3.3.31 \(\int e^{-x^4} x^5 \, dx\) [231]

Optimal. Leaf size=28 \[ -\frac {1}{4} e^{-x^4} x^2+\frac {1}{8} \sqrt {\pi } \text {erf}\left (x^2\right ) \]

[Out]

-1/4*x^2/exp(x^4)+1/8*Pi^(1/2)*erf(x^2)

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Rubi [A]
time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2243, 2242, 2236} \begin {gather*} \frac {1}{8} \sqrt {\pi } \text {erf}\left (x^2\right )-\frac {1}{4} e^{-x^4} x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5/E^x^4,x]

[Out]

-1/4*x^2/E^x^4 + (Sqrt[Pi]*Erf[x^2])/8

Rule 2236

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

Rule 2242

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dist[1/(d*(m + 1))
, Subst[Int[F^(a + b*x^2), x], x, (c + d*x)^(m + 1)], x] /; FreeQ[{F, a, b, c, d, m, n}, x] && EqQ[n, 2*(m + 1
)]

Rule 2243

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m
- n + 1)*(F^(a + b*(c + d*x)^n)/(b*d*n*Log[F])), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2*((m + 1)/n)] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=-\frac {1}{4} e^{-x^4} x^2+\frac {1}{2} \int e^{-x^4} x \, dx\\ &=-\frac {1}{4} e^{-x^4} x^2+\frac {1}{4} \text {Subst}\left (\int e^{-x^2} \, dx,x,x^2\right )\\ &=-\frac {1}{4} e^{-x^4} x^2+\frac {1}{8} \sqrt {\pi } \text {erf}\left (x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.02, size = 28, normalized size = 1.00 \begin {gather*} -\frac {1}{4} e^{-x^4} x^2+\frac {1}{8} \sqrt {\pi } \text {erf}\left (x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5/E^x^4,x]

[Out]

-1/4*x^2/E^x^4 + (Sqrt[Pi]*Erf[x^2])/8

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Maple [A]
time = 0.02, size = 22, normalized size = 0.79

method	result	size
meijerg	\(-\frac {x^{2} {\mathrm e}^{-x^{4}}}{4}+\frac {\sqrt {\pi }\, \erf \left (x^{2}\right )}{8}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*exp(-x^4),x,method=_RETURNVERBOSE)

[Out]

-1/4*x^2*exp(-x^4)+1/8*Pi^(1/2)*erf(x^2)

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Maxima [A]
time = 0.35, size = 21, normalized size = 0.75 \begin {gather*} -\frac {1}{4} \, x^{2} e^{\left (-x^{4}\right )} + \frac {1}{8} \, \sqrt {\pi } \operatorname {erf}\left (x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*exp(-x^4),x, algorithm="maxima")

[Out]

-1/4*x^2*e^(-x^4) + 1/8*sqrt(pi)*erf(x^2)

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Fricas [A]
time = 0.59, size = 21, normalized size = 0.75 \begin {gather*} -\frac {1}{4} \, x^{2} e^{\left (-x^{4}\right )} + \frac {1}{8} \, \sqrt {\pi } \operatorname {erf}\left (x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*exp(-x^4),x, algorithm="fricas")

[Out]

-1/4*x^2*e^(-x^4) + 1/8*sqrt(pi)*erf(x^2)

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Sympy [A]
time = 0.58, size = 20, normalized size = 0.71 \begin {gather*} - \frac {x^{2} e^{- x^{4}}}{4} + \frac {\sqrt {\pi } \operatorname {erf}{\left (x^{2} \right )}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*exp(-x**4),x)

[Out]

-x**2*exp(-x**4)/4 + sqrt(pi)*erf(x**2)/8

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Giac [A]
time = 0.50, size = 35, normalized size = 1.25 \begin {gather*} -\frac {\sqrt {\pi } {\left (\frac {2 \, \sqrt {x^{4}} e^{\left (-x^{4}\right )}}{\sqrt {\pi }} - \operatorname {erf}\left (\sqrt {x^{4}}\right )\right )} {\left | x \right |}}{8 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*exp(-x^4),x, algorithm="giac")

[Out]

-1/8*sqrt(pi)*(2*sqrt(x^4)*e^(-x^4)/sqrt(pi) - erf(sqrt(x^4)))*abs(x)/x

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Mupad [B]
time = 0.11, size = 21, normalized size = 0.75 \begin {gather*} \frac {\sqrt {\pi }\,\mathrm {erf}\left (x^2\right )}{8}-\frac {x^2\,{\mathrm {e}}^{-x^4}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*exp(-x^4),x)

[Out]

(pi^(1/2)*erf(x^2))/8 - (x^2*exp(-x^4))/4

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Chatgpt [F] Failed to verify
time = 1.00, size = 23, normalized size = 0.82 \begin {gather*} -\frac {x^{4} {\mathrm e}^{-x^{4}}}{4}-\frac {5 x^{\frac {1}{4}} {\mathrm e}^{-x^{4}}}{16} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int(x^5*exp(-x^4),x)

[Out]

-1/4*x^4*exp(-x^4)-5/16*x^(1/4)*exp(-x^4)

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