3.5.68 \(\int -\frac {1}{(2 x \log (x)+x \log (x) \log (\log (x))) \log (-2-\log (\log (x)))} \, dx\) [468]

Optimal. Leaf size=17 \[ e^2-\log (2 \log (-2-\log (\log (x)))) \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 11, normalized size of antiderivative = 0.65, number of steps used = 5, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2437, 2339, 29} \begin {gather*} -\log (\log (-\log (\log (x))-2)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-Log[Log[-2 - Log[Log[x]]]]

Rule 29

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

Rule 2339

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 2437

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[(f*(x/d))^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rubi steps

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

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Mathematica [A]
time = 0.04, size = 11, normalized size = 0.65 \begin {gather*} -\log (\log (-2-\log (\log (x)))) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-Log[Log[-2 - Log[Log[x]]]]

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Maple [A]
time = 0.26, size = 12, normalized size = 0.71

method	result	size
default	\(-\ln \left (\ln \left (-\ln \left (\ln \left (x \right )\right )-2\right )\right )\)	\(12\)
risch	\(-\ln \left (\ln \left (-\ln \left (\ln \left (x \right )\right )-2\right )\right )\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 11, normalized size = 0.65 \begin {gather*} -\log \left (\log \left (-\log \left (\log \left (x\right )\right ) - 2\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(log(-log(log(x)) - 2))

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Fricas [A]
time = 0.31, size = 11, normalized size = 0.65 \begin {gather*} -\log \left (\log \left (-\log \left (\log \left (x\right )\right ) - 2\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(log(-log(log(x)) - 2))

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Sympy [A]
time = 0.12, size = 12, normalized size = 0.71 \begin {gather*} - \log {\left (\log {\left (- \log {\left (\log {\left (x \right )} \right )} - 2 \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(log(-log(log(x)) - 2))

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Giac [A]
time = 0.39, size = 12, normalized size = 0.71 \begin {gather*} -\log \left ({\left | \log \left (-\log \left (\log \left (x\right )\right ) - 2\right ) \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(abs(log(-log(log(x)) - 2)))

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Mupad [B]
time = 0.97, size = 11, normalized size = 0.65 \begin {gather*} -\ln \left (\ln \left (-\ln \left (\ln \left (x\right )\right )-2\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(log(- log(log(x)) - 2))

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