3.77.88 \(\int \frac {e^{-2 x} (-18 e^x x^2+e^{2 x} (-36 x-18 x^2-114 x^3)+(-18 x^3+e^x (-36 x-36 x^2+96 x^3-114 x^4)+e^{2 x} (-72-36 x+114 x^3+722 x^4)) \log (x)+(18 e^{2 x} x^2+(18 e^x x^3+e^{2 x} (36 x-114 x^3)) \log (x)) \log (\log (x)))}{9 x^3 \log (x)} \, dx\)

Optimal. Leaf size=25 \[ \left (e^{-x}+\frac {19 x}{3}+\frac {2+x}{x}-\log (\log (x))\right )^2 \]

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Rubi [F]  time = 4.03, 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^{-2 x} \left (-18 e^x x^2+e^{2 x} \left (-36 x-18 x^2-114 x^3\right )+\left (-18 x^3+e^x \left (-36 x-36 x^2+96 x^3-114 x^4\right )+e^{2 x} \left (-72-36 x+114 x^3+722 x^4\right )\right ) \log (x)+\left (18 e^{2 x} x^2+\left (18 e^x x^3+e^{2 x} \left (36 x-114 x^3\right )\right ) \log (x)\right ) \log (\log (x))\right )}{9 x^3 \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-18*E^x*x^2 + E^(2*x)*(-36*x - 18*x^2 - 114*x^3) + (-18*x^3 + E^x*(-36*x - 36*x^2 + 96*x^3 - 114*x^4) + E
^(2*x)*(-72 - 36*x + 114*x^3 + 722*x^4))*Log[x] + (18*E^(2*x)*x^2 + (18*E^x*x^3 + E^(2*x)*(36*x - 114*x^3))*Lo
g[x])*Log[Log[x]])/(9*E^(2*x)*x^3*Log[x]),x]

[Out]

E^(-2*x) + 2/E^x + 4/x^2 + 4/x + 4/(E^x*x) + (38*x)/3 + (38*x)/(3*E^x) + (361*x^2)/9 + 4*ExpIntegralEi[-Log[x]
] - (4*Log[Log[x]])/x - (38*x*Log[Log[x]])/3 + Log[Log[x]]^2 + (38*LogIntegral[x])/3 - 2*Defer[Int][1/(E^x*x*L
og[x]), x] - (2*Defer[Int][(6 + 3*x + 19*x^2)/(x^2*Log[x]), x])/3 + 2*Defer[Int][Log[Log[x]]/E^x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{9} \int \frac {e^{-2 x} \left (-18 e^x x^2+e^{2 x} \left (-36 x-18 x^2-114 x^3\right )+\left (-18 x^3+e^x \left (-36 x-36 x^2+96 x^3-114 x^4\right )+e^{2 x} \left (-72-36 x+114 x^3+722 x^4\right )\right ) \log (x)+\left (18 e^{2 x} x^2+\left (18 e^x x^3+e^{2 x} \left (36 x-114 x^3\right )\right ) \log (x)\right ) \log (\log (x))\right )}{x^3 \log (x)} \, dx\\ &=\frac {1}{9} \int \frac {2 e^{-2 x} \left (3 e^x x-\left (-3 x^2+e^x \left (-6+19 x^2\right )\right ) \log (x)\right ) \left (-3 x-e^x \left (6+3 x+19 x^2\right )+3 e^x x \log (\log (x))\right )}{x^3 \log (x)} \, dx\\ &=\frac {2}{9} \int \frac {e^{-2 x} \left (3 e^x x-\left (-3 x^2+e^x \left (-6+19 x^2\right )\right ) \log (x)\right ) \left (-3 x-e^x \left (6+3 x+19 x^2\right )+3 e^x x \log (\log (x))\right )}{x^3 \log (x)} \, dx\\ &=\frac {2}{9} \int \left (-9 e^{-2 x}+\frac {\left (-3 x-6 \log (x)+19 x^2 \log (x)\right ) \left (6+3 x+19 x^2-3 x \log (\log (x))\right )}{x^3 \log (x)}-\frac {3 e^{-x} \left (3 x+6 \log (x)+6 x \log (x)-16 x^2 \log (x)+19 x^3 \log (x)-3 x^2 \log (x) \log (\log (x))\right )}{x^2 \log (x)}\right ) \, dx\\ &=\frac {2}{9} \int \frac {\left (-3 x-6 \log (x)+19 x^2 \log (x)\right ) \left (6+3 x+19 x^2-3 x \log (\log (x))\right )}{x^3 \log (x)} \, dx-\frac {2}{3} \int \frac {e^{-x} \left (3 x+6 \log (x)+6 x \log (x)-16 x^2 \log (x)+19 x^3 \log (x)-3 x^2 \log (x) \log (\log (x))\right )}{x^2 \log (x)} \, dx-2 \int e^{-2 x} \, dx\\ &=e^{-2 x}+\frac {2}{9} \int \left (\frac {\left (6+3 x+19 x^2\right ) \left (-3 x-6 \log (x)+19 x^2 \log (x)\right )}{x^3 \log (x)}-\frac {3 \left (-3 x-6 \log (x)+19 x^2 \log (x)\right ) \log (\log (x))}{x^2 \log (x)}\right ) \, dx-\frac {2}{3} \int \left (\frac {e^{-x} \left (3 x+6 \log (x)+6 x \log (x)-16 x^2 \log (x)+19 x^3 \log (x)\right )}{x^2 \log (x)}-3 e^{-x} \log (\log (x))\right ) \, dx\\ &=e^{-2 x}+\frac {2}{9} \int \frac {\left (6+3 x+19 x^2\right ) \left (-3 x-6 \log (x)+19 x^2 \log (x)\right )}{x^3 \log (x)} \, dx-\frac {2}{3} \int \frac {e^{-x} \left (3 x+6 \log (x)+6 x \log (x)-16 x^2 \log (x)+19 x^3 \log (x)\right )}{x^2 \log (x)} \, dx-\frac {2}{3} \int \frac {\left (-3 x-6 \log (x)+19 x^2 \log (x)\right ) \log (\log (x))}{x^2 \log (x)} \, dx+2 \int e^{-x} \log (\log (x)) \, dx\\ &=e^{-2 x}+\frac {2}{9} \int \left (\frac {-36-18 x+57 x^3+361 x^4}{x^3}-\frac {3 \left (6+3 x+19 x^2\right )}{x^2 \log (x)}\right ) \, dx-\frac {2}{3} \int \left (\frac {e^{-x} \left (6+6 x-16 x^2+19 x^3\right )}{x^2}+\frac {3 e^{-x}}{x \log (x)}\right ) \, dx-\frac {2}{3} \int \left (19 \log (\log (x))-\frac {6 \log (\log (x))}{x^2}-\frac {3 \log (\log (x))}{x \log (x)}\right ) \, dx+2 \int e^{-x} \log (\log (x)) \, dx\\ &=e^{-2 x}+\frac {2}{9} \int \frac {-36-18 x+57 x^3+361 x^4}{x^3} \, dx-\frac {2}{3} \int \frac {e^{-x} \left (6+6 x-16 x^2+19 x^3\right )}{x^2} \, dx-\frac {2}{3} \int \frac {6+3 x+19 x^2}{x^2 \log (x)} \, dx-2 \int \frac {e^{-x}}{x \log (x)} \, dx+2 \int e^{-x} \log (\log (x)) \, dx+2 \int \frac {\log (\log (x))}{x \log (x)} \, dx+4 \int \frac {\log (\log (x))}{x^2} \, dx-\frac {38}{3} \int \log (\log (x)) \, dx\\ &=e^{-2 x}-\frac {4 \log (\log (x))}{x}-\frac {38}{3} x \log (\log (x))+\frac {2}{9} \int \left (57-\frac {36}{x^3}-\frac {18}{x^2}+361 x\right ) \, dx-\frac {2}{3} \int \left (-16 e^{-x}+\frac {6 e^{-x}}{x^2}+\frac {6 e^{-x}}{x}+19 e^{-x} x\right ) \, dx-\frac {2}{3} \int \frac {6+3 x+19 x^2}{x^2 \log (x)} \, dx-2 \int \frac {e^{-x}}{x \log (x)} \, dx+2 \int e^{-x} \log (\log (x)) \, dx+2 \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,\log (x)\right )+4 \int \frac {1}{x^2 \log (x)} \, dx+\frac {38}{3} \int \frac {1}{\log (x)} \, dx\\ &=e^{-2 x}+\frac {4}{x^2}+\frac {4}{x}+\frac {38 x}{3}+\frac {361 x^2}{9}-\frac {4 \log (\log (x))}{x}-\frac {38}{3} x \log (\log (x))+\log ^2(\log (x))+\frac {38 \text {li}(x)}{3}-\frac {2}{3} \int \frac {6+3 x+19 x^2}{x^2 \log (x)} \, dx-2 \int \frac {e^{-x}}{x \log (x)} \, dx+2 \int e^{-x} \log (\log (x)) \, dx-4 \int \frac {e^{-x}}{x^2} \, dx-4 \int \frac {e^{-x}}{x} \, dx+4 \operatorname {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (x)\right )+\frac {32}{3} \int e^{-x} \, dx-\frac {38}{3} \int e^{-x} x \, dx\\ &=e^{-2 x}-\frac {32 e^{-x}}{3}+\frac {4}{x^2}+\frac {4}{x}+\frac {4 e^{-x}}{x}+\frac {38 x}{3}+\frac {38 e^{-x} x}{3}+\frac {361 x^2}{9}-4 \text {Ei}(-x)+4 \text {Ei}(-\log (x))-\frac {4 \log (\log (x))}{x}-\frac {38}{3} x \log (\log (x))+\log ^2(\log (x))+\frac {38 \text {li}(x)}{3}-\frac {2}{3} \int \frac {6+3 x+19 x^2}{x^2 \log (x)} \, dx-2 \int \frac {e^{-x}}{x \log (x)} \, dx+2 \int e^{-x} \log (\log (x)) \, dx+4 \int \frac {e^{-x}}{x} \, dx-\frac {38}{3} \int e^{-x} \, dx\\ &=e^{-2 x}+2 e^{-x}+\frac {4}{x^2}+\frac {4}{x}+\frac {4 e^{-x}}{x}+\frac {38 x}{3}+\frac {38 e^{-x} x}{3}+\frac {361 x^2}{9}+4 \text {Ei}(-\log (x))-\frac {4 \log (\log (x))}{x}-\frac {38}{3} x \log (\log (x))+\log ^2(\log (x))+\frac {38 \text {li}(x)}{3}-\frac {2}{3} \int \frac {6+3 x+19 x^2}{x^2 \log (x)} \, dx-2 \int \frac {e^{-x}}{x \log (x)} \, dx+2 \int e^{-x} \log (\log (x)) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.79, size = 87, normalized size = 3.48 \begin {gather*} \frac {2}{9} \left (\frac {9 e^{-2 x}}{2}+\frac {18}{x^2}+\frac {18}{x}+57 x+\frac {361 x^2}{2}+e^{-x} \left (9+\frac {18}{x}+57 x\right )-9 \log (\log (x))-\frac {3 \left (6+3 e^{-x} x+19 x^2\right ) \log (\log (x))}{x}+\frac {9}{2} \log ^2(\log (x))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-18*E^x*x^2 + E^(2*x)*(-36*x - 18*x^2 - 114*x^3) + (-18*x^3 + E^x*(-36*x - 36*x^2 + 96*x^3 - 114*x^
4) + E^(2*x)*(-72 - 36*x + 114*x^3 + 722*x^4))*Log[x] + (18*E^(2*x)*x^2 + (18*E^x*x^3 + E^(2*x)*(36*x - 114*x^
3))*Log[x])*Log[Log[x]])/(9*E^(2*x)*x^3*Log[x]),x]

[Out]

(2*(9/(2*E^(2*x)) + 18/x^2 + 18/x + 57*x + (361*x^2)/2 + (9 + 18/x + 57*x)/E^x - 9*Log[Log[x]] - (3*(6 + (3*x)
/E^x + 19*x^2)*Log[Log[x]])/x + (9*Log[Log[x]]^2)/2))/9

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fricas [B]  time = 1.03, size = 99, normalized size = 3.96 \begin {gather*} \frac {{\left (9 \, x^{2} e^{\left (2 \, x\right )} \log \left (\log \relax (x)\right )^{2} + 9 \, x^{2} + {\left (361 \, x^{4} + 114 \, x^{3} + 36 \, x + 36\right )} e^{\left (2 \, x\right )} + 6 \, {\left (19 \, x^{3} + 3 \, x^{2} + 6 \, x\right )} e^{x} - 6 \, {\left (3 \, x^{2} e^{x} + {\left (19 \, x^{3} + 3 \, x^{2} + 6 \, x\right )} e^{\left (2 \, x\right )}\right )} \log \left (\log \relax (x)\right )\right )} e^{\left (-2 \, x\right )}}{9 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((((-114*x^3+36*x)*exp(x)^2+18*exp(x)*x^3)*log(x)+18*exp(x)^2*x^2)*log(log(x))+((722*x^4+114*x^3
-36*x-72)*exp(x)^2+(-114*x^4+96*x^3-36*x^2-36*x)*exp(x)-18*x^3)*log(x)+(-114*x^3-18*x^2-36*x)*exp(x)^2-18*exp(
x)*x^2)/x^3/exp(x)^2/log(x),x, algorithm="fricas")

[Out]

1/9*(9*x^2*e^(2*x)*log(log(x))^2 + 9*x^2 + (361*x^4 + 114*x^3 + 36*x + 36)*e^(2*x) + 6*(19*x^3 + 3*x^2 + 6*x)*
e^x - 6*(3*x^2*e^x + (19*x^3 + 3*x^2 + 6*x)*e^(2*x))*log(log(x)))*e^(-2*x)/x^2

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giac [B]  time = 0.14, size = 98, normalized size = 3.92 \begin {gather*} \frac {361 \, x^{4} + 114 \, x^{3} e^{\left (-x\right )} - 114 \, x^{3} \log \left (\log \relax (x)\right ) - 18 \, x^{2} e^{\left (-x\right )} \log \left (\log \relax (x)\right ) + 9 \, x^{2} \log \left (\log \relax (x)\right )^{2} + 114 \, x^{3} + 18 \, x^{2} e^{\left (-x\right )} + 9 \, x^{2} e^{\left (-2 \, x\right )} - 18 \, x^{2} \log \left (\log \relax (x)\right ) + 36 \, x e^{\left (-x\right )} - 36 \, x \log \left (\log \relax (x)\right ) + 36 \, x + 36}{9 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((((-114*x^3+36*x)*exp(x)^2+18*exp(x)*x^3)*log(x)+18*exp(x)^2*x^2)*log(log(x))+((722*x^4+114*x^3
-36*x-72)*exp(x)^2+(-114*x^4+96*x^3-36*x^2-36*x)*exp(x)-18*x^3)*log(x)+(-114*x^3-18*x^2-36*x)*exp(x)^2-18*exp(
x)*x^2)/x^3/exp(x)^2/log(x),x, algorithm="giac")

[Out]

1/9*(361*x^4 + 114*x^3*e^(-x) - 114*x^3*log(log(x)) - 18*x^2*e^(-x)*log(log(x)) + 9*x^2*log(log(x))^2 + 114*x^
3 + 18*x^2*e^(-x) + 9*x^2*e^(-2*x) - 18*x^2*log(log(x)) + 36*x*e^(-x) - 36*x*log(log(x)) + 36*x + 36)/x^2

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maple [B]  time = 0.08, size = 111, normalized size = 4.44

method	result	size
risch	\(\ln \left (\ln \relax (x )\right )^{2}-\frac {2 \left (19 \,{\mathrm e}^{x} x^{2}+3 x +6 \,{\mathrm e}^{x}\right ) {\mathrm e}^{-x} \ln \left (\ln \relax (x )\right )}{3 x}-\frac {\left (-361 \,{\mathrm e}^{2 x} x^{4}+18 \ln \left (\ln \relax (x )\right ) {\mathrm e}^{2 x} x^{2}-114 \,{\mathrm e}^{2 x} x^{3}-114 \,{\mathrm e}^{x} x^{3}-18 \,{\mathrm e}^{x} x^{2}-36 x \,{\mathrm e}^{2 x}-9 x^{2}-36 \,{\mathrm e}^{x} x -36 \,{\mathrm e}^{2 x}\right ) {\mathrm e}^{-2 x}}{9 x^{2}}\)	\(111\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/9*((((-114*x^3+36*x)*exp(x)^2+18*exp(x)*x^3)*ln(x)+18*exp(x)^2*x^2)*ln(ln(x))+((722*x^4+114*x^3-36*x-72)
*exp(x)^2+(-114*x^4+96*x^3-36*x^2-36*x)*exp(x)-18*x^3)*ln(x)+(-114*x^3-18*x^2-36*x)*exp(x)^2-18*exp(x)*x^2)/x^
3/exp(x)^2/ln(x),x,method=_RETURNVERBOSE)

[Out]

ln(ln(x))^2-2/3*(19*exp(x)*x^2+3*x+6*exp(x))*exp(-x)/x*ln(ln(x))-1/9*(-361*exp(2*x)*x^4+18*ln(ln(x))*exp(2*x)*
x^2-114*exp(2*x)*x^3-114*exp(x)*x^3-18*exp(x)*x^2-36*x*exp(2*x)-9*x^2-36*exp(x)*x-36*exp(2*x))*exp(-2*x)/x^2

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {361}{9} \, x^{2} + \frac {38}{3} \, {\left (x + 1\right )} e^{\left (-x\right )} + \frac {38}{3} \, x + \frac {{\left (3 \, x e^{x} \log \left (\log \relax (x)\right )^{2} - 2 \, {\left ({\left (19 \, x^{2} + 6\right )} e^{x} + 3 \, x\right )} \log \left (\log \relax (x)\right )\right )} e^{\left (-x\right )}}{3 \, x} + \frac {4}{x} + \frac {4}{x^{2}} - 4 \, {\rm Ei}\left (-x\right ) - 4 \, {\rm Ei}\left (-\log \relax (x)\right ) - \frac {38}{3} \, {\rm Ei}\left (\log \relax (x)\right ) - \frac {32}{3} \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 4 \, \Gamma \left (-1, x\right ) + \frac {2}{9} \, \int \frac {3 \, {\left (19 \, x^{2} + 6\right )}}{x^{2} \log \relax (x)}\,{d x} - 2 \, \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((((-114*x^3+36*x)*exp(x)^2+18*exp(x)*x^3)*log(x)+18*exp(x)^2*x^2)*log(log(x))+((722*x^4+114*x^3
-36*x-72)*exp(x)^2+(-114*x^4+96*x^3-36*x^2-36*x)*exp(x)-18*x^3)*log(x)+(-114*x^3-18*x^2-36*x)*exp(x)^2-18*exp(
x)*x^2)/x^3/exp(x)^2/log(x),x, algorithm="maxima")

[Out]

361/9*x^2 + 38/3*(x + 1)*e^(-x) + 38/3*x + 1/3*(3*x*e^x*log(log(x))^2 - 2*((19*x^2 + 6)*e^x + 3*x)*log(log(x))
)*e^(-x)/x + 4/x + 4/x^2 - 4*Ei(-x) - 4*Ei(-log(x)) - 38/3*Ei(log(x)) - 32/3*e^(-x) + e^(-2*x) + 4*gamma(-1, x
) + 2/9*integrate(3*(19*x^2 + 6)/(x^2*log(x)), x) - 2*log(log(x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {{\mathrm {e}}^{-2\,x}\,\left (2\,x^2\,{\mathrm {e}}^x+\frac {\ln \relax (x)\,\left ({\mathrm {e}}^x\,\left (114\,x^4-96\,x^3+36\,x^2+36\,x\right )+{\mathrm {e}}^{2\,x}\,\left (-722\,x^4-114\,x^3+36\,x+72\right )+18\,x^3\right )}{9}+\frac {{\mathrm {e}}^{2\,x}\,\left (114\,x^3+18\,x^2+36\,x\right )}{9}-\frac {\ln \left (\ln \relax (x)\right )\,\left (18\,x^2\,{\mathrm {e}}^{2\,x}+\ln \relax (x)\,\left ({\mathrm {e}}^{2\,x}\,\left (36\,x-114\,x^3\right )+18\,x^3\,{\mathrm {e}}^x\right )\right )}{9}\right )}{x^3\,\ln \relax (x)} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-2*x)*(2*x^2*exp(x) + (log(x)*(exp(x)*(36*x + 36*x^2 - 96*x^3 + 114*x^4) + exp(2*x)*(36*x - 114*x^3
- 722*x^4 + 72) + 18*x^3))/9 + (exp(2*x)*(36*x + 18*x^2 + 114*x^3))/9 - (log(log(x))*(18*x^2*exp(2*x) + log(x)
*(exp(2*x)*(36*x - 114*x^3) + 18*x^3*exp(x))))/9))/(x^3*log(x)),x)

[Out]

int(-(exp(-2*x)*(2*x^2*exp(x) + (log(x)*(exp(x)*(36*x + 36*x^2 - 96*x^3 + 114*x^4) + exp(2*x)*(36*x - 114*x^3
- 722*x^4 + 72) + 18*x^3))/9 + (exp(2*x)*(36*x + 18*x^2 + 114*x^3))/9 - (log(log(x))*(18*x^2*exp(2*x) + log(x)
*(exp(2*x)*(36*x - 114*x^3) + 18*x^3*exp(x))))/9))/(x^3*log(x)), x)

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sympy [B]  time = 0.66, size = 85, normalized size = 3.40 \begin {gather*} \frac {361 x^{2}}{9} + \frac {38 x}{3} + \log {\left (\log {\relax (x )} \right )}^{2} - 2 \log {\left (\log {\relax (x )} \right )} + \frac {\left (- 38 x^{2} - 12\right ) \log {\left (\log {\relax (x )} \right )}}{3 x} + \frac {3 x e^{- 2 x} + \left (38 x^{2} - 6 x \log {\left (\log {\relax (x )} \right )} + 6 x + 12\right ) e^{- x}}{3 x} + \frac {36 x + 36}{9 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((((-114*x**3+36*x)*exp(x)**2+18*exp(x)*x**3)*ln(x)+18*exp(x)**2*x**2)*ln(ln(x))+((722*x**4+114*
x**3-36*x-72)*exp(x)**2+(-114*x**4+96*x**3-36*x**2-36*x)*exp(x)-18*x**3)*ln(x)+(-114*x**3-18*x**2-36*x)*exp(x)
**2-18*exp(x)*x**2)/x**3/exp(x)**2/ln(x),x)

[Out]

361*x**2/9 + 38*x/3 + log(log(x))**2 - 2*log(log(x)) + (-38*x**2 - 12)*log(log(x))/(3*x) + (3*x*exp(-2*x) + (3
8*x**2 - 6*x*log(log(x)) + 6*x + 12)*exp(-x))/(3*x) + (36*x + 36)/(9*x**2)

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