3.694 \(\int x^{-2-\frac{1}{x}} (1-\log (x)) \, dx\)

Optimal. Leaf size=9 \[ -x^{-1/x} \]

[Out]

-x^(-x^(-1))

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Rubi [F]  time = 0.0726652, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int x^{-2-\frac{1}{x}} (1-\log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^(-2 - x^(-1))*(1 - Log[x]),x]

[Out]

Defer[Int][x^(-2 - x^(-1)), x] - Log[x]*Defer[Int][x^(-2 - x^(-1)), x] + Defer[Int][Defer[Int][x^(-2 - x^(-1))
, x]/x, x]

Rubi steps

\begin{align*} \int x^{-2-\frac{1}{x}} (1-\log (x)) \, dx &=\int \left (x^{-2-\frac{1}{x}}-x^{-2-\frac{1}{x}} \log (x)\right ) \, dx\\ &=\int x^{-2-\frac{1}{x}} \, dx-\int x^{-2-\frac{1}{x}} \log (x) \, dx\\ &=-\left (\log (x) \int x^{-2-\frac{1}{x}} \, dx\right )+\int x^{-2-\frac{1}{x}} \, dx+\int \frac{\int x^{-2-\frac{1}{x}} \, dx}{x} \, dx\\ \end{align*}

Mathematica [A]  time = 0.0199442, size = 9, normalized size = 1. \[ -x^{-1/x} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-2 - x^(-1))*(1 - Log[x]),x]

[Out]

-x^(-x^(-1))

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Maple [A]  time = 0.032, size = 18, normalized size = 2. \begin{align*} -{x}^{2}{x}^{-{\frac{1+2\,x}{x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-2-1/x)*(1-ln(x)),x)

[Out]

-x^2*x^(-(1+2*x)/x)

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Maxima [A]  time = 1.16794, size = 12, normalized size = 1.33 \begin{align*} -\frac{1}{x^{\left (\frac{1}{x}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-2-1/x)*(1-log(x)),x, algorithm="maxima")

[Out]

-1/x^(1/x)

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Fricas [A]  time = 0.815074, size = 30, normalized size = 3.33 \begin{align*} -\frac{x^{2}}{x^{\frac{2 \, x + 1}{x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-2-1/x)*(1-log(x)),x, algorithm="fricas")

[Out]

-x^2/x^((2*x + 1)/x)

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Sympy [A]  time = 0.338044, size = 12, normalized size = 1.33 \begin{align*} - x^{2} x^{-2 - \frac{1}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-2-1/x)*(1-ln(x)),x)

[Out]

-x**2*x**(-2 - 1/x)

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Giac [A]  time = 1.29806, size = 22, normalized size = 2.44 \begin{align*} -x e^{\left (-\frac{x \log \left (x\right ) + \log \left (x\right )}{x}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-2-1/x)*(1-log(x)),x, algorithm="giac")

[Out]

-x*e^(-(x*log(x) + log(x))/x)