3.18 \(\int \text {erf}(a+b x) \, dx\)

Optimal. Leaf size=36 \[ \frac {(a+b x) \text {erf}(a+b x)}{b}+\frac {e^{-(a+b x)^2}}{\sqrt {\pi } b} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6349} \[ \frac {(a+b x) \text {Erf}(a+b x)}{b}+\frac {e^{-(a+b x)^2}}{\sqrt {\pi } b} \]

Antiderivative was successfully verified.

[In]

Int[Erf[a + b*x],x]

[Out]

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

Rule 6349

Int[Erf[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[((a + b*x)*Erf[a + b*x])/b, x] + Simp[1/(b*Sqrt[Pi]*E^(a + b*x)
^2), x] /; FreeQ[{a, b}, x]

Rubi steps

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

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Mathematica [A]  time = 0.05, size = 35, normalized size = 0.97 \[ \left (\frac {a}{b}+x\right ) \text {erf}(a+b x)+\frac {e^{-(a+b x)^2}}{\sqrt {\pi } b} \]

Antiderivative was successfully verified.

[In]

Integrate[Erf[a + b*x],x]

[Out]

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

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fricas [A]  time = 0.44, size = 47, normalized size = 1.31 \[ \frac {{\left (\pi b x + \pi a\right )} \operatorname {erf}\left (b x + a\right ) + \sqrt {\pi } e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{\pi b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.19, size = 59, normalized size = 1.64 \[ x \operatorname {erf}\left (b x + a\right ) - \frac {\frac {\sqrt {\pi } a \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} - \frac {e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}}{\sqrt {\pi }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 32, normalized size = 0.89 \[ \frac {\left (b x +a \right ) \erf \left (b x +a \right )+\frac {{\mathrm e}^{-\left (b x +a \right )^{2}}}{\sqrt {\pi }}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.33, size = 31, normalized size = 0.86 \[ \frac {{\left (b x + a\right )} \operatorname {erf}\left (b x + a\right ) + \frac {e^{\left (-{\left (b x + a\right )}^{2}\right )}}{\sqrt {\pi }}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.22, size = 48, normalized size = 1.33 \[ x\,\mathrm {erf}\left (a+b\,x\right )+\frac {a\,\mathrm {erf}\left (a+b\,x\right )}{b}+\frac {{\mathrm {e}}^{-b^2\,x^2}\,{\mathrm {e}}^{-a^2}\,{\mathrm {e}}^{-2\,a\,b\,x}}{b\,\sqrt {\pi }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erf(a + b*x),x)

[Out]

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

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sympy [A]  time = 0.62, size = 53, normalized size = 1.47 \[ \begin {cases} \frac {a \operatorname {erf}{\left (a + b x \right )}}{b} + x \operatorname {erf}{\left (a + b x \right )} + \frac {e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{\sqrt {\pi } b} & \text {for}\: b \neq 0 \\x \operatorname {erf}{\relax (a )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*erf(a + b*x)/b + x*erf(a + b*x) + exp(-a**2)*exp(-b**2*x**2)*exp(-2*a*b*x)/(sqrt(pi)*b), Ne(b, 0)
), (x*erf(a), True))

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