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

Optimal result

Integrand size = 5, antiderivative size = 11 \[ \int e^{x^2} \, dx=\frac {1}{2} \sqrt {\pi } \text {erfi}(x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2235} \[ \int e^{x^2} \, dx=\frac {1}{2} \sqrt {\pi } \text {erfi}(x) \]

[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*} \text {integral}& = \frac {1}{2} \sqrt {\pi } \text {erfi}(x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int e^{x^2} \, dx=\frac {1}{2} \sqrt {\pi } \text {erfi}(x) \]

[In]

Integrate[E^x^2,x]

[Out]

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

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.73

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.64 \[ \int e^{x^2} \, dx=\frac {1}{2} \, \sqrt {\pi } \operatorname {erfi}\left (x\right ) \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.73 \[ \int e^{x^2} \, dx=\frac {\sqrt {\pi } \operatorname {erfi}{\left (x \right )}}{2} \]

[In]

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

[Out]

sqrt(pi)*erfi(x)/2

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.17 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82 \[ \int e^{x^2} \, dx=-\frac {1}{2} i \, \sqrt {\pi } \operatorname {erf}\left (i \, x\right ) \]

[In]

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

[Out]

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

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82 \[ \int e^{x^2} \, dx=\frac {1}{2} i \, \sqrt {\pi } \operatorname {erf}\left (-i \, x\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.64 \[ \int e^{x^2} \, dx=\frac {\sqrt {\pi }\,\mathrm {erfi}\left (x\right )}{2} \]

[In]

int(exp(x^2),x)

[Out]

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