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

Optimal. Leaf size=13 \[ \log \left (\frac {e^{-2 x} x}{\log (\log (x))}\right ) \]

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Rubi [A]  time = 0.16, antiderivative size = 12, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {6688, 2302, 29} \begin {gather*} -2 x+\log (x)-\log (\log (\log (x))) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2*x + Log[x] - Log[Log[Log[x]]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 6688

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

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 12, normalized size = 0.92 \begin {gather*} -2 x+\log (x)-\log (\log (\log (x))) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2*x + Log[x] - Log[Log[Log[x]]]

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fricas [A]  time = 0.64, size = 12, normalized size = 0.92 \begin {gather*} -2 \, x + \log \relax (x) - \log \left (\log \left (\log \relax (x)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x + log(x) - log(log(log(x)))

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giac [A]  time = 0.13, size = 12, normalized size = 0.92 \begin {gather*} -2 \, x + \log \relax (x) - \log \left (\log \left (\log \relax (x)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x + log(x) - log(log(log(x)))

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maple [A]  time = 0.04, size = 13, normalized size = 1.00

method	result	size
default	\(\ln \relax (x )-2 x -\ln \left (\ln \left (\ln \relax (x )\right )\right )\)	\(13\)
norman	\(\ln \relax (x )-2 x -\ln \left (\ln \left (\ln \relax (x )\right )\right )\)	\(13\)
risch	\(\ln \relax (x )-2 x -\ln \left (\ln \left (\ln \relax (x )\right )\right )\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(x)-2*x-ln(ln(ln(x)))

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maxima [A]  time = 0.37, size = 12, normalized size = 0.92 \begin {gather*} -2 \, x + \log \relax (x) - \log \left (\log \left (\log \relax (x)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x + log(x) - log(log(log(x)))

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mupad [B]  time = 4.30, size = 12, normalized size = 0.92 \begin {gather*} \ln \relax (x)-\ln \left (\ln \left (\ln \relax (x)\right )\right )-2\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x) - log(log(log(x))) - 2*x

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sympy [A]  time = 0.25, size = 12, normalized size = 0.92 \begin {gather*} - 2 x + \log {\relax (x )} - \log {\left (\log {\left (\log {\relax (x )} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x + log(x) - log(log(log(x)))

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