3.73.27 \(\int \frac {8-2 x+((8-4 x) \log (x) \log (\frac {1}{\log (x)}) \log (\log (\frac {1}{\log (x)}))+(4-2 x) \log (3) \log (x) \log (\frac {1}{\log (x)}) \log ^2(\log (\frac {1}{\log (x)}))) \log (\frac {2+\log (3) \log (\log (\frac {1}{\log (x)}))}{\log (\log (\frac {1}{\log (x)}))})}{2 \log (x) \log (\frac {1}{\log (x)}) \log (\log (\frac {1}{\log (x)}))+\log (3) \log (x) \log (\frac {1}{\log (x)}) \log ^2(\log (\frac {1}{\log (x)}))} \, dx\)

Optimal. Leaf size=21 \[ (4-x) x \log \left (\log (3)+\frac {2}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right ) \]

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Rubi [F]  time = 3.57, 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 {8-2 x+\left ((8-4 x) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+(4-2 x) \log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )\right ) \log \left (\frac {2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )}{2 \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+\log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(8 - 2*x + ((8 - 4*x)*Log[x]*Log[Log[x]^(-1)]*Log[Log[Log[x]^(-1)]] + (4 - 2*x)*Log[3]*Log[x]*Log[Log[x]^(
-1)]*Log[Log[Log[x]^(-1)]]^2)*Log[(2 + Log[3]*Log[Log[Log[x]^(-1)]])/Log[Log[Log[x]^(-1)]]])/(2*Log[x]*Log[Log
[x]^(-1)]*Log[Log[Log[x]^(-1)]] + Log[3]*Log[x]*Log[Log[x]^(-1)]*Log[Log[Log[x]^(-1)]]^2),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8-2 x+\left ((8-4 x) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+(4-2 x) \log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )\right ) \log \left (\frac {2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )}{\log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right ) \left (2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )\right )} \, dx\\ &=\int \frac {2 \left (4-x-(-2+x) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right ) \left (2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )\right ) \log \left (\log (3)+\frac {2}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )\right )}{\log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right ) \left (2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )\right )} \, dx\\ &=2 \int \frac {4-x-(-2+x) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right ) \left (2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )\right ) \log \left (\log (3)+\frac {2}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )}{\log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right ) \left (2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )\right )} \, dx\\ &=2 \int \left (\frac {4-x}{\log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right ) \left (2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )\right )}-(-2+x) \log \left (\log (3)+\frac {2}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )\right ) \, dx\\ &=2 \int \frac {4-x}{\log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right ) \left (2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )\right )} \, dx-2 \int (-2+x) \log \left (\log (3)+\frac {2}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right ) \, dx\\ &=2 \int \left (\frac {4-x}{2 \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )}+\frac {(-4+x) \log (3)}{2 \log (x) \log \left (\frac {1}{\log (x)}\right ) \left (2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )\right )}\right ) \, dx-2 \int \left (-2 \log \left (\log (3)+\frac {2}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )+x \log \left (\log (3)+\frac {2}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )\right ) \, dx\\ &=-\left (2 \int x \log \left (\log (3)+\frac {2}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right ) \, dx\right )+4 \int \log \left (\log (3)+\frac {2}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right ) \, dx+\log (3) \int \frac {-4+x}{\log (x) \log \left (\frac {1}{\log (x)}\right ) \left (2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )\right )} \, dx+\int \frac {4-x}{\log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )} \, dx\\ &=-\left (2 \int x \log \left (\log (3)+\frac {2}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right ) \, dx\right )+4 \int \log \left (\log (3)+\frac {2}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right ) \, dx+\log (3) \int \left (-\frac {4}{\log (x) \log \left (\frac {1}{\log (x)}\right ) \left (2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )\right )}+\frac {x}{\log (x) \log \left (\frac {1}{\log (x)}\right ) \left (2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )\right )}\right ) \, dx+\int \left (\frac {4}{\log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )}-\frac {x}{\log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right ) \, dx\\ &=-\left (2 \int x \log \left (\log (3)+\frac {2}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right ) \, dx\right )+4 \int \frac {1}{\log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )} \, dx+4 \int \log \left (\log (3)+\frac {2}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right ) \, dx+\log (3) \int \frac {x}{\log (x) \log \left (\frac {1}{\log (x)}\right ) \left (2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )\right )} \, dx-(4 \log (3)) \int \frac {1}{\log (x) \log \left (\frac {1}{\log (x)}\right ) \left (2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )\right )} \, dx-\int \frac {x}{\log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.37, size = 20, normalized size = 0.95 \begin {gather*} -\left ((-4+x) x \log \left (\log (3)+\frac {2}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8 - 2*x + ((8 - 4*x)*Log[x]*Log[Log[x]^(-1)]*Log[Log[Log[x]^(-1)]] + (4 - 2*x)*Log[3]*Log[x]*Log[Lo
g[x]^(-1)]*Log[Log[Log[x]^(-1)]]^2)*Log[(2 + Log[3]*Log[Log[Log[x]^(-1)]])/Log[Log[Log[x]^(-1)]]])/(2*Log[x]*L
og[Log[x]^(-1)]*Log[Log[Log[x]^(-1)]] + Log[3]*Log[x]*Log[Log[x]^(-1)]*Log[Log[Log[x]^(-1)]]^2),x]

[Out]

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

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fricas [A]  time = 1.12, size = 30, normalized size = 1.43 \begin {gather*} -{\left (x^{2} - 4 \, x\right )} \log \left (\frac {\log \relax (3) \log \left (\log \left (\frac {1}{\log \relax (x)}\right )\right ) + 2}{\log \left (\log \left (\frac {1}{\log \relax (x)}\right )\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4-2*x)*log(3)*log(x)*log(1/log(x))*log(log(1/log(x)))^2+(-4*x+8)*log(x)*log(1/log(x))*log(log(1/l
og(x))))*log((log(3)*log(log(1/log(x)))+2)/log(log(1/log(x))))-2*x+8)/(log(3)*log(x)*log(1/log(x))*log(log(1/l
og(x)))^2+2*log(x)*log(1/log(x))*log(log(1/log(x)))),x, algorithm="fricas")

[Out]

-(x^2 - 4*x)*log((log(3)*log(log(1/log(x))) + 2)/log(log(1/log(x))))

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giac [B]  time = 1.94, size = 54, normalized size = 2.57 \begin {gather*} -x^{2} \log \left (\log \relax (3) \log \left (-\log \left (\log \relax (x)\right )\right ) + 2\right ) + x^{2} \log \left (\log \left (-\log \left (\log \relax (x)\right )\right )\right ) + 4 \, x \log \left (\log \relax (3) \log \left (-\log \left (\log \relax (x)\right )\right ) + 2\right ) - 4 \, x \log \left (\log \left (-\log \left (\log \relax (x)\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4-2*x)*log(3)*log(x)*log(1/log(x))*log(log(1/log(x)))^2+(-4*x+8)*log(x)*log(1/log(x))*log(log(1/l
og(x))))*log((log(3)*log(log(1/log(x)))+2)/log(log(1/log(x))))-2*x+8)/(log(3)*log(x)*log(1/log(x))*log(log(1/l
og(x)))^2+2*log(x)*log(1/log(x))*log(log(1/log(x)))),x, algorithm="giac")

[Out]

-x^2*log(log(3)*log(-log(log(x))) + 2) + x^2*log(log(-log(log(x)))) + 4*x*log(log(3)*log(-log(log(x))) + 2) -
4*x*log(log(-log(log(x))))

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maple [C]  time = 0.30, size = 397, normalized size = 18.90

method	result	size
risch	\(\left (-x^{2}+4 x \right ) \ln \left (\ln \relax (3) \ln \left (-\ln \left (\ln \relax (x )\right )\right )+2\right )+x^{2} \ln \left (\ln \left (-\ln \left (\ln \relax (x )\right )\right )\right )-4 x \ln \left (\ln \left (-\ln \left (\ln \relax (x )\right )\right )\right )+\frac {i \pi \,x^{2} \mathrm {csgn}\left (i \left (\ln \relax (3) \ln \left (-\ln \left (\ln \relax (x )\right )\right )+2\right )\right ) \mathrm {csgn}\left (\frac {i}{\ln \left (-\ln \left (\ln \relax (x )\right )\right )}\right ) \mathrm {csgn}\left (\frac {i \left (\ln \relax (3) \ln \left (-\ln \left (\ln \relax (x )\right )\right )+2\right )}{\ln \left (-\ln \left (\ln \relax (x )\right )\right )}\right )}{2}-\frac {i \pi \,x^{2} \mathrm {csgn}\left (i \left (\ln \relax (3) \ln \left (-\ln \left (\ln \relax (x )\right )\right )+2\right )\right ) \mathrm {csgn}\left (\frac {i \left (\ln \relax (3) \ln \left (-\ln \left (\ln \relax (x )\right )\right )+2\right )}{\ln \left (-\ln \left (\ln \relax (x )\right )\right )}\right )^{2}}{2}-\frac {i \pi \,x^{2} \mathrm {csgn}\left (\frac {i}{\ln \left (-\ln \left (\ln \relax (x )\right )\right )}\right ) \mathrm {csgn}\left (\frac {i \left (\ln \relax (3) \ln \left (-\ln \left (\ln \relax (x )\right )\right )+2\right )}{\ln \left (-\ln \left (\ln \relax (x )\right )\right )}\right )^{2}}{2}+\frac {i \pi \,x^{2} \mathrm {csgn}\left (\frac {i \left (\ln \relax (3) \ln \left (-\ln \left (\ln \relax (x )\right )\right )+2\right )}{\ln \left (-\ln \left (\ln \relax (x )\right )\right )}\right )^{3}}{2}-2 i \pi x \,\mathrm {csgn}\left (i \left (\ln \relax (3) \ln \left (-\ln \left (\ln \relax (x )\right )\right )+2\right )\right ) \mathrm {csgn}\left (\frac {i}{\ln \left (-\ln \left (\ln \relax (x )\right )\right )}\right ) \mathrm {csgn}\left (\frac {i \left (\ln \relax (3) \ln \left (-\ln \left (\ln \relax (x )\right )\right )+2\right )}{\ln \left (-\ln \left (\ln \relax (x )\right )\right )}\right )+2 i \pi x \,\mathrm {csgn}\left (i \left (\ln \relax (3) \ln \left (-\ln \left (\ln \relax (x )\right )\right )+2\right )\right ) \mathrm {csgn}\left (\frac {i \left (\ln \relax (3) \ln \left (-\ln \left (\ln \relax (x )\right )\right )+2\right )}{\ln \left (-\ln \left (\ln \relax (x )\right )\right )}\right )^{2}+2 i \pi x \,\mathrm {csgn}\left (\frac {i}{\ln \left (-\ln \left (\ln \relax (x )\right )\right )}\right ) \mathrm {csgn}\left (\frac {i \left (\ln \relax (3) \ln \left (-\ln \left (\ln \relax (x )\right )\right )+2\right )}{\ln \left (-\ln \left (\ln \relax (x )\right )\right )}\right )^{2}-2 i \pi x \mathrm {csgn}\left (\frac {i \left (\ln \relax (3) \ln \left (-\ln \left (\ln \relax (x )\right )\right )+2\right )}{\ln \left (-\ln \left (\ln \relax (x )\right )\right )}\right )^{3}\)	\(397\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((4-2*x)*ln(3)*ln(x)*ln(1/ln(x))*ln(ln(1/ln(x)))^2+(-4*x+8)*ln(x)*ln(1/ln(x))*ln(ln(1/ln(x))))*ln((ln(3)*
ln(ln(1/ln(x)))+2)/ln(ln(1/ln(x))))-2*x+8)/(ln(3)*ln(x)*ln(1/ln(x))*ln(ln(1/ln(x)))^2+2*ln(x)*ln(1/ln(x))*ln(l
n(1/ln(x)))),x,method=_RETURNVERBOSE)

[Out]

(-x^2+4*x)*ln(ln(3)*ln(-ln(ln(x)))+2)+x^2*ln(ln(-ln(ln(x))))-4*x*ln(ln(-ln(ln(x))))+1/2*I*Pi*x^2*csgn(I*(ln(3)
*ln(-ln(ln(x)))+2))*csgn(I/ln(-ln(ln(x))))*csgn(I/ln(-ln(ln(x)))*(ln(3)*ln(-ln(ln(x)))+2))-1/2*I*Pi*x^2*csgn(I
*(ln(3)*ln(-ln(ln(x)))+2))*csgn(I/ln(-ln(ln(x)))*(ln(3)*ln(-ln(ln(x)))+2))^2-1/2*I*Pi*x^2*csgn(I/ln(-ln(ln(x))
))*csgn(I/ln(-ln(ln(x)))*(ln(3)*ln(-ln(ln(x)))+2))^2+1/2*I*Pi*x^2*csgn(I/ln(-ln(ln(x)))*(ln(3)*ln(-ln(ln(x)))+
2))^3-2*I*Pi*x*csgn(I*(ln(3)*ln(-ln(ln(x)))+2))*csgn(I/ln(-ln(ln(x))))*csgn(I/ln(-ln(ln(x)))*(ln(3)*ln(-ln(ln(
x)))+2))+2*I*Pi*x*csgn(I*(ln(3)*ln(-ln(ln(x)))+2))*csgn(I/ln(-ln(ln(x)))*(ln(3)*ln(-ln(ln(x)))+2))^2+2*I*Pi*x*
csgn(I/ln(-ln(ln(x))))*csgn(I/ln(-ln(ln(x)))*(ln(3)*ln(-ln(ln(x)))+2))^2-2*I*Pi*x*csgn(I/ln(-ln(ln(x)))*(ln(3)
*ln(-ln(ln(x)))+2))^3

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maxima [A]  time = 0.58, size = 37, normalized size = 1.76 \begin {gather*} -{\left (x^{2} - 4 \, x\right )} \log \left (\log \relax (3) \log \left (-\log \left (\log \relax (x)\right )\right ) + 2\right ) + {\left (x^{2} - 4 \, x\right )} \log \left (\log \left (-\log \left (\log \relax (x)\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4-2*x)*log(3)*log(x)*log(1/log(x))*log(log(1/log(x)))^2+(-4*x+8)*log(x)*log(1/log(x))*log(log(1/l
og(x))))*log((log(3)*log(log(1/log(x)))+2)/log(log(1/log(x))))-2*x+8)/(log(3)*log(x)*log(1/log(x))*log(log(1/l
og(x)))^2+2*log(x)*log(1/log(x))*log(log(1/log(x)))),x, algorithm="maxima")

[Out]

-(x^2 - 4*x)*log(log(3)*log(-log(log(x))) + 2) + (x^2 - 4*x)*log(log(-log(log(x))))

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mupad [B]  time = 6.28, size = 27, normalized size = 1.29 \begin {gather*} -x\,\ln \left (\frac {\ln \left (\ln \left (\frac {1}{\ln \relax (x)}\right )\right )\,\ln \relax (3)+2}{\ln \left (\ln \left (\frac {1}{\ln \relax (x)}\right )\right )}\right )\,\left (x-4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x + log((log(log(1/log(x)))*log(3) + 2)/log(log(1/log(x))))*(log(log(1/log(x)))*log(1/log(x))*log(x)*(
4*x - 8) + log(log(1/log(x)))^2*log(3)*log(1/log(x))*log(x)*(2*x - 4)) - 8)/(2*log(log(1/log(x)))*log(1/log(x)
)*log(x) + log(log(1/log(x)))^2*log(3)*log(1/log(x))*log(x)),x)

[Out]

-x*log((log(log(1/log(x)))*log(3) + 2)/log(log(1/log(x))))*(x - 4)

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sympy [A]  time = 2.45, size = 29, normalized size = 1.38 \begin {gather*} \left (- x^{2} + 4 x\right ) \log {\left (\frac {\log {\relax (3 )} \log {\left (\log {\left (\frac {1}{\log {\relax (x )}} \right )} \right )} + 2}{\log {\left (\log {\left (\frac {1}{\log {\relax (x )}} \right )} \right )}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4-2*x)*ln(3)*ln(x)*ln(1/ln(x))*ln(ln(1/ln(x)))**2+(-4*x+8)*ln(x)*ln(1/ln(x))*ln(ln(1/ln(x))))*ln(
(ln(3)*ln(ln(1/ln(x)))+2)/ln(ln(1/ln(x))))-2*x+8)/(ln(3)*ln(x)*ln(1/ln(x))*ln(ln(1/ln(x)))**2+2*ln(x)*ln(1/ln(
x))*ln(ln(1/ln(x)))),x)

[Out]

(-x**2 + 4*x)*log((log(3)*log(log(1/log(x))) + 2)/log(log(1/log(x))))

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