Integrand size = 192, antiderivative size = 31 \[ \int \frac {-32-112 x-152 x^2-100 x^3-96 x^4-164 x^5-128 x^6-32 x^7+\left (128 x^3+320 x^4+256 x^5+64 x^6\right ) \log (3)+\left (-64 x^2-160 x^3-128 x^4-32 x^5\right ) \log ^2(3)+\left (-64 x-160 x^2-144 x^3-312 x^4-712 x^5-576 x^6-128 x^7+\left (384 x^3+1088 x^4+896 x^5+192 x^6\right ) \log (3)+\left (-128 x^2-384 x^3-320 x^4-64 x^5\right ) \log ^2(3)\right ) \log (x)}{8 x+12 x^2+6 x^3+x^4} \, dx=3-2 (1+x)^2 \left (2+\frac {16 x^2 (-x+\log (3))^2}{(2+x)^2}\right ) \log (x) \]
Leaf count is larger than twice the leaf count of optimal. \(84\) vs. \(2(31)=62\).
Time = 0.13 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.71 \[ \int \frac {-32-112 x-152 x^2-100 x^3-96 x^4-164 x^5-128 x^6-32 x^7+\left (128 x^3+320 x^4+256 x^5+64 x^6\right ) \log (3)+\left (-64 x^2-160 x^3-128 x^4-32 x^5\right ) \log ^2(3)+\left (-64 x-160 x^2-144 x^3-312 x^4-712 x^5-576 x^6-128 x^7+\left (384 x^3+1088 x^4+896 x^5+192 x^6\right ) \log (3)+\left (-128 x^2-384 x^3-320 x^4-64 x^5\right ) \log ^2(3)\right ) \log (x)}{8 x+12 x^2+6 x^3+x^4} \, dx=-\frac {4 \left (12+36 x+24 x^6+3 x^2 \left (13+8 \log ^2(3)\right )+2 x^3 \left (9+648 \log (3)+24 \log ^2(3)-224 \log (27)\right )-16 x^5 (-3+\log (27))+x^4 \left (27-264 \log (3)+24 \log ^2(3)+56 \log (27)\right )\right ) \log (x)}{3 (2+x)^2} \]
Integrate[(-32 - 112*x - 152*x^2 - 100*x^3 - 96*x^4 - 164*x^5 - 128*x^6 - 32*x^7 + (128*x^3 + 320*x^4 + 256*x^5 + 64*x^6)*Log[3] + (-64*x^2 - 160*x^ 3 - 128*x^4 - 32*x^5)*Log[3]^2 + (-64*x - 160*x^2 - 144*x^3 - 312*x^4 - 71 2*x^5 - 576*x^6 - 128*x^7 + (384*x^3 + 1088*x^4 + 896*x^5 + 192*x^6)*Log[3 ] + (-128*x^2 - 384*x^3 - 320*x^4 - 64*x^5)*Log[3]^2)*Log[x])/(8*x + 12*x^ 2 + 6*x^3 + x^4),x]
(-4*(12 + 36*x + 24*x^6 + 3*x^2*(13 + 8*Log[3]^2) + 2*x^3*(9 + 648*Log[3] + 24*Log[3]^2 - 224*Log[27]) - 16*x^5*(-3 + Log[27]) + x^4*(27 - 264*Log[3 ] + 24*Log[3]^2 + 56*Log[27]))*Log[x])/(3*(2 + x)^2)
Leaf count is larger than twice the leaf count of optimal. \(352\) vs. \(2(31)=62\).
Time = 1.53 (sec) , antiderivative size = 352, normalized size of antiderivative = 11.35, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {2026, 2007, 7293, 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.
| \(\displaystyle \int \frac {-32 x^7-128 x^6-164 x^5-96 x^4-100 x^3-152 x^2+\left (64 x^6+256 x^5+320 x^4+128 x^3\right ) \log (3)+\left (-32 x^5-128 x^4-160 x^3-64 x^2\right ) \log ^2(3)+\left (-128 x^7-576 x^6-712 x^5-312 x^4-144 x^3-160 x^2+\left (192 x^6+896 x^5+1088 x^4+384 x^3\right ) \log (3)+\left (-64 x^5-320 x^4-384 x^3-128 x^2\right ) \log ^2(3)-64 x\right ) \log (x)-112 x-32}{x^4+6 x^3+12 x^2+8 x} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {-32 x^7-128 x^6-164 x^5-96 x^4-100 x^3-152 x^2+\left (64 x^6+256 x^5+320 x^4+128 x^3\right ) \log (3)+\left (-32 x^5-128 x^4-160 x^3-64 x^2\right ) \log ^2(3)+\left (-128 x^7-576 x^6-712 x^5-312 x^4-144 x^3-160 x^2+\left (192 x^6+896 x^5+1088 x^4+384 x^3\right ) \log (3)+\left (-64 x^5-320 x^4-384 x^3-128 x^2\right ) \log ^2(3)-64 x\right ) \log (x)-112 x-32}{x \left (x^3+6 x^2+12 x+8\right )}dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {-32 x^7-128 x^6-164 x^5-96 x^4-100 x^3-152 x^2+\left (64 x^6+256 x^5+320 x^4+128 x^3\right ) \log (3)+\left (-32 x^5-128 x^4-160 x^3-64 x^2\right ) \log ^2(3)+\left (-128 x^7-576 x^6-712 x^5-312 x^4-144 x^3-160 x^2+\left (192 x^6+896 x^5+1088 x^4+384 x^3\right ) \log (3)+\left (-64 x^5-320 x^4-384 x^3-128 x^2\right ) \log ^2(3)-64 x\right ) \log (x)-112 x-32}{x (x+2)^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {32 x^6}{(x+2)^3}-\frac {128 x^5}{(x+2)^3}-\frac {164 x^4}{(x+2)^3}-\frac {96 x^3}{(x+2)^3}-\frac {100 x^2}{(x+2)^3}+\frac {64 (x+1)^2 x^2 \log (3)}{(x+2)^2}+\frac {8 (x+1) \left (-16 x^5-56 x^4 \left (1-\frac {3 \log (3)}{7}\right )-33 x^3 \left (1+\frac {8}{33} (\log (3)-11) \log (3)\right )-6 x^2 \left (1+\frac {8}{3} \log (3) (\log (9)-3)\right )-12 x \left (1+\frac {4 \log ^2(3)}{3}\right )-8\right ) \log (x)}{(x+2)^3}-\frac {152 x}{(x+2)^3}-\frac {112}{(x+2)^3}-\frac {32}{(x+2)^3 x}-\frac {32 (x+1)^2 x \log ^2(3)}{(x+2)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -32 x^4 \log (x)+\frac {64 x^3}{3}+64 x^3 (1+\log (3)) \log (x)-\frac {64}{3} x^3 (1+\log (3))+\frac {64}{3} x^3 \log (3)-\frac {38 x^2}{(x+2)^2}-82 x^2-4 x^2 \left (41+8 \log ^2(3)+32 \log (3)\right ) \log (x)+2 x^2 \left (41+8 \log ^2(3)+32 \log (3)\right )-16 x^2 \log ^2(3)-64 x^2 \log (3)+376 x-\frac {408}{x+2}+\frac {152}{(x+2)^2}-\frac {64 x \left (16-13 \log ^2(3)+2 \log (3) (7+4 \log (9))\right ) \log (x)}{x+2}+64 x \log ^2(3)-\frac {128 \left (4-3 \log ^2(3)+\log ^2(9)+\log (81)\right ) \log (x)}{(x+2)^2}+32 \left (4-3 \log ^2(3)+\log ^2(9)+\log (81)\right ) \log (x)-32 \left (4-3 \log ^2(3)+\log ^2(9)+\log (81)\right ) \log (x+2)+64 \left (16-13 \log ^2(3)+2 \log (3) (7+4 \log (9))\right ) \log (x+2)-160 \log ^2(3) \log (x+2)+\frac {64 \left (4-3 \log ^2(3)+\log ^2(9)+\log (81)\right )}{x+2}-\frac {64 \log ^2(3)}{x+2}+8 x (47+8 \log (3) (5+\log (3))) \log (x)-8 x (47+8 \log (3) (5+\log (3)))+320 x \log (3)-4 \log (x)-768 \log (3) \log (x+2)-896 \log (x+2)-\frac {256 \log (3)}{x+2}\) |
Int[(-32 - 112*x - 152*x^2 - 100*x^3 - 96*x^4 - 164*x^5 - 128*x^6 - 32*x^7 + (128*x^3 + 320*x^4 + 256*x^5 + 64*x^6)*Log[3] + (-64*x^2 - 160*x^3 - 12 8*x^4 - 32*x^5)*Log[3]^2 + (-64*x - 160*x^2 - 144*x^3 - 312*x^4 - 712*x^5 - 576*x^6 - 128*x^7 + (384*x^3 + 1088*x^4 + 896*x^5 + 192*x^6)*Log[3] + (- 128*x^2 - 384*x^3 - 320*x^4 - 64*x^5)*Log[3]^2)*Log[x])/(8*x + 12*x^2 + 6* x^3 + x^4),x]
376*x - 82*x^2 + (64*x^3)/3 + 152/(2 + x)^2 - (38*x^2)/(2 + x)^2 - 408/(2 + x) + 320*x*Log[3] - 64*x^2*Log[3] + (64*x^3*Log[3])/3 - (256*Log[3])/(2 + x) + 64*x*Log[3]^2 - 16*x^2*Log[3]^2 - (64*Log[3]^2)/(2 + x) - (64*x^3*( 1 + Log[3]))/3 + 2*x^2*(41 + 32*Log[3] + 8*Log[3]^2) - 8*x*(47 + 8*Log[3]* (5 + Log[3])) + (64*(4 - 3*Log[3]^2 + Log[9]^2 + Log[81]))/(2 + x) - 4*Log [x] - 32*x^4*Log[x] + 64*x^3*(1 + Log[3])*Log[x] - 4*x^2*(41 + 32*Log[3] + 8*Log[3]^2)*Log[x] + 8*x*(47 + 8*Log[3]*(5 + Log[3]))*Log[x] - (64*x*(16 - 13*Log[3]^2 + 2*Log[3]*(7 + 4*Log[9]))*Log[x])/(2 + x) + 32*(4 - 3*Log[3 ]^2 + Log[9]^2 + Log[81])*Log[x] - (128*(4 - 3*Log[3]^2 + Log[9]^2 + Log[8 1])*Log[x])/(2 + x)^2 - 896*Log[2 + x] - 768*Log[3]*Log[2 + x] - 160*Log[3 ]^2*Log[2 + x] + 64*(16 - 13*Log[3]^2 + 2*Log[3]*(7 + 4*Log[9]))*Log[2 + x ] - 32*(4 - 3*Log[3]^2 + Log[9]^2 + Log[81])*Log[2 + x]
3.28.28.3.1 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])
Leaf count of result is larger than twice the leaf count of optimal. \(85\) vs. \(2(31)=62\).
Time = 0.30 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.77
| method | result | size |
| norman | \(\frac {-16 \ln \left (x \right )+\left (-64+64 \ln \left (3\right )\right ) x^{5} \ln \left (x \right )-48 x \ln \left (x \right )+\left (-52-32 \ln \left (3\right )^{2}\right ) x^{2} \ln \left (x \right )+\left (-36+128 \ln \left (3\right )-32 \ln \left (3\right )^{2}\right ) x^{4} \ln \left (x \right )+\left (-24+64 \ln \left (3\right )-64 \ln \left (3\right )^{2}\right ) x^{3} \ln \left (x \right )-32 x^{6} \ln \left (x \right )}{\left (2+x \right )^{2}}\) | \(86\) |
| parallelrisch | \(-\frac {32 \ln \left (x \right ) \ln \left (3\right )^{2} x^{4}-64 \ln \left (x \right ) \ln \left (3\right ) x^{5}+32 x^{6} \ln \left (x \right )+64 x^{3} \ln \left (3\right )^{2} \ln \left (x \right )-128 \ln \left (x \right ) \ln \left (3\right ) x^{4}+64 x^{5} \ln \left (x \right )+32 x^{2} \ln \left (3\right )^{2} \ln \left (x \right )-64 \ln \left (3\right ) \ln \left (x \right ) x^{3}+36 x^{4} \ln \left (x \right )+24 x^{3} \ln \left (x \right )+52 x^{2} \ln \left (x \right )+48 x \ln \left (x \right )+16 \ln \left (x \right )}{x^{2}+4 x +4}\) | \(118\) |
| risch | \(-\frac {4 \left (8 x^{4} \ln \left (3\right )^{2}-16 x^{5} \ln \left (3\right )+8 x^{6}+16 x^{3} \ln \left (3\right )^{2}-32 x^{4} \ln \left (3\right )+16 x^{5}-32 x^{2} \ln \left (3\right )^{2}-16 x^{3} \ln \left (3\right )+9 x^{4}-160 x \ln \left (3\right )^{2}-192 x^{2} \ln \left (3\right )+6 x^{3}-160 \ln \left (3\right )^{2}-768 x \ln \left (3\right )-212 x^{2}-768 \ln \left (3\right )-888 x -896\right ) \ln \left (x \right )}{x^{2}+4 x +4}-160 \ln \left (3\right )^{2} \ln \left (x \right )-768 \ln \left (3\right ) \ln \left (x \right )-900 \ln \left (x \right )\) | \(141\) |
| default | \(32 \ln \left (3\right )^{2} \left (2 x \ln \left (x \right )-x^{2} \ln \left (x \right )-\frac {6 \ln \left (x \right ) x}{2+x}+\frac {\ln \left (x \right ) x \left (4+x \right )}{\left (2+x \right )^{2}}\right )+64 \ln \left (3\right ) \left (x^{3} \ln \left (x \right )-2 x^{2} \ln \left (x \right )+5 x \ln \left (x \right )-\frac {14 \ln \left (x \right ) x}{2+x}+\frac {2 \ln \left (x \right ) x \left (4+x \right )}{\left (2+x \right )^{2}}\right )-4 \ln \left (x \right )-32 x^{4} \ln \left (x \right )+64 x^{3} \ln \left (x \right )-164 x^{2} \ln \left (x \right )+376 x \ln \left (x \right )-\frac {1024 \ln \left (x \right ) x}{2+x}+\frac {128 \ln \left (x \right ) x \left (4+x \right )}{\left (2+x \right )^{2}}\) | \(142\) |
| parts | \(-16 x^{2} \ln \left (3\right )^{2}+\frac {64 x^{3} \ln \left (3\right )}{3}+64 x \ln \left (3\right )^{2}-64 x^{2} \ln \left (3\right )+\frac {64 x^{3}}{3}+320 x \ln \left (3\right )-82 x^{2}-4 \ln \left (x \right )+\frac {-64 \ln \left (3\right )^{2}-256 \ln \left (3\right )-256}{2+x}-4 \left (40 \ln \left (3\right )^{2}+192 \ln \left (3\right )+224\right ) \ln \left (2+x \right )-32 x^{4} \ln \left (x \right )-8 \left (\frac {x^{3} \ln \left (x \right )}{3}-\frac {x^{3}}{9}\right ) \left (-24-24 \ln \left (3\right )\right )-8 \left (\frac {x^{2} \ln \left (x \right )}{2}-\frac {x^{2}}{4}\right ) \left (41+32 \ln \left (3\right )+8 \ln \left (3\right )^{2}\right )+376 x \ln \left (x \right )+320 \left (x \ln \left (x \right )-x \right ) \ln \left (3\right )+64 \left (x \ln \left (x \right )-x \right ) \ln \left (3\right )^{2}-8 \left (-\frac {\ln \left (2+x \right )}{2}+\frac {x \ln \left (x \right )}{4+2 x}\right ) \left (256+224 \ln \left (3\right )+48 \ln \left (3\right )^{2}\right )-8 \left (\frac {1}{4 x +8}-\frac {\ln \left (2+x \right )}{8}+\frac {\ln \left (x \right ) x \left (4+x \right )}{8 \left (2+x \right )^{2}}\right ) \left (-32 \ln \left (3\right )^{2}-128 \ln \left (3\right )-128\right )\) | \(246\) |
int((((-64*x^5-320*x^4-384*x^3-128*x^2)*ln(3)^2+(192*x^6+896*x^5+1088*x^4+ 384*x^3)*ln(3)-128*x^7-576*x^6-712*x^5-312*x^4-144*x^3-160*x^2-64*x)*ln(x) +(-32*x^5-128*x^4-160*x^3-64*x^2)*ln(3)^2+(64*x^6+256*x^5+320*x^4+128*x^3) *ln(3)-32*x^7-128*x^6-164*x^5-96*x^4-100*x^3-152*x^2-112*x-32)/(x^4+6*x^3+ 12*x^2+8*x),x,method=_RETURNVERBOSE)
(-16*ln(x)+(-64+64*ln(3))*x^5*ln(x)-48*x*ln(x)+(-52-32*ln(3)^2)*x^2*ln(x)+ (-36+128*ln(3)-32*ln(3)^2)*x^4*ln(x)+(-24+64*ln(3)-64*ln(3)^2)*x^3*ln(x)-3 2*x^6*ln(x))/(2+x)^2
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (31) = 62\).
Time = 0.30 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.52 \[ \int \frac {-32-112 x-152 x^2-100 x^3-96 x^4-164 x^5-128 x^6-32 x^7+\left (128 x^3+320 x^4+256 x^5+64 x^6\right ) \log (3)+\left (-64 x^2-160 x^3-128 x^4-32 x^5\right ) \log ^2(3)+\left (-64 x-160 x^2-144 x^3-312 x^4-712 x^5-576 x^6-128 x^7+\left (384 x^3+1088 x^4+896 x^5+192 x^6\right ) \log (3)+\left (-128 x^2-384 x^3-320 x^4-64 x^5\right ) \log ^2(3)\right ) \log (x)}{8 x+12 x^2+6 x^3+x^4} \, dx=-\frac {4 \, {\left (8 \, x^{6} + 16 \, x^{5} + 9 \, x^{4} + 6 \, x^{3} + 8 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} \log \left (3\right )^{2} + 13 \, x^{2} - 16 \, {\left (x^{5} + 2 \, x^{4} + x^{3}\right )} \log \left (3\right ) + 12 \, x + 4\right )} \log \left (x\right )}{x^{2} + 4 \, x + 4} \]
integrate((((-64*x^5-320*x^4-384*x^3-128*x^2)*log(3)^2+(192*x^6+896*x^5+10 88*x^4+384*x^3)*log(3)-128*x^7-576*x^6-712*x^5-312*x^4-144*x^3-160*x^2-64* x)*log(x)+(-32*x^5-128*x^4-160*x^3-64*x^2)*log(3)^2+(64*x^6+256*x^5+320*x^ 4+128*x^3)*log(3)-32*x^7-128*x^6-164*x^5-96*x^4-100*x^3-152*x^2-112*x-32)/ (x^4+6*x^3+12*x^2+8*x),x, algorithm=\
-4*(8*x^6 + 16*x^5 + 9*x^4 + 6*x^3 + 8*(x^4 + 2*x^3 + x^2)*log(3)^2 + 13*x ^2 - 16*(x^5 + 2*x^4 + x^3)*log(3) + 12*x + 4)*log(x)/(x^2 + 4*x + 4)
Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (27) = 54\).
Time = 0.27 (sec) , antiderivative size = 151, normalized size of antiderivative = 4.87 \[ \int \frac {-32-112 x-152 x^2-100 x^3-96 x^4-164 x^5-128 x^6-32 x^7+\left (128 x^3+320 x^4+256 x^5+64 x^6\right ) \log (3)+\left (-64 x^2-160 x^3-128 x^4-32 x^5\right ) \log ^2(3)+\left (-64 x-160 x^2-144 x^3-312 x^4-712 x^5-576 x^6-128 x^7+\left (384 x^3+1088 x^4+896 x^5+192 x^6\right ) \log (3)+\left (-128 x^2-384 x^3-320 x^4-64 x^5\right ) \log ^2(3)\right ) \log (x)}{8 x+12 x^2+6 x^3+x^4} \, dx=\left (-900 - 768 \log {\left (3 \right )} - 160 \log {\left (3 \right )}^{2}\right ) \log {\left (x \right )} + \frac {\left (- 32 x^{6} - 64 x^{5} + 64 x^{5} \log {\left (3 \right )} - 32 x^{4} \log {\left (3 \right )}^{2} - 36 x^{4} + 128 x^{4} \log {\left (3 \right )} - 64 x^{3} \log {\left (3 \right )}^{2} - 24 x^{3} + 64 x^{3} \log {\left (3 \right )} + 128 x^{2} \log {\left (3 \right )}^{2} + 768 x^{2} \log {\left (3 \right )} + 848 x^{2} + 640 x \log {\left (3 \right )}^{2} + 3072 x \log {\left (3 \right )} + 3552 x + 640 \log {\left (3 \right )}^{2} + 3072 \log {\left (3 \right )} + 3584\right ) \log {\left (x \right )}}{x^{2} + 4 x + 4} \]
integrate((((-64*x**5-320*x**4-384*x**3-128*x**2)*ln(3)**2+(192*x**6+896*x **5+1088*x**4+384*x**3)*ln(3)-128*x**7-576*x**6-712*x**5-312*x**4-144*x**3 -160*x**2-64*x)*ln(x)+(-32*x**5-128*x**4-160*x**3-64*x**2)*ln(3)**2+(64*x* *6+256*x**5+320*x**4+128*x**3)*ln(3)-32*x**7-128*x**6-164*x**5-96*x**4-100 *x**3-152*x**2-112*x-32)/(x**4+6*x**3+12*x**2+8*x),x)
(-900 - 768*log(3) - 160*log(3)**2)*log(x) + (-32*x**6 - 64*x**5 + 64*x**5 *log(3) - 32*x**4*log(3)**2 - 36*x**4 + 128*x**4*log(3) - 64*x**3*log(3)** 2 - 24*x**3 + 64*x**3*log(3) + 128*x**2*log(3)**2 + 768*x**2*log(3) + 848* x**2 + 640*x*log(3)**2 + 3072*x*log(3) + 3552*x + 640*log(3)**2 + 3072*log (3) + 3584)*log(x)/(x**2 + 4*x + 4)
Leaf count of result is larger than twice the leaf count of optimal. 638 vs. \(2 (31) = 62\).
Time = 0.32 (sec) , antiderivative size = 638, normalized size of antiderivative = 20.58 \[ \int \frac {-32-112 x-152 x^2-100 x^3-96 x^4-164 x^5-128 x^6-32 x^7+\left (128 x^3+320 x^4+256 x^5+64 x^6\right ) \log (3)+\left (-64 x^2-160 x^3-128 x^4-32 x^5\right ) \log ^2(3)+\left (-64 x-160 x^2-144 x^3-312 x^4-712 x^5-576 x^6-128 x^7+\left (384 x^3+1088 x^4+896 x^5+192 x^6\right ) \log (3)+\left (-128 x^2-384 x^3-320 x^4-64 x^5\right ) \log ^2(3)\right ) \log (x)}{8 x+12 x^2+6 x^3+x^4} \, dx =\text {Too large to display} \]
integrate((((-64*x^5-320*x^4-384*x^3-128*x^2)*log(3)^2+(192*x^6+896*x^5+10 88*x^4+384*x^3)*log(3)-128*x^7-576*x^6-712*x^5-312*x^4-144*x^3-160*x^2-64* x)*log(x)+(-32*x^5-128*x^4-160*x^3-64*x^2)*log(3)^2+(64*x^6+256*x^5+320*x^ 4+128*x^3)*log(3)-32*x^7-128*x^6-164*x^5-96*x^4-100*x^3-152*x^2-112*x-32)/ (x^4+6*x^3+12*x^2+8*x),x, algorithm=\
-8*x^4 + 64/3*x^3 - 16*(x^2 - 12*x + 16*(4*x + 7)/(x^2 + 4*x + 4) + 48*log (x + 2))*log(3)^2 - 128*(x - 4*(3*x + 5)/(x^2 + 4*x + 4) - 6*log(x + 2))*l og(3)^2 + 32*(4*(x + 1)*log(x)/(x^2 + 4*x + 4) + 2/(x + 2) + log(x + 2) - log(x))*log(3)^2 - 160*(2*(2*x + 3)/(x^2 + 4*x + 4) + log(x + 2))*log(3)^2 - 82*x^2 + 64/3*(x^3 - 9*x^2 + 72*x - 48*(5*x + 9)/(x^2 + 4*x + 4) - 240* log(x + 2))*log(3) + 128*(x^2 - 12*x + 16*(4*x + 7)/(x^2 + 4*x + 4) + 48*l og(x + 2))*log(3) + 320*(x - 4*(3*x + 5)/(x^2 + 4*x + 4) - 6*log(x + 2))*l og(3) + 128*(2*(2*x + 3)/(x^2 + 4*x + 4) + log(x + 2))*log(3) + 64*(x + 1) *log(3)^2/(x^2 + 4*x + 4) + 16*(8*log(3)^2 + 48*log(3) + 53)*log(x + 2) + 376*x + 160*(x + 1)*log(x)/(x^2 + 4*x + 4) + 2/3*(12*x^6 - 16*x^5*(2*log(3 ) - 1) + (24*log(3)^2 - 32*log(3) + 43)*x^4 - 8*x^3*(28*log(3) + 25) - 12* (24*log(3)^2 + 128*log(3) + 147)*x^2 - 48*(8*log(3)^2 + 32*log(3) + 41)*x - 6*(8*x^6 - 16*x^5*(log(3) - 1) + (8*log(3)^2 - 32*log(3) + 9)*x^4 + 2*(8 *log(3)^2 - 8*log(3) + 3)*x^3)*log(x) + 768*log(3) + 576)/(x^2 + 4*x + 4) - 1024*(6*x + 11)/(x^2 + 4*x + 4) + 2048*(5*x + 9)/(x^2 + 4*x + 4) - 1312* (4*x + 7)/(x^2 + 4*x + 4) + 384*(3*x + 5)/(x^2 + 4*x + 4) - 200*(2*x + 3)/ (x^2 + 4*x + 4) - 8*(x + 3)/(x^2 + 4*x + 4) + 152*(x + 1)/(x^2 + 4*x + 4) + 32*log(x)/(x^2 + 4*x + 4) + 56/(x^2 + 4*x + 4) + 64/(x + 2) - 848*log(x + 2) - 52*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (31) = 62\).
Time = 0.27 (sec) , antiderivative size = 106, normalized size of antiderivative = 3.42 \[ \int \frac {-32-112 x-152 x^2-100 x^3-96 x^4-164 x^5-128 x^6-32 x^7+\left (128 x^3+320 x^4+256 x^5+64 x^6\right ) \log (3)+\left (-64 x^2-160 x^3-128 x^4-32 x^5\right ) \log ^2(3)+\left (-64 x-160 x^2-144 x^3-312 x^4-712 x^5-576 x^6-128 x^7+\left (384 x^3+1088 x^4+896 x^5+192 x^6\right ) \log (3)+\left (-128 x^2-384 x^3-320 x^4-64 x^5\right ) \log ^2(3)\right ) \log (x)}{8 x+12 x^2+6 x^3+x^4} \, dx=-4 \, {\left (8 \, x^{4} - 16 \, x^{3} {\left (\log \left (3\right ) + 1\right )} + {\left (8 \, \log \left (3\right )^{2} + 32 \, \log \left (3\right ) + 41\right )} x^{2} - 2 \, {\left (8 \, \log \left (3\right )^{2} + 40 \, \log \left (3\right ) + 47\right )} x - \frac {32 \, {\left (3 \, x \log \left (3\right )^{2} + 14 \, x \log \left (3\right ) + 5 \, \log \left (3\right )^{2} + 16 \, x + 24 \, \log \left (3\right ) + 28\right )}}{x^{2} + 4 \, x + 4}\right )} \log \left (x\right ) - 4 \, {\left (40 \, \log \left (3\right )^{2} + 192 \, \log \left (3\right ) + 225\right )} \log \left (x\right ) \]
integrate((((-64*x^5-320*x^4-384*x^3-128*x^2)*log(3)^2+(192*x^6+896*x^5+10 88*x^4+384*x^3)*log(3)-128*x^7-576*x^6-712*x^5-312*x^4-144*x^3-160*x^2-64* x)*log(x)+(-32*x^5-128*x^4-160*x^3-64*x^2)*log(3)^2+(64*x^6+256*x^5+320*x^ 4+128*x^3)*log(3)-32*x^7-128*x^6-164*x^5-96*x^4-100*x^3-152*x^2-112*x-32)/ (x^4+6*x^3+12*x^2+8*x),x, algorithm=\
-4*(8*x^4 - 16*x^3*(log(3) + 1) + (8*log(3)^2 + 32*log(3) + 41)*x^2 - 2*(8 *log(3)^2 + 40*log(3) + 47)*x - 32*(3*x*log(3)^2 + 14*x*log(3) + 5*log(3)^ 2 + 16*x + 24*log(3) + 28)/(x^2 + 4*x + 4))*log(x) - 4*(40*log(3)^2 + 192* log(3) + 225)*log(x)
Time = 11.55 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.39 \[ \int \frac {-32-112 x-152 x^2-100 x^3-96 x^4-164 x^5-128 x^6-32 x^7+\left (128 x^3+320 x^4+256 x^5+64 x^6\right ) \log (3)+\left (-64 x^2-160 x^3-128 x^4-32 x^5\right ) \log ^2(3)+\left (-64 x-160 x^2-144 x^3-312 x^4-712 x^5-576 x^6-128 x^7+\left (384 x^3+1088 x^4+896 x^5+192 x^6\right ) \log (3)+\left (-128 x^2-384 x^3-320 x^4-64 x^5\right ) \log ^2(3)\right ) \log (x)}{8 x+12 x^2+6 x^3+x^4} \, dx=-\frac {4\,\ln \left (x\right )\,{\left (x+1\right )}^2\,\left (4\,x+8\,x^2\,{\ln \left (3\right )}^2-16\,x^3\,\ln \left (3\right )+x^2+8\,x^4+4\right )}{{\left (x+2\right )}^2} \]
int(-(112*x - log(3)*(128*x^3 + 320*x^4 + 256*x^5 + 64*x^6) + log(x)*(64*x - log(3)*(384*x^3 + 1088*x^4 + 896*x^5 + 192*x^6) + log(3)^2*(128*x^2 + 3 84*x^3 + 320*x^4 + 64*x^5) + 160*x^2 + 144*x^3 + 312*x^4 + 712*x^5 + 576*x ^6 + 128*x^7) + log(3)^2*(64*x^2 + 160*x^3 + 128*x^4 + 32*x^5) + 152*x^2 + 100*x^3 + 96*x^4 + 164*x^5 + 128*x^6 + 32*x^7 + 32)/(8*x + 12*x^2 + 6*x^3 + x^4),x)