3.25.11 \(\int \frac {4+(4 x-2 x^3) \log (x)-4 x^2 \log (x) \log (\log (x))-2 x \log (x) \log ^2(\log (x))}{((4 x^2+x^3+2 x^4) \log (x)+(4 x+2 x^2+4 x^3) \log (x) \log (\log (x))+(x+2 x^2) \log (x) \log ^2(\log (x))) \log ^2(\frac {-4-x-2 x^2+(-1-2 x) \log (\log (x))}{x+\log (\log (x))})} \, dx\) [2411]

Optimal. Leaf size=17 \[ \frac {1}{\log \left (-1-2 x-\frac {4}{x+\log (\log (x))}\right )} \]

[Out]

1/ln(-2*x-4/(ln(ln(x))+x)-1)

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Rubi [F]
time = 3.42, 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 {4+\left (4 x-2 x^3\right ) \log (x)-4 x^2 \log (x) \log (\log (x))-2 x \log (x) \log ^2(\log (x))}{\left (\left (4 x^2+x^3+2 x^4\right ) \log (x)+\left (4 x+2 x^2+4 x^3\right ) \log (x) \log (\log (x))+\left (x+2 x^2\right ) \log (x) \log ^2(\log (x))\right ) \log ^2\left (\frac {-4-x-2 x^2+(-1-2 x) \log (\log (x))}{x+\log (\log (x))}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4 + (4*x - 2*x^3)*Log[x] - 4*x^2*Log[x]*Log[Log[x]] - 2*x*Log[x]*Log[Log[x]]^2)/(((4*x^2 + x^3 + 2*x^4)*L
og[x] + (4*x + 2*x^2 + 4*x^3)*Log[x]*Log[Log[x]] + (x + 2*x^2)*Log[x]*Log[Log[x]]^2)*Log[(-4 - x - 2*x^2 + (-1
- 2*x)*Log[Log[x]])/(x + Log[Log[x]])]^2),x]

[Out]

4*Defer[Int][1/((4*x + x^2 + 2*x^3 + 4*Log[Log[x]] + 2*x*Log[Log[x]] + 4*x^2*Log[Log[x]] + Log[Log[x]]^2 + 2*x
*Log[Log[x]]^2)*Log[-((4 + x + 2*x^2 + Log[Log[x]] + 2*x*Log[Log[x]])/(x + Log[Log[x]]))]^2), x] - 2*Defer[Int
][x^2/((4*x + x^2 + 2*x^3 + 4*Log[Log[x]] + 2*x*Log[Log[x]] + 4*x^2*Log[Log[x]] + Log[Log[x]]^2 + 2*x*Log[Log[
x]]^2)*Log[-((4 + x + 2*x^2 + Log[Log[x]] + 2*x*Log[Log[x]])/(x + Log[Log[x]]))]^2), x] + 4*Defer[Int][1/(x*Lo
g[x]*(4*x + x^2 + 2*x^3 + 4*Log[Log[x]] + 2*x*Log[Log[x]] + 4*x^2*Log[Log[x]] + Log[Log[x]]^2 + 2*x*Log[Log[x]
]^2)*Log[-((4 + x + 2*x^2 + Log[Log[x]] + 2*x*Log[Log[x]])/(x + Log[Log[x]]))]^2), x] - 4*Defer[Int][(x*Log[Lo
g[x]])/((4*x + x^2 + 2*x^3 + 4*Log[Log[x]] + 2*x*Log[Log[x]] + 4*x^2*Log[Log[x]] + Log[Log[x]]^2 + 2*x*Log[Log
[x]]^2)*Log[-((4 + x + 2*x^2 + Log[Log[x]] + 2*x*Log[Log[x]])/(x + Log[Log[x]]))]^2), x] - 2*Defer[Int][Log[Lo
g[x]]^2/((4*x + x^2 + 2*x^3 + 4*Log[Log[x]] + 2*x*Log[Log[x]] + 4*x^2*Log[Log[x]] + Log[Log[x]]^2 + 2*x*Log[Lo
g[x]]^2)*Log[-((4 + x + 2*x^2 + Log[Log[x]] + 2*x*Log[Log[x]])/(x + Log[Log[x]]))]^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (2+2 x \log (x)-x^3 \log (x)-2 x^2 \log (x) \log (\log (x))-x \log (x) \log ^2(\log (x))\right )}{x \log (x) \left (4 x+x^2+2 x^3+4 \log (\log (x))+2 x \log (\log (x))+4 x^2 \log (\log (x))+\log ^2(\log (x))+2 x \log ^2(\log (x))\right ) \log ^2\left (\frac {-4-x-2 x^2+(-1-2 x) \log (\log (x))}{x+\log (\log (x))}\right )} \, dx\\ &=2 \int \frac {2+2 x \log (x)-x^3 \log (x)-2 x^2 \log (x) \log (\log (x))-x \log (x) \log ^2(\log (x))}{x \log (x) \left (4 x+x^2+2 x^3+4 \log (\log (x))+2 x \log (\log (x))+4 x^2 \log (\log (x))+\log ^2(\log (x))+2 x \log ^2(\log (x))\right ) \log ^2\left (\frac {-4-x-2 x^2+(-1-2 x) \log (\log (x))}{x+\log (\log (x))}\right )} \, dx\\ &=2 \int \left (\frac {2}{\left (4 x+x^2+2 x^3+4 \log (\log (x))+2 x \log (\log (x))+4 x^2 \log (\log (x))+\log ^2(\log (x))+2 x \log ^2(\log (x))\right ) \log ^2\left (-\frac {4+x+2 x^2+\log (\log (x))+2 x \log (\log (x))}{x+\log (\log (x))}\right )}-\frac {x^2}{\left (4 x+x^2+2 x^3+4 \log (\log (x))+2 x \log (\log (x))+4 x^2 \log (\log (x))+\log ^2(\log (x))+2 x \log ^2(\log (x))\right ) \log ^2\left (-\frac {4+x+2 x^2+\log (\log (x))+2 x \log (\log (x))}{x+\log (\log (x))}\right )}+\frac {2}{x \log (x) \left (4 x+x^2+2 x^3+4 \log (\log (x))+2 x \log (\log (x))+4 x^2 \log (\log (x))+\log ^2(\log (x))+2 x \log ^2(\log (x))\right ) \log ^2\left (-\frac {4+x+2 x^2+\log (\log (x))+2 x \log (\log (x))}{x+\log (\log (x))}\right )}-\frac {2 x \log (\log (x))}{\left (4 x+x^2+2 x^3+4 \log (\log (x))+2 x \log (\log (x))+4 x^2 \log (\log (x))+\log ^2(\log (x))+2 x \log ^2(\log (x))\right ) \log ^2\left (-\frac {4+x+2 x^2+\log (\log (x))+2 x \log (\log (x))}{x+\log (\log (x))}\right )}-\frac {\log ^2(\log (x))}{\left (4 x+x^2+2 x^3+4 \log (\log (x))+2 x \log (\log (x))+4 x^2 \log (\log (x))+\log ^2(\log (x))+2 x \log ^2(\log (x))\right ) \log ^2\left (-\frac {4+x+2 x^2+\log (\log (x))+2 x \log (\log (x))}{x+\log (\log (x))}\right )}\right ) \, dx\\ &=-\left (2 \int \frac {x^2}{\left (4 x+x^2+2 x^3+4 \log (\log (x))+2 x \log (\log (x))+4 x^2 \log (\log (x))+\log ^2(\log (x))+2 x \log ^2(\log (x))\right ) \log ^2\left (-\frac {4+x+2 x^2+\log (\log (x))+2 x \log (\log (x))}{x+\log (\log (x))}\right )} \, dx\right )-2 \int \frac {\log ^2(\log (x))}{\left (4 x+x^2+2 x^3+4 \log (\log (x))+2 x \log (\log (x))+4 x^2 \log (\log (x))+\log ^2(\log (x))+2 x \log ^2(\log (x))\right ) \log ^2\left (-\frac {4+x+2 x^2+\log (\log (x))+2 x \log (\log (x))}{x+\log (\log (x))}\right )} \, dx+4 \int \frac {1}{\left (4 x+x^2+2 x^3+4 \log (\log (x))+2 x \log (\log (x))+4 x^2 \log (\log (x))+\log ^2(\log (x))+2 x \log ^2(\log (x))\right ) \log ^2\left (-\frac {4+x+2 x^2+\log (\log (x))+2 x \log (\log (x))}{x+\log (\log (x))}\right )} \, dx+4 \int \frac {1}{x \log (x) \left (4 x+x^2+2 x^3+4 \log (\log (x))+2 x \log (\log (x))+4 x^2 \log (\log (x))+\log ^2(\log (x))+2 x \log ^2(\log (x))\right ) \log ^2\left (-\frac {4+x+2 x^2+\log (\log (x))+2 x \log (\log (x))}{x+\log (\log (x))}\right )} \, dx-4 \int \frac {x \log (\log (x))}{\left (4 x+x^2+2 x^3+4 \log (\log (x))+2 x \log (\log (x))+4 x^2 \log (\log (x))+\log ^2(\log (x))+2 x \log ^2(\log (x))\right ) \log ^2\left (-\frac {4+x+2 x^2+\log (\log (x))+2 x \log (\log (x))}{x+\log (\log (x))}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.05, size = 29, normalized size = 1.71 \begin {gather*} \frac {1}{\log \left (-\frac {4+x+2 x^2+\log (\log (x))+2 x \log (\log (x))}{x+\log (\log (x))}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 + (4*x - 2*x^3)*Log[x] - 4*x^2*Log[x]*Log[Log[x]] - 2*x*Log[x]*Log[Log[x]]^2)/(((4*x^2 + x^3 + 2*
x^4)*Log[x] + (4*x + 2*x^2 + 4*x^3)*Log[x]*Log[Log[x]] + (x + 2*x^2)*Log[x]*Log[Log[x]]^2)*Log[(-4 - x - 2*x^2
+ (-1 - 2*x)*Log[Log[x]])/(x + Log[Log[x]])]^2),x]

[Out]

Log[-((4 + x + 2*x^2 + Log[Log[x]] + 2*x*Log[Log[x]])/(x + Log[Log[x]]))]^(-1)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 2.58, size = 274, normalized size = 16.12

method	result	size
risch	\(\frac {2}{2 \ln \left (2\right )+2 i \pi -2 \ln \left (\ln \left (\ln \left (x \right )\right )+x \right )+2 \ln \left (x^{2}+\left (\ln \left (\ln \left (x \right )\right )+\frac {1}{2}\right ) x +\frac {\ln \left (\ln \left (x \right )\right )}{2}+2\right )-i \pi \,\mathrm {csgn}\left (\frac {i}{\ln \left (\ln \left (x \right )\right )+x}\right ) \mathrm {csgn}\left (i \left (x^{2}+\left (\ln \left (\ln \left (x \right )\right )+\frac {1}{2}\right ) x +\frac {\ln \left (\ln \left (x \right )\right )}{2}+2\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+\left (\ln \left (\ln \left (x \right )\right )+\frac {1}{2}\right ) x +\frac {\ln \left (\ln \left (x \right )\right )}{2}+2\right )}{\ln \left (\ln \left (x \right )\right )+x}\right )+i \pi \,\mathrm {csgn}\left (\frac {i}{\ln \left (\ln \left (x \right )\right )+x}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+\left (\ln \left (\ln \left (x \right )\right )+\frac {1}{2}\right ) x +\frac {\ln \left (\ln \left (x \right )\right )}{2}+2\right )}{\ln \left (\ln \left (x \right )\right )+x}\right )^{2}+i \pi \,\mathrm {csgn}\left (i \left (x^{2}+\left (\ln \left (\ln \left (x \right )\right )+\frac {1}{2}\right ) x +\frac {\ln \left (\ln \left (x \right )\right )}{2}+2\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+\left (\ln \left (\ln \left (x \right )\right )+\frac {1}{2}\right ) x +\frac {\ln \left (\ln \left (x \right )\right )}{2}+2\right )}{\ln \left (\ln \left (x \right )\right )+x}\right )^{2}+i \pi \mathrm {csgn}\left (\frac {i \left (x^{2}+\left (\ln \left (\ln \left (x \right )\right )+\frac {1}{2}\right ) x +\frac {\ln \left (\ln \left (x \right )\right )}{2}+2\right )}{\ln \left (\ln \left (x \right )\right )+x}\right )^{3}-2 i \pi \mathrm {csgn}\left (\frac {i \left (x^{2}+\left (\ln \left (\ln \left (x \right )\right )+\frac {1}{2}\right ) x +\frac {\ln \left (\ln \left (x \right )\right )}{2}+2\right )}{\ln \left (\ln \left (x \right )\right )+x}\right )^{2}}\)	\(274\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x*ln(x)*ln(ln(x))^2-4*x^2*ln(x)*ln(ln(x))+(-2*x^3+4*x)*ln(x)+4)/((2*x^2+x)*ln(x)*ln(ln(x))^2+(4*x^3+2*
x^2+4*x)*ln(x)*ln(ln(x))+(2*x^4+x^3+4*x^2)*ln(x))/ln(((-2*x-1)*ln(ln(x))-2*x^2-x-4)/(ln(ln(x))+x))^2,x,method=
_RETURNVERBOSE)

[Out]

2/(2*ln(2)+2*I*Pi-2*ln(ln(ln(x))+x)+2*ln(x^2+(ln(ln(x))+1/2)*x+1/2*ln(ln(x))+2)-I*Pi*csgn(I/(ln(ln(x))+x))*csg
n(I*(x^2+(ln(ln(x))+1/2)*x+1/2*ln(ln(x))+2))*csgn(I/(ln(ln(x))+x)*(x^2+(ln(ln(x))+1/2)*x+1/2*ln(ln(x))+2))+I*P
i*csgn(I/(ln(ln(x))+x))*csgn(I/(ln(ln(x))+x)*(x^2+(ln(ln(x))+1/2)*x+1/2*ln(ln(x))+2))^2+I*Pi*csgn(I*(x^2+(ln(l
n(x))+1/2)*x+1/2*ln(ln(x))+2))*csgn(I/(ln(ln(x))+x)*(x^2+(ln(ln(x))+1/2)*x+1/2*ln(ln(x))+2))^2+I*Pi*csgn(I/(ln
(ln(x))+x)*(x^2+(ln(ln(x))+1/2)*x+1/2*ln(ln(x))+2))^3-2*I*Pi*csgn(I/(ln(ln(x))+x)*(x^2+(ln(ln(x))+1/2)*x+1/2*l
n(ln(x))+2))^2)

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Maxima [A]
time = 0.35, size = 32, normalized size = 1.88 \begin {gather*} \frac {1}{\log \left (-2 \, x^{2} - {\left (2 \, x + 1\right )} \log \left (\log \left (x\right )\right ) - x - 4\right ) - \log \left (x + \log \left (\log \left (x\right )\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*log(x)*log(log(x))^2-4*x^2*log(x)*log(log(x))+(-2*x^3+4*x)*log(x)+4)/((2*x^2+x)*log(x)*log(log
(x))^2+(4*x^3+2*x^2+4*x)*log(x)*log(log(x))+(2*x^4+x^3+4*x^2)*log(x))/log(((-1-2*x)*log(log(x))-2*x^2-x-4)/(lo
g(log(x))+x))^2,x, algorithm="maxima")

[Out]

1/(log(-2*x^2 - (2*x + 1)*log(log(x)) - x - 4) - log(x + log(log(x))))

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Fricas [A]
time = 0.36, size = 29, normalized size = 1.71 \begin {gather*} \frac {1}{\log \left (-\frac {2 \, x^{2} + {\left (2 \, x + 1\right )} \log \left (\log \left (x\right )\right ) + x + 4}{x + \log \left (\log \left (x\right )\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*log(x)*log(log(x))^2-4*x^2*log(x)*log(log(x))+(-2*x^3+4*x)*log(x)+4)/((2*x^2+x)*log(x)*log(log
(x))^2+(4*x^3+2*x^2+4*x)*log(x)*log(log(x))+(2*x^4+x^3+4*x^2)*log(x))/log(((-1-2*x)*log(log(x))-2*x^2-x-4)/(lo
g(log(x))+x))^2,x, algorithm="fricas")

[Out]

1/log(-(2*x^2 + (2*x + 1)*log(log(x)) + x + 4)/(x + log(log(x))))

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Sympy [A]
time = 1.76, size = 29, normalized size = 1.71 \begin {gather*} \frac {1}{\log {\left (\frac {- 2 x^{2} - x + \left (- 2 x - 1\right ) \log {\left (\log {\left (x \right )} \right )} - 4}{x + \log {\left (\log {\left (x \right )} \right )}} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*ln(x)*ln(ln(x))**2-4*x**2*ln(x)*ln(ln(x))+(-2*x**3+4*x)*ln(x)+4)/((2*x**2+x)*ln(x)*ln(ln(x))**
2+(4*x**3+2*x**2+4*x)*ln(x)*ln(ln(x))+(2*x**4+x**3+4*x**2)*ln(x))/ln(((-1-2*x)*ln(ln(x))-2*x**2-x-4)/(ln(ln(x)
)+x))**2,x)

[Out]

1/log((-2*x**2 - x + (-2*x - 1)*log(log(x)) - 4)/(x + log(log(x))))

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Giac [A]
time = 0.62, size = 33, normalized size = 1.94 \begin {gather*} \frac {1}{\log \left (-2 \, x^{2} - 2 \, x \log \left (\log \left (x\right )\right ) - x - \log \left (\log \left (x\right )\right ) - 4\right ) - \log \left (x + \log \left (\log \left (x\right )\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*log(x)*log(log(x))^2-4*x^2*log(x)*log(log(x))+(-2*x^3+4*x)*log(x)+4)/((2*x^2+x)*log(x)*log(log
(x))^2+(4*x^3+2*x^2+4*x)*log(x)*log(log(x))+(2*x^4+x^3+4*x^2)*log(x))/log(((-1-2*x)*log(log(x))-2*x^2-x-4)/(lo
g(log(x))+x))^2,x, algorithm="giac")

[Out]

1/(log(-2*x^2 - 2*x*log(log(x)) - x - log(log(x)) - 4) - log(x + log(log(x))))

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Mupad [B]
time = 2.83, size = 29, normalized size = 1.71 \begin {gather*} \frac {1}{\ln \left (-\frac {x+2\,x^2+\ln \left (\ln \left (x\right )\right )\,\left (2\,x+1\right )+4}{x+\ln \left (\ln \left (x\right )\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(4*x - 2*x^3) - 2*x*log(log(x))^2*log(x) - 4*x^2*log(log(x))*log(x) + 4)/(log(-(x + 2*x^2 + log(lo
g(x))*(2*x + 1) + 4)/(x + log(log(x))))^2*(log(x)*(4*x^2 + x^3 + 2*x^4) + log(log(x))^2*log(x)*(x + 2*x^2) + l
og(log(x))*log(x)*(4*x + 2*x^2 + 4*x^3))),x)

[Out]

1/log(-(x + 2*x^2 + log(log(x))*(2*x + 1) + 4)/(x + log(log(x))))

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