3.37 \(\int \text {erf}(a+b x)^2 \, dx\)

Optimal. Leaf size=71 \[ \frac {(a+b x) \text {erf}(a+b x)^2}{b}+\frac {2 e^{-(a+b x)^2} \text {erf}(a+b x)}{\sqrt {\pi } b}-\frac {\sqrt {\frac {2}{\pi }} \text {erf}\left (\sqrt {2} (a+b x)\right )}{b} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.18, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6352, 6382, 2205} \[ \frac {(a+b x) \text {Erf}(a+b x)^2}{b}+\frac {2 e^{-(a+b x)^2} \text {Erf}(a+b x)}{\sqrt {\pi } b}-\frac {\sqrt {\frac {2}{\pi }} \text {Erf}\left (\sqrt {2} (a+b x)\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[Erf[a + b*x]^2,x]

[Out]

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

Rule 2205

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 6352

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 6382

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 66, normalized size = 0.93 \[ \frac {(a+b x) \text {erf}(a+b x)^2+\frac {2 e^{-(a+b x)^2} \text {erf}(a+b x)}{\sqrt {\pi }}-\sqrt {\frac {2}{\pi }} \text {erf}\left (\sqrt {2} (a+b x)\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Erf[a + b*x]^2,x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.47, size = 91, normalized size = 1.28 \[ \frac {2 \, \sqrt {\pi } b \operatorname {erf}\left (b x + a\right ) e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )} + {\left (\pi b^{2} x + \pi a b\right )} \operatorname {erf}\left (b x + a\right )^{2} - \sqrt {2} \sqrt {\pi } \sqrt {b^{2}} \operatorname {erf}\left (\frac {\sqrt {2} \sqrt {b^{2}} {\left (b x + a\right )}}{b}\right )}{\pi b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {erf}\left (b x + a\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(erf(b*x + a)^2, x)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 59, normalized size = 0.83 \[ \frac {\left (b x +a \right ) \erf \left (b x +a \right )^{2}+\frac {2 \erf \left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{\sqrt {\pi }}-\frac {\sqrt {2}\, \erf \left (\left (b x +a \right ) \sqrt {2}\right )}{\sqrt {\pi }}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ x \operatorname {erf}\left (b x + a\right )^{2} - \frac {4 \, b e^{\left (-a^{2}\right )} \int x \operatorname {erf}\left (b x + a\right ) e^{\left (-b^{2} x^{2} - 2 \, a b x\right )}\,{d x}}{\sqrt {\pi }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.13, size = 79, normalized size = 1.11 \[ x\,{\mathrm {erf}\left (a+b\,x\right )}^2+\frac {a\,{\mathrm {erf}\left (a+b\,x\right )}^2}{b}-\frac {\sqrt {2}\,\mathrm {erf}\left (\sqrt {2}\,\left (a+b\,x\right )\right )}{b\,\sqrt {\pi }}+\frac {2\,\mathrm {erf}\left (a+b\,x\right )\,{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2}}{b\,\sqrt {\pi }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erf(a + b*x)^2,x)

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {erf}^{2}{\left (a + b x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(erf(a + b*x)**2, x)

________________________________________________________________________________________