3.7.94 \(\int x^{-2-\frac {1}{x}} (1-\log (x)) \, dx\) [694]

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

[Out]

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

________________________________________________________________________________________

Rubi [F]
time = 0.05, 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^{-2-\frac {1}{x}} (1-\log (x)) \, dx \end {gather*}

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 = 0.01, size = 9, normalized size = 1.00 \begin {gather*} -x^{-1/x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.01, size = 18, normalized size = 2.00

method	result	size
risch	\(-x^{2} x^{-\frac {2 x +1}{x}}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.33, size = 9, normalized size = 1.00 \begin {gather*} -\frac {1}{x^{\left (\frac {1}{x}\right )}} \end {gather*}

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)

________________________________________________________________________________________

Fricas [A]
time = 0.35, size = 18, normalized size = 2.00 \begin {gather*} -\frac {x^{2}}{x^{\frac {2 \, x + 1}{x}}} \end {gather*}

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)

________________________________________________________________________________________

Sympy [A]
time = 0.06, size = 12, normalized size = 1.33 \begin {gather*} - x^{2} x^{-2 - \frac {1}{x}} \end {gather*}

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)

________________________________________________________________________________________

Giac [A]
time = 2.70, size = 16, normalized size = 1.78 \begin {gather*} -x e^{\left (-\frac {x \log \left (x\right ) + \log \left (x\right )}{x}\right )} \end {gather*}

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)

________________________________________________________________________________________

Mupad [B]
time = 3.58, size = 9, normalized size = 1.00 \begin {gather*} -\frac {1}{x^{1/x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x) - 1)/x^(1/x + 2),x)

[Out]

-1/x^(1/x)

________________________________________________________________________________________