3.89.16 $\int \frac{-2x+x^2-2\log (\frac{e^x}{2})\log (\log (\frac{e^x}{2}))\log (\log (\log (\frac{e^x}{2})))+(-1+x^2)\log (\frac{e^x}{2})\log (\log (\frac{e^x}{2})){\log }^2(\log (\log (\frac{e^x}{2})))}{x^2\log (\frac{e^x}{2})\log (\log (\frac{e^x}{2})){\log }^2(\log (\log (\frac{e^x}{2})))}dx$

Optimal. Leaf size=25 $$x+\frac{1-\frac{-2+x}{\log (\log (\log \left(\frac{e^x}{2}\right)))}}{x}$$

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Rubi [F]  time = 1.26, 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 = {} $$\begin{array}{c}\int \frac{-2x+x^2-2\log \left(\frac{e^x}{2}\right)\log (\log \left(\frac{e^x}{2}\right))\log (\log (\log \left(\frac{e^x}{2}\right)))+(-1+x^2)\log \left(\frac{e^x}{2}\right)\log (\log \left(\frac{e^x}{2}\right)){\log }^2(\log (\log \left(\frac{e^x}{2}\right)))}{x^2\log \left(\frac{e^x}{2}\right)\log (\log \left(\frac{e^x}{2}\right)){\log }^2(\log (\log \left(\frac{e^x}{2}\right)))}dx\end{array}$$

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{\frac{(-2+x)x}{\log \left(\frac{e^x}{2}\right)\log (\log \left(\frac{e^x}{2}\right))}+\log (\log (\log \left(\frac{e^x}{2}\right)))(-2+(-1+x^2)\log (\log (\log \left(\frac{e^x}{2}\right))))}{x^2{\log }^2(\log (\log \left(\frac{e^x}{2}\right)))}dx\\ & =\int (\frac{-1+x^2}{x^2}+\frac{-2+x}{x\log \left(\frac{e^x}{2}\right)\log (\log \left(\frac{e^x}{2}\right)){\log }^2(\log (\log \left(\frac{e^x}{2}\right)))}-\frac{2}{x^2\log (\log (\log \left(\frac{e^x}{2}\right)))})dx\\ & =-(2\int \frac{1}{x^2\log (\log (\log \left(\frac{e^x}{2}\right)))}dx)+\int \frac{-1+x^2}{x^2}dx+\int \frac{-2+x}{x\log \left(\frac{e^x}{2}\right)\log (\log \left(\frac{e^x}{2}\right)){\log }^2(\log (\log \left(\frac{e^x}{2}\right)))}dx\\ & =-(2\int \frac{1}{x^2\log (\log (\log \left(\frac{e^x}{2}\right)))}dx)+\int (1-\frac{1}{x^2})dx+\int (\frac{1}{\log \left(\frac{e^x}{2}\right)\log (\log \left(\frac{e^x}{2}\right)){\log }^2(\log (\log \left(\frac{e^x}{2}\right)))}-\frac{2}{x\log \left(\frac{e^x}{2}\right)\log (\log \left(\frac{e^x}{2}\right)){\log }^2(\log (\log \left(\frac{e^x}{2}\right)))})dx\\ & =\frac{1}{x}+x-2\int \frac{1}{x\log \left(\frac{e^x}{2}\right)\log (\log \left(\frac{e^x}{2}\right)){\log }^2(\log (\log \left(\frac{e^x}{2}\right)))}dx-2\int \frac{1}{x^2\log (\log (\log \left(\frac{e^x}{2}\right)))}dx+\int \frac{1}{\log \left(\frac{e^x}{2}\right)\log (\log \left(\frac{e^x}{2}\right)){\log }^2(\log (\log \left(\frac{e^x}{2}\right)))}dx\\ & =\frac{1}{x}+x-2\int \frac{1}{x\log \left(\frac{e^x}{2}\right)\log (\log \left(\frac{e^x}{2}\right)){\log }^2(\log (\log \left(\frac{e^x}{2}\right)))}dx-2\int \frac{1}{x^2\log (\log (\log \left(\frac{e^x}{2}\right)))}dx+\mathrm{Subst}(\int \frac{1}{x\log \left(\frac{x}{2}\right)\log (\log \left(\frac{x}{2}\right)){\log }^2(\log (\log \left(\frac{x}{2}\right)))}dx,x,e^x)\\ & =\frac{1}{x}+x-2\int \frac{1}{x\log \left(\frac{e^x}{2}\right)\log (\log \left(\frac{e^x}{2}\right)){\log }^2(\log (\log \left(\frac{e^x}{2}\right)))}dx-2\int \frac{1}{x^2\log (\log (\log \left(\frac{e^x}{2}\right)))}dx+\mathrm{Subst}(\int \frac{1}{x\log (x){\log }^2(\log (x))}dx,x,\log \left(\frac{e^x}{2}\right))\\ & =\frac{1}{x}+x-2\int \frac{1}{x\log \left(\frac{e^x}{2}\right)\log (\log \left(\frac{e^x}{2}\right)){\log }^2(\log (\log \left(\frac{e^x}{2}\right)))}dx-2\int \frac{1}{x^2\log (\log (\log \left(\frac{e^x}{2}\right)))}dx+\mathrm{Subst}(\int \frac{1}{x{\log }^2(x)}dx,x,\log (\log \left(\frac{e^x}{2}\right)))\\ & =\frac{1}{x}+x-2\int \frac{1}{x\log \left(\frac{e^x}{2}\right)\log (\log \left(\frac{e^x}{2}\right)){\log }^2(\log (\log \left(\frac{e^x}{2}\right)))}dx-2\int \frac{1}{x^2\log (\log (\log \left(\frac{e^x}{2}\right)))}dx+\mathrm{Subst}(\int \frac{1}{x^2}dx,x,\log (\log (\log \left(\frac{e^x}{2}\right))))\\ & =\frac{1}{x}+x-\frac{1}{\log (\log (\log \left(\frac{e^x}{2}\right)))}-2\int \frac{1}{x\log \left(\frac{e^x}{2}\right)\log (\log \left(\frac{e^x}{2}\right)){\log }^2(\log (\log \left(\frac{e^x}{2}\right)))}dx-2\int \frac{1}{x^2\log (\log (\log \left(\frac{e^x}{2}\right)))}dx\end{array}\end{array}$$

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Mathematica [A]  time = 0.08, size = 26, normalized size = 1.04 $$\begin{array}{c}\frac{1}{x}+x+\frac{2-x}{x\log (\log (\log \left(\frac{e^x}{2}\right)))}\end{array}$$

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.47, size = 33, normalized size = 1.32 $$\begin{array}{c}\frac{(x^2+1)\log (\log (x-\log (2)))-x+2}{x\log (\log (x-\log (2)))}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.16, size = 23, normalized size = 0.92 $$\begin{array}{c}x+\frac{1}{x}-\frac{x-2}{x\log (\log (x-\log (2)))}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.37, size = 31, normalized size = 1.24

method	result	size
risch	$\frac{x^2+1}{x}-\frac{x-2}{x\ln (\ln (-\ln (2)+\ln \left({\mathrm{e}}^x\right)))}$	$31$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2-1)*ln(1/2*exp(x))*ln(ln(1/2*exp(x)))*ln(ln(ln(1/2*exp(x))))^2-2*ln(1/2*exp(x))*ln(ln(1/2*exp(x)))*ln
(ln(ln(1/2*exp(x))))+x^2-2*x)/x^2/ln(1/2*exp(x))/ln(ln(1/2*exp(x)))/ln(ln(ln(1/2*exp(x))))^2,x,method=_RETURNV
ERBOSE)

[Out]

(x^2+1)/x-(x-2)/x/ln(ln(-ln(2)+ln(exp(x))))

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maxima [A]  time = 0.54, size = 31, normalized size = 1.24 $$\begin{array}{c}x+\frac{1}{x}-\frac{1}{\log (\log (\log \left(\frac{1}{2}{e}^x\right)))}+\frac{2}{x\log (\log (x-\log (2)))}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 5.69, size = 32, normalized size = 1.28 $$\begin{array}{c}x+\frac{2}{x\ln (\ln (x-\ln (2)))}+\frac{1}{x}-\frac{1}{\ln (\ln (x-\ln (2)))}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 $$\begin{array}{c}\text{Timed out}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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