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 = 0.00665885, 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 \[ \frac{1}{2} \sqrt{\pi } \text{Erfi}(x) \]

Antiderivative was successfully verified.

[In]  Int[E^x^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 0.486834, size = 8, normalized size = 0.73 \[ \frac{\sqrt{\pi } \operatorname{erfi}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(x**2),x)

[Out]

sqrt(pi)*erfi(x)/2

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Mathematica [A]  time = 0.00213845, 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 = 0.002, size = 8, normalized size = 0.7 \[{\frac{{\it erfi} \left ( x \right ) \sqrt{\pi }}{2}} \]

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 [A]  time = 1.45931, size = 12, normalized size = 1.09 \[ -\frac{1}{2} i \, \sqrt{\pi } \operatorname{erf}\left (i \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(x^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.237532, size = 9, normalized size = 0.82 \[ \frac{1}{2} \, \sqrt{\pi } \operatorname{erfi}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(x^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.288247, size = 8, normalized size = 0.73 \[ \frac{\sqrt{\pi } \operatorname{erfi}{\left (x \right )}}{2} \]

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/XCAS [A]  time = 0.201048, size = 12, normalized size = 1.09 \[ \frac{1}{2} i \, \sqrt{\pi } \operatorname{erf}\left (-i \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(x^2),x, algorithm="giac")

[Out]

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