\(\int \text {erf}(b x)^2 \, dx\) [31]

Optimal result

Integrand size = 6, antiderivative size = 56 \[ \int \text {erf}(b x)^2 \, dx=\frac {2 e^{-b^2 x^2} \text {erf}(b x)}{b \sqrt {\pi }}+x \text {erf}(b x)^2-\frac {\sqrt {\frac {2}{\pi }} \text {erf}\left (\sqrt {2} b x\right )}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6487, 12, 6517, 2236} \[ \int \text {erf}(b x)^2 \, dx=\frac {2 e^{-b^2 x^2} \text {erf}(b x)}{\sqrt {\pi } b}+x \text {erf}(b x)^2-\frac {\sqrt {\frac {2}{\pi }} \text {erf}\left (\sqrt {2} b x\right )}{b} \]

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 6487

Int[Erf[(a_.) + (b_.)*(x_)]^2, x_Symbol] :> Simp[(a + b*x)*(Erf[a + b*x]^2/b), x] - Dist[4/Sqrt[Pi], Int[(a +
b*x)*(Erf[a + b*x]/E^(a + b*x)^2), x], x] /; FreeQ[{a, b}, x]

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]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00 \[ \int \text {erf}(b x)^2 \, dx=\frac {2 e^{-b^2 x^2} \text {erf}(b x)}{b \sqrt {\pi }}+x \text {erf}(b x)^2-\frac {\sqrt {\frac {2}{\pi }} \text {erf}\left (\sqrt {2} b x\right )}{b} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.86

method	result	size
derivativedivides	\(\frac {\operatorname {erf}\left (b x \right )^{2} b x +\frac {2 \,\operatorname {erf}\left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}}{\sqrt {\pi }}-\frac {\sqrt {2}\, \operatorname {erf}\left (b x \sqrt {2}\right )}{\sqrt {\pi }}}{b}\)	\(48\)
default	\(\frac {\operatorname {erf}\left (b x \right )^{2} b x +\frac {2 \,\operatorname {erf}\left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}}{\sqrt {\pi }}-\frac {\sqrt {2}\, \operatorname {erf}\left (b x \sqrt {2}\right )}{\sqrt {\pi }}}{b}\)	\(48\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.12 \[ \int \text {erf}(b x)^2 \, dx=\frac {\pi b^{2} x \operatorname {erf}\left (b x\right )^{2} + 2 \, \sqrt {\pi } b \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} - \sqrt {2} \sqrt {\pi } \sqrt {b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {b^{2}} x\right )}{\pi b^{2}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(erf(b*x)**2, x)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.11 \[ \int \text {erf}(b x)^2 \, dx=\frac {{\left (\sqrt {\pi } b x \operatorname {erf}\left (b x\right )^{2} e^{\left (b^{2} x^{2}\right )} + 2 \, \operatorname {erf}\left (b x\right )\right )} e^{\left (-b^{2} x^{2}\right )}}{\sqrt {\pi } b} - \frac {\sqrt {2} \operatorname {erf}\left (\sqrt {2} b x\right )}{\sqrt {\pi } b} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.86 \[ \int \text {erf}(b x)^2 \, dx=x \operatorname {erf}\left (b x\right )^{2} + \frac {b {\left (\frac {2 \, \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{b^{2}} + \frac {\sqrt {2} \operatorname {erf}\left (-\sqrt {2} b x\right )}{b^{2}}\right )}}{\sqrt {\pi }} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.79 \[ \int \text {erf}(b x)^2 \, dx=x\,{\mathrm {erf}\left (b\,x\right )}^2+\frac {2\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erf}\left (b\,x\right )-\sqrt {2}\,\mathrm {erf}\left (\sqrt {2}\,b\,x\right )}{b\,\sqrt {\pi }} \]

[In]

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

[Out]

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