\(\int \frac {-2 x+x^2-2 \log (\frac {e^x}{2}) \log (\log (\frac {e^x}{2})) \log (\log (\log (\frac {e^x}{2})))+(-1+x^2) \log (\frac {e^x}{2}) \log (\log (\frac {e^x}{2})) \log ^2(\log (\log (\frac {e^x}{2})))}{x^2 \log (\frac {e^x}{2}) \log (\log (\frac {e^x}{2})) \log ^2(\log (\log (\frac {e^x}{2})))} \, dx\) [8816]

Optimal result

Integrand size = 108, antiderivative size = 25 \[ \int \frac {-2 x+x^2-2 \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )+\left (-1+x^2\right ) \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )}{x^2 \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx=x+\frac {1-\frac {-2+x}{\log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )}}{x} \]

[Out]

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

Rubi [F]

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

[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{align*} \text {integral}& = \int \frac {\frac {(-2+x) x}{\log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right )}+\log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right ) \left (-2+\left (-1+x^2\right ) \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )\right )}{x^2 \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx \\ & = \int \left (\frac {-1+x^2}{x^2}+\frac {-2+x}{x \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )}-\frac {2}{x^2 \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )}\right ) \, dx \\ & = -\left (2 \int \frac {1}{x^2 \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx\right )+\int \frac {-1+x^2}{x^2} \, dx+\int \frac {-2+x}{x \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx \\ & = -\left (2 \int \frac {1}{x^2 \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx\right )+\int \left (1-\frac {1}{x^2}\right ) \, dx+\int \left (\frac {1}{\log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )}-\frac {2}{x \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )}\right ) \, dx \\ & = \frac {1}{x}+x-2 \int \frac {1}{x \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx-2 \int \frac {1}{x^2 \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx+\int \frac {1}{\log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx \\ & = \frac {1}{x}+x-2 \int \frac {1}{x \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx-2 \int \frac {1}{x^2 \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx+\text {Subst}\left (\int \frac {1}{x \log \left (\frac {x}{2}\right ) \log \left (\log \left (\frac {x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {x}{2}\right )\right )\right )} \, dx,x,e^x\right ) \\ & = \frac {1}{x}+x-2 \int \frac {1}{x \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx-2 \int \frac {1}{x^2 \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx+\text {Subst}\left (\int \frac {1}{x \log (x) \log ^2(\log (x))} \, dx,x,\log \left (\frac {e^x}{2}\right )\right ) \\ & = \frac {1}{x}+x-2 \int \frac {1}{x \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx-2 \int \frac {1}{x^2 \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx+\text {Subst}\left (\int \frac {1}{x \log ^2(x)} \, dx,x,\log \left (\log \left (\frac {e^x}{2}\right )\right )\right ) \\ & = \frac {1}{x}+x-2 \int \frac {1}{x \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx-2 \int \frac {1}{x^2 \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx+\text {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )\right ) \\ & = \frac {1}{x}+x-\frac {1}{\log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )}-2 \int \frac {1}{x \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx-2 \int \frac {1}{x^2 \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {-2 x+x^2-2 \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )+\left (-1+x^2\right ) \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )}{x^2 \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx=\frac {1}{x}+x+\frac {2-x}{x \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \]

[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]]])

Maple [A] (verified)

Time = 4.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24

method	result	size
risch	\(\frac {x^{2}+1}{x}-\frac {-2+x}{x \ln \left (\ln \left (-\ln \left (2\right )+\ln \left ({\mathrm e}^{x}\right )\right )\right )}\)	\(31\)
parallelrisch	\(-\frac {-12-6 \ln \left (\ln \left (\ln \left (\frac {{\mathrm e}^{x}}{2}\right )\right )\right ) x^{2}+6 x -6 \ln \left (\ln \left (\ln \left (\frac {{\mathrm e}^{x}}{2}\right )\right )\right )}{6 \ln \left (\ln \left (\ln \left (\frac {{\mathrm e}^{x}}{2}\right )\right )\right ) x}\)	\(41\)

[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]

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

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \frac {-2 x+x^2-2 \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )+\left (-1+x^2\right ) \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )}{x^2 \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx=\frac {{\left (x^{2} + 1\right )} \log \left (\log \left (x - \log \left (2\right )\right )\right ) - x + 2}{x \log \left (\log \left (x - \log \left (2\right )\right )\right )} \]

[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))))

Sympy [F(-1)]

Timed out. \[ \int \frac {-2 x+x^2-2 \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )+\left (-1+x^2\right ) \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )}{x^2 \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx=\text {Timed out} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {-2 x+x^2-2 \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )+\left (-1+x^2\right ) \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )}{x^2 \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx=x + \frac {1}{x} - \frac {1}{\log \left (\log \left (\log \left (\frac {1}{2} \, e^{x}\right )\right )\right )} + \frac {2}{x \log \left (\log \left (x - \log \left (2\right )\right )\right )} \]

[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))))

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {-2 x+x^2-2 \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )+\left (-1+x^2\right ) \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )}{x^2 \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx=x + \frac {1}{x} - \frac {x - 2}{x \log \left (\log \left (x - \log \left (2\right )\right )\right )} \]

[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))))

Mupad [B] (verification not implemented)

Time = 10.43 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {-2 x+x^2-2 \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )+\left (-1+x^2\right ) \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )}{x^2 \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx=x+\frac {2}{x\,\ln \left (\ln \left (x-\ln \left (2\right )\right )\right )}+\frac {1}{x}-\frac {1}{\ln \left (\ln \left (x-\ln \left (2\right )\right )\right )} \]

[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)))