3.14.78 \(\int \frac {-10+12 x-6 x^2+e^{20+x} (-6+12 x-6 x^2)+(-9+23 x-17 x^2+3 x^3+e^{20+x} (3-6 x+3 x^2)) \log (\frac {36-72 x+36 x^2}{81-252 x+250 x^2-84 x^3+9 x^4+e^{40+2 x} (9-18 x+9 x^2)+e^{20+x} (-54+138 x-102 x^2+18 x^3)})}{(-9+23 x-17 x^2+3 x^3+e^{20+x} (3-6 x+3 x^2)) \log (\frac {36-72 x+36 x^2}{81-252 x+250 x^2-84 x^3+9 x^4+e^{40+2 x} (9-18 x+9 x^2)+e^{20+x} (-54+138 x-102 x^2+18 x^3)})} \, dx\)

Optimal. Leaf size=27 \[ x+\log \left (\log \left (\frac {4}{\left (-3+e^{20+x}+\frac {2}{-3+\frac {3}{x}}+x\right )^2}\right )\right ) \]

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Rubi [F]  time = 4.48, 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 {-10+12 x-6 x^2+e^{20+x} \left (-6+12 x-6 x^2\right )+\left (-9+23 x-17 x^2+3 x^3+e^{20+x} \left (3-6 x+3 x^2\right )\right ) \log \left (\frac {36-72 x+36 x^2}{81-252 x+250 x^2-84 x^3+9 x^4+e^{40+2 x} \left (9-18 x+9 x^2\right )+e^{20+x} \left (-54+138 x-102 x^2+18 x^3\right )}\right )}{\left (-9+23 x-17 x^2+3 x^3+e^{20+x} \left (3-6 x+3 x^2\right )\right ) \log \left (\frac {36-72 x+36 x^2}{81-252 x+250 x^2-84 x^3+9 x^4+e^{40+2 x} \left (9-18 x+9 x^2\right )+e^{20+x} \left (-54+138 x-102 x^2+18 x^3\right )}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-10 + 12*x - 6*x^2 + E^(20 + x)*(-6 + 12*x - 6*x^2) + (-9 + 23*x - 17*x^2 + 3*x^3 + E^(20 + x)*(3 - 6*x +
3*x^2))*Log[(36 - 72*x + 36*x^2)/(81 - 252*x + 250*x^2 - 84*x^3 + 9*x^4 + E^(40 + 2*x)*(9 - 18*x + 9*x^2) + E
^(20 + x)*(-54 + 138*x - 102*x^2 + 18*x^3))])/((-9 + 23*x - 17*x^2 + 3*x^3 + E^(20 + x)*(3 - 6*x + 3*x^2))*Log
[(36 - 72*x + 36*x^2)/(81 - 252*x + 250*x^2 - 84*x^3 + 9*x^4 + E^(40 + 2*x)*(9 - 18*x + 9*x^2) + E^(20 + x)*(-
54 + 138*x - 102*x^2 + 18*x^3))]),x]

[Out]

x - 2*Defer[Int][Log[(36*(-1 + x)^2)/(9 + 3*E^(20 + x)*(-1 + x) - 14*x + 3*x^2)^2]^(-1), x] + 24*Defer[Int][1/
((9 - 3*E^(20 + x) - 14*x + 3*E^(20 + x)*x + 3*x^2)*Log[(36*(-1 + x)^2)/(9 + 3*E^(20 + x)*(-1 + x) - 14*x + 3*
x^2)^2]), x] - 4*Defer[Int][1/((-1 + x)*(9 - 3*E^(20 + x) - 14*x + 3*E^(20 + x)*x + 3*x^2)*Log[(36*(-1 + x)^2)
/(9 + 3*E^(20 + x)*(-1 + x) - 14*x + 3*x^2)^2]), x] - 34*Defer[Int][x/((9 - 3*E^(20 + x) - 14*x + 3*E^(20 + x)
*x + 3*x^2)*Log[(36*(-1 + x)^2)/(9 + 3*E^(20 + x)*(-1 + x) - 14*x + 3*x^2)^2]), x] + 6*Defer[Int][x^2/((9 - 3*
E^(20 + x) - 14*x + 3*E^(20 + x)*x + 3*x^2)*Log[(36*(-1 + x)^2)/(9 + 3*E^(20 + x)*(-1 + x) - 14*x + 3*x^2)^2])
, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {10+6 e^{20+x} (-1+x)^2-12 x+6 x^2-(-1+x) \left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right ) \log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )}{(1-x) \left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right ) \log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )} \, dx\\ &=\int \left (\frac {2 \left (-14+29 x-20 x^2+3 x^3\right )}{(-1+x) \left (9-3 e^{20+x}-14 x+3 e^{20+x} x+3 x^2\right ) \log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )}+\frac {-2+\log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )}{\log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )}\right ) \, dx\\ &=2 \int \frac {-14+29 x-20 x^2+3 x^3}{(-1+x) \left (9-3 e^{20+x}-14 x+3 e^{20+x} x+3 x^2\right ) \log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )} \, dx+\int \frac {-2+\log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )}{\log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )} \, dx\\ &=2 \int \left (\frac {12}{\left (9-3 e^{20+x}-14 x+3 e^{20+x} x+3 x^2\right ) \log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )}-\frac {2}{(-1+x) \left (9-3 e^{20+x}-14 x+3 e^{20+x} x+3 x^2\right ) \log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )}-\frac {17 x}{\left (9-3 e^{20+x}-14 x+3 e^{20+x} x+3 x^2\right ) \log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )}+\frac {3 x^2}{\left (9-3 e^{20+x}-14 x+3 e^{20+x} x+3 x^2\right ) \log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )}\right ) \, dx+\int \left (1-\frac {2}{\log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )}\right ) \, dx\\ &=x-2 \int \frac {1}{\log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )} \, dx-4 \int \frac {1}{(-1+x) \left (9-3 e^{20+x}-14 x+3 e^{20+x} x+3 x^2\right ) \log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )} \, dx+6 \int \frac {x^2}{\left (9-3 e^{20+x}-14 x+3 e^{20+x} x+3 x^2\right ) \log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )} \, dx+24 \int \frac {1}{\left (9-3 e^{20+x}-14 x+3 e^{20+x} x+3 x^2\right ) \log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )} \, dx-34 \int \frac {x}{\left (9-3 e^{20+x}-14 x+3 e^{20+x} x+3 x^2\right ) \log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 33, normalized size = 1.22 \begin {gather*} x+\log \left (\log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-10 + 12*x - 6*x^2 + E^(20 + x)*(-6 + 12*x - 6*x^2) + (-9 + 23*x - 17*x^2 + 3*x^3 + E^(20 + x)*(3 -
6*x + 3*x^2))*Log[(36 - 72*x + 36*x^2)/(81 - 252*x + 250*x^2 - 84*x^3 + 9*x^4 + E^(40 + 2*x)*(9 - 18*x + 9*x^
2) + E^(20 + x)*(-54 + 138*x - 102*x^2 + 18*x^3))])/((-9 + 23*x - 17*x^2 + 3*x^3 + E^(20 + x)*(3 - 6*x + 3*x^2
))*Log[(36 - 72*x + 36*x^2)/(81 - 252*x + 250*x^2 - 84*x^3 + 9*x^4 + E^(40 + 2*x)*(9 - 18*x + 9*x^2) + E^(20 +
x)*(-54 + 138*x - 102*x^2 + 18*x^3))]),x]

[Out]

x + Log[Log[(36*(-1 + x)^2)/(9 + 3*E^(20 + x)*(-1 + x) - 14*x + 3*x^2)^2]]

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fricas [B]  time = 1.14, size = 73, normalized size = 2.70 \begin {gather*} x + \log \left (\log \left (\frac {36 \, {\left (x^{2} - 2 \, x + 1\right )}}{9 \, x^{4} - 84 \, x^{3} + 250 \, x^{2} + 9 \, {\left (x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x + 40\right )} + 6 \, {\left (3 \, x^{3} - 17 \, x^{2} + 23 \, x - 9\right )} e^{\left (x + 20\right )} - 252 \, x + 81}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x^2-6*x+3)*exp(20+x)+3*x^3-17*x^2+23*x-9)*log((36*x^2-72*x+36)/((9*x^2-18*x+9)*exp(20+x)^2+(18*
x^3-102*x^2+138*x-54)*exp(20+x)+9*x^4-84*x^3+250*x^2-252*x+81))+(-6*x^2+12*x-6)*exp(20+x)-6*x^2+12*x-10)/((3*x
^2-6*x+3)*exp(20+x)+3*x^3-17*x^2+23*x-9)/log((36*x^2-72*x+36)/((9*x^2-18*x+9)*exp(20+x)^2+(18*x^3-102*x^2+138*
x-54)*exp(20+x)+9*x^4-84*x^3+250*x^2-252*x+81)),x, algorithm="fricas")

[Out]

x + log(log(36*(x^2 - 2*x + 1)/(9*x^4 - 84*x^3 + 250*x^2 + 9*(x^2 - 2*x + 1)*e^(2*x + 40) + 6*(3*x^3 - 17*x^2
+ 23*x - 9)*e^(x + 20) - 252*x + 81)))

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giac [B]  time = 4.83, size = 97, normalized size = 3.59 \begin {gather*} x + \log \left (-\log \left (9 \, x^{4} + 18 \, x^{3} e^{\left (x + 20\right )} - 84 \, x^{3} + 9 \, x^{2} e^{\left (2 \, x + 40\right )} - 102 \, x^{2} e^{\left (x + 20\right )} + 250 \, x^{2} - 18 \, x e^{\left (2 \, x + 40\right )} + 138 \, x e^{\left (x + 20\right )} - 252 \, x + 9 \, e^{\left (2 \, x + 40\right )} - 54 \, e^{\left (x + 20\right )} + 81\right ) + \log \left (36 \, x^{2} - 72 \, x + 36\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x^2-6*x+3)*exp(20+x)+3*x^3-17*x^2+23*x-9)*log((36*x^2-72*x+36)/((9*x^2-18*x+9)*exp(20+x)^2+(18*
x^3-102*x^2+138*x-54)*exp(20+x)+9*x^4-84*x^3+250*x^2-252*x+81))+(-6*x^2+12*x-6)*exp(20+x)-6*x^2+12*x-10)/((3*x
^2-6*x+3)*exp(20+x)+3*x^3-17*x^2+23*x-9)/log((36*x^2-72*x+36)/((9*x^2-18*x+9)*exp(20+x)^2+(18*x^3-102*x^2+138*
x-54)*exp(20+x)+9*x^4-84*x^3+250*x^2-252*x+81)),x, algorithm="giac")

[Out]

x + log(-log(9*x^4 + 18*x^3*e^(x + 20) - 84*x^3 + 9*x^2*e^(2*x + 40) - 102*x^2*e^(x + 20) + 250*x^2 - 18*x*e^(
2*x + 40) + 138*x*e^(x + 20) - 252*x + 9*e^(2*x + 40) - 54*e^(x + 20) + 81) + log(36*x^2 - 72*x + 36))

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maple [B]  time = 0.50, size = 75, normalized size = 2.78

method	result	size
norman	\(x +\ln \left (\ln \left (\frac {36 x^{2}-72 x +36}{\left (9 x^{2}-18 x +9\right ) {\mathrm e}^{40+2 x}+\left (18 x^{3}-102 x^{2}+138 x -54\right ) {\mathrm e}^{20+x}+9 x^{4}-84 x^{3}+250 x^{2}-252 x +81}\right )\right )\)	\(75\)
risch	\(x +\ln \left (\ln \left (x^{2}+\left ({\mathrm e}^{20+x}-\frac {14}{3}\right ) x -{\mathrm e}^{20+x}+3\right )+\frac {i \left (\pi \mathrm {csgn}\left (i \left (x -1\right )\right )^{2} \mathrm {csgn}\left (i \left (x -1\right )^{2}\right )-2 \pi \,\mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (i \left (x -1\right )^{2}\right )^{2}+\pi \mathrm {csgn}\left (i \left (x -1\right )^{2}\right )^{3}+\pi \,\mathrm {csgn}\left (i \left (x -1\right )^{2}\right ) \mathrm {csgn}\left (\frac {i}{\left (x^{2}+\left ({\mathrm e}^{20+x}-\frac {14}{3}\right ) x -{\mathrm e}^{20+x}+3\right )^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (x -1\right )^{2}}{\left (x^{2}+\left ({\mathrm e}^{20+x}-\frac {14}{3}\right ) x -{\mathrm e}^{20+x}+3\right )^{2}}\right )-\pi \,\mathrm {csgn}\left (i \left (x -1\right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (x -1\right )^{2}}{\left (x^{2}+\left ({\mathrm e}^{20+x}-\frac {14}{3}\right ) x -{\mathrm e}^{20+x}+3\right )^{2}}\right )^{2}-\pi \,\mathrm {csgn}\left (\frac {i}{\left (x^{2}+\left ({\mathrm e}^{20+x}-\frac {14}{3}\right ) x -{\mathrm e}^{20+x}+3\right )^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (x -1\right )^{2}}{\left (x^{2}+\left ({\mathrm e}^{20+x}-\frac {14}{3}\right ) x -{\mathrm e}^{20+x}+3\right )^{2}}\right )^{2}-\pi \mathrm {csgn}\left (i \left (x^{2}+\left ({\mathrm e}^{20+x}-\frac {14}{3}\right ) x -{\mathrm e}^{20+x}+3\right )\right )^{2} \mathrm {csgn}\left (i \left (x^{2}+\left ({\mathrm e}^{20+x}-\frac {14}{3}\right ) x -{\mathrm e}^{20+x}+3\right )^{2}\right )+2 \pi \,\mathrm {csgn}\left (i \left (x^{2}+\left ({\mathrm e}^{20+x}-\frac {14}{3}\right ) x -{\mathrm e}^{20+x}+3\right )\right ) \mathrm {csgn}\left (i \left (x^{2}+\left ({\mathrm e}^{20+x}-\frac {14}{3}\right ) x -{\mathrm e}^{20+x}+3\right )^{2}\right )^{2}-\pi \mathrm {csgn}\left (i \left (x^{2}+\left ({\mathrm e}^{20+x}-\frac {14}{3}\right ) x -{\mathrm e}^{20+x}+3\right )^{2}\right )^{3}+\pi \mathrm {csgn}\left (\frac {i \left (x -1\right )^{2}}{\left (x^{2}+\left ({\mathrm e}^{20+x}-\frac {14}{3}\right ) x -{\mathrm e}^{20+x}+3\right )^{2}}\right )^{3}+4 i \ln \left (x -1\right )+4 i \ln \relax (2)\right )}{4}\right )\)	\(435\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((3*x^2-6*x+3)*exp(20+x)+3*x^3-17*x^2+23*x-9)*ln((36*x^2-72*x+36)/((9*x^2-18*x+9)*exp(20+x)^2+(18*x^3-102
*x^2+138*x-54)*exp(20+x)+9*x^4-84*x^3+250*x^2-252*x+81))+(-6*x^2+12*x-6)*exp(20+x)-6*x^2+12*x-10)/((3*x^2-6*x+
3)*exp(20+x)+3*x^3-17*x^2+23*x-9)/ln((36*x^2-72*x+36)/((9*x^2-18*x+9)*exp(20+x)^2+(18*x^3-102*x^2+138*x-54)*ex
p(20+x)+9*x^4-84*x^3+250*x^2-252*x+81)),x,method=_RETURNVERBOSE)

[Out]

x+ln(ln((36*x^2-72*x+36)/((9*x^2-18*x+9)*exp(20+x)^2+(18*x^3-102*x^2+138*x-54)*exp(20+x)+9*x^4-84*x^3+250*x^2-
252*x+81)))

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maxima [A]  time = 15.40, size = 42, normalized size = 1.56 \begin {gather*} x + \log \left (-\log \relax (3) - \log \relax (2) + \log \left (3 \, x^{2} + 3 \, {\left (x e^{20} - e^{20}\right )} e^{x} - 14 \, x + 9\right ) - \log \left (x - 1\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x^2-6*x+3)*exp(20+x)+3*x^3-17*x^2+23*x-9)*log((36*x^2-72*x+36)/((9*x^2-18*x+9)*exp(20+x)^2+(18*
x^3-102*x^2+138*x-54)*exp(20+x)+9*x^4-84*x^3+250*x^2-252*x+81))+(-6*x^2+12*x-6)*exp(20+x)-6*x^2+12*x-10)/((3*x
^2-6*x+3)*exp(20+x)+3*x^3-17*x^2+23*x-9)/log((36*x^2-72*x+36)/((9*x^2-18*x+9)*exp(20+x)^2+(18*x^3-102*x^2+138*
x-54)*exp(20+x)+9*x^4-84*x^3+250*x^2-252*x+81)),x, algorithm="maxima")

[Out]

x + log(-log(3) - log(2) + log(3*x^2 + 3*(x*e^20 - e^20)*e^x - 14*x + 9) - log(x - 1))

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mupad [B]  time = 2.45, size = 74, normalized size = 2.74 \begin {gather*} x+\ln \left (\ln \left (\frac {36\,x^2-72\,x+36}{250\,x^2-252\,x-84\,x^3+9\,x^4+{\mathrm {e}}^{20}\,{\mathrm {e}}^x\,\left (18\,x^3-102\,x^2+138\,x-54\right )+{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{40}\,\left (9\,x^2-18\,x+9\right )+81}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x + 20)*(6*x^2 - 12*x + 6) - 12*x - log((36*x^2 - 72*x + 36)/(exp(x + 20)*(138*x - 102*x^2 + 18*x^3
- 54) - 252*x + exp(2*x + 40)*(9*x^2 - 18*x + 9) + 250*x^2 - 84*x^3 + 9*x^4 + 81))*(23*x + exp(x + 20)*(3*x^2
- 6*x + 3) - 17*x^2 + 3*x^3 - 9) + 6*x^2 + 10)/(log((36*x^2 - 72*x + 36)/(exp(x + 20)*(138*x - 102*x^2 + 18*x^
3 - 54) - 252*x + exp(2*x + 40)*(9*x^2 - 18*x + 9) + 250*x^2 - 84*x^3 + 9*x^4 + 81))*(23*x + exp(x + 20)*(3*x^
2 - 6*x + 3) - 17*x^2 + 3*x^3 - 9)),x)

[Out]

x + log(log((36*x^2 - 72*x + 36)/(250*x^2 - 252*x - 84*x^3 + 9*x^4 + exp(20)*exp(x)*(138*x - 102*x^2 + 18*x^3
- 54) + exp(2*x)*exp(40)*(9*x^2 - 18*x + 9) + 81)))

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sympy [B]  time = 1.25, size = 71, normalized size = 2.63 \begin {gather*} x + \log {\left (\log {\left (\frac {36 x^{2} - 72 x + 36}{9 x^{4} - 84 x^{3} + 250 x^{2} - 252 x + \left (9 x^{2} - 18 x + 9\right ) e^{2 x + 40} + \left (18 x^{3} - 102 x^{2} + 138 x - 54\right ) e^{x + 20} + 81} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x**2-6*x+3)*exp(20+x)+3*x**3-17*x**2+23*x-9)*ln((36*x**2-72*x+36)/((9*x**2-18*x+9)*exp(20+x)**2
+(18*x**3-102*x**2+138*x-54)*exp(20+x)+9*x**4-84*x**3+250*x**2-252*x+81))+(-6*x**2+12*x-6)*exp(20+x)-6*x**2+12
*x-10)/((3*x**2-6*x+3)*exp(20+x)+3*x**3-17*x**2+23*x-9)/ln((36*x**2-72*x+36)/((9*x**2-18*x+9)*exp(20+x)**2+(18
*x**3-102*x**2+138*x-54)*exp(20+x)+9*x**4-84*x**3+250*x**2-252*x+81)),x)

[Out]

x + log(log((36*x**2 - 72*x + 36)/(9*x**4 - 84*x**3 + 250*x**2 - 252*x + (9*x**2 - 18*x + 9)*exp(2*x + 40) + (
18*x**3 - 102*x**2 + 138*x - 54)*exp(x + 20) + 81)))

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