3.48.36 \(\int \frac {-972-648 x+(3888+3564 x+756 x^2) \log (3)+(-5832-6804 x-2592 x^2-324 x^3) \log ^2(3)+(3888+5508 x+2916 x^2+684 x^3+60 x^4) \log ^3(3)+(-972-1620 x-1080 x^2-360 x^3-60 x^4-4 x^5) \log ^4(3)}{243 x^5+405 x^6+270 x^7+90 x^8+15 x^9+x^{10}} \, dx\)

Optimal. Leaf size=18 \[ \frac {\left (\frac {3}{3+x}-\log (3)\right )^4}{x^4} \]

________________________________________________________________________________________

Rubi [B]  time = 0.20, antiderivative size = 138, normalized size of antiderivative = 7.67, number of steps used = 2, number of rules used = 1, integrand size = 125, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.008, Rules used = {2074} \begin {gather*} \frac {(1-\log (3))^4}{x^4}-\frac {4 (1-\log (3))^3}{3 x^3}+\frac {2 (1-\log (3))^2 (5-\log (9))}{9 x^2}+\frac {1}{(x+3)^4}+\frac {4 (1-\log (3)) \left (5+\log ^2(3)-5 \log (3)\right )}{27 (x+3)}-\frac {4 (1-\log (3)) \left (5+\log ^2(3)-5 \log (3)\right )}{27 x}+\frac {2 (1-\log (3)) (5-\log (27))}{9 (x+3)^2}+\frac {4 (1-\log (3))}{3 (x+3)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-972 - 648*x + (3888 + 3564*x + 756*x^2)*Log[3] + (-5832 - 6804*x - 2592*x^2 - 324*x^3)*Log[3]^2 + (3888
+ 5508*x + 2916*x^2 + 684*x^3 + 60*x^4)*Log[3]^3 + (-972 - 1620*x - 1080*x^2 - 360*x^3 - 60*x^4 - 4*x^5)*Log[3
]^4)/(243*x^5 + 405*x^6 + 270*x^7 + 90*x^8 + 15*x^9 + x^10),x]

[Out]

(3 + x)^(-4) + (4*(1 - Log[3]))/(3*(3 + x)^3) - (4*(1 - Log[3])^3)/(3*x^3) + (1 - Log[3])^4/x^4 - (4*(1 - Log[
3])*(5 - 5*Log[3] + Log[3]^2))/(27*x) + (4*(1 - Log[3])*(5 - 5*Log[3] + Log[3]^2))/(27*(3 + x)) + (2*(1 - Log[
3])^2*(5 - Log[9]))/(9*x^2) + (2*(1 - Log[3])*(5 - Log[27]))/(9*(3 + x)^2)

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {4}{(3+x)^5}+\frac {4 (-1+\log (3))}{(3+x)^4}+\frac {4 (5-3 \log (3)) (-1+\log (3))}{9 (3+x)^3}-\frac {4 (-1+\log (3))^3}{x^4}-\frac {4 (-1+\log (3))^4}{x^5}+\frac {4 (1-\log (3)) \left (-5+5 \log (3)-\log ^2(3)\right )}{27 (3+x)^2}+\frac {4 (1-\log (3)) \left (5-5 \log (3)+\log ^2(3)\right )}{27 x^2}+\frac {4 (-1+\log (3))^2 (-5+\log (9))}{9 x^3}\right ) \, dx\\ &=\frac {1}{(3+x)^4}+\frac {4 (1-\log (3))}{3 (3+x)^3}-\frac {4 (1-\log (3))^3}{3 x^3}+\frac {(1-\log (3))^4}{x^4}-\frac {4 (1-\log (3)) \left (5-5 \log (3)+\log ^2(3)\right )}{27 x}+\frac {4 (1-\log (3)) \left (5-5 \log (3)+\log ^2(3)\right )}{27 (3+x)}+\frac {2 (1-\log (3))^2 (5-\log (9))}{9 x^2}+\frac {2 (1-\log (3)) (5-\log (27))}{9 (3+x)^2}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [B]  time = 0.23, size = 312, normalized size = 17.33 \begin {gather*} \frac {2916 x (-1+\log (3)) \log (3) (-3+\log (27))^2+729 (-1+\log (3)) (-3+\log (27))^3+324 x^3 \log (3) \left (9 \log ^3(3)-9 \log ^2(3) (-2+\log (27))-(-3+\log (27))^2 \log (27)+3 \log (3) \left (9-9 \log (27)+2 \log ^2(27)\right )\right )+9 x^4 \log (3) \left (702 \log ^3(3)-125 (-3+\log (27))^2 \log (27)+375 \log (3) \left (9-9 \log (27)+2 \log ^2(27)\right )-675 \log ^2(3) (-5+\log (729))\right )+156 x^5 \log (3) \left (27 \log ^3(3)-5 (-3+\log (27))^2 \log (27)+15 \log (3) \left (9-9 \log (27)+2 \log ^2(27)\right )-27 \log ^2(3) (-5+\log (729))\right )+42 x^6 \log (3) \left (27 \log ^3(3)-5 (-3+\log (27))^2 \log (27)+15 \log (3) \left (9-9 \log (27)+2 \log ^2(27)\right )-27 \log ^2(3) (-5+\log (729))\right )+4 x^7 \log (3) \left (27 \log ^3(3)-5 (-3+\log (27))^2 \log (27)+15 \log (3) \left (9-9 \log (27)+2 \log ^2(27)\right )-27 \log ^2(3) (-5+\log (729))\right )+486 x^2 \log (3) (-3+\log (27)) \left (9 \log ^2(3)-\log (19683)\right )}{243 x^4 (3+x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-972 - 648*x + (3888 + 3564*x + 756*x^2)*Log[3] + (-5832 - 6804*x - 2592*x^2 - 324*x^3)*Log[3]^2 +
(3888 + 5508*x + 2916*x^2 + 684*x^3 + 60*x^4)*Log[3]^3 + (-972 - 1620*x - 1080*x^2 - 360*x^3 - 60*x^4 - 4*x^5)
*Log[3]^4)/(243*x^5 + 405*x^6 + 270*x^7 + 90*x^8 + 15*x^9 + x^10),x]

[Out]

(2916*x*(-1 + Log[3])*Log[3]*(-3 + Log[27])^2 + 729*(-1 + Log[3])*(-3 + Log[27])^3 + 324*x^3*Log[3]*(9*Log[3]^
3 - 9*Log[3]^2*(-2 + Log[27]) - (-3 + Log[27])^2*Log[27] + 3*Log[3]*(9 - 9*Log[27] + 2*Log[27]^2)) + 9*x^4*Log
[3]*(702*Log[3]^3 - 125*(-3 + Log[27])^2*Log[27] + 375*Log[3]*(9 - 9*Log[27] + 2*Log[27]^2) - 675*Log[3]^2*(-5
+ Log[729])) + 156*x^5*Log[3]*(27*Log[3]^3 - 5*(-3 + Log[27])^2*Log[27] + 15*Log[3]*(9 - 9*Log[27] + 2*Log[27
]^2) - 27*Log[3]^2*(-5 + Log[729])) + 42*x^6*Log[3]*(27*Log[3]^3 - 5*(-3 + Log[27])^2*Log[27] + 15*Log[3]*(9 -
9*Log[27] + 2*Log[27]^2) - 27*Log[3]^2*(-5 + Log[729])) + 4*x^7*Log[3]*(27*Log[3]^3 - 5*(-3 + Log[27])^2*Log[
27] + 15*Log[3]*(9 - 9*Log[27] + 2*Log[27]^2) - 27*Log[3]^2*(-5 + Log[729])) + 486*x^2*Log[3]*(-3 + Log[27])*(
9*Log[3]^2 - Log[19683]))/(243*x^4*(3 + x)^4)

________________________________________________________________________________________

fricas [B]  time = 1.02, size = 92, normalized size = 5.11 \begin {gather*} \frac {{\left (x^{4} + 12 \, x^{3} + 54 \, x^{2} + 108 \, x + 81\right )} \log \relax (3)^{4} - 12 \, {\left (x^{3} + 9 \, x^{2} + 27 \, x + 27\right )} \log \relax (3)^{3} + 54 \, {\left (x^{2} + 6 \, x + 9\right )} \log \relax (3)^{2} - 108 \, {\left (x + 3\right )} \log \relax (3) + 81}{x^{8} + 12 \, x^{7} + 54 \, x^{6} + 108 \, x^{5} + 81 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^5-60*x^4-360*x^3-1080*x^2-1620*x-972)*log(3)^4+(60*x^4+684*x^3+2916*x^2+5508*x+3888)*log(3)^3
+(-324*x^3-2592*x^2-6804*x-5832)*log(3)^2+(756*x^2+3564*x+3888)*log(3)-648*x-972)/(x^10+15*x^9+90*x^8+270*x^7+
405*x^6+243*x^5),x, algorithm="fricas")

[Out]

((x^4 + 12*x^3 + 54*x^2 + 108*x + 81)*log(3)^4 - 12*(x^3 + 9*x^2 + 27*x + 27)*log(3)^3 + 54*(x^2 + 6*x + 9)*lo
g(3)^2 - 108*(x + 3)*log(3) + 81)/(x^8 + 12*x^7 + 54*x^6 + 108*x^5 + 81*x^4)

________________________________________________________________________________________

giac [B]  time = 2.22, size = 113, normalized size = 6.28 \begin {gather*} \frac {x^{4} \log \relax (3)^{4} + 12 \, x^{3} \log \relax (3)^{4} - 12 \, x^{3} \log \relax (3)^{3} + 54 \, x^{2} \log \relax (3)^{4} - 108 \, x^{2} \log \relax (3)^{3} + 108 \, x \log \relax (3)^{4} + 54 \, x^{2} \log \relax (3)^{2} - 324 \, x \log \relax (3)^{3} + 81 \, \log \relax (3)^{4} + 324 \, x \log \relax (3)^{2} - 324 \, \log \relax (3)^{3} - 108 \, x \log \relax (3) + 486 \, \log \relax (3)^{2} - 324 \, \log \relax (3) + 81}{{\left (x^{2} + 3 \, x\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^5-60*x^4-360*x^3-1080*x^2-1620*x-972)*log(3)^4+(60*x^4+684*x^3+2916*x^2+5508*x+3888)*log(3)^3
+(-324*x^3-2592*x^2-6804*x-5832)*log(3)^2+(756*x^2+3564*x+3888)*log(3)-648*x-972)/(x^10+15*x^9+90*x^8+270*x^7+
405*x^6+243*x^5),x, algorithm="giac")

[Out]

(x^4*log(3)^4 + 12*x^3*log(3)^4 - 12*x^3*log(3)^3 + 54*x^2*log(3)^4 - 108*x^2*log(3)^3 + 108*x*log(3)^4 + 54*x
^2*log(3)^2 - 324*x*log(3)^3 + 81*log(3)^4 + 324*x*log(3)^2 - 324*log(3)^3 - 108*x*log(3) + 486*log(3)^2 - 324
*log(3) + 81)/(x^2 + 3*x)^4

________________________________________________________________________________________

maple [B]  time = 0.17, size = 107, normalized size = 5.94

method	result	size
norman	\(\frac {x^{4} \ln \relax (3)^{4}+\left (12 \ln \relax (3)^{4}-12 \ln \relax (3)^{3}\right ) x^{3}+\left (54 \ln \relax (3)^{4}-108 \ln \relax (3)^{3}+54 \ln \relax (3)^{2}\right ) x^{2}+\left (108 \ln \relax (3)^{4}-324 \ln \relax (3)^{3}+324 \ln \relax (3)^{2}-108 \ln \relax (3)\right ) x +81+81 \ln \relax (3)^{4}-324 \ln \relax (3)^{3}+486 \ln \relax (3)^{2}-324 \ln \relax (3)}{x^{4} \left (3+x \right )^{4}}\)	\(107\)
risch	\(\frac {x^{4} \ln \relax (3)^{4}+\left (12 \ln \relax (3)^{4}-12 \ln \relax (3)^{3}\right ) x^{3}+\left (54 \ln \relax (3)^{4}-108 \ln \relax (3)^{3}+54 \ln \relax (3)^{2}\right ) x^{2}+\left (108 \ln \relax (3)^{4}-324 \ln \relax (3)^{3}+324 \ln \relax (3)^{2}-108 \ln \relax (3)\right ) x +81+81 \ln \relax (3)^{4}-324 \ln \relax (3)^{3}+486 \ln \relax (3)^{2}-324 \ln \relax (3)}{x^{4} \left (x^{4}+12 x^{3}+54 x^{2}+108 x +81\right )}\)	\(122\)
gosper	\(\frac {x^{4} \ln \relax (3)^{4}+12 \ln \relax (3)^{4} x^{3}+54 x^{2} \ln \relax (3)^{4}-12 x^{3} \ln \relax (3)^{3}+108 x \ln \relax (3)^{4}-108 \ln \relax (3)^{3} x^{2}+81 \ln \relax (3)^{4}-324 x \ln \relax (3)^{3}+54 x^{2} \ln \relax (3)^{2}-324 \ln \relax (3)^{3}+324 x \ln \relax (3)^{2}+486 \ln \relax (3)^{2}-108 x \ln \relax (3)-324 \ln \relax (3)+81}{x^{4} \left (x^{4}+12 x^{3}+54 x^{2}+108 x +81\right )}\)	\(128\)
default	\(-\frac {4 \left (\ln \relax (3)-1\right )}{3 \left (3+x \right )^{3}}+\frac {1}{\left (3+x \right )^{4}}-\frac {2 \left (-\frac {5}{9}+\frac {8 \ln \relax (3)}{9}-\frac {\ln \relax (3)^{2}}{3}\right )}{\left (3+x \right )^{2}}-\frac {4 \left (-\frac {5}{27}+\frac {10 \ln \relax (3)}{27}-\frac {2 \ln \relax (3)^{2}}{9}+\frac {\ln \relax (3)^{3}}{27}\right )}{3+x}-\frac {4 \left (-\ln \relax (3)^{3}+3 \ln \relax (3)^{2}-3 \ln \relax (3)+1\right )}{3 x^{3}}-\frac {-\ln \relax (3)^{4}+4 \ln \relax (3)^{3}-6 \ln \relax (3)^{2}+4 \ln \relax (3)-1}{x^{4}}-\frac {4 \left (-\frac {\ln \relax (3)^{3}}{27}+\frac {2 \ln \relax (3)^{2}}{9}-\frac {10 \ln \relax (3)}{27}+\frac {5}{27}\right )}{x}-\frac {2 \left (\frac {2 \ln \relax (3)^{3}}{9}-\ln \relax (3)^{2}+\frac {4 \ln \relax (3)}{3}-\frac {5}{9}\right )}{x^{2}}\)	\(160\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x^5-60*x^4-360*x^3-1080*x^2-1620*x-972)*ln(3)^4+(60*x^4+684*x^3+2916*x^2+5508*x+3888)*ln(3)^3+(-324*x
^3-2592*x^2-6804*x-5832)*ln(3)^2+(756*x^2+3564*x+3888)*ln(3)-648*x-972)/(x^10+15*x^9+90*x^8+270*x^7+405*x^6+24
3*x^5),x,method=_RETURNVERBOSE)

[Out]

(x^4*ln(3)^4+(12*ln(3)^4-12*ln(3)^3)*x^3+(54*ln(3)^4-108*ln(3)^3+54*ln(3)^2)*x^2+(108*ln(3)^4-324*ln(3)^3+324*
ln(3)^2-108*ln(3))*x+81+81*ln(3)^4-324*ln(3)^3+486*ln(3)^2-324*ln(3))/x^4/(3+x)^4

________________________________________________________________________________________

maxima [B]  time = 0.36, size = 119, normalized size = 6.61 \begin {gather*} \frac {x^{4} \log \relax (3)^{4} + 12 \, {\left (\log \relax (3)^{4} - \log \relax (3)^{3}\right )} x^{3} + 81 \, \log \relax (3)^{4} + 54 \, {\left (\log \relax (3)^{4} - 2 \, \log \relax (3)^{3} + \log \relax (3)^{2}\right )} x^{2} - 324 \, \log \relax (3)^{3} + 108 \, {\left (\log \relax (3)^{4} - 3 \, \log \relax (3)^{3} + 3 \, \log \relax (3)^{2} - \log \relax (3)\right )} x + 486 \, \log \relax (3)^{2} - 324 \, \log \relax (3) + 81}{x^{8} + 12 \, x^{7} + 54 \, x^{6} + 108 \, x^{5} + 81 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^5-60*x^4-360*x^3-1080*x^2-1620*x-972)*log(3)^4+(60*x^4+684*x^3+2916*x^2+5508*x+3888)*log(3)^3
+(-324*x^3-2592*x^2-6804*x-5832)*log(3)^2+(756*x^2+3564*x+3888)*log(3)-648*x-972)/(x^10+15*x^9+90*x^8+270*x^7+
405*x^6+243*x^5),x, algorithm="maxima")

[Out]

(x^4*log(3)^4 + 12*(log(3)^4 - log(3)^3)*x^3 + 81*log(3)^4 + 54*(log(3)^4 - 2*log(3)^3 + log(3)^2)*x^2 - 324*l
og(3)^3 + 108*(log(3)^4 - 3*log(3)^3 + 3*log(3)^2 - log(3))*x + 486*log(3)^2 - 324*log(3) + 81)/(x^8 + 12*x^7
+ 54*x^6 + 108*x^5 + 81*x^4)

________________________________________________________________________________________

mupad [B]  time = 3.56, size = 494, normalized size = 27.44 \begin {gather*} \frac {81}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}-\frac {324\,\ln \relax (3)}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}+\frac {486\,{\ln \relax (3)}^2}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}-\frac {324\,{\ln \relax (3)}^3}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}+\frac {81\,{\ln \relax (3)}^4}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}+\frac {324\,x\,{\ln \relax (3)}^2}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}-\frac {324\,x\,{\ln \relax (3)}^3}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}+\frac {108\,x\,{\ln \relax (3)}^4}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}+\frac {54\,x^2\,{\ln \relax (3)}^2}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}-\frac {108\,x^2\,{\ln \relax (3)}^3}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}+\frac {54\,x^2\,{\ln \relax (3)}^4}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}-\frac {12\,x^3\,{\ln \relax (3)}^3}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}+\frac {12\,x^3\,{\ln \relax (3)}^4}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}+\frac {x^4\,{\ln \relax (3)}^4}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}-\frac {108\,x\,\ln \relax (3)}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(648*x - log(3)^3*(5508*x + 2916*x^2 + 684*x^3 + 60*x^4 + 3888) - log(3)*(3564*x + 756*x^2 + 3888) + log(
3)^4*(1620*x + 1080*x^2 + 360*x^3 + 60*x^4 + 4*x^5 + 972) + log(3)^2*(6804*x + 2592*x^2 + 324*x^3 + 5832) + 97
2)/(243*x^5 + 405*x^6 + 270*x^7 + 90*x^8 + 15*x^9 + x^10),x)

[Out]

81/(81*x^4 + 108*x^5 + 54*x^6 + 12*x^7 + x^8) - (324*log(3))/(81*x^4 + 108*x^5 + 54*x^6 + 12*x^7 + x^8) + (486
*log(3)^2)/(81*x^4 + 108*x^5 + 54*x^6 + 12*x^7 + x^8) - (324*log(3)^3)/(81*x^4 + 108*x^5 + 54*x^6 + 12*x^7 + x
^8) + (81*log(3)^4)/(81*x^4 + 108*x^5 + 54*x^6 + 12*x^7 + x^8) + (324*x*log(3)^2)/(81*x^4 + 108*x^5 + 54*x^6 +
12*x^7 + x^8) - (324*x*log(3)^3)/(81*x^4 + 108*x^5 + 54*x^6 + 12*x^7 + x^8) + (108*x*log(3)^4)/(81*x^4 + 108*
x^5 + 54*x^6 + 12*x^7 + x^8) + (54*x^2*log(3)^2)/(81*x^4 + 108*x^5 + 54*x^6 + 12*x^7 + x^8) - (108*x^2*log(3)^
3)/(81*x^4 + 108*x^5 + 54*x^6 + 12*x^7 + x^8) + (54*x^2*log(3)^4)/(81*x^4 + 108*x^5 + 54*x^6 + 12*x^7 + x^8) -
(12*x^3*log(3)^3)/(81*x^4 + 108*x^5 + 54*x^6 + 12*x^7 + x^8) + (12*x^3*log(3)^4)/(81*x^4 + 108*x^5 + 54*x^6 +
12*x^7 + x^8) + (x^4*log(3)^4)/(81*x^4 + 108*x^5 + 54*x^6 + 12*x^7 + x^8) - (108*x*log(3))/(81*x^4 + 108*x^5
+ 54*x^6 + 12*x^7 + x^8)

________________________________________________________________________________________

sympy [B]  time = 7.54, size = 128, normalized size = 7.11 \begin {gather*} - \frac {- x^{4} \log {\relax (3 )}^{4} + x^{3} \left (- 12 \log {\relax (3 )}^{4} + 12 \log {\relax (3 )}^{3}\right ) + x^{2} \left (- 54 \log {\relax (3 )}^{4} - 54 \log {\relax (3 )}^{2} + 108 \log {\relax (3 )}^{3}\right ) + x \left (- 324 \log {\relax (3 )}^{2} - 108 \log {\relax (3 )}^{4} + 108 \log {\relax (3 )} + 324 \log {\relax (3 )}^{3}\right ) - 486 \log {\relax (3 )}^{2} - 81 \log {\relax (3 )}^{4} - 81 + 324 \log {\relax (3 )} + 324 \log {\relax (3 )}^{3}}{x^{8} + 12 x^{7} + 54 x^{6} + 108 x^{5} + 81 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x**5-60*x**4-360*x**3-1080*x**2-1620*x-972)*ln(3)**4+(60*x**4+684*x**3+2916*x**2+5508*x+3888)*l
n(3)**3+(-324*x**3-2592*x**2-6804*x-5832)*ln(3)**2+(756*x**2+3564*x+3888)*ln(3)-648*x-972)/(x**10+15*x**9+90*x
**8+270*x**7+405*x**6+243*x**5),x)

[Out]

-(-x**4*log(3)**4 + x**3*(-12*log(3)**4 + 12*log(3)**3) + x**2*(-54*log(3)**4 - 54*log(3)**2 + 108*log(3)**3)
+ x*(-324*log(3)**2 - 108*log(3)**4 + 108*log(3) + 324*log(3)**3) - 486*log(3)**2 - 81*log(3)**4 - 81 + 324*lo
g(3) + 324*log(3)**3)/(x**8 + 12*x**7 + 54*x**6 + 108*x**5 + 81*x**4)

________________________________________________________________________________________