3.3.77 \(\int \text {erf}(a+x) \, dx\) [277]

Optimal. Leaf size=24 \[ \frac {e^{-(a+x)^2}}{\sqrt {\pi }}+(a+x) \text {erf}(a+x) \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6484} \begin {gather*} (a+x) \text {erf}(a+x)+\frac {e^{-(a+x)^2}}{\sqrt {\pi }} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Erf[a + x],x]

[Out]

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

Rule 6484

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

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Mathematica [A]
time = 0.02, size = 24, normalized size = 1.00 \begin {gather*} \frac {e^{-(a+x)^2}}{\sqrt {\pi }}+(a+x) \text {erf}(a+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Erf[a + x],x]

[Out]

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

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Mathics [A]
time = 2.07, size = 34, normalized size = 1.42 \begin {gather*} a \text {Erf}\left [a+x\right ]+x \text {Erf}\left [a+x\right ]+\frac {E^{-a^2-2 a x-x^2}}{\sqrt {\text {Pi}}} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[Erf[x+a],x]')

[Out]

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

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Maple [A]
time = 0.02, size = 22, normalized size = 0.92

method	result	size
derivativedivides	\(\left (a +x \right ) \erf \left (a +x \right )+\frac {{\mathrm e}^{-\left (a +x \right )^{2}}}{\sqrt {\pi }}\)	\(22\)
default	\(\left (a +x \right ) \erf \left (a +x \right )+\frac {{\mathrm e}^{-\left (a +x \right )^{2}}}{\sqrt {\pi }}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erf(a+x),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.33, size = 21, normalized size = 0.88 \begin {gather*} {\left (a + x\right )} \operatorname {erf}\left (a + x\right ) + \frac {e^{\left (-{\left (a + x\right )}^{2}\right )}}{\sqrt {\pi }} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.30, size = 37, normalized size = 1.54 \begin {gather*} \frac {{\left (\pi a + \pi x\right )} \operatorname {erf}\left (a + x\right ) + \sqrt {\pi } e^{\left (-a^{2} - 2 \, a x - x^{2}\right )}}{\pi } \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.24, size = 36, normalized size = 1.50 \begin {gather*} a \operatorname {erf}{\left (a + x \right )} + x \operatorname {erf}{\left (a + x \right )} + \frac {e^{- a^{2}} e^{- x^{2}} e^{- 2 a x}}{\sqrt {\pi }} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(a+x),x)

[Out]

a*erf(a + x) + x*erf(a + x) + exp(-a**2)*exp(-x**2)*exp(-2*a*x)/sqrt(pi)

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Giac [A]
time = 0.00, size = 50, normalized size = 2.08 \begin {gather*} x \mathrm {erf}\left (a+x\right )-\frac {2 \sqrt {\pi } \left (-\frac {\mathrm {e}^{-\left (a^{2}+2 a x+x^{2}\right )}}{2}-\frac {1}{2} a \sqrt {\pi } \mathrm {erf}\left (x+\frac {2}{2} a\right )\right )}{\pi } \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(a+x),x)

[Out]

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

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Mupad [B]
time = 0.07, size = 21, normalized size = 0.88 \begin {gather*} \mathrm {erf}\left (a+x\right )\,\left (a+x\right )+\frac {{\mathrm {e}}^{-{\left (a+x\right )}^2}}{\sqrt {\pi }} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erf(a + x),x)

[Out]

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

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