3.166 \(\int e^{x^2} \, dx\)

Optimal. Leaf size=11 \[ \frac{1}{2} \sqrt{\pi } \text{Erfi}(x) \]

[Out]

(Sqrt[Pi]*Erfi[x])/2

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Rubi [A]  time = 0.0027621, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 5, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2204} \[ \frac{1}{2} \sqrt{\pi } \text{Erfi}(x) \]

Antiderivative was successfully verified.

[In]

Int[E^x^2,x]

[Out]

(Sqrt[Pi]*Erfi[x])/2

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rubi steps

\begin{align*} \int e^{x^2} \, dx &=\frac{1}{2} \sqrt{\pi } \text{erfi}(x)\\ \end{align*}

Mathematica [A]  time = 0.0014612, size = 11, normalized size = 1. \[ \frac{1}{2} \sqrt{\pi } \text{Erfi}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[E^x^2,x]

[Out]

(Sqrt[Pi]*Erfi[x])/2

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Maple [A]  time = 0.003, size = 8, normalized size = 0.7 \begin{align*}{\frac{{\it erfi} \left ( x \right ) \sqrt{\pi }}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x^2),x)

[Out]

1/2*erfi(x)*Pi^(1/2)

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Maxima [C]  time = 0.94179, size = 12, normalized size = 1.09 \begin{align*} -\frac{1}{2} i \, \sqrt{\pi } \operatorname{erf}\left (i \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x^2),x, algorithm="maxima")

[Out]

-1/2*I*sqrt(pi)*erf(I*x)

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Fricas [A]  time = 1.60826, size = 30, normalized size = 2.73 \begin{align*} \frac{1}{2} \, \sqrt{\pi } \operatorname{erfi}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x^2),x, algorithm="fricas")

[Out]

1/2*sqrt(pi)*erfi(x)

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Sympy [A]  time = 0.242371, size = 8, normalized size = 0.73 \begin{align*} \frac{\sqrt{\pi } \operatorname{erfi}{\left (x \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x**2),x)

[Out]

sqrt(pi)*erfi(x)/2

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Giac [C]  time = 1.10665, size = 12, normalized size = 1.09 \begin{align*} \frac{1}{2} i \, \sqrt{\pi } \operatorname{erf}\left (-i \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x^2),x, algorithm="giac")

[Out]

1/2*I*sqrt(pi)*erf(-I*x)