3.61.55 \(\int \frac {e^{-x} (9 x+3 x^2+(-9 x^2-3 x^3+(-24 x^2+3 x^4) \log (x)+(9 x+3 x^2+(18 x-3 x^2-3 x^3) \log (x)) \log (3+x)) \log (\log (x))+(9 x+3 x^2+(18 x-3 x^2-3 x^3) \log (x)) \log (\log (x)) \log (\log (\log (x))))}{(3+x) \log (\log (x))} \, dx\)

Optimal. Leaf size=24 \[ 3 e^{-x} x^2 \log (x) (-x+\log (3+x)+\log (\log (\log (x)))) \]

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Rubi [F]  time = 10.95, 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^{-x} \left (9 x+3 x^2+\left (-9 x^2-3 x^3+\left (-24 x^2+3 x^4\right ) \log (x)+\left (9 x+3 x^2+\left (18 x-3 x^2-3 x^3\right ) \log (x)\right ) \log (3+x)\right ) \log (\log (x))+\left (9 x+3 x^2+\left (18 x-3 x^2-3 x^3\right ) \log (x)\right ) \log (\log (x)) \log (\log (\log (x)))\right )}{(3+x) \log (\log (x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(9*x + 3*x^2 + (-9*x^2 - 3*x^3 + (-24*x^2 + 3*x^4)*Log[x] + (9*x + 3*x^2 + (18*x - 3*x^2 - 3*x^3)*Log[x])*
Log[3 + x])*Log[Log[x]] + (9*x + 3*x^2 + (18*x - 3*x^2 - 3*x^3)*Log[x])*Log[Log[x]]*Log[Log[Log[x]]])/(E^x*(3
+ x)*Log[Log[x]]),x]

[Out]

(-3*x^3*Log[x])/E^x + (3*x^2*Log[x]*Log[3 + x])/E^x + 3*Defer[Int][x/(E^x*Log[Log[x]]), x] + 3*Defer[Int][(x*L
og[Log[Log[x]]])/E^x, x] + 6*Defer[Int][(x*Log[x]*Log[Log[Log[x]]])/E^x, x] - 3*Defer[Int][(x^2*Log[x]*Log[Log
[Log[x]]])/E^x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 e^{-x} x \left (3+x+\log (\log (x)) \left (-((3+x) (x-\log (3+x)-\log (\log (\log (x)))))+\log (x) \left (x \left (-8+x^2\right )-\left (-6+x+x^2\right ) \log (3+x)-\left (-6+x+x^2\right ) \log (\log (\log (x)))\right )\right )\right )}{(3+x) \log (\log (x))} \, dx\\ &=3 \int \frac {e^{-x} x \left (3+x+\log (\log (x)) \left (-((3+x) (x-\log (3+x)-\log (\log (\log (x)))))+\log (x) \left (x \left (-8+x^2\right )-\left (-6+x+x^2\right ) \log (3+x)-\left (-6+x+x^2\right ) \log (\log (\log (x)))\right )\right )\right )}{(3+x) \log (\log (x))} \, dx\\ &=3 \int \left (\frac {e^{-x} x \left (3+x-3 x \log (\log (x))-x^2 \log (\log (x))-8 x \log (x) \log (\log (x))+x^3 \log (x) \log (\log (x))+3 \log (3+x) \log (\log (x))+x \log (3+x) \log (\log (x))+6 \log (x) \log (3+x) \log (\log (x))-x \log (x) \log (3+x) \log (\log (x))-x^2 \log (x) \log (3+x) \log (\log (x))\right )}{(3+x) \log (\log (x))}-e^{-x} x (-1-2 \log (x)+x \log (x)) \log (\log (\log (x)))\right ) \, dx\\ &=3 \int \frac {e^{-x} x \left (3+x-3 x \log (\log (x))-x^2 \log (\log (x))-8 x \log (x) \log (\log (x))+x^3 \log (x) \log (\log (x))+3 \log (3+x) \log (\log (x))+x \log (3+x) \log (\log (x))+6 \log (x) \log (3+x) \log (\log (x))-x \log (x) \log (3+x) \log (\log (x))-x^2 \log (x) \log (3+x) \log (\log (x))\right )}{(3+x) \log (\log (x))} \, dx-3 \int e^{-x} x (-1-2 \log (x)+x \log (x)) \log (\log (\log (x))) \, dx\\ &=3 \int \frac {e^{-x} x \left (3+x+\left (-((3+x) (x-\log (3+x)))+\log (x) \left (x \left (-8+x^2\right )-\left (-6+x+x^2\right ) \log (3+x)\right )\right ) \log (\log (x))\right )}{(3+x) \log (\log (x))} \, dx-3 \int \left (-e^{-x} x \log (\log (\log (x)))-2 e^{-x} x \log (x) \log (\log (\log (x)))+e^{-x} x^2 \log (x) \log (\log (\log (x)))\right ) \, dx\\ &=3 \int \left (\frac {e^{-x} x \left (-3 x-x^2-8 x \log (x)+x^3 \log (x)+3 \log (3+x)+x \log (3+x)+6 \log (x) \log (3+x)-x \log (x) \log (3+x)-x^2 \log (x) \log (3+x)\right )}{3+x}+\frac {e^{-x} x}{\log (\log (x))}\right ) \, dx+3 \int e^{-x} x \log (\log (\log (x))) \, dx-3 \int e^{-x} x^2 \log (x) \log (\log (\log (x))) \, dx+6 \int e^{-x} x \log (x) \log (\log (\log (x))) \, dx\\ &=3 \int \frac {e^{-x} x \left (-3 x-x^2-8 x \log (x)+x^3 \log (x)+3 \log (3+x)+x \log (3+x)+6 \log (x) \log (3+x)-x \log (x) \log (3+x)-x^2 \log (x) \log (3+x)\right )}{3+x} \, dx+3 \int \frac {e^{-x} x}{\log (\log (x))} \, dx+3 \int e^{-x} x \log (\log (\log (x))) \, dx-3 \int e^{-x} x^2 \log (x) \log (\log (\log (x))) \, dx+6 \int e^{-x} x \log (x) \log (\log (\log (x))) \, dx\\ &=3 \int \frac {e^{-x} x \left (-((3+x) (x-\log (3+x)))+\log (x) \left (x \left (-8+x^2\right )-\left (-6+x+x^2\right ) \log (3+x)\right )\right )}{3+x} \, dx+3 \int \frac {e^{-x} x}{\log (\log (x))} \, dx+3 \int e^{-x} x \log (\log (\log (x))) \, dx-3 \int e^{-x} x^2 \log (x) \log (\log (\log (x))) \, dx+6 \int e^{-x} x \log (x) \log (\log (\log (x))) \, dx\\ &=3 \int \left (\frac {e^{-x} x^2 \left (-3-x-8 \log (x)+x^2 \log (x)\right )}{3+x}-e^{-x} x (-1-2 \log (x)+x \log (x)) \log (3+x)\right ) \, dx+3 \int \frac {e^{-x} x}{\log (\log (x))} \, dx+3 \int e^{-x} x \log (\log (\log (x))) \, dx-3 \int e^{-x} x^2 \log (x) \log (\log (\log (x))) \, dx+6 \int e^{-x} x \log (x) \log (\log (\log (x))) \, dx\\ &=3 \int \frac {e^{-x} x^2 \left (-3-x-8 \log (x)+x^2 \log (x)\right )}{3+x} \, dx-3 \int e^{-x} x (-1-2 \log (x)+x \log (x)) \log (3+x) \, dx+3 \int \frac {e^{-x} x}{\log (\log (x))} \, dx+3 \int e^{-x} x \log (\log (\log (x))) \, dx-3 \int e^{-x} x^2 \log (x) \log (\log (\log (x))) \, dx+6 \int e^{-x} x \log (x) \log (\log (\log (x))) \, dx\\ &=3 \int \left (-e^{-x} x^2+\frac {e^{-x} x^2 \left (-8+x^2\right ) \log (x)}{3+x}\right ) \, dx-3 \int \left (-e^{-x} x \log (3+x)-2 e^{-x} x \log (x) \log (3+x)+e^{-x} x^2 \log (x) \log (3+x)\right ) \, dx+3 \int \frac {e^{-x} x}{\log (\log (x))} \, dx+3 \int e^{-x} x \log (\log (\log (x))) \, dx-3 \int e^{-x} x^2 \log (x) \log (\log (\log (x))) \, dx+6 \int e^{-x} x \log (x) \log (\log (\log (x))) \, dx\\ &=-\left (3 \int e^{-x} x^2 \, dx\right )+3 \int \frac {e^{-x} x^2 \left (-8+x^2\right ) \log (x)}{3+x} \, dx+3 \int e^{-x} x \log (3+x) \, dx-3 \int e^{-x} x^2 \log (x) \log (3+x) \, dx+3 \int \frac {e^{-x} x}{\log (\log (x))} \, dx+3 \int e^{-x} x \log (\log (\log (x))) \, dx-3 \int e^{-x} x^2 \log (x) \log (\log (\log (x))) \, dx+6 \int e^{-x} x \log (x) \log (3+x) \, dx+6 \int e^{-x} x \log (x) \log (\log (\log (x))) \, dx\\ &=3 e^{-x} x^2+6 e^{-x} \log (x)-3 e^{-x} x \log (x)-3 e^{-x} x^3 \log (x)+27 e^3 \text {Ei}(-3-x) \log (x)-3 e^{-x} \log (3+x)-3 e^{-x} x \log (3+x)+3 e^{-x} x^2 \log (x) \log (3+x)-3 \int \frac {e^{-x} (-1-x)}{3+x} \, dx-3 \int \frac {e^{-x} \left (2-x-x^3+9 e^{3+x} \text {Ei}(-3-x)\right )}{x} \, dx+3 \int \frac {e^{-x} \left (-2-2 x-x^2\right ) \log (x)}{3+x} \, dx+3 \int \frac {e^{-x} \left (-2-2 x-x^2\right ) \log (3+x)}{x} \, dx+3 \int \frac {e^{-x} x}{\log (\log (x))} \, dx+3 \int e^{-x} x \log (\log (\log (x))) \, dx-3 \int e^{-x} x^2 \log (x) \log (\log (\log (x))) \, dx-6 \int e^{-x} x \, dx-6 \int \frac {e^{-x} (-1-x) \log (x)}{3+x} \, dx-6 \int \frac {e^{-x} (-1-x) \log (3+x)}{x} \, dx+6 \int e^{-x} x \log (x) \log (\log (\log (x))) \, dx\\ &=6 e^{-x} x+3 e^{-x} x^2-3 e^{-x} x^3 \log (x)+3 e^{-x} x^2 \log (x) \log (3+x)-3 \int \left (-e^{-x}+\frac {2 e^{-x}}{3+x}\right ) \, dx-3 \int \left (e^{-x}-\frac {5 e^3 \text {Ei}(-3-x)}{x}\right ) \, dx-3 \int \left (\frac {e^{-x} \left (2-x-x^3\right )}{x}+\frac {9 e^3 \text {Ei}(-3-x)}{x}\right ) \, dx-3 \int \frac {e^{-x} \left (3+x-2 e^x \text {Ei}(-x)\right )}{3+x} \, dx+3 \int \frac {e^{-x} x}{\log (\log (x))} \, dx+3 \int e^{-x} x \log (\log (\log (x))) \, dx-3 \int e^{-x} x^2 \log (x) \log (\log (\log (x))) \, dx-6 \int e^{-x} \, dx+6 \int \frac {e^{-x}+2 e^3 \text {Ei}(-3-x)}{x} \, dx+6 \int \frac {e^{-x}-\text {Ei}(-x)}{3+x} \, dx+6 \int e^{-x} x \log (x) \log (\log (\log (x))) \, dx\\ &=6 e^{-x}+6 e^{-x} x+3 e^{-x} x^2-3 e^{-x} x^3 \log (x)+3 e^{-x} x^2 \log (x) \log (3+x)-3 \int \frac {e^{-x} \left (2-x-x^3\right )}{x} \, dx-3 \int \left (e^{-x}-\frac {2 \text {Ei}(-x)}{3+x}\right ) \, dx+3 \int \frac {e^{-x} x}{\log (\log (x))} \, dx+3 \int e^{-x} x \log (\log (\log (x))) \, dx-3 \int e^{-x} x^2 \log (x) \log (\log (\log (x))) \, dx-6 \int \frac {e^{-x}}{3+x} \, dx+6 \int \left (\frac {e^{-x}}{x}+\frac {2 e^3 \text {Ei}(-3-x)}{x}\right ) \, dx+6 \int \left (\frac {e^{-x}}{3+x}-\frac {\text {Ei}(-x)}{3+x}\right ) \, dx+6 \int e^{-x} x \log (x) \log (\log (\log (x))) \, dx+\left (15 e^3\right ) \int \frac {\text {Ei}(-3-x)}{x} \, dx-\left (27 e^3\right ) \int \frac {\text {Ei}(-3-x)}{x} \, dx\\ &=6 e^{-x}+6 e^{-x} x+3 e^{-x} x^2-6 e^3 \text {Ei}(-3-x)-3 e^{-x} x^3 \log (x)+3 e^{-x} x^2 \log (x) \log (3+x)-3 \int e^{-x} \, dx-3 \int \left (-e^{-x}+\frac {2 e^{-x}}{x}-e^{-x} x^2\right ) \, dx+3 \int \frac {e^{-x} x}{\log (\log (x))} \, dx+3 \int e^{-x} x \log (\log (\log (x))) \, dx-3 \int e^{-x} x^2 \log (x) \log (\log (\log (x))) \, dx+6 \int \frac {e^{-x}}{x} \, dx+6 \int \frac {e^{-x}}{3+x} \, dx+6 \int e^{-x} x \log (x) \log (\log (\log (x))) \, dx+\left (12 e^3\right ) \int \frac {\text {Ei}(-3-x)}{x} \, dx+\left (15 e^3\right ) \int \frac {\text {Ei}(-3-x)}{x} \, dx-\left (27 e^3\right ) \int \frac {\text {Ei}(-3-x)}{x} \, dx\\ &=9 e^{-x}+6 e^{-x} x+3 e^{-x} x^2+6 \text {Ei}(-x)-3 e^{-x} x^3 \log (x)+3 e^{-x} x^2 \log (x) \log (3+x)+3 \int e^{-x} \, dx+3 \int e^{-x} x^2 \, dx+3 \int \frac {e^{-x} x}{\log (\log (x))} \, dx+3 \int e^{-x} x \log (\log (\log (x))) \, dx-3 \int e^{-x} x^2 \log (x) \log (\log (\log (x))) \, dx-6 \int \frac {e^{-x}}{x} \, dx+6 \int e^{-x} x \log (x) \log (\log (\log (x))) \, dx+\left (12 e^3\right ) \int \frac {\text {Ei}(-3-x)}{x} \, dx+\left (15 e^3\right ) \int \frac {\text {Ei}(-3-x)}{x} \, dx-\left (27 e^3\right ) \int \frac {\text {Ei}(-3-x)}{x} \, dx\\ &=6 e^{-x}+6 e^{-x} x-3 e^{-x} x^3 \log (x)+3 e^{-x} x^2 \log (x) \log (3+x)+3 \int \frac {e^{-x} x}{\log (\log (x))} \, dx+3 \int e^{-x} x \log (\log (\log (x))) \, dx-3 \int e^{-x} x^2 \log (x) \log (\log (\log (x))) \, dx+6 \int e^{-x} x \, dx+6 \int e^{-x} x \log (x) \log (\log (\log (x))) \, dx+\left (12 e^3\right ) \int \frac {\text {Ei}(-3-x)}{x} \, dx+\left (15 e^3\right ) \int \frac {\text {Ei}(-3-x)}{x} \, dx-\left (27 e^3\right ) \int \frac {\text {Ei}(-3-x)}{x} \, dx\\ &=6 e^{-x}-3 e^{-x} x^3 \log (x)+3 e^{-x} x^2 \log (x) \log (3+x)+3 \int \frac {e^{-x} x}{\log (\log (x))} \, dx+3 \int e^{-x} x \log (\log (\log (x))) \, dx-3 \int e^{-x} x^2 \log (x) \log (\log (\log (x))) \, dx+6 \int e^{-x} \, dx+6 \int e^{-x} x \log (x) \log (\log (\log (x))) \, dx+\left (12 e^3\right ) \int \frac {\text {Ei}(-3-x)}{x} \, dx+\left (15 e^3\right ) \int \frac {\text {Ei}(-3-x)}{x} \, dx-\left (27 e^3\right ) \int \frac {\text {Ei}(-3-x)}{x} \, dx\\ &=-3 e^{-x} x^3 \log (x)+3 e^{-x} x^2 \log (x) \log (3+x)+3 \int \frac {e^{-x} x}{\log (\log (x))} \, dx+3 \int e^{-x} x \log (\log (\log (x))) \, dx-3 \int e^{-x} x^2 \log (x) \log (\log (\log (x))) \, dx+6 \int e^{-x} x \log (x) \log (\log (\log (x))) \, dx+\left (12 e^3\right ) \int \frac {\text {Ei}(-3-x)}{x} \, dx+\left (15 e^3\right ) \int \frac {\text {Ei}(-3-x)}{x} \, dx-\left (27 e^3\right ) \int \frac {\text {Ei}(-3-x)}{x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.69, size = 26, normalized size = 1.08 \begin {gather*} -3 e^{-x} x^2 \log (x) (x-\log (3+x)-\log (\log (\log (x)))) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(9*x + 3*x^2 + (-9*x^2 - 3*x^3 + (-24*x^2 + 3*x^4)*Log[x] + (9*x + 3*x^2 + (18*x - 3*x^2 - 3*x^3)*Lo
g[x])*Log[3 + x])*Log[Log[x]] + (9*x + 3*x^2 + (18*x - 3*x^2 - 3*x^3)*Log[x])*Log[Log[x]]*Log[Log[Log[x]]])/(E
^x*(3 + x)*Log[Log[x]]),x]

[Out]

(-3*x^2*Log[x]*(x - Log[3 + x] - Log[Log[Log[x]]]))/E^x

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fricas [A]  time = 0.53, size = 42, normalized size = 1.75 \begin {gather*} -3 \, x^{3} e^{\left (-x\right )} \log \relax (x) + 3 \, x^{2} e^{\left (-x\right )} \log \left (x + 3\right ) \log \relax (x) + 3 \, x^{2} e^{\left (-x\right )} \log \relax (x) \log \left (\log \left (\log \relax (x)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3*x^3-3*x^2+18*x)*log(x)+3*x^2+9*x)*log(log(x))*log(log(log(x)))+(((-3*x^3-3*x^2+18*x)*log(x)+3*
x^2+9*x)*log(3+x)+(3*x^4-24*x^2)*log(x)-3*x^3-9*x^2)*log(log(x))+3*x^2+9*x)/(3+x)/exp(x)/log(log(x)),x, algori
thm="fricas")

[Out]

-3*x^3*e^(-x)*log(x) + 3*x^2*e^(-x)*log(x + 3)*log(x) + 3*x^2*e^(-x)*log(x)*log(log(log(x)))

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giac [A]  time = 0.27, size = 42, normalized size = 1.75 \begin {gather*} -3 \, x^{3} e^{\left (-x\right )} \log \relax (x) + 3 \, x^{2} e^{\left (-x\right )} \log \left (x + 3\right ) \log \relax (x) + 3 \, x^{2} e^{\left (-x\right )} \log \relax (x) \log \left (\log \left (\log \relax (x)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3*x^3-3*x^2+18*x)*log(x)+3*x^2+9*x)*log(log(x))*log(log(log(x)))+(((-3*x^3-3*x^2+18*x)*log(x)+3*
x^2+9*x)*log(3+x)+(3*x^4-24*x^2)*log(x)-3*x^3-9*x^2)*log(log(x))+3*x^2+9*x)/(3+x)/exp(x)/log(log(x)),x, algori
thm="giac")

[Out]

-3*x^3*e^(-x)*log(x) + 3*x^2*e^(-x)*log(x + 3)*log(x) + 3*x^2*e^(-x)*log(x)*log(log(log(x)))

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maple [A]  time = 0.30, size = 36, normalized size = 1.50

method	result	size
risch	\(3 x^{2} {\mathrm e}^{-x} \ln \relax (x ) \ln \left (\ln \left (\ln \relax (x )\right )\right )-3 x^{2} \ln \relax (x ) \left (-\ln \left (3+x \right )+x \right ) {\mathrm e}^{-x}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-3*x^3-3*x^2+18*x)*ln(x)+3*x^2+9*x)*ln(ln(x))*ln(ln(ln(x)))+(((-3*x^3-3*x^2+18*x)*ln(x)+3*x^2+9*x)*ln(3
+x)+(3*x^4-24*x^2)*ln(x)-3*x^3-9*x^2)*ln(ln(x))+3*x^2+9*x)/(3+x)/exp(x)/ln(ln(x)),x,method=_RETURNVERBOSE)

[Out]

3*x^2*exp(-x)*ln(x)*ln(ln(ln(x)))-3*x^2*ln(x)*(-ln(3+x)+x)*exp(-x)

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maxima [A]  time = 0.40, size = 42, normalized size = 1.75 \begin {gather*} -3 \, x^{3} e^{\left (-x\right )} \log \relax (x) + 3 \, x^{2} e^{\left (-x\right )} \log \left (x + 3\right ) \log \relax (x) + 3 \, x^{2} e^{\left (-x\right )} \log \relax (x) \log \left (\log \left (\log \relax (x)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3*x^3-3*x^2+18*x)*log(x)+3*x^2+9*x)*log(log(x))*log(log(log(x)))+(((-3*x^3-3*x^2+18*x)*log(x)+3*
x^2+9*x)*log(3+x)+(3*x^4-24*x^2)*log(x)-3*x^3-9*x^2)*log(log(x))+3*x^2+9*x)/(3+x)/exp(x)/log(log(x)),x, algori
thm="maxima")

[Out]

-3*x^3*e^(-x)*log(x) + 3*x^2*e^(-x)*log(x + 3)*log(x) + 3*x^2*e^(-x)*log(x)*log(log(log(x)))

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mupad [B]  time = 5.26, size = 23, normalized size = 0.96 \begin {gather*} 3\,x^2\,{\mathrm {e}}^{-x}\,\ln \relax (x)\,\left (\ln \left (x+3\right )-x+\ln \left (\ln \left (\ln \relax (x)\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-x)*(9*x - log(log(x))*(log(x)*(24*x^2 - 3*x^4) + 9*x^2 + 3*x^3 - log(x + 3)*(9*x + 3*x^2 - log(x)*(3
*x^2 - 18*x + 3*x^3))) + 3*x^2 + log(log(x))*log(log(log(x)))*(9*x + 3*x^2 - log(x)*(3*x^2 - 18*x + 3*x^3))))/
(log(log(x))*(x + 3)),x)

[Out]

3*x^2*exp(-x)*log(x)*(log(x + 3) - x + log(log(log(x))))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3*x**3-3*x**2+18*x)*ln(x)+3*x**2+9*x)*ln(ln(x))*ln(ln(ln(x)))+(((-3*x**3-3*x**2+18*x)*ln(x)+3*x*
*2+9*x)*ln(3+x)+(3*x**4-24*x**2)*ln(x)-3*x**3-9*x**2)*ln(ln(x))+3*x**2+9*x)/(3+x)/exp(x)/ln(ln(x)),x)

[Out]

Timed out

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