\(\int \frac {8-2 x+((8-4 x) \log (x) \log (\frac {1}{\log (x)}) \log (\log (\frac {1}{\log (x)}))+(4-2 x) \log (3) \log (x) \log (\frac {1}{\log (x)}) \log ^2(\log (\frac {1}{\log (x)}))) \log (\frac {2+\log (3) \log (\log (\frac {1}{\log (x)}))}{\log (\log (\frac {1}{\log (x)}))})}{2 \log (x) \log (\frac {1}{\log (x)}) \log (\log (\frac {1}{\log (x)}))+\log (3) \log (x) \log (\frac {1}{\log (x)}) \log ^2(\log (\frac {1}{\log (x)}))} \, dx\) [7227]

Optimal result

Integrand size = 107, antiderivative size = 21 \[ \int \frac {8-2 x+\left ((8-4 x) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+(4-2 x) \log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )\right ) \log \left (\frac {2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )}{2 \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+\log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )} \, dx=(4-x) x \log \left (\log (3)+\frac {2}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right ) \]

[Out]

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

Rubi [F]

\[ \int \frac {8-2 x+\left ((8-4 x) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+(4-2 x) \log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )\right ) \log \left (\frac {2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )}{2 \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+\log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )} \, dx=\int \frac {8-2 x+\left ((8-4 x) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+(4-2 x) \log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )\right ) \log \left (\frac {2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )}{2 \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+\log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )} \, dx \]

[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{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {8-2 x+\left ((8-4 x) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+(4-2 x) \log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )\right ) \log \left (\frac {2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )}{2 \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+\log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )} \, dx=-\left ((-4+x) x \log \left (\log (3)+\frac {2}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )\right ) \]

[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)]]])

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.22 (sec) , antiderivative size = 397, normalized size of antiderivative = 18.90

\[\left (-x^{2}+4 x \right ) \ln \left (\ln \left (3\right ) \ln \left (-\ln \left (\ln \left (x \right )\right )\right )+2\right )+x^{2} \ln \left (\ln \left (-\ln \left (\ln \left (x \right )\right )\right )\right )-4 x \ln \left (\ln \left (-\ln \left (\ln \left (x \right )\right )\right )\right )+\frac {i \pi \,x^{2} \operatorname {csgn}\left (i \left (\ln \left (3\right ) \ln \left (-\ln \left (\ln \left (x \right )\right )\right )+2\right )\right ) \operatorname {csgn}\left (\frac {i}{\ln \left (-\ln \left (\ln \left (x \right )\right )\right )}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (3\right ) \ln \left (-\ln \left (\ln \left (x \right )\right )\right )+2\right )}{\ln \left (-\ln \left (\ln \left (x \right )\right )\right )}\right )}{2}-\frac {i \pi \,x^{2} \operatorname {csgn}\left (i \left (\ln \left (3\right ) \ln \left (-\ln \left (\ln \left (x \right )\right )\right )+2\right )\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (3\right ) \ln \left (-\ln \left (\ln \left (x \right )\right )\right )+2\right )}{\ln \left (-\ln \left (\ln \left (x \right )\right )\right )}\right )^{2}}{2}-\frac {i \pi \,x^{2} \operatorname {csgn}\left (\frac {i}{\ln \left (-\ln \left (\ln \left (x \right )\right )\right )}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (3\right ) \ln \left (-\ln \left (\ln \left (x \right )\right )\right )+2\right )}{\ln \left (-\ln \left (\ln \left (x \right )\right )\right )}\right )^{2}}{2}+\frac {i \pi \,x^{2} \operatorname {csgn}\left (\frac {i \left (\ln \left (3\right ) \ln \left (-\ln \left (\ln \left (x \right )\right )\right )+2\right )}{\ln \left (-\ln \left (\ln \left (x \right )\right )\right )}\right )^{3}}{2}-2 i \pi x \,\operatorname {csgn}\left (i \left (\ln \left (3\right ) \ln \left (-\ln \left (\ln \left (x \right )\right )\right )+2\right )\right ) \operatorname {csgn}\left (\frac {i}{\ln \left (-\ln \left (\ln \left (x \right )\right )\right )}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (3\right ) \ln \left (-\ln \left (\ln \left (x \right )\right )\right )+2\right )}{\ln \left (-\ln \left (\ln \left (x \right )\right )\right )}\right )+2 i \pi x \,\operatorname {csgn}\left (i \left (\ln \left (3\right ) \ln \left (-\ln \left (\ln \left (x \right )\right )\right )+2\right )\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (3\right ) \ln \left (-\ln \left (\ln \left (x \right )\right )\right )+2\right )}{\ln \left (-\ln \left (\ln \left (x \right )\right )\right )}\right )^{2}+2 i \pi x \,\operatorname {csgn}\left (\frac {i}{\ln \left (-\ln \left (\ln \left (x \right )\right )\right )}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (3\right ) \ln \left (-\ln \left (\ln \left (x \right )\right )\right )+2\right )}{\ln \left (-\ln \left (\ln \left (x \right )\right )\right )}\right )^{2}-2 i \pi x \operatorname {csgn}\left (\frac {i \left (\ln \left (3\right ) \ln \left (-\ln \left (\ln \left (x \right )\right )\right )+2\right )}{\ln \left (-\ln \left (\ln \left (x \right )\right )\right )}\right )^{3}\]

[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)

[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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.43 \[ \int \frac {8-2 x+\left ((8-4 x) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+(4-2 x) \log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )\right ) \log \left (\frac {2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )}{2 \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+\log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )} \, dx=-{\left (x^{2} - 4 \, x\right )} \log \left (\frac {\log \left (3\right ) \log \left (\log \left (\frac {1}{\log \left (x\right )}\right )\right ) + 2}{\log \left (\log \left (\frac {1}{\log \left (x\right )}\right )\right )}\right ) \]

[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))))

Sympy [A] (verification not implemented)

Time = 1.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.38 \[ \int \frac {8-2 x+\left ((8-4 x) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+(4-2 x) \log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )\right ) \log \left (\frac {2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )}{2 \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+\log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )} \, dx=\left (- x^{2} + 4 x\right ) \log {\left (\frac {\log {\left (3 \right )} \log {\left (\log {\left (\frac {1}{\log {\left (x \right )}} \right )} \right )} + 2}{\log {\left (\log {\left (\frac {1}{\log {\left (x \right )}} \right )} \right )}} \right )} \]

[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))))

Maxima [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.76 \[ \int \frac {8-2 x+\left ((8-4 x) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+(4-2 x) \log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )\right ) \log \left (\frac {2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )}{2 \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+\log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )} \, dx=-{\left (x^{2} - 4 \, x\right )} \log \left (\log \left (3\right ) \log \left (-\log \left (\log \left (x\right )\right )\right ) + 2\right ) + {\left (x^{2} - 4 \, x\right )} \log \left (\log \left (-\log \left (\log \left (x\right )\right )\right )\right ) \]

[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))))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (20) = 40\).

Time = 1.37 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.57 \[ \int \frac {8-2 x+\left ((8-4 x) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+(4-2 x) \log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )\right ) \log \left (\frac {2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )}{2 \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+\log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )} \, dx=-x^{2} \log \left (\log \left (3\right ) \log \left (-\log \left (\log \left (x\right )\right )\right ) + 2\right ) + x^{2} \log \left (\log \left (-\log \left (\log \left (x\right )\right )\right )\right ) + 4 \, x \log \left (\log \left (3\right ) \log \left (-\log \left (\log \left (x\right )\right )\right ) + 2\right ) - 4 \, x \log \left (\log \left (-\log \left (\log \left (x\right )\right )\right )\right ) \]

[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))))

Mupad [B] (verification not implemented)

Time = 15.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.29 \[ \int \frac {8-2 x+\left ((8-4 x) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+(4-2 x) \log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )\right ) \log \left (\frac {2+\log (3) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )}{\log \left (\log \left (\frac {1}{\log (x)}\right )\right )}\right )}{2 \log (x) \log \left (\frac {1}{\log (x)}\right ) \log \left (\log \left (\frac {1}{\log (x)}\right )\right )+\log (3) \log (x) \log \left (\frac {1}{\log (x)}\right ) \log ^2\left (\log \left (\frac {1}{\log (x)}\right )\right )} \, dx=-x\,\ln \left (\frac {\ln \left (\ln \left (\frac {1}{\ln \left (x\right )}\right )\right )\,\ln \left (3\right )+2}{\ln \left (\ln \left (\frac {1}{\ln \left (x\right )}\right )\right )}\right )\,\left (x-4\right ) \]

[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)