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

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

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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 = {} $$\begin{array}{c}\int \frac{8-2x+((8-4x)\log (x)\log \left(\frac{1}{\log (x)}\right)\log (\log \left(\frac{1}{\log (x)}\right))+(4-2x)\log (3)\log (x)\log \left(\frac{1}{\log (x)}\right){\log }^2(\log \left(\frac{1}{\log (x)}\right)))\log \left(\frac{2+\log (3)\log (\log \left(\frac{1}{\log (x)}\right))}{\log (\log \left(\frac{1}{\log (x)}\right))}\right)}{2\log (x)\log \left(\frac{1}{\log (x)}\right)\log (\log \left(\frac{1}{\log (x)}\right))+\log (3)\log (x)\log \left(\frac{1}{\log (x)}\right){\log }^2(\log \left(\frac{1}{\log (x)}\right))}dx\end{array}$$

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{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{8-2x+((8-4x)\log (x)\log \left(\frac{1}{\log (x)}\right)\log (\log \left(\frac{1}{\log (x)}\right))+(4-2x)\log (3)\log (x)\log \left(\frac{1}{\log (x)}\right){\log }^2(\log \left(\frac{1}{\log (x)}\right)))\log \left(\frac{2+\log (3)\log (\log \left(\frac{1}{\log (x)}\right))}{\log (\log \left(\frac{1}{\log (x)}\right))}\right)}{\log (x)\log \left(\frac{1}{\log (x)}\right)\log (\log \left(\frac{1}{\log (x)}\right))(2+\log (3)\log (\log \left(\frac{1}{\log (x)}\right)))}dx\\ & =\int \frac{2(4-x-(-2+x)\log (x)\log \left(\frac{1}{\log (x)}\right)\log (\log \left(\frac{1}{\log (x)}\right))(2+\log (3)\log (\log \left(\frac{1}{\log (x)}\right)))\log (\log (3)+\frac{2}{\log (\log \left(\frac{1}{\log (x)}\right))}))}{\log (x)\log \left(\frac{1}{\log (x)}\right)\log (\log \left(\frac{1}{\log (x)}\right))(2+\log (3)\log (\log \left(\frac{1}{\log (x)}\right)))}dx\\ & =2\int \frac{4-x-(-2+x)\log (x)\log \left(\frac{1}{\log (x)}\right)\log (\log \left(\frac{1}{\log (x)}\right))(2+\log (3)\log (\log \left(\frac{1}{\log (x)}\right)))\log (\log (3)+\frac{2}{\log (\log \left(\frac{1}{\log (x)}\right))})}{\log (x)\log \left(\frac{1}{\log (x)}\right)\log (\log \left(\frac{1}{\log (x)}\right))(2+\log (3)\log (\log \left(\frac{1}{\log (x)}\right)))}dx\\ & =2\int (\frac{4-x}{\log (x)\log \left(\frac{1}{\log (x)}\right)\log (\log \left(\frac{1}{\log (x)}\right))(2+\log (3)\log (\log \left(\frac{1}{\log (x)}\right)))}-(-2+x)\log (\log (3)+\frac{2}{\log (\log \left(\frac{1}{\log (x)}\right))}))dx\\ & =2\int \frac{4-x}{\log (x)\log \left(\frac{1}{\log (x)}\right)\log (\log \left(\frac{1}{\log (x)}\right))(2+\log (3)\log (\log \left(\frac{1}{\log (x)}\right)))}dx-2\int (-2+x)\log (\log (3)+\frac{2}{\log (\log \left(\frac{1}{\log (x)}\right))})dx\\ & =2\int (\frac{4-x}{2\log (x)\log \left(\frac{1}{\log (x)}\right)\log (\log \left(\frac{1}{\log (x)}\right))}+\frac{(-4+x)\log (3)}{2\log (x)\log \left(\frac{1}{\log (x)}\right)(2+\log (3)\log (\log \left(\frac{1}{\log (x)}\right)))})dx-2\int (-2\log (\log (3)+\frac{2}{\log (\log \left(\frac{1}{\log (x)}\right))})+x\log (\log (3)+\frac{2}{\log (\log \left(\frac{1}{\log (x)}\right))}))dx\\ & =-(2\int x\log (\log (3)+\frac{2}{\log (\log \left(\frac{1}{\log (x)}\right))})dx)+4\int \log (\log (3)+\frac{2}{\log (\log \left(\frac{1}{\log (x)}\right))})dx+\log (3)\int \frac{-4+x}{\log (x)\log \left(\frac{1}{\log (x)}\right)(2+\log (3)\log (\log \left(\frac{1}{\log (x)}\right)))}dx+\int \frac{4-x}{\log (x)\log \left(\frac{1}{\log (x)}\right)\log (\log \left(\frac{1}{\log (x)}\right))}dx\\ & =-(2\int x\log (\log (3)+\frac{2}{\log (\log \left(\frac{1}{\log (x)}\right))})dx)+4\int \log (\log (3)+\frac{2}{\log (\log \left(\frac{1}{\log (x)}\right))})dx+\log (3)\int (-\frac{4}{\log (x)\log \left(\frac{1}{\log (x)}\right)(2+\log (3)\log (\log \left(\frac{1}{\log (x)}\right)))}+\frac{x}{\log (x)\log \left(\frac{1}{\log (x)}\right)(2+\log (3)\log (\log \left(\frac{1}{\log (x)}\right)))})dx+\int (\frac{4}{\log (x)\log \left(\frac{1}{\log (x)}\right)\log (\log \left(\frac{1}{\log (x)}\right))}-\frac{x}{\log (x)\log \left(\frac{1}{\log (x)}\right)\log (\log \left(\frac{1}{\log (x)}\right))})dx\\ & =-(2\int x\log (\log (3)+\frac{2}{\log (\log \left(\frac{1}{\log (x)}\right))})dx)+4\int \frac{1}{\log (x)\log \left(\frac{1}{\log (x)}\right)\log (\log \left(\frac{1}{\log (x)}\right))}dx+4\int \log (\log (3)+\frac{2}{\log (\log \left(\frac{1}{\log (x)}\right))})dx+\log (3)\int \frac{x}{\log (x)\log \left(\frac{1}{\log (x)}\right)(2+\log (3)\log (\log \left(\frac{1}{\log (x)}\right)))}dx-(4\log (3))\int \frac{1}{\log (x)\log \left(\frac{1}{\log (x)}\right)(2+\log (3)\log (\log \left(\frac{1}{\log (x)}\right)))}dx-\int \frac{x}{\log (x)\log \left(\frac{1}{\log (x)}\right)\log (\log \left(\frac{1}{\log (x)}\right))}dx\end{array}\end{array}$$

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

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 $$\begin{array}{c}-(x^2-4x)\log \left(\frac{\log (3)\log (\log \left(\frac{1}{\log (x)}\right))+2}{\log (\log \left(\frac{1}{\log (x)}\right))}\right)\end{array}$$

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 $$\begin{array}{c}-x^2\log (\log (3)\log (-\log (\log (x)))+2)+x^2\log (\log (-\log (\log (x))))+4x\log (\log (3)\log (-\log (\log (x)))+2)-4x\log (\log (-\log (\log (x))))\end{array}$$

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	$(-x^2+4x)\ln (\ln (3)\ln (-\ln (\ln (x)))+2)+x^2\ln (\ln (-\ln (\ln (x))))-4x\ln (\ln (-\ln (\ln (x))))+\frac{i\pi x^2\mathrm{csgn}\left(i(\ln (3)\ln (-\ln (\ln (x)))+2)\right)\mathrm{csgn}\left(\frac{i}{\ln (-\ln (\ln (x)))}\right)\mathrm{csgn}\left(\frac{i(\ln (3)\ln (-\ln (\ln (x)))+2)}{\ln (-\ln (\ln (x)))}\right)}{2}-\frac{i\pi x^2\mathrm{csgn}\left(i(\ln (3)\ln (-\ln (\ln (x)))+2)\right)\mathrm{csgn}{\left(\frac{i(\ln (3)\ln (-\ln (\ln (x)))+2)}{\ln (-\ln (\ln (x)))}\right)}^2}{2}-\frac{i\pi x^2\mathrm{csgn}\left(\frac{i}{\ln (-\ln (\ln (x)))}\right)\mathrm{csgn}{\left(\frac{i(\ln (3)\ln (-\ln (\ln (x)))+2)}{\ln (-\ln (\ln (x)))}\right)}^2}{2}+\frac{i\pi x^2\mathrm{csgn}{\left(\frac{i(\ln (3)\ln (-\ln (\ln (x)))+2)}{\ln (-\ln (\ln (x)))}\right)}^3}{2}-2i\pi x\mathrm{csgn}\left(i(\ln (3)\ln (-\ln (\ln (x)))+2)\right)\mathrm{csgn}\left(\frac{i}{\ln (-\ln (\ln (x)))}\right)\mathrm{csgn}\left(\frac{i(\ln (3)\ln (-\ln (\ln (x)))+2)}{\ln (-\ln (\ln (x)))}\right)+2i\pi x\mathrm{csgn}\left(i(\ln (3)\ln (-\ln (\ln (x)))+2)\right)\mathrm{csgn}{\left(\frac{i(\ln (3)\ln (-\ln (\ln (x)))+2)}{\ln (-\ln (\ln (x)))}\right)}^2+2i\pi x\mathrm{csgn}\left(\frac{i}{\ln (-\ln (\ln (x)))}\right)\mathrm{csgn}{\left(\frac{i(\ln (3)\ln (-\ln (\ln (x)))+2)}{\ln (-\ln (\ln (x)))}\right)}^2-2i\pi x\mathrm{csgn}{\left(\frac{i(\ln (3)\ln (-\ln (\ln (x)))+2)}{\ln (-\ln (\ln (x)))}\right)}^3$	$397$

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 $$\begin{array}{c}-(x^2-4x)\log (\log (3)\log (-\log (\log (x)))+2)+(x^2-4x)\log (\log (-\log (\log (x))))\end{array}$$

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 $$\begin{array}{c}-x\ln \left(\frac{\ln (\ln \left(\frac{1}{\ln (x)}\right))\ln (3)+2}{\ln (\ln \left(\frac{1}{\ln (x)}\right))}\right)(x-4)\end{array}$$

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 $$\begin{array}{c}(-x^2+4x)\log \left(\frac{\log (3)\log (\log \left(\frac{1}{\log (x)}\right))+2}{\log (\log \left(\frac{1}{\log (x)}\right))}\right)\end{array}$$

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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