\(\int \frac {16 e^{e^x}-16 e^{e^x} \log (x) \log (\log (x)) \log (\log (\log (x)))-16 e^{e^x+x} x \log (x) \log (\log (x)) \log (\log (\log (x))) \log (\frac {\log (\log (\log (x)))}{x})}{x \log (x) \log (\log (x)) \log (\log (\log (x))) \log ^2(\frac {\log (\log (\log (x)))}{x})} \, dx\) [327]

Optimal result

Integrand size = 82, antiderivative size = 23 \[ \int \frac {16 e^{e^x}-16 e^{e^x} \log (x) \log (\log (x)) \log (\log (\log (x)))-16 e^{e^x+x} x \log (x) \log (\log (x)) \log (\log (\log (x))) \log \left (\frac {\log (\log (\log (x)))}{x}\right )}{x \log (x) \log (\log (x)) \log (\log (\log (x))) \log ^2\left (\frac {\log (\log (\log (x)))}{x}\right )} \, dx=-2-16 \left (5+\frac {e^{e^x}}{\log \left (\frac {\log (\log (\log (x)))}{x}\right )}\right ) \]

[Out]

-82-16*exp(exp(x))/ln(ln(ln(ln(x)))/x)

Rubi [F]

\[ \int \frac {16 e^{e^x}-16 e^{e^x} \log (x) \log (\log (x)) \log (\log (\log (x)))-16 e^{e^x+x} x \log (x) \log (\log (x)) \log (\log (\log (x))) \log \left (\frac {\log (\log (\log (x)))}{x}\right )}{x \log (x) \log (\log (x)) \log (\log (\log (x))) \log ^2\left (\frac {\log (\log (\log (x)))}{x}\right )} \, dx=\int \frac {16 e^{e^x}-16 e^{e^x} \log (x) \log (\log (x)) \log (\log (\log (x)))-16 e^{e^x+x} x \log (x) \log (\log (x)) \log (\log (\log (x))) \log \left (\frac {\log (\log (\log (x)))}{x}\right )}{x \log (x) \log (\log (x)) \log (\log (\log (x))) \log ^2\left (\frac {\log (\log (\log (x)))}{x}\right )} \, dx \]

[In]

Int[(16*E^E^x - 16*E^E^x*Log[x]*Log[Log[x]]*Log[Log[Log[x]]] - 16*E^(E^x + x)*x*Log[x]*Log[Log[x]]*Log[Log[Log
[x]]]*Log[Log[Log[Log[x]]]/x])/(x*Log[x]*Log[Log[x]]*Log[Log[Log[x]]]*Log[Log[Log[Log[x]]]/x]^2),x]

[Out]

-16*Defer[Int][E^E^x/(x*Log[Log[Log[Log[x]]]/x]^2), x] + 16*Defer[Int][E^E^x/(x*Log[x]*Log[Log[x]]*Log[Log[Log
[x]]]*Log[Log[Log[Log[x]]]/x]^2), x] - 16*Defer[Int][E^(E^x + x)/Log[Log[Log[Log[x]]]/x], x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {16 e^{e^x} \left (1-\log (x) \log (\log (x)) \log (\log (\log (x))) \left (1+e^x x \log \left (\frac {\log (\log (\log (x)))}{x}\right )\right )\right )}{x \log (x) \log (\log (x)) \log (\log (\log (x))) \log ^2\left (\frac {\log (\log (\log (x)))}{x}\right )} \, dx \\ & = 16 \int \frac {e^{e^x} \left (1-\log (x) \log (\log (x)) \log (\log (\log (x))) \left (1+e^x x \log \left (\frac {\log (\log (\log (x)))}{x}\right )\right )\right )}{x \log (x) \log (\log (x)) \log (\log (\log (x))) \log ^2\left (\frac {\log (\log (\log (x)))}{x}\right )} \, dx \\ & = 16 \int \left (\frac {e^{e^x} (1-\log (x) \log (\log (x)) \log (\log (\log (x))))}{x \log (x) \log (\log (x)) \log (\log (\log (x))) \log ^2\left (\frac {\log (\log (\log (x)))}{x}\right )}-\frac {e^{e^x+x}}{\log \left (\frac {\log (\log (\log (x)))}{x}\right )}\right ) \, dx \\ & = 16 \int \frac {e^{e^x} (1-\log (x) \log (\log (x)) \log (\log (\log (x))))}{x \log (x) \log (\log (x)) \log (\log (\log (x))) \log ^2\left (\frac {\log (\log (\log (x)))}{x}\right )} \, dx-16 \int \frac {e^{e^x+x}}{\log \left (\frac {\log (\log (\log (x)))}{x}\right )} \, dx \\ & = 16 \int \left (-\frac {e^{e^x}}{x \log ^2\left (\frac {\log (\log (\log (x)))}{x}\right )}+\frac {e^{e^x}}{x \log (x) \log (\log (x)) \log (\log (\log (x))) \log ^2\left (\frac {\log (\log (\log (x)))}{x}\right )}\right ) \, dx-16 \int \frac {e^{e^x+x}}{\log \left (\frac {\log (\log (\log (x)))}{x}\right )} \, dx \\ & = -\left (16 \int \frac {e^{e^x}}{x \log ^2\left (\frac {\log (\log (\log (x)))}{x}\right )} \, dx\right )+16 \int \frac {e^{e^x}}{x \log (x) \log (\log (x)) \log (\log (\log (x))) \log ^2\left (\frac {\log (\log (\log (x)))}{x}\right )} \, dx-16 \int \frac {e^{e^x+x}}{\log \left (\frac {\log (\log (\log (x)))}{x}\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78 \[ \int \frac {16 e^{e^x}-16 e^{e^x} \log (x) \log (\log (x)) \log (\log (\log (x)))-16 e^{e^x+x} x \log (x) \log (\log (x)) \log (\log (\log (x))) \log \left (\frac {\log (\log (\log (x)))}{x}\right )}{x \log (x) \log (\log (x)) \log (\log (\log (x))) \log ^2\left (\frac {\log (\log (\log (x)))}{x}\right )} \, dx=-\frac {16 e^{e^x}}{\log \left (\frac {\log (\log (\log (x)))}{x}\right )} \]

[In]

Integrate[(16*E^E^x - 16*E^E^x*Log[x]*Log[Log[x]]*Log[Log[Log[x]]] - 16*E^(E^x + x)*x*Log[x]*Log[Log[x]]*Log[L
og[Log[x]]]*Log[Log[Log[Log[x]]]/x])/(x*Log[x]*Log[Log[x]]*Log[Log[Log[x]]]*Log[Log[Log[Log[x]]]/x]^2),x]

[Out]

(-16*E^E^x)/Log[Log[Log[Log[x]]]/x]

Maple [C] (warning: unable to verify)

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

Time = 0.03 (sec) , antiderivative size = 113, normalized size of antiderivative = 4.91

\[-\frac {32 i {\mathrm e}^{{\mathrm e}^{x}}}{\pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (i \ln \left (\ln \left (\ln \left (x \right )\right )\right )\right ) \operatorname {csgn}\left (\frac {i \ln \left (\ln \left (\ln \left (x \right )\right )\right )}{x}\right )-\pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i \ln \left (\ln \left (\ln \left (x \right )\right )\right )}{x}\right )^{2}-\pi \,\operatorname {csgn}\left (i \ln \left (\ln \left (\ln \left (x \right )\right )\right )\right ) \operatorname {csgn}\left (\frac {i \ln \left (\ln \left (\ln \left (x \right )\right )\right )}{x}\right )^{2}+\pi \operatorname {csgn}\left (\frac {i \ln \left (\ln \left (\ln \left (x \right )\right )\right )}{x}\right )^{3}-2 i \ln \left (x \right )+2 i \ln \left (\ln \left (\ln \left (\ln \left (x \right )\right )\right )\right )}\]

[In]

int((-16*x*exp(x)*ln(x)*exp(exp(x))*ln(ln(x))*ln(ln(ln(x)))*ln(ln(ln(ln(x)))/x)-16*ln(x)*exp(exp(x))*ln(ln(x))
*ln(ln(ln(x)))+16*exp(exp(x)))/x/ln(x)/ln(ln(x))/ln(ln(ln(x)))/ln(ln(ln(ln(x)))/x)^2,x)

[Out]

-32*I*exp(exp(x))/(Pi*csgn(I/x)*csgn(I*ln(ln(ln(x))))*csgn(I/x*ln(ln(ln(x))))-Pi*csgn(I/x)*csgn(I/x*ln(ln(ln(x
))))^2-Pi*csgn(I*ln(ln(ln(x))))*csgn(I/x*ln(ln(ln(x))))^2+Pi*csgn(I/x*ln(ln(ln(x))))^3-2*I*ln(x)+2*I*ln(ln(ln(
ln(x)))))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.70 \[ \int \frac {16 e^{e^x}-16 e^{e^x} \log (x) \log (\log (x)) \log (\log (\log (x)))-16 e^{e^x+x} x \log (x) \log (\log (x)) \log (\log (\log (x))) \log \left (\frac {\log (\log (\log (x)))}{x}\right )}{x \log (x) \log (\log (x)) \log (\log (\log (x))) \log ^2\left (\frac {\log (\log (\log (x)))}{x}\right )} \, dx=-\frac {16 \, e^{\left (e^{x}\right )}}{\log \left (\frac {\log \left (\log \left (\log \left (x\right )\right )\right )}{x}\right )} \]

[In]

integrate((-16*x*exp(x)*log(x)*exp(exp(x))*log(log(x))*log(log(log(x)))*log(log(log(log(x)))/x)-16*log(x)*exp(
exp(x))*log(log(x))*log(log(log(x)))+16*exp(exp(x)))/x/log(x)/log(log(x))/log(log(log(x)))/log(log(log(log(x))
)/x)^2,x, algorithm="fricas")

[Out]

-16*e^(e^x)/log(log(log(log(x)))/x)

Sympy [A] (verification not implemented)

Time = 95.22 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {16 e^{e^x}-16 e^{e^x} \log (x) \log (\log (x)) \log (\log (\log (x)))-16 e^{e^x+x} x \log (x) \log (\log (x)) \log (\log (\log (x))) \log \left (\frac {\log (\log (\log (x)))}{x}\right )}{x \log (x) \log (\log (x)) \log (\log (\log (x))) \log ^2\left (\frac {\log (\log (\log (x)))}{x}\right )} \, dx=- \frac {16 e^{e^{x}}}{\log {\left (\frac {\log {\left (\log {\left (\log {\left (x \right )} \right )} \right )}}{x} \right )}} \]

[In]

integrate((-16*x*exp(x)*ln(x)*exp(exp(x))*ln(ln(x))*ln(ln(ln(x)))*ln(ln(ln(ln(x)))/x)-16*ln(x)*exp(exp(x))*ln(
ln(x))*ln(ln(ln(x)))+16*exp(exp(x)))/x/ln(x)/ln(ln(x))/ln(ln(ln(x)))/ln(ln(ln(ln(x)))/x)**2,x)

[Out]

-16*exp(exp(x))/log(log(log(log(x)))/x)

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {16 e^{e^x}-16 e^{e^x} \log (x) \log (\log (x)) \log (\log (\log (x)))-16 e^{e^x+x} x \log (x) \log (\log (x)) \log (\log (\log (x))) \log \left (\frac {\log (\log (\log (x)))}{x}\right )}{x \log (x) \log (\log (x)) \log (\log (\log (x))) \log ^2\left (\frac {\log (\log (\log (x)))}{x}\right )} \, dx=\frac {16 \, e^{\left (e^{x}\right )}}{\log \left (x\right ) - \log \left (\log \left (\log \left (\log \left (x\right )\right )\right )\right )} \]

[In]

integrate((-16*x*exp(x)*log(x)*exp(exp(x))*log(log(x))*log(log(log(x)))*log(log(log(log(x)))/x)-16*log(x)*exp(
exp(x))*log(log(x))*log(log(log(x)))+16*exp(exp(x)))/x/log(x)/log(log(x))/log(log(log(x)))/log(log(log(log(x))
)/x)^2,x, algorithm="maxima")

[Out]

16*e^(e^x)/(log(x) - log(log(log(log(x)))))

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {16 e^{e^x}-16 e^{e^x} \log (x) \log (\log (x)) \log (\log (\log (x)))-16 e^{e^x+x} x \log (x) \log (\log (x)) \log (\log (\log (x))) \log \left (\frac {\log (\log (\log (x)))}{x}\right )}{x \log (x) \log (\log (x)) \log (\log (\log (x))) \log ^2\left (\frac {\log (\log (\log (x)))}{x}\right )} \, dx=\frac {16 \, e^{\left (e^{x}\right )}}{\log \left (x\right ) - \log \left (\log \left (\log \left (\log \left (x\right )\right )\right )\right )} \]

[In]

integrate((-16*x*exp(x)*log(x)*exp(exp(x))*log(log(x))*log(log(log(x)))*log(log(log(log(x)))/x)-16*log(x)*exp(
exp(x))*log(log(x))*log(log(log(x)))+16*exp(exp(x)))/x/log(x)/log(log(x))/log(log(log(x)))/log(log(log(log(x))
)/x)^2,x, algorithm="giac")

[Out]

16*e^(e^x)/(log(x) - log(log(log(log(x)))))

Mupad [F(-1)]

Timed out. \[ \int \frac {16 e^{e^x}-16 e^{e^x} \log (x) \log (\log (x)) \log (\log (\log (x)))-16 e^{e^x+x} x \log (x) \log (\log (x)) \log (\log (\log (x))) \log \left (\frac {\log (\log (\log (x)))}{x}\right )}{x \log (x) \log (\log (x)) \log (\log (\log (x))) \log ^2\left (\frac {\log (\log (\log (x)))}{x}\right )} \, dx=\int -\frac {16\,\ln \left (\ln \left (x\right )\right )\,{\mathrm {e}}^{{\mathrm {e}}^x}\,\ln \left (\ln \left (\ln \left (x\right )\right )\right )\,\ln \left (x\right )-16\,{\mathrm {e}}^{{\mathrm {e}}^x}+16\,x\,\ln \left (\ln \left (x\right )\right )\,\ln \left (\frac {\ln \left (\ln \left (\ln \left (x\right )\right )\right )}{x}\right )\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^x\,\ln \left (\ln \left (\ln \left (x\right )\right )\right )\,\ln \left (x\right )}{x\,\ln \left (\ln \left (x\right )\right )\,{\ln \left (\frac {\ln \left (\ln \left (\ln \left (x\right )\right )\right )}{x}\right )}^2\,\ln \left (\ln \left (\ln \left (x\right )\right )\right )\,\ln \left (x\right )} \,d x \]

[In]

int(-(16*log(log(x))*exp(exp(x))*log(log(log(x)))*log(x) - 16*exp(exp(x)) + 16*x*log(log(x))*log(log(log(log(x
)))/x)*exp(exp(x))*exp(x)*log(log(log(x)))*log(x))/(x*log(log(x))*log(log(log(log(x)))/x)^2*log(log(log(x)))*l
og(x)),x)

[Out]

int(-(16*log(log(x))*exp(exp(x))*log(log(log(x)))*log(x) - 16*exp(exp(x)) + 16*x*log(log(x))*log(log(log(log(x
)))/x)*exp(exp(x))*exp(x)*log(log(log(x)))*log(x))/(x*log(log(x))*log(log(log(log(x)))/x)^2*log(log(log(x)))*l
og(x)), x)