3.2.66 \(\int e^{x^2} \, dx\) [166]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2235} \begin {gather*} \frac {1}{2} \sqrt {\pi } \text {Erfi}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^x^2,x]

[Out]

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

Rule 2235

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*}

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Mathematica [A]
time = 0.01, size = 11, normalized size = 1.00 \begin {gather*} \frac {1}{2} \sqrt {\pi } \text {erfi}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^x^2,x]

[Out]

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

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Maple [A]
time = 0.02, size = 8, normalized size = 0.73

method	result	size
default	\(\frac {\erfi \left (x \right ) \sqrt {\pi }}{2}\)	\(8\)
meijerg	\(\frac {\erfi \left (x \right ) \sqrt {\pi }}{2}\)	\(8\)
risch	\(\frac {\erfi \left (x \right ) \sqrt {\pi }}{2}\)	\(8\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x^2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [C] Result contains complex when optimal does not.
time = 1.70, size = 9, normalized size = 0.82 \begin {gather*} -\frac {1}{2} i \, \sqrt {\pi } \operatorname {erf}\left (i \, x\right ) \end {gather*}

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 = 0.51, size = 7, normalized size = 0.64 \begin {gather*} \frac {1}{2} \, \sqrt {\pi } \operatorname {erfi}\left (x\right ) \end {gather*}

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 = 0.07, size = 8, normalized size = 0.73 \begin {gather*} \frac {\sqrt {\pi } \operatorname {erfi}{\left (x \right )}}{2} \end {gather*}

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] Result contains complex when optimal does not.
time = 0.94, size = 9, normalized size = 0.82 \begin {gather*} \frac {1}{2} i \, \sqrt {\pi } \operatorname {erf}\left (-i \, x\right ) \end {gather*}

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)

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Mupad [B]
time = 0.02, size = 7, normalized size = 0.64 \begin {gather*} \frac {\sqrt {\pi }\,\mathrm {erfi}\left (x\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x^2),x)

[Out]

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

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