3.99.27 \(\int \frac {e^{e^{2 x}} (-1+2 e^{2 x} x) \log (x) \log ^3(\log (x))+e^{\frac {256+x \log ^2(\log (x))}{\log ^2(\log (x))}} (512+(1-x) \log (x) \log ^3(\log (x)))}{e^{\frac {2 (256+x \log ^2(\log (x)))}{\log ^2(\log (x))}} \log (x) \log ^3(\log (x))+e^{\frac {256+x \log ^2(\log (x))}{\log ^2(\log (x))}} (-2 e^{e^{2 x}} \log (x)+2 x \log (x)) \log ^3(\log (x))+(e^{2 e^{2 x}} \log (x)-2 e^{e^{2 x}} x \log (x)+x^2 \log (x)) \log ^3(\log (x))} \, dx\)

Optimal. Leaf size=26 \[ \frac {x}{-e^{e^{2 x}}+e^{x+\frac {256}{\log ^2(\log (x))}}+x} \]

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Rubi [F]  time = 8.23, 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 {gather*} \int \frac {e^{e^{2 x}} \left (-1+2 e^{2 x} x\right ) \log (x) \log ^3(\log (x))+e^{\frac {256+x \log ^2(\log (x))}{\log ^2(\log (x))}} \left (512+(1-x) \log (x) \log ^3(\log (x))\right )}{e^{\frac {2 \left (256+x \log ^2(\log (x))\right )}{\log ^2(\log (x))}} \log (x) \log ^3(\log (x))+e^{\frac {256+x \log ^2(\log (x))}{\log ^2(\log (x))}} \left (-2 e^{e^{2 x}} \log (x)+2 x \log (x)\right ) \log ^3(\log (x))+\left (e^{2 e^{2 x}} \log (x)-2 e^{e^{2 x}} x \log (x)+x^2 \log (x)\right ) \log ^3(\log (x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^E^(2*x)*(-1 + 2*E^(2*x)*x)*Log[x]*Log[Log[x]]^3 + E^((256 + x*Log[Log[x]]^2)/Log[Log[x]]^2)*(512 + (1 -
x)*Log[x]*Log[Log[x]]^3))/(E^((2*(256 + x*Log[Log[x]]^2))/Log[Log[x]]^2)*Log[x]*Log[Log[x]]^3 + E^((256 + x*L
og[Log[x]]^2)/Log[Log[x]]^2)*(-2*E^E^(2*x)*Log[x] + 2*x*Log[x])*Log[Log[x]]^3 + (E^(2*E^(2*x))*Log[x] - 2*E^E^
(2*x)*x*Log[x] + x^2*Log[x])*Log[Log[x]]^3),x]

[Out]

-Defer[Int][(E^E^(2*x) - E^(x + 256/Log[Log[x]]^2) - x)^(-1), x] - Defer[Int][x/(E^E^(2*x) - E^(x + 256/Log[Lo
g[x]]^2) - x)^2, x] - Defer[Int][(E^E^(2*x)*x)/(E^E^(2*x) - E^(x + 256/Log[Log[x]]^2) - x)^2, x] + Defer[Int][
x/(E^E^(2*x) - E^(x + 256/Log[Log[x]]^2) - x), x] + Defer[Int][x^2/(E^E^(2*x) - E^(x + 256/Log[Log[x]]^2) - x)
^2, x] + 2*Defer[Int][(E^(E^(2*x) + 2*x)*x)/(-E^E^(2*x) + E^(x + 256/Log[Log[x]]^2) + x)^2, x] + 512*Defer[Int
][E^E^(2*x)/((E^E^(2*x) - E^(x + 256/Log[Log[x]]^2) - x)^2*Log[x]*Log[Log[x]]^3), x] - 512*Defer[Int][1/((E^E^
(2*x) - E^(x + 256/Log[Log[x]]^2) - x)*Log[x]*Log[Log[x]]^3), x] - 512*Defer[Int][x/((E^E^(2*x) - E^(x + 256/L
og[Log[x]]^2) - x)^2*Log[x]*Log[Log[x]]^3), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {512 e^{x+\frac {256}{\log ^2(\log (x))}}+\left (-e^{e^{2 x}}-e^{x+\frac {256}{\log ^2(\log (x))}} (-1+x)+2 e^{e^{2 x}+2 x} x\right ) \log (x) \log ^3(\log (x))}{\left (e^{e^{2 x}}-e^{x+\frac {256}{\log ^2(\log (x))}}-x\right )^2 \log (x) \log ^3(\log (x))} \, dx\\ &=\int \left (\frac {-512-\log (x) \log ^3(\log (x))+x \log (x) \log ^3(\log (x))}{\left (e^{e^{2 x}}-e^{x+\frac {256}{\log ^2(\log (x))}}-x\right ) \log (x) \log ^3(\log (x))}+\frac {512 e^{e^{2 x}}-512 x-x \log (x) \log ^3(\log (x))-e^{e^{2 x}} x \log (x) \log ^3(\log (x))+2 e^{e^{2 x}+2 x} x \log (x) \log ^3(\log (x))+x^2 \log (x) \log ^3(\log (x))}{\left (e^{e^{2 x}}-e^{x+\frac {256}{\log ^2(\log (x))}}-x\right )^2 \log (x) \log ^3(\log (x))}\right ) \, dx\\ &=\int \frac {-512-\log (x) \log ^3(\log (x))+x \log (x) \log ^3(\log (x))}{\left (e^{e^{2 x}}-e^{x+\frac {256}{\log ^2(\log (x))}}-x\right ) \log (x) \log ^3(\log (x))} \, dx+\int \frac {512 e^{e^{2 x}}-512 x-x \log (x) \log ^3(\log (x))-e^{e^{2 x}} x \log (x) \log ^3(\log (x))+2 e^{e^{2 x}+2 x} x \log (x) \log ^3(\log (x))+x^2 \log (x) \log ^3(\log (x))}{\left (e^{e^{2 x}}-e^{x+\frac {256}{\log ^2(\log (x))}}-x\right )^2 \log (x) \log ^3(\log (x))} \, dx\\ &=\int \frac {-512+(-1+x) \log (x) \log ^3(\log (x))}{\left (e^{e^{2 x}}-e^{x+\frac {256}{\log ^2(\log (x))}}-x\right ) \log (x) \log ^3(\log (x))} \, dx+\int \frac {512 \left (e^{e^{2 x}}-x\right )+x \left (-1-e^{e^{2 x}}+2 e^{e^{2 x}+2 x}+x\right ) \log (x) \log ^3(\log (x))}{\left (e^{e^{2 x}}-e^{x+\frac {256}{\log ^2(\log (x))}}-x\right )^2 \log (x) \log ^3(\log (x))} \, dx\\ &=\int \left (-\frac {1}{e^{e^{2 x}}-e^{x+\frac {256}{\log ^2(\log (x))}}-x}+\frac {x}{e^{e^{2 x}}-e^{x+\frac {256}{\log ^2(\log (x))}}-x}-\frac {512}{\left (e^{e^{2 x}}-e^{x+\frac {256}{\log ^2(\log (x))}}-x\right ) \log (x) \log ^3(\log (x))}\right ) \, dx+\int \left (-\frac {x}{\left (e^{e^{2 x}}-e^{x+\frac {256}{\log ^2(\log (x))}}-x\right )^2}-\frac {e^{e^{2 x}} x}{\left (e^{e^{2 x}}-e^{x+\frac {256}{\log ^2(\log (x))}}-x\right )^2}+\frac {x^2}{\left (e^{e^{2 x}}-e^{x+\frac {256}{\log ^2(\log (x))}}-x\right )^2}+\frac {2 e^{e^{2 x}+2 x} x}{\left (-e^{e^{2 x}}+e^{x+\frac {256}{\log ^2(\log (x))}}+x\right )^2}+\frac {512 e^{e^{2 x}}}{\left (e^{e^{2 x}}-e^{x+\frac {256}{\log ^2(\log (x))}}-x\right )^2 \log (x) \log ^3(\log (x))}-\frac {512 x}{\left (e^{e^{2 x}}-e^{x+\frac {256}{\log ^2(\log (x))}}-x\right )^2 \log (x) \log ^3(\log (x))}\right ) \, dx\\ &=2 \int \frac {e^{e^{2 x}+2 x} x}{\left (-e^{e^{2 x}}+e^{x+\frac {256}{\log ^2(\log (x))}}+x\right )^2} \, dx+512 \int \frac {e^{e^{2 x}}}{\left (e^{e^{2 x}}-e^{x+\frac {256}{\log ^2(\log (x))}}-x\right )^2 \log (x) \log ^3(\log (x))} \, dx-512 \int \frac {1}{\left (e^{e^{2 x}}-e^{x+\frac {256}{\log ^2(\log (x))}}-x\right ) \log (x) \log ^3(\log (x))} \, dx-512 \int \frac {x}{\left (e^{e^{2 x}}-e^{x+\frac {256}{\log ^2(\log (x))}}-x\right )^2 \log (x) \log ^3(\log (x))} \, dx-\int \frac {1}{e^{e^{2 x}}-e^{x+\frac {256}{\log ^2(\log (x))}}-x} \, dx-\int \frac {x}{\left (e^{e^{2 x}}-e^{x+\frac {256}{\log ^2(\log (x))}}-x\right )^2} \, dx-\int \frac {e^{e^{2 x}} x}{\left (e^{e^{2 x}}-e^{x+\frac {256}{\log ^2(\log (x))}}-x\right )^2} \, dx+\int \frac {x}{e^{e^{2 x}}-e^{x+\frac {256}{\log ^2(\log (x))}}-x} \, dx+\int \frac {x^2}{\left (e^{e^{2 x}}-e^{x+\frac {256}{\log ^2(\log (x))}}-x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.18, size = 26, normalized size = 1.00 \begin {gather*} \frac {x}{-e^{e^{2 x}}+e^{x+\frac {256}{\log ^2(\log (x))}}+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^E^(2*x)*(-1 + 2*E^(2*x)*x)*Log[x]*Log[Log[x]]^3 + E^((256 + x*Log[Log[x]]^2)/Log[Log[x]]^2)*(512
+ (1 - x)*Log[x]*Log[Log[x]]^3))/(E^((2*(256 + x*Log[Log[x]]^2))/Log[Log[x]]^2)*Log[x]*Log[Log[x]]^3 + E^((256
+ x*Log[Log[x]]^2)/Log[Log[x]]^2)*(-2*E^E^(2*x)*Log[x] + 2*x*Log[x])*Log[Log[x]]^3 + (E^(2*E^(2*x))*Log[x] -
2*E^E^(2*x)*x*Log[x] + x^2*Log[x])*Log[Log[x]]^3),x]

[Out]

x/(-E^E^(2*x) + E^(x + 256/Log[Log[x]]^2) + x)

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fricas [A]  time = 0.63, size = 29, normalized size = 1.12 \begin {gather*} \frac {x}{x + e^{\left (\frac {x \log \left (\log \relax (x)\right )^{2} + 256}{\log \left (\log \relax (x)\right )^{2}}\right )} - e^{\left (e^{\left (2 \, x\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x+1)*log(x)*log(log(x))^3+512)*exp((x*log(log(x))^2+256)/log(log(x))^2)+(2*x*exp(x)^2-1)*log(x)*
exp(exp(x)^2)*log(log(x))^3)/(log(x)*log(log(x))^3*exp((x*log(log(x))^2+256)/log(log(x))^2)^2+(-2*log(x)*exp(e
xp(x)^2)+2*x*log(x))*log(log(x))^3*exp((x*log(log(x))^2+256)/log(log(x))^2)+(log(x)*exp(exp(x)^2)^2-2*x*log(x)
*exp(exp(x)^2)+x^2*log(x))*log(log(x))^3),x, algorithm="fricas")

[Out]

x/(x + e^((x*log(log(x))^2 + 256)/log(log(x))^2) - e^(e^(2*x)))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {undef} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x+1)*log(x)*log(log(x))^3+512)*exp((x*log(log(x))^2+256)/log(log(x))^2)+(2*x*exp(x)^2-1)*log(x)*
exp(exp(x)^2)*log(log(x))^3)/(log(x)*log(log(x))^3*exp((x*log(log(x))^2+256)/log(log(x))^2)^2+(-2*log(x)*exp(e
xp(x)^2)+2*x*log(x))*log(log(x))^3*exp((x*log(log(x))^2+256)/log(log(x))^2)+(log(x)*exp(exp(x)^2)^2-2*x*log(x)
*exp(exp(x)^2)+x^2*log(x))*log(log(x))^3),x, algorithm="giac")

[Out]

undef

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maple [A]  time = 0.10, size = 30, normalized size = 1.15

method	result	size
risch	\(\frac {x}{x +{\mathrm e}^{\frac {x \ln \left (\ln \relax (x )\right )^{2}+256}{\ln \left (\ln \relax (x )\right )^{2}}}-{\mathrm e}^{{\mathrm e}^{2 x}}}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((1-x)*ln(x)*ln(ln(x))^3+512)*exp((x*ln(ln(x))^2+256)/ln(ln(x))^2)+(2*x*exp(x)^2-1)*ln(x)*exp(exp(x)^2)*l
n(ln(x))^3)/(ln(x)*ln(ln(x))^3*exp((x*ln(ln(x))^2+256)/ln(ln(x))^2)^2+(-2*ln(x)*exp(exp(x)^2)+2*x*ln(x))*ln(ln
(x))^3*exp((x*ln(ln(x))^2+256)/ln(ln(x))^2)+(ln(x)*exp(exp(x)^2)^2-2*x*ln(x)*exp(exp(x)^2)+x^2*ln(x))*ln(ln(x)
)^3),x,method=_RETURNVERBOSE)

[Out]

x/(x+exp((x*ln(ln(x))^2+256)/ln(ln(x))^2)-exp(exp(2*x)))

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maxima [A]  time = 0.52, size = 23, normalized size = 0.88 \begin {gather*} \frac {x}{x + e^{\left (x + \frac {256}{\log \left (\log \relax (x)\right )^{2}}\right )} - e^{\left (e^{\left (2 \, x\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x+1)*log(x)*log(log(x))^3+512)*exp((x*log(log(x))^2+256)/log(log(x))^2)+(2*x*exp(x)^2-1)*log(x)*
exp(exp(x)^2)*log(log(x))^3)/(log(x)*log(log(x))^3*exp((x*log(log(x))^2+256)/log(log(x))^2)^2+(-2*log(x)*exp(e
xp(x)^2)+2*x*log(x))*log(log(x))^3*exp((x*log(log(x))^2+256)/log(log(x))^2)+(log(x)*exp(exp(x)^2)^2-2*x*log(x)
*exp(exp(x)^2)+x^2*log(x))*log(log(x))^3),x, algorithm="maxima")

[Out]

x/(x + e^(x + 256/log(log(x))^2) - e^(e^(2*x)))

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mupad [B]  time = 6.14, size = 215, normalized size = 8.27 \begin {gather*} \frac {{\mathrm {e}}^{{\mathrm {e}}^{2\,x}}\,\left (x^3\,{\ln \left (\ln \relax (x)\right )}^6\,{\ln \relax (x)}^2-512\,x^2\,{\ln \left (\ln \relax (x)\right )}^3\,\ln \relax (x)\right )+x^3\,{\ln \left (\ln \relax (x)\right )}^6\,{\ln \relax (x)}^2-x^4\,{\ln \left (\ln \relax (x)\right )}^6\,{\ln \relax (x)}^2+512\,x^3\,{\ln \left (\ln \relax (x)\right )}^3\,\ln \relax (x)-2\,x^3\,{\ln \left (\ln \relax (x)\right )}^6\,{\mathrm {e}}^{2\,x+{\mathrm {e}}^{2\,x}}\,{\ln \relax (x)}^2}{\left (x+{\mathrm {e}}^{x+\frac {256}{{\ln \left (\ln \relax (x)\right )}^2}}-{\mathrm {e}}^{{\mathrm {e}}^{2\,x}}\right )\,\left (x^2\,{\ln \left (\ln \relax (x)\right )}^6\,{\ln \relax (x)}^2-x^3\,{\ln \left (\ln \relax (x)\right )}^6\,{\ln \relax (x)}^2+512\,x^2\,{\ln \left (\ln \relax (x)\right )}^3\,\ln \relax (x)-2\,x^2\,{\ln \left (\ln \relax (x)\right )}^6\,{\mathrm {e}}^{2\,x+{\mathrm {e}}^{2\,x}}\,{\ln \relax (x)}^2-512\,x\,{\ln \left (\ln \relax (x)\right )}^3\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x}}\,\ln \relax (x)+x^2\,{\ln \left (\ln \relax (x)\right )}^6\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x}}\,{\ln \relax (x)}^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((x*log(log(x))^2 + 256)/log(log(x))^2)*(log(log(x))^3*log(x)*(x - 1) - 512) - log(log(x))^3*exp(exp(
2*x))*log(x)*(2*x*exp(2*x) - 1))/(log(log(x))^3*(x^2*log(x) + exp(2*exp(2*x))*log(x) - 2*x*exp(exp(2*x))*log(x
)) + log(log(x))^3*exp((2*(x*log(log(x))^2 + 256))/log(log(x))^2)*log(x) + log(log(x))^3*exp((x*log(log(x))^2
+ 256)/log(log(x))^2)*(2*x*log(x) - 2*exp(exp(2*x))*log(x))),x)

[Out]

(exp(exp(2*x))*(x^3*log(log(x))^6*log(x)^2 - 512*x^2*log(log(x))^3*log(x)) + x^3*log(log(x))^6*log(x)^2 - x^4*
log(log(x))^6*log(x)^2 + 512*x^3*log(log(x))^3*log(x) - 2*x^3*log(log(x))^6*exp(2*x + exp(2*x))*log(x)^2)/((x
+ exp(x + 256/log(log(x))^2) - exp(exp(2*x)))*(x^2*log(log(x))^6*log(x)^2 - x^3*log(log(x))^6*log(x)^2 + 512*x
^2*log(log(x))^3*log(x) - 2*x^2*log(log(x))^6*exp(2*x + exp(2*x))*log(x)^2 - 512*x*log(log(x))^3*exp(exp(2*x))
*log(x) + x^2*log(log(x))^6*exp(exp(2*x))*log(x)^2))

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sympy [A]  time = 3.77, size = 29, normalized size = 1.12 \begin {gather*} - \frac {x}{- x - e^{\frac {x \log {\left (\log {\relax (x )} \right )}^{2} + 256}{\log {\left (\log {\relax (x )} \right )}^{2}}} + e^{e^{2 x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x+1)*ln(x)*ln(ln(x))**3+512)*exp((x*ln(ln(x))**2+256)/ln(ln(x))**2)+(2*x*exp(x)**2-1)*ln(x)*exp(
exp(x)**2)*ln(ln(x))**3)/(ln(x)*ln(ln(x))**3*exp((x*ln(ln(x))**2+256)/ln(ln(x))**2)**2+(-2*ln(x)*exp(exp(x)**2
)+2*x*ln(x))*ln(ln(x))**3*exp((x*ln(ln(x))**2+256)/ln(ln(x))**2)+(ln(x)*exp(exp(x)**2)**2-2*x*ln(x)*exp(exp(x)
**2)+x**2*ln(x))*ln(ln(x))**3),x)

[Out]

-x/(-x - exp((x*log(log(x))**2 + 256)/log(log(x))**2) + exp(exp(2*x)))

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