Integrand size = 125, antiderivative size = 18 \[ \int \frac {-972-648 x+\left (3888+3564 x+756 x^2\right ) \log (3)+\left (-5832-6804 x-2592 x^2-324 x^3\right ) \log ^2(3)+\left (3888+5508 x+2916 x^2+684 x^3+60 x^4\right ) \log ^3(3)+\left (-972-1620 x-1080 x^2-360 x^3-60 x^4-4 x^5\right ) \log ^4(3)}{243 x^5+405 x^6+270 x^7+90 x^8+15 x^9+x^{10}} \, dx=\frac {\left (\frac {3}{3+x}-\log (3)\right )^4}{x^4} \]
Leaf count is larger than twice the leaf count of optimal. \(312\) vs. \(2(18)=36\).
Time = 0.21 (sec) , antiderivative size = 312, normalized size of antiderivative = 17.33 \[ \int \frac {-972-648 x+\left (3888+3564 x+756 x^2\right ) \log (3)+\left (-5832-6804 x-2592 x^2-324 x^3\right ) \log ^2(3)+\left (3888+5508 x+2916 x^2+684 x^3+60 x^4\right ) \log ^3(3)+\left (-972-1620 x-1080 x^2-360 x^3-60 x^4-4 x^5\right ) \log ^4(3)}{243 x^5+405 x^6+270 x^7+90 x^8+15 x^9+x^{10}} \, dx=\frac {2916 x (-1+\log (3)) \log (3) (-3+\log (27))^2+729 (-1+\log (3)) (-3+\log (27))^3+324 x^3 \log (3) \left (9 \log ^3(3)-9 \log ^2(3) (-2+\log (27))-(-3+\log (27))^2 \log (27)+3 \log (3) \left (9-9 \log (27)+2 \log ^2(27)\right )\right )+9 x^4 \log (3) \left (702 \log ^3(3)-125 (-3+\log (27))^2 \log (27)+375 \log (3) \left (9-9 \log (27)+2 \log ^2(27)\right )-675 \log ^2(3) (-5+\log (729))\right )+156 x^5 \log (3) \left (27 \log ^3(3)-5 (-3+\log (27))^2 \log (27)+15 \log (3) \left (9-9 \log (27)+2 \log ^2(27)\right )-27 \log ^2(3) (-5+\log (729))\right )+42 x^6 \log (3) \left (27 \log ^3(3)-5 (-3+\log (27))^2 \log (27)+15 \log (3) \left (9-9 \log (27)+2 \log ^2(27)\right )-27 \log ^2(3) (-5+\log (729))\right )+4 x^7 \log (3) \left (27 \log ^3(3)-5 (-3+\log (27))^2 \log (27)+15 \log (3) \left (9-9 \log (27)+2 \log ^2(27)\right )-27 \log ^2(3) (-5+\log (729))\right )+486 x^2 \log (3) (-3+\log (27)) \left (9 \log ^2(3)-\log (19683)\right )}{243 x^4 (3+x)^4} \]
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)*L og[3]^4)/(243*x^5 + 405*x^6 + 270*x^7 + 90*x^8 + 15*x^9 + x^10),x]
(2916*x*(-1 + Log[3])*Log[3]*(-3 + Log[27])^2 + 729*(-1 + Log[3])*(-3 + Lo g[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])) + 4 86*x^2*Log[3]*(-3 + Log[27])*(9*Log[3]^2 - Log[19683]))/(243*x^4*(3 + x)^4 )
Leaf count is larger than twice the leaf count of optimal. \(138\) vs. \(2(18)=36\).
Time = 0.52 (sec) , antiderivative size = 138, normalized size of antiderivative = 7.67, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {2026, 2007, 2123, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (756 x^2+3564 x+3888\right ) \log (3)+\left (-324 x^3-2592 x^2-6804 x-5832\right ) \log ^2(3)+\left (60 x^4+684 x^3+2916 x^2+5508 x+3888\right ) \log ^3(3)+\left (-4 x^5-60 x^4-360 x^3-1080 x^2-1620 x-972\right ) \log ^4(3)-648 x-972}{x^{10}+15 x^9+90 x^8+270 x^7+405 x^6+243 x^5} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {\left (756 x^2+3564 x+3888\right ) \log (3)+\left (-324 x^3-2592 x^2-6804 x-5832\right ) \log ^2(3)+\left (60 x^4+684 x^3+2916 x^2+5508 x+3888\right ) \log ^3(3)+\left (-4 x^5-60 x^4-360 x^3-1080 x^2-1620 x-972\right ) \log ^4(3)-648 x-972}{x^5 \left (x^5+15 x^4+90 x^3+270 x^2+405 x+243\right )}dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {\left (756 x^2+3564 x+3888\right ) \log (3)+\left (-324 x^3-2592 x^2-6804 x-5832\right ) \log ^2(3)+\left (60 x^4+684 x^3+2916 x^2+5508 x+3888\right ) \log ^3(3)+\left (-4 x^5-60 x^4-360 x^3-1080 x^2-1620 x-972\right ) \log ^4(3)-648 x-972}{x^5 (x+3)^5}dx\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle \int \left (-\frac {4 (\log (3)-1)^4}{x^5}-\frac {4 (\log (3)-1)^3}{x^4}+\frac {4 (\log (3)-1)^2 (\log (9)-5)}{9 x^3}+\frac {4 (1-\log (3)) \left (5+\log ^2(3)-5 \log (3)\right )}{27 x^2}-\frac {4}{(x+3)^5}+\frac {4 (1-\log (3)) \left (-5-\log ^2(3)+5 \log (3)\right )}{27 (x+3)^2}+\frac {4 (\log (3)-1)}{(x+3)^4}+\frac {4 (5-3 \log (3)) (\log (3)-1)}{9 (x+3)^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(1-\log (3))^4}{x^4}-\frac {4 (1-\log (3))^3}{3 x^3}+\frac {2 (1-\log (3))^2 (5-\log (9))}{9 x^2}+\frac {1}{(x+3)^4}+\frac {4 (1-\log (3)) \left (5+\log ^2(3)-5 \log (3)\right )}{27 (x+3)}-\frac {4 (1-\log (3)) \left (5+\log ^2(3)-5 \log (3)\right )}{27 x}+\frac {2 (1-\log (3)) (5-\log (27))}{9 (x+3)^2}+\frac {4 (1-\log (3))}{3 (x+3)^3}\) |
Int[(-972 - 648*x + (3888 + 3564*x + 756*x^2)*Log[3] + (-5832 - 6804*x - 2 592*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]
(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)
3.18.36.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c , d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
Leaf count of result is larger than twice the leaf count of optimal. \(106\) vs. \(2(18)=36\).
Time = 0.16 (sec) , antiderivative size = 107, normalized size of antiderivative = 5.94
| method | result | size |
| norman | \(\frac {x^{4} \ln \left (3\right )^{4}+\left (12 \ln \left (3\right )^{4}-12 \ln \left (3\right )^{3}\right ) x^{3}+\left (54 \ln \left (3\right )^{4}-108 \ln \left (3\right )^{3}+54 \ln \left (3\right )^{2}\right ) x^{2}+\left (108 \ln \left (3\right )^{4}-324 \ln \left (3\right )^{3}+324 \ln \left (3\right )^{2}-108 \ln \left (3\right )\right ) x +81+81 \ln \left (3\right )^{4}-324 \ln \left (3\right )^{3}+486 \ln \left (3\right )^{2}-324 \ln \left (3\right )}{x^{4} \left (3+x \right )^{4}}\) | \(107\) |
| risch | \(\frac {x^{4} \ln \left (3\right )^{4}+\left (12 \ln \left (3\right )^{4}-12 \ln \left (3\right )^{3}\right ) x^{3}+\left (54 \ln \left (3\right )^{4}-108 \ln \left (3\right )^{3}+54 \ln \left (3\right )^{2}\right ) x^{2}+\left (108 \ln \left (3\right )^{4}-324 \ln \left (3\right )^{3}+324 \ln \left (3\right )^{2}-108 \ln \left (3\right )\right ) x +81+81 \ln \left (3\right )^{4}-324 \ln \left (3\right )^{3}+486 \ln \left (3\right )^{2}-324 \ln \left (3\right )}{x^{4} \left (x^{4}+12 x^{3}+54 x^{2}+108 x +81\right )}\) | \(122\) |
| gosper | \(\frac {x^{4} \ln \left (3\right )^{4}+12 \ln \left (3\right )^{4} x^{3}+54 x^{2} \ln \left (3\right )^{4}-12 x^{3} \ln \left (3\right )^{3}+108 x \ln \left (3\right )^{4}-108 x^{2} \ln \left (3\right )^{3}+81 \ln \left (3\right )^{4}-324 x \ln \left (3\right )^{3}+54 x^{2} \ln \left (3\right )^{2}-324 \ln \left (3\right )^{3}+324 x \ln \left (3\right )^{2}+486 \ln \left (3\right )^{2}-108 x \ln \left (3\right )-324 \ln \left (3\right )+81}{x^{4} \left (x^{4}+12 x^{3}+54 x^{2}+108 x +81\right )}\) | \(128\) |
| parallelrisch | \(\frac {x^{4} \ln \left (3\right )^{4}+12 \ln \left (3\right )^{4} x^{3}+54 x^{2} \ln \left (3\right )^{4}-12 x^{3} \ln \left (3\right )^{3}+108 x \ln \left (3\right )^{4}-108 x^{2} \ln \left (3\right )^{3}+81 \ln \left (3\right )^{4}-324 x \ln \left (3\right )^{3}+54 x^{2} \ln \left (3\right )^{2}-324 \ln \left (3\right )^{3}+324 x \ln \left (3\right )^{2}+486 \ln \left (3\right )^{2}-108 x \ln \left (3\right )-324 \ln \left (3\right )+81}{x^{4} \left (x^{4}+12 x^{3}+54 x^{2}+108 x +81\right )}\) | \(128\) |
| default | \(-\frac {4 \left (-\ln \left (3\right )^{3}+3 \ln \left (3\right )^{2}-3 \ln \left (3\right )+1\right )}{3 x^{3}}-\frac {-\ln \left (3\right )^{4}+4 \ln \left (3\right )^{3}-6 \ln \left (3\right )^{2}+4 \ln \left (3\right )-1}{x^{4}}-\frac {4 \left (-\frac {\ln \left (3\right )^{3}}{27}+\frac {2 \ln \left (3\right )^{2}}{9}-\frac {10 \ln \left (3\right )}{27}+\frac {5}{27}\right )}{x}-\frac {2 \left (\frac {2 \ln \left (3\right )^{3}}{9}-\ln \left (3\right )^{2}+\frac {4 \ln \left (3\right )}{3}-\frac {5}{9}\right )}{x^{2}}-\frac {4 \left (\ln \left (3\right )-1\right )}{3 \left (3+x \right )^{3}}+\frac {1}{\left (3+x \right )^{4}}-\frac {2 \left (-\frac {5}{9}+\frac {8 \ln \left (3\right )}{9}-\frac {\ln \left (3\right )^{2}}{3}\right )}{\left (3+x \right )^{2}}-\frac {4 \left (-\frac {5}{27}+\frac {10 \ln \left (3\right )}{27}-\frac {2 \ln \left (3\right )^{2}}{9}+\frac {\ln \left (3\right )^{3}}{27}\right )}{3+x}\) | \(160\) |
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+2 916*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+243* x^5),x,method=_RETURNVERBOSE)
(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
Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (18) = 36\).
Time = 0.24 (sec) , antiderivative size = 92, normalized size of antiderivative = 5.11 \[ \int \frac {-972-648 x+\left (3888+3564 x+756 x^2\right ) \log (3)+\left (-5832-6804 x-2592 x^2-324 x^3\right ) \log ^2(3)+\left (3888+5508 x+2916 x^2+684 x^3+60 x^4\right ) \log ^3(3)+\left (-972-1620 x-1080 x^2-360 x^3-60 x^4-4 x^5\right ) \log ^4(3)}{243 x^5+405 x^6+270 x^7+90 x^8+15 x^9+x^{10}} \, dx=\frac {{\left (x^{4} + 12 \, x^{3} + 54 \, x^{2} + 108 \, x + 81\right )} \log \left (3\right )^{4} - 12 \, {\left (x^{3} + 9 \, x^{2} + 27 \, x + 27\right )} \log \left (3\right )^{3} + 54 \, {\left (x^{2} + 6 \, x + 9\right )} \log \left (3\right )^{2} - 108 \, {\left (x + 3\right )} \log \left (3\right ) + 81}{x^{8} + 12 \, x^{7} + 54 \, x^{6} + 108 \, x^{5} + 81 \, x^{4}} \]
integrate(((-4*x^5-60*x^4-360*x^3-1080*x^2-1620*x-972)*log(3)^4+(60*x^4+68 4*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+40 5*x^6+243*x^5),x, algorithm=\
((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)*log(3)^2 - 108*(x + 3)*log(3) + 81)/(x^8 + 12*x^7 + 54*x^6 + 108*x^5 + 81*x^4)
Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (12) = 24\).
Time = 3.97 (sec) , antiderivative size = 128, normalized size of antiderivative = 7.11 \[ \int \frac {-972-648 x+\left (3888+3564 x+756 x^2\right ) \log (3)+\left (-5832-6804 x-2592 x^2-324 x^3\right ) \log ^2(3)+\left (3888+5508 x+2916 x^2+684 x^3+60 x^4\right ) \log ^3(3)+\left (-972-1620 x-1080 x^2-360 x^3-60 x^4-4 x^5\right ) \log ^4(3)}{243 x^5+405 x^6+270 x^7+90 x^8+15 x^9+x^{10}} \, dx=- \frac {- x^{4} \log {\left (3 \right )}^{4} + x^{3} \left (- 12 \log {\left (3 \right )}^{4} + 12 \log {\left (3 \right )}^{3}\right ) + x^{2} \left (- 54 \log {\left (3 \right )}^{4} - 54 \log {\left (3 \right )}^{2} + 108 \log {\left (3 \right )}^{3}\right ) + x \left (- 324 \log {\left (3 \right )}^{2} - 108 \log {\left (3 \right )}^{4} + 108 \log {\left (3 \right )} + 324 \log {\left (3 \right )}^{3}\right ) - 486 \log {\left (3 \right )}^{2} - 81 \log {\left (3 \right )}^{4} - 81 + 324 \log {\left (3 \right )} + 324 \log {\left (3 \right )}^{3}}{x^{8} + 12 x^{7} + 54 x^{6} + 108 x^{5} + 81 x^{4}} \]
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)*ln(3)**3+(-324*x**3-2592*x**2-6804*x-58 32)*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)
-(-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*log( 3) + 324*log(3)**3)/(x**8 + 12*x**7 + 54*x**6 + 108*x**5 + 81*x**4)
Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (18) = 36\).
Time = 0.20 (sec) , antiderivative size = 119, normalized size of antiderivative = 6.61 \[ \int \frac {-972-648 x+\left (3888+3564 x+756 x^2\right ) \log (3)+\left (-5832-6804 x-2592 x^2-324 x^3\right ) \log ^2(3)+\left (3888+5508 x+2916 x^2+684 x^3+60 x^4\right ) \log ^3(3)+\left (-972-1620 x-1080 x^2-360 x^3-60 x^4-4 x^5\right ) \log ^4(3)}{243 x^5+405 x^6+270 x^7+90 x^8+15 x^9+x^{10}} \, dx=\frac {x^{4} \log \left (3\right )^{4} + 12 \, {\left (\log \left (3\right )^{4} - \log \left (3\right )^{3}\right )} x^{3} + 81 \, \log \left (3\right )^{4} + 54 \, {\left (\log \left (3\right )^{4} - 2 \, \log \left (3\right )^{3} + \log \left (3\right )^{2}\right )} x^{2} - 324 \, \log \left (3\right )^{3} + 108 \, {\left (\log \left (3\right )^{4} - 3 \, \log \left (3\right )^{3} + 3 \, \log \left (3\right )^{2} - \log \left (3\right )\right )} x + 486 \, \log \left (3\right )^{2} - 324 \, \log \left (3\right ) + 81}{x^{8} + 12 \, x^{7} + 54 \, x^{6} + 108 \, x^{5} + 81 \, x^{4}} \]
integrate(((-4*x^5-60*x^4-360*x^3-1080*x^2-1620*x-972)*log(3)^4+(60*x^4+68 4*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+40 5*x^6+243*x^5),x, algorithm=\
(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*log(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)
Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (18) = 36\).
Time = 0.28 (sec) , antiderivative size = 113, normalized size of antiderivative = 6.28 \[ \int \frac {-972-648 x+\left (3888+3564 x+756 x^2\right ) \log (3)+\left (-5832-6804 x-2592 x^2-324 x^3\right ) \log ^2(3)+\left (3888+5508 x+2916 x^2+684 x^3+60 x^4\right ) \log ^3(3)+\left (-972-1620 x-1080 x^2-360 x^3-60 x^4-4 x^5\right ) \log ^4(3)}{243 x^5+405 x^6+270 x^7+90 x^8+15 x^9+x^{10}} \, dx=\frac {x^{4} \log \left (3\right )^{4} + 12 \, x^{3} \log \left (3\right )^{4} - 12 \, x^{3} \log \left (3\right )^{3} + 54 \, x^{2} \log \left (3\right )^{4} - 108 \, x^{2} \log \left (3\right )^{3} + 108 \, x \log \left (3\right )^{4} + 54 \, x^{2} \log \left (3\right )^{2} - 324 \, x \log \left (3\right )^{3} + 81 \, \log \left (3\right )^{4} + 324 \, x \log \left (3\right )^{2} - 324 \, \log \left (3\right )^{3} - 108 \, x \log \left (3\right ) + 486 \, \log \left (3\right )^{2} - 324 \, \log \left (3\right ) + 81}{{\left (x^{2} + 3 \, x\right )}^{4}} \]
integrate(((-4*x^5-60*x^4-360*x^3-1080*x^2-1620*x-972)*log(3)^4+(60*x^4+68 4*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+40 5*x^6+243*x^5),x, algorithm=\
(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*l og(3) + 81)/(x^2 + 3*x)^4
Time = 0.45 (sec) , antiderivative size = 494, normalized size of antiderivative = 27.44 \[ \int \frac {-972-648 x+\left (3888+3564 x+756 x^2\right ) \log (3)+\left (-5832-6804 x-2592 x^2-324 x^3\right ) \log ^2(3)+\left (3888+5508 x+2916 x^2+684 x^3+60 x^4\right ) \log ^3(3)+\left (-972-1620 x-1080 x^2-360 x^3-60 x^4-4 x^5\right ) \log ^4(3)}{243 x^5+405 x^6+270 x^7+90 x^8+15 x^9+x^{10}} \, dx=\frac {81}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}-\frac {324\,\ln \left (3\right )}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}+\frac {486\,{\ln \left (3\right )}^2}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}-\frac {324\,{\ln \left (3\right )}^3}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}+\frac {81\,{\ln \left (3\right )}^4}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}+\frac {324\,x\,{\ln \left (3\right )}^2}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}-\frac {324\,x\,{\ln \left (3\right )}^3}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}+\frac {108\,x\,{\ln \left (3\right )}^4}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}+\frac {54\,x^2\,{\ln \left (3\right )}^2}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}-\frac {108\,x^2\,{\ln \left (3\right )}^3}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}+\frac {54\,x^2\,{\ln \left (3\right )}^4}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}-\frac {12\,x^3\,{\ln \left (3\right )}^3}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}+\frac {12\,x^3\,{\ln \left (3\right )}^4}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}+\frac {x^4\,{\ln \left (3\right )}^4}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}-\frac {108\,x\,\ln \left (3\right )}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4} \]
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) + 972) /(243*x^5 + 405*x^6 + 270*x^7 + 90*x^8 + 15*x^9 + x^10),x)
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 + 10 8*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)