\(\int \text {erf}(a+b x)^2 \, dx\) [37]

Optimal result

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

[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] (verified)

Time = 0.18 (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 = {6487, 6517, 2236} \[ \int \text {erf}(a+b x)^2 \, dx=\frac {(a+b x) \text {erf}(a+b x)^2}{b}+\frac {2 e^{-(a+b x)^2} \text {erf}(a+b x)}{\sqrt {\pi } b}-\frac {\sqrt {\frac {2}{\pi }} \text {erf}\left (\sqrt {2} (a+b x)\right )}{b} \]

[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 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}& = \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 \text {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 \text {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] (verified)

Time = 0.01 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.93 \[ \int \text {erf}(a+b x)^2 \, dx=\frac {\frac {2 e^{-(a+b x)^2} \text {erf}(a+b x)}{\sqrt {\pi }}+(a+b x) \text {erf}(a+b x)^2-\sqrt {\frac {2}{\pi }} \text {erf}\left (\sqrt {2} (a+b x)\right )}{b} \]

[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

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(\frac {\operatorname {erf}\left (b x +a \right )^{2} \left (b x +a \right )+\frac {2 \,\operatorname {erf}\left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{\sqrt {\pi }}-\frac {\sqrt {2}\, \operatorname {erf}\left (\left (b x +a \right ) \sqrt {2}\right )}{\sqrt {\pi }}}{b}\)	\(59\)
default	\(\frac {\operatorname {erf}\left (b x +a \right )^{2} \left (b x +a \right )+\frac {2 \,\operatorname {erf}\left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{\sqrt {\pi }}-\frac {\sqrt {2}\, \operatorname {erf}\left (\left (b x +a \right ) \sqrt {2}\right )}{\sqrt {\pi }}}{b}\)	\(59\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.28 \[ \int \text {erf}(a+b x)^2 \, dx=\frac {2 \, \sqrt {\pi } b \operatorname {erf}\left (b x + a\right ) e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )} + {\left (\pi b^{2} x + \pi a b\right )} \operatorname {erf}\left (b x + a\right )^{2} - \sqrt {2} \sqrt {\pi } \sqrt {b^{2}} \operatorname {erf}\left (\frac {\sqrt {2} \sqrt {b^{2}} {\left (b x + a\right )}}{b}\right )}{\pi b^{2}} \]

[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)

Sympy [F]

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

[In]

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

[Out]

Integral(erf(a + b*x)**2, x)

Maxima [F]

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

[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)

Giac [F]

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

[In]

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

[Out]

integrate(erf(b*x + a)^2, x)

Mupad [B] (verification not implemented)

Time = 5.08 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.11 \[ \int \text {erf}(a+b x)^2 \, dx=x\,{\mathrm {erf}\left (a+b\,x\right )}^2+\frac {a\,{\mathrm {erf}\left (a+b\,x\right )}^2}{b}-\frac {\sqrt {2}\,\mathrm {erf}\left (\sqrt {2}\,\left (a+b\,x\right )\right )}{b\,\sqrt {\pi }}+\frac {2\,\mathrm {erf}\left (a+b\,x\right )\,{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2}}{b\,\sqrt {\pi }} \]

[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))