3.4.7 \(\int x^{-\log (x)} \, dx\) [307]

Optimal. Leaf size=23 \[ -\frac {1}{2} \sqrt [4]{e} \sqrt {\pi } \text {erf}\left (\frac {1}{2}-\log (x)\right ) \]

[Out]

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

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Rubi [F]
time = 0.01, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int x^{-\log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[x^(-Log[x]),x]

[Out]

Defer[Int][x^(-Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\int x^{-\log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.03, size = 25, normalized size = 1.09 \begin {gather*} \frac {1}{2} \sqrt [4]{e} \sqrt {\pi } \text {erf}\left (\frac {1}{2} (-1+2 \log (x))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(-Log[x]),x]

[Out]

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

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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int x^{-\ln \left (x \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-ln(x)),x)

[Out]

int(x^(-ln(x)),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-log(x)),x, algorithm="maxima")

[Out]

integrate(1/(x^log(x)), x)

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Fricas [A]
time = 0.59, size = 12, normalized size = 0.52 \begin {gather*} \frac {1}{2} \, \sqrt {\pi } \operatorname {erf}\left (\log \left (x\right ) - \frac {1}{2}\right ) e^{\frac {1}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-log(x)),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{- \log {\left (x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-ln(x)),x)

[Out]

Integral(x**(-log(x)), x)

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Giac [A]
time = 0.44, size = 12, normalized size = 0.52 \begin {gather*} \frac {1}{2} \, \sqrt {\pi } \operatorname {erf}\left (\log \left (x\right ) - \frac {1}{2}\right ) e^{\frac {1}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-log(x)),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^log(x),x)

[Out]

int(exp(-log(x)^2), x)

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Chatgpt [F] Failed to verify
time = 1.00, size = 9, normalized size = 0.39 \begin {gather*} \frac {\expIntegral \left (-\ln \left (x \right )^{2}\right )}{2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int(x^(-ln(x)),x)

[Out]

1/2*Ei(-ln(x)^2)

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