3.3.5 \(\int e^{-x^{2 n}} \, dx\) [205]

Optimal. Leaf size=23 \[ -\frac {x \operatorname {ExpIntegralE}\left (1-\frac {1}{2 n},x^{2 n}\right )}{2 n} \]

[Out]

-1/2*x*Ei(1-1/2/n,x^(2*n))/n

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Rubi [A]
time = 0.00, antiderivative size = 34, normalized size of antiderivative = 1.48, number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2239} \begin {gather*} -\frac {x \left (x^{2 n}\right )^{\left .-\frac {1}{2}\right /n} \Gamma \left (\frac {1}{2 n},x^{2 n}\right )}{2 n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(-x^(2*n)),x]

[Out]

-1/2*(x*Gamma[1/(2*n), x^(2*n)])/(n*(x^(2*n))^(1/(2*n)))

Rule 2239

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> Simp[(-F^a)*(c + d*x)*(Gamma[1/n, (-b)*(c + d
*x)^n*Log[F]]/(d*n*((-b)*(c + d*x)^n*Log[F])^(1/n))), x] /; FreeQ[{F, a, b, c, d, n}, x] &&  !IntegerQ[2/n]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=-\frac {x \left (x^{2 n}\right )^{\left .-\frac {1}{2}\right /n} \Gamma \left (\frac {1}{2 n},x^{2 n}\right )}{2 n}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.00, size = 34, normalized size = 1.48 \begin {gather*} -\frac {x \left (x^{2 n}\right )^{\left .-\frac {1}{2}\right /n} \Gamma \left (\frac {1}{2 n},x^{2 n}\right )}{2 n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-x^(2*n)),x]

[Out]

-1/2*(x*Gamma[1/(2*n), x^(2*n)])/(n*(x^(2*n))^(1/(2*n)))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.04, size = 185, normalized size = 8.04

method	result	size
meijerg	\(\frac {\frac {4 n^{2} x^{-2 n +1} \left (2 n \,x^{2 n}+2 n +1\right ) \left (x^{2 n}\right )^{-\frac {2 n +1}{4 n}} {\mathrm e}^{-\frac {x^{2 n}}{2}} M_{\frac {1}{2 n}-\frac {2 n +1}{4 n},\frac {2 n +1}{4 n}+\frac {1}{2}}\left (x^{2 n}\right )}{\left (2 n +1\right ) \left (4 n +1\right )}+\frac {2 n \,x^{-2 n +1} \left (2 n +1\right ) \left (x^{2 n}\right )^{-\frac {2 n +1}{4 n}} {\mathrm e}^{-\frac {x^{2 n}}{2}} M_{\frac {1}{2 n}-\frac {2 n +1}{4 n}+1,\frac {2 n +1}{4 n}+\frac {1}{2}}\left (x^{2 n}\right )}{4 n +1}}{2 n}\)	\(185\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-x^(2*n)),x,method=_RETURNVERBOSE)

[Out]

1/2/n*(4*n^2*x^(-2*n+1)*(2*n*x^(2*n)+2*n+1)/(2*n+1)/(4*n+1)*(x^(2*n))^(-1/4*(2*n+1)/n)*exp(-1/2*x^(2*n))*Whitt
akerM(1/2/n-1/4*(2*n+1)/n,1/4*(2*n+1)/n+1/2,x^(2*n))+2*n*x^(-2*n+1)*(2*n+1)/(4*n+1)*(x^(2*n))^(-1/4*(2*n+1)/n)
*exp(-1/2*x^(2*n))*WhittakerM(1/2/n-1/4*(2*n+1)/n+1,1/4*(2*n+1)/n+1/2,x^(2*n)))

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Maxima [A]
time = 0.39, size = 30, normalized size = 1.30 \begin {gather*} -\frac {x \Gamma \left (\frac {1}{2 \, n}, x^{2 \, n}\right )}{2 \, n {\left (x^{2 \, n}\right )}^{\frac {1}{2 \, n}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*x*gamma(1/2/n, x^(2*n))/(n*(x^(2*n))^(1/2/n))

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Fricas [F]
time = 0.59, size = 10, normalized size = 0.43 \begin {gather*} {\rm integral}\left (e^{\left (-x^{2 \, n}\right )}, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(e^(-x^(2*n)), x)

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Sympy [A]
time = 0.44, size = 29, normalized size = 1.26 \begin {gather*} \frac {\Gamma \left (\frac {1}{2 n}\right ) \gamma \left (\frac {1}{2 n}, x^{2 n}\right )}{4 n^{2} \Gamma \left (1 + \frac {1}{2 n}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

gamma(1/(2*n))*lowergamma(1/(2*n), x**(2*n))/(4*n**2*gamma(1 + 1/(2*n)))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(e^(-x^(2*n)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int {\mathrm {e}}^{-x^{2\,n}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-x^(2*n)),x)

[Out]

int(exp(-x^(2*n)), x)

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Chatgpt [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {not solved} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int(exp(-x^(2*n)),x)

[Out]

not solved

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