 b^2 x^2} \, dx}{5 b^4 \pi }-\frac {8 \int e^{-2 b^2 x^2} \, dx}{5 b^4 \pi } \\ & = \frac {11 e^{-2 b^2 x^2} x}{20 b^4 \pi }+\frac {e^{-2 b^2 x^2} x^3}{5 b^2 \pi }+\frac {4 e^{-b^2 x^2} \text {erf}(b x)}{5 b^5 \sqrt {\pi }}+\frac {4 e^{-b^2 x^2} x^2 \text {erf}(b x)}{5 b^3 \sqrt {\pi }}+\frac {2 e^{-b^2 x^2} x^4 \text {erf}(b x)}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erf}(b x)^2-\frac {2 \sqrt {\frac {2}{\pi }} \text {erf}\left (\sqrt {2} b x\right )}{5 b^5}-\frac {11 \text {erf}\left (\sqrt {2} b x\right )}{40 b^5 \sqrt {2 \pi }} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.64 \[ \int x^4 \text {erf}(b x)^2 \, dx=\frac {4 b e^{-2 b^2 x^2} x \left (11+4 b^2 x^2\right )+32 e^{-b^2 x^2} \sqrt {\pi } \left (2+2 b^2 x^2+b^4 x^4\right ) \text {erf}(b x)+16 b^5 \pi x^5 \text {erf}(b x)^2-43 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} b x\right )}{80 b^5 \pi } \]

[In]

Integrate[x^4*Erf[b*x]^2,x]

[Out]

((4*b*x*(11 + 4*b^2*x^2))/E^(2*b^2*x^2) + (32*Sqrt[Pi]*(2 + 2*b^2*x^2 + b^4*x^4)*Erf[b*x])/E^(b^2*x^2) + 16*b^
5*Pi*x^5*Erf[b*x]^2 - 43*Sqrt[2*Pi]*Erf[Sqrt[2]*b*x])/(80*b^5*Pi)

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.79

method	result	size
derivativedivides	\(\frac {\frac {\operatorname {erf}\left (b x \right )^{2} b^{5} x^{5}}{5}-\frac {4 \,\operatorname {erf}\left (b x \right ) \left (-\frac {{\mathrm e}^{-b^{2} x^{2}} x^{4} b^{4}}{2}-x^{2} {\mathrm e}^{-b^{2} x^{2}} b^{2}-{\mathrm e}^{-b^{2} x^{2}}\right )}{5 \sqrt {\pi }}+\frac {-\frac {43 \sqrt {2}\, \sqrt {\pi }\, \operatorname {erf}\left (b x \sqrt {2}\right )}{80}+\frac {11 \,{\mathrm e}^{-2 b^{2} x^{2}} b x}{20}+\frac {{\mathrm e}^{-2 b^{2} x^{2}} b^{3} x^{3}}{5}}{\pi }}{b^{5}}\)	\(131\)
default	\(\frac {\frac {\operatorname {erf}\left (b x \right )^{2} b^{5} x^{5}}{5}-\frac {4 \,\operatorname {erf}\left (b x \right ) \left (-\frac {{\mathrm e}^{-b^{2} x^{2}} x^{4} b^{4}}{2}-x^{2} {\mathrm e}^{-b^{2} x^{2}} b^{2}-{\mathrm e}^{-b^{2} x^{2}}\right )}{5 \sqrt {\pi }}+\frac {-\frac {43 \sqrt {2}\, \sqrt {\pi }\, \operatorname {erf}\left (b x \sqrt {2}\right )}{80}+\frac {11 \,{\mathrm e}^{-2 b^{2} x^{2}} b x}{20}+\frac {{\mathrm e}^{-2 b^{2} x^{2}} b^{3} x^{3}}{5}}{\pi }}{b^{5}}\)	\(131\)

[In]

int(x^4*erf(b*x)^2,x,method=_RETURNVERBOSE)

[Out]

1/b^5*(1/5*erf(b*x)^2*b^5*x^5-4/5*erf(b*x)/Pi^(1/2)*(-1/2/exp(b^2*x^2)*b^4*x^4-b^2*x^2/exp(b^2*x^2)-1/exp(b^2*
x^2))+4/5/Pi*(-43/64*2^(1/2)*Pi^(1/2)*erf(b*x*2^(1/2))+11/16/exp(b^2*x^2)^2*b*x+1/4/exp(b^2*x^2)^2*b^3*x^3))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.67 \[ \int x^4 \text {erf}(b x)^2 \, dx=\frac {16 \, \pi b^{6} x^{5} \operatorname {erf}\left (b x\right )^{2} + 32 \, \sqrt {\pi } {\left (b^{5} x^{4} + 2 \, b^{3} x^{2} + 2 \, b\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} - 43 \, \sqrt {2} \sqrt {\pi } \sqrt {b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {b^{2}} x\right ) + 4 \, {\left (4 \, b^{4} x^{3} + 11 \, b^{2} x\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{80 \, \pi b^{6}} \]

[In]

integrate(x^4*erf(b*x)^2,x, algorithm="fricas")

[Out]

1/80*(16*pi*b^6*x^5*erf(b*x)^2 + 32*sqrt(pi)*(b^5*x^4 + 2*b^3*x^2 + 2*b)*erf(b*x)*e^(-b^2*x^2) - 43*sqrt(2)*sq
rt(pi)*sqrt(b^2)*erf(sqrt(2)*sqrt(b^2)*x) + 4*(4*b^4*x^3 + 11*b^2*x)*e^(-2*b^2*x^2))/(pi*b^6)

Sympy [F]

\[ \int x^4 \text {erf}(b x)^2 \, dx=\int x^{4} \operatorname {erf}^{2}{\left (b x \right )}\, dx \]

[In]

integrate(x**4*erf(b*x)**2,x)

[Out]

Integral(x**4*erf(b*x)**2, x)

Maxima [F]

\[ \int x^4 \text {erf}(b x)^2 \, dx=\int { x^{4} \operatorname {erf}\left (b x\right )^{2} \,d x } \]

[In]

integrate(x^4*erf(b*x)^2,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.03 \[ \int x^4 \text {erf}(b x)^2 \, dx=\frac {1}{5} \, x^{5} \operatorname {erf}\left (b x\right )^{2} + \frac {b {\left (\frac {32 \, {\left (b^{4} x^{4} + 2 \, b^{2} x^{2} + 2\right )} \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{b^{6}} + \frac {b^{4} {\left (\frac {4 \, {\left (4 \, b^{2} x^{3} + 3 \, x\right )} e^{\left (-2 \, b^{2} x^{2}\right )}}{b^{4}} + \frac {3 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} b x\right )}{b^{5}}\right )} + 8 \, b^{2} {\left (\frac {4 \, x e^{\left (-2 \, b^{2} x^{2}\right )}}{b^{2}} + \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} b x\right )}{b^{3}}\right )} + \frac {32 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {2} b x\right )}{b}}{\sqrt {\pi } b^{5}}\right )}}{80 \, \sqrt {\pi }} \]

[In]

integrate(x^4*erf(b*x)^2,x, algorithm="giac")

[Out]

1/5*x^5*erf(b*x)^2 + 1/80*b*(32*(b^4*x^4 + 2*b^2*x^2 + 2)*erf(b*x)*e^(-b^2*x^2)/b^6 + (b^4*(4*(4*b^2*x^3 + 3*x
)*e^(-2*b^2*x^2)/b^4 + 3*sqrt(2)*sqrt(pi)*erf(-sqrt(2)*b*x)/b^5) + 8*b^2*(4*x*e^(-2*b^2*x^2)/b^2 + sqrt(2)*sqr
t(pi)*erf(-sqrt(2)*b*x)/b^3) + 32*sqrt(2)*sqrt(pi)*erf(-sqrt(2)*b*x)/b)/(sqrt(pi)*b^5))/sqrt(pi)

Mupad [B] (verification not implemented)

Time = 5.50 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.79 \[ \int x^4 \text {erf}(b x)^2 \, dx=\frac {x^5\,{\mathrm {erf}\left (b\,x\right )}^2}{5}+\frac {\frac {4\,\sqrt {\pi }\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{5}+\frac {b^3\,x^3\,{\mathrm {e}}^{-2\,b^2\,x^2}}{5}-\frac {43\,\sqrt {2}\,\sqrt {\pi }\,\mathrm {erf}\left (\sqrt {2}\,b\,x\right )}{80}+\frac {11\,b\,x\,{\mathrm {e}}^{-2\,b^2\,x^2}}{20}+\frac {4\,b^2\,x^2\,\sqrt {\pi }\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{5}+\frac {2\,b^4\,x^4\,\sqrt {\pi }\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )}{5}}{b^5\,\pi } \]

[In]

int(x^4*erf(b*x)^2,x)

[Out]

(x^5*erf(b*x)^2)/5 + ((4*pi^(1/2)*exp(-b^2*x^2)*erf(b*x))/5 + (b^3*x^3*exp(-2*b^2*x^2))/5 - (43*2^(1/2)*pi^(1/
2)*erf(2^(1/2)*b*x))/80 + (11*b*x*exp(-2*b^2*x^2))/20 + (4*b^2*x^2*pi^(1/2)*exp(-b^2*x^2)*erf(b*x))/5 + (2*b^4
*x^4*pi^(1/2)*exp(-b^2*x^2)*erf(b*x))/5)/(b^5*pi)