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]

(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

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Rubi [A]  time = 0.179626, antiderivative size = 71, normalized size of antiderivative = 1., 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 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]

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]

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 = 0.00847, 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

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Maple [A]  time = 0.044, size = 59, normalized size = 0.8 \begin{align*}{\frac{1}{b} \left ( \left ( bx+a \right ) \left ({\it Erf} \left ( bx+a \right ) \right ) ^{2}+2\,{\frac{{\it Erf} \left ( bx+a \right ){{\rm e}^{- \left ( bx+a \right ) ^{2}}}}{\sqrt{\pi }}}-{\frac{\sqrt{2}{\it Erf} \left ( \left ( bx+a \right ) \sqrt{2} \right ) }{\sqrt{\pi }}} \right ) } \end{align*}

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)))

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [A]  time = 2.59558, size = 227, normalized size = 3.2 \begin{align*} \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}} \end{align*}

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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{erf}^{2}{\left (a + b x \right )}\, dx \end{align*}

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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{erf}\left (b x + a\right )^{2}\,{d x} \end{align*}

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)