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

Optimal. Leaf size=17 \[ -3+e^2-\frac {-1+x^{-1+x}}{x} \]

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Rubi [F]  time = 0.12, 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 \frac {-1+x^{-1+x} (2-x-x \log (x))}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

x^(-1) + 2*Defer[Int][x^(-3 + x), x] - Defer[Int][x^(-2 + x), x] - Log[x]*Defer[Int][x^(-2 + x), x] + Defer[In
t][Defer[Int][x^(-2 + x), x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {1}{x^2}-x^{-3+x} (-2+x+x \log (x))\right ) \, dx\\ &=\frac {1}{x}-\int x^{-3+x} (-2+x+x \log (x)) \, dx\\ &=\frac {1}{x}-\int \left (-2 x^{-3+x}+x^{-2+x}+x^{-2+x} \log (x)\right ) \, dx\\ &=\frac {1}{x}+2 \int x^{-3+x} \, dx-\int x^{-2+x} \, dx-\int x^{-2+x} \log (x) \, dx\\ &=\frac {1}{x}+2 \int x^{-3+x} \, dx-\log (x) \int x^{-2+x} \, dx-\int x^{-2+x} \, dx+\int \frac {\int x^{-2+x} \, dx}{x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 12, normalized size = 0.71 \begin {gather*} -\frac {-x+x^x}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((-x + x^x)/x^2)

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fricas [A]  time = 0.56, size = 12, normalized size = 0.71 \begin {gather*} -\frac {x^{x - 1} - 1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x^(x - 1) - 1)/x

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giac [A]  time = 0.47, size = 12, normalized size = 0.71 \begin {gather*} \frac {1}{x} - \frac {x^{x}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/x - x^x/x^2

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maple [A]  time = 0.06, size = 15, normalized size = 0.88

method	result	size
risch	\(-\frac {x^{x -1}}{x}+\frac {1}{x}\)	\(15\)
norman	\(\frac {1-{\mathrm e}^{\left (x -1\right ) \ln \relax (x )}}{x}\)	\(16\)
default	\(-\frac {{\mathrm e}^{\left (x -1\right ) \ln \relax (x )}}{x}+\frac {1}{x}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.62, size = 12, normalized size = 0.71 \begin {gather*} \frac {1}{x} - \frac {x^{x}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/x - x^x/x^2

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mupad [B]  time = 0.40, size = 11, normalized size = 0.65 \begin {gather*} \frac {x-x^x}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(log(x)*(x - 1))*(x + x*log(x) - 2) + 1)/x^2,x)

[Out]

(x - x^x)/x^2

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sympy [A]  time = 0.24, size = 12, normalized size = 0.71 \begin {gather*} - \frac {e^{\left (x - 1\right ) \log {\relax (x )}}}{x} + \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-exp((x - 1)*log(x))/x + 1/x

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