3.99.16 \(\int (-10+x^x (-1-\log (x))) \, dx\)

Optimal. Leaf size=17 \[ \frac {1}{e^2}-x^x-5 \log \left (e^{2 x}\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 9, normalized size of antiderivative = 0.53, number of steps used = 4, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6742, 2553} \begin {gather*} -x^x-10 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-10 + x^x*(-1 - Log[x]),x]

[Out]

-10*x - x^x

Rule 2553

Int[Log[u_]*(u_)^((a_.)*(x_)), x_Symbol] :> Simp[u^(a*x)/a, x] - Int[SimplifyIntegrand[x*u^(a*x - 1)*D[u, x],
x], x] /; FreeQ[a, x] && InverseFunctionFreeQ[u, x]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-10 x+\int x^x (-1-\log (x)) \, dx\\ &=-10 x+\int \left (-x^x-x^x \log (x)\right ) \, dx\\ &=-10 x-\int x^x \, dx-\int x^x \log (x) \, dx\\ &=-10 x-x^x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 9, normalized size = 0.53 \begin {gather*} -10 x-x^x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-10 + x^x*(-1 - Log[x]),x]

[Out]

-10*x - x^x

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fricas [A]  time = 0.79, size = 9, normalized size = 0.53 \begin {gather*} -x^{x} - 10 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(x)-1)*exp(x*log(x))-10,x, algorithm="fricas")

[Out]

-x^x - 10*x

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giac [A]  time = 0.13, size = 9, normalized size = 0.53 \begin {gather*} -x^{x} - 10 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(x)-1)*exp(x*log(x))-10,x, algorithm="giac")

[Out]

-x^x - 10*x

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maple [A]  time = 0.03, size = 10, normalized size = 0.59

method	result	size
risch	\(-10 x -x^{x}\)	\(10\)
default	\(-10 x -{\mathrm e}^{x \ln \relax (x )}\)	\(12\)
norman	\(-10 x -{\mathrm e}^{x \ln \relax (x )}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-ln(x)-1)*exp(x*ln(x))-10,x,method=_RETURNVERBOSE)

[Out]

-10*x-x^x

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maxima [A]  time = 0.38, size = 9, normalized size = 0.53 \begin {gather*} -x^{x} - 10 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(x)-1)*exp(x*log(x))-10,x, algorithm="maxima")

[Out]

-x^x - 10*x

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mupad [B]  time = 5.51, size = 9, normalized size = 0.53 \begin {gather*} -10\,x-x^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(- exp(x*log(x))*(log(x) + 1) - 10,x)

[Out]

- 10*x - x^x

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sympy [A]  time = 0.23, size = 10, normalized size = 0.59 \begin {gather*} - 10 x - e^{x \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-ln(x)-1)*exp(x*ln(x))-10,x)

[Out]

-10*x - exp(x*log(x))

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