Integrand size = 153, antiderivative size = 29 \[ \int \frac {-16-16 x+60 x^2+156 x^3-124 x^4-399 x^5+243 x^6+1050 x^7+912 x^8+333 x^9+45 x^{10}+\left (16+32 x+80 x^2-256 x^3-687 x^4+245 x^5+1650 x^6+1542 x^7+585 x^8+81 x^9\right ) \log (3 x)+\left (16 x-72 x^2-276 x^3+2 x^4+600 x^5+630 x^6+252 x^7+36 x^8\right ) \log ^2(3 x)}{8 x} \, dx=\left (1+\left (3-\frac {1}{x^2}\right ) \left (x+\frac {x^2}{2}\right )^2 (x+\log (3 x))\right )^2 \]
Leaf count is larger than twice the leaf count of optimal. \(114\) vs. \(2(29)=58\).
Time = 0.08 (sec) , antiderivative size = 114, normalized size of antiderivative = 3.93 \[ \int \frac {-16-16 x+60 x^2+156 x^3-124 x^4-399 x^5+243 x^6+1050 x^7+912 x^8+333 x^9+45 x^{10}+\left (16+32 x+80 x^2-256 x^3-687 x^4+245 x^5+1650 x^6+1542 x^7+585 x^8+81 x^9\right ) \log (3 x)+\left (16 x-72 x^2-276 x^3+2 x^4+600 x^5+630 x^6+252 x^7+36 x^8\right ) \log ^2(3 x)}{8 x} \, dx=\frac {1}{16} \left (x (2+x)^2 \left (-8+4 x+28 x^2-23 x^3-24 x^4+30 x^5+36 x^6+9 x^7\right )-32 \log (x)+2 x^2 \left (76-24 x-172 x^2+x^3+240 x^4+210 x^5+72 x^6+9 x^7\right ) \log (3 x)+(2+x)^4 \left (1-3 x^2\right )^2 \log ^2(3 x)\right ) \]
Integrate[(-16 - 16*x + 60*x^2 + 156*x^3 - 124*x^4 - 399*x^5 + 243*x^6 + 1 050*x^7 + 912*x^8 + 333*x^9 + 45*x^10 + (16 + 32*x + 80*x^2 - 256*x^3 - 68 7*x^4 + 245*x^5 + 1650*x^6 + 1542*x^7 + 585*x^8 + 81*x^9)*Log[3*x] + (16*x - 72*x^2 - 276*x^3 + 2*x^4 + 600*x^5 + 630*x^6 + 252*x^7 + 36*x^8)*Log[3* x]^2)/(8*x),x]
(x*(2 + x)^2*(-8 + 4*x + 28*x^2 - 23*x^3 - 24*x^4 + 30*x^5 + 36*x^6 + 9*x^ 7) - 32*Log[x] + 2*x^2*(76 - 24*x - 172*x^2 + x^3 + 240*x^4 + 210*x^5 + 72 *x^6 + 9*x^7)*Log[3*x] + (2 + x)^4*(1 - 3*x^2)^2*Log[3*x]^2)/16
Leaf count is larger than twice the leaf count of optimal. \(230\) vs. \(2(29)=58\).
Time = 0.77 (sec) , antiderivative size = 230, normalized size of antiderivative = 7.93, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {27, 25, 2010, 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.
| \(\displaystyle \int \frac {45 x^{10}+333 x^9+912 x^8+1050 x^7+243 x^6-399 x^5-124 x^4+156 x^3+60 x^2+\left (36 x^8+252 x^7+630 x^6+600 x^5+2 x^4-276 x^3-72 x^2+16 x\right ) \log ^2(3 x)+\left (81 x^9+585 x^8+1542 x^7+1650 x^6+245 x^5-687 x^4-256 x^3+80 x^2+32 x+16\right ) \log (3 x)-16 x-16}{8 x} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \int -\frac {-45 x^{10}-333 x^9-912 x^8-1050 x^7-243 x^6+399 x^5+124 x^4-156 x^3-60 x^2+16 x-2 \left (18 x^8+126 x^7+315 x^6+300 x^5+x^4-138 x^3-36 x^2+8 x\right ) \log ^2(3 x)-\left (81 x^9+585 x^8+1542 x^7+1650 x^6+245 x^5-687 x^4-256 x^3+80 x^2+32 x+16\right ) \log (3 x)+16}{x}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{8} \int \frac {-45 x^{10}-333 x^9-912 x^8-1050 x^7-243 x^6+399 x^5+124 x^4-156 x^3-60 x^2+16 x-2 \left (18 x^8+126 x^7+315 x^6+300 x^5+x^4-138 x^3-36 x^2+8 x\right ) \log ^2(3 x)-\left (81 x^9+585 x^8+1542 x^7+1650 x^6+245 x^5-687 x^4-256 x^3+80 x^2+32 x+16\right ) \log (3 x)+16}{x}dx\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle -\frac {1}{8} \int \left (-2 \left (3 x^2-1\right ) \left (6 x^2+6 x-1\right ) \log ^2(3 x) (x+2)^3-\frac {(x+1) \left (81 x^7+342 x^6+354 x^5-96 x^4-175 x^3+30 x^2+4 x+8\right ) \log (3 x) (x+2)}{x}+\frac {-45 x^{10}-333 x^9-912 x^8-1050 x^7-243 x^6+399 x^5+124 x^4-156 x^3-60 x^2+16 x+16}{x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{8} \left (\frac {9 x^{10}}{2}+36 x^9+9 x^9 \log (3 x)+105 x^8+\frac {9}{2} x^8 \log ^2(3 x)+72 x^8 \log (3 x)+120 x^7+36 x^7 \log ^2(3 x)+210 x^7 \log (3 x)+\frac {x^6}{2}+105 x^6 \log ^2(3 x)+240 x^6 \log (3 x)-80 x^5+120 x^5 \log ^2(3 x)+x^5 \log (3 x)+12 x^4+\frac {1}{2} x^4 \log ^2(3 x)-172 x^4 \log (3 x)+60 x^3-92 x^3 \log ^2(3 x)-24 x^3 \log (3 x)-8 x^2-36 x^2 \log ^2(3 x)+76 x^2 \log (3 x)-16 x+16 x \log ^2(3 x)+8 \log ^2(3 x)-16 \log (x)\right )\) |
Int[(-16 - 16*x + 60*x^2 + 156*x^3 - 124*x^4 - 399*x^5 + 243*x^6 + 1050*x^ 7 + 912*x^8 + 333*x^9 + 45*x^10 + (16 + 32*x + 80*x^2 - 256*x^3 - 687*x^4 + 245*x^5 + 1650*x^6 + 1542*x^7 + 585*x^8 + 81*x^9)*Log[3*x] + (16*x - 72* x^2 - 276*x^3 + 2*x^4 + 600*x^5 + 630*x^6 + 252*x^7 + 36*x^8)*Log[3*x]^2)/ (8*x),x]
(-16*x - 8*x^2 + 60*x^3 + 12*x^4 - 80*x^5 + x^6/2 + 120*x^7 + 105*x^8 + 36 *x^9 + (9*x^10)/2 - 16*Log[x] + 76*x^2*Log[3*x] - 24*x^3*Log[3*x] - 172*x^ 4*Log[3*x] + x^5*Log[3*x] + 240*x^6*Log[3*x] + 210*x^7*Log[3*x] + 72*x^8*L og[3*x] + 9*x^9*Log[3*x] + 8*Log[3*x]^2 + 16*x*Log[3*x]^2 - 36*x^2*Log[3*x ]^2 - 92*x^3*Log[3*x]^2 + (x^4*Log[3*x]^2)/2 + 120*x^5*Log[3*x]^2 + 105*x^ 6*Log[3*x]^2 + 36*x^7*Log[3*x]^2 + (9*x^8*Log[3*x]^2)/2)/8
3.4.63.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Leaf count of result is larger than twice the leaf count of optimal. \(147\) vs. \(2(27)=54\).
Time = 0.07 (sec) , antiderivative size = 148, normalized size of antiderivative = 5.10
| method | result | size |
| risch | \(\frac {\left (\frac {9}{2} x^{8}+36 x^{7}+105 x^{6}+120 x^{5}+\frac {1}{2} x^{4}-92 x^{3}-36 x^{2}+16 x +8\right ) \ln \left (3 x \right )^{2}}{8}+\frac {\left (9 x^{9}+72 x^{8}+210 x^{7}+240 x^{6}+x^{5}-172 x^{4}-24 x^{3}+76 x^{2}\right ) \ln \left (3 x \right )}{8}+\frac {9 x^{10}}{16}+\frac {9 x^{9}}{2}+\frac {105 x^{8}}{8}+15 x^{7}+\frac {x^{6}}{16}-10 x^{5}+\frac {3 x^{4}}{2}+\frac {15 x^{3}}{2}-x^{2}-2 x +1-2 \ln \left (x \right )\) | \(148\) |
| parallelrisch | \(-2 x +\frac {9 \ln \left (3 x \right )^{2} x^{8}}{16}+\frac {9 \ln \left (3 x \right ) x^{9}}{8}+\frac {9 \ln \left (3 x \right )^{2} x^{7}}{2}+9 \ln \left (3 x \right ) x^{8}+\frac {105 \ln \left (3 x \right )^{2} x^{6}}{8}+\frac {105 \ln \left (3 x \right ) x^{7}}{4}+15 \ln \left (3 x \right )^{2} x^{5}+30 \ln \left (3 x \right ) x^{6}+\frac {\ln \left (3 x \right )^{2} x^{4}}{16}+\frac {19 x^{2} \ln \left (3 x \right )}{2}-\frac {23 x^{3} \ln \left (3 x \right )^{2}}{2}-\frac {9 x^{2} \ln \left (3 x \right )^{2}}{2}+\ln \left (3 x \right )^{2}+2 x \ln \left (3 x \right )^{2}+\frac {9 x^{10}}{16}+\frac {9 x^{9}}{2}+15 x^{7}+\frac {105 x^{8}}{8}+\frac {3 x^{4}}{2}+\frac {15 x^{3}}{2}-x^{2}+\frac {x^{6}}{16}-10 x^{5}-2 \ln \left (x \right )-3 x^{3} \ln \left (3 x \right )+\frac {\ln \left (3 x \right ) x^{5}}{8}-\frac {43 \ln \left (3 x \right ) x^{4}}{2}\) | \(218\) |
| parts | \(-2 x +\frac {9 \ln \left (3 x \right )^{2} x^{8}}{16}+\frac {9 \ln \left (3 x \right ) x^{9}}{8}+\frac {9 \ln \left (3 x \right )^{2} x^{7}}{2}+9 \ln \left (3 x \right ) x^{8}+\frac {105 \ln \left (3 x \right )^{2} x^{6}}{8}+\frac {105 \ln \left (3 x \right ) x^{7}}{4}+15 \ln \left (3 x \right )^{2} x^{5}+30 \ln \left (3 x \right ) x^{6}+\frac {\ln \left (3 x \right )^{2} x^{4}}{16}+\frac {19 x^{2} \ln \left (3 x \right )}{2}-\frac {23 x^{3} \ln \left (3 x \right )^{2}}{2}-\frac {9 x^{2} \ln \left (3 x \right )^{2}}{2}+\ln \left (3 x \right )^{2}+2 x \ln \left (3 x \right )^{2}+\frac {9 x^{10}}{16}+\frac {9 x^{9}}{2}+15 x^{7}+\frac {105 x^{8}}{8}+\frac {3 x^{4}}{2}+\frac {15 x^{3}}{2}-x^{2}+\frac {x^{6}}{16}-10 x^{5}-2 \ln \left (x \right )-3 x^{3} \ln \left (3 x \right )+\frac {\ln \left (3 x \right ) x^{5}}{8}-\frac {43 \ln \left (3 x \right ) x^{4}}{2}\) | \(218\) |
| derivativedivides | \(-2 x +\frac {9 \ln \left (3 x \right )^{2} x^{8}}{16}+\frac {9 \ln \left (3 x \right ) x^{9}}{8}+\frac {9 \ln \left (3 x \right )^{2} x^{7}}{2}+9 \ln \left (3 x \right ) x^{8}+\frac {105 \ln \left (3 x \right )^{2} x^{6}}{8}+\frac {105 \ln \left (3 x \right ) x^{7}}{4}+15 \ln \left (3 x \right )^{2} x^{5}+30 \ln \left (3 x \right ) x^{6}+\frac {\ln \left (3 x \right )^{2} x^{4}}{16}+\frac {19 x^{2} \ln \left (3 x \right )}{2}-\frac {23 x^{3} \ln \left (3 x \right )^{2}}{2}-\frac {9 x^{2} \ln \left (3 x \right )^{2}}{2}-2 \ln \left (3 x \right )+\ln \left (3 x \right )^{2}+2 x \ln \left (3 x \right )^{2}+\frac {9 x^{10}}{16}+\frac {9 x^{9}}{2}+15 x^{7}+\frac {105 x^{8}}{8}+\frac {3 x^{4}}{2}+\frac {15 x^{3}}{2}-x^{2}+\frac {x^{6}}{16}-10 x^{5}-3 x^{3} \ln \left (3 x \right )+\frac {\ln \left (3 x \right ) x^{5}}{8}-\frac {43 \ln \left (3 x \right ) x^{4}}{2}\) | \(220\) |
| default | \(-2 x +\frac {9 \ln \left (3 x \right )^{2} x^{8}}{16}+\frac {9 \ln \left (3 x \right ) x^{9}}{8}+\frac {9 \ln \left (3 x \right )^{2} x^{7}}{2}+9 \ln \left (3 x \right ) x^{8}+\frac {105 \ln \left (3 x \right )^{2} x^{6}}{8}+\frac {105 \ln \left (3 x \right ) x^{7}}{4}+15 \ln \left (3 x \right )^{2} x^{5}+30 \ln \left (3 x \right ) x^{6}+\frac {\ln \left (3 x \right )^{2} x^{4}}{16}+\frac {19 x^{2} \ln \left (3 x \right )}{2}-\frac {23 x^{3} \ln \left (3 x \right )^{2}}{2}-\frac {9 x^{2} \ln \left (3 x \right )^{2}}{2}-2 \ln \left (3 x \right )+\ln \left (3 x \right )^{2}+2 x \ln \left (3 x \right )^{2}+\frac {9 x^{10}}{16}+\frac {9 x^{9}}{2}+15 x^{7}+\frac {105 x^{8}}{8}+\frac {3 x^{4}}{2}+\frac {15 x^{3}}{2}-x^{2}+\frac {x^{6}}{16}-10 x^{5}-3 x^{3} \ln \left (3 x \right )+\frac {\ln \left (3 x \right ) x^{5}}{8}-\frac {43 \ln \left (3 x \right ) x^{4}}{2}\) | \(220\) |
int(1/8*((36*x^8+252*x^7+630*x^6+600*x^5+2*x^4-276*x^3-72*x^2+16*x)*ln(3*x )^2+(81*x^9+585*x^8+1542*x^7+1650*x^6+245*x^5-687*x^4-256*x^3+80*x^2+32*x+ 16)*ln(3*x)+45*x^10+333*x^9+912*x^8+1050*x^7+243*x^6-399*x^5-124*x^4+156*x ^3+60*x^2-16*x-16)/x,x,method=_RETURNVERBOSE)
1/8*(9/2*x^8+36*x^7+105*x^6+120*x^5+1/2*x^4-92*x^3-36*x^2+16*x+8)*ln(3*x)^ 2+1/8*(9*x^9+72*x^8+210*x^7+240*x^6+x^5-172*x^4-24*x^3+76*x^2)*ln(3*x)+9/1 6*x^10+9/2*x^9+105/8*x^8+15*x^7+1/16*x^6-10*x^5+3/2*x^4+15/2*x^3-x^2-2*x+1 -2*ln(x)
Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (27) = 54\).
Time = 0.24 (sec) , antiderivative size = 141, normalized size of antiderivative = 4.86 \[ \int \frac {-16-16 x+60 x^2+156 x^3-124 x^4-399 x^5+243 x^6+1050 x^7+912 x^8+333 x^9+45 x^{10}+\left (16+32 x+80 x^2-256 x^3-687 x^4+245 x^5+1650 x^6+1542 x^7+585 x^8+81 x^9\right ) \log (3 x)+\left (16 x-72 x^2-276 x^3+2 x^4+600 x^5+630 x^6+252 x^7+36 x^8\right ) \log ^2(3 x)}{8 x} \, dx=\frac {9}{16} \, x^{10} + \frac {9}{2} \, x^{9} + \frac {105}{8} \, x^{8} + 15 \, x^{7} + \frac {1}{16} \, x^{6} - 10 \, x^{5} + \frac {3}{2} \, x^{4} + \frac {15}{2} \, x^{3} + \frac {1}{16} \, {\left (9 \, x^{8} + 72 \, x^{7} + 210 \, x^{6} + 240 \, x^{5} + x^{4} - 184 \, x^{3} - 72 \, x^{2} + 32 \, x + 16\right )} \log \left (3 \, x\right )^{2} - x^{2} + \frac {1}{8} \, {\left (9 \, x^{9} + 72 \, x^{8} + 210 \, x^{7} + 240 \, x^{6} + x^{5} - 172 \, x^{4} - 24 \, x^{3} + 76 \, x^{2} - 16\right )} \log \left (3 \, x\right ) - 2 \, x \]
integrate(1/8*((36*x^8+252*x^7+630*x^6+600*x^5+2*x^4-276*x^3-72*x^2+16*x)* log(3*x)^2+(81*x^9+585*x^8+1542*x^7+1650*x^6+245*x^5-687*x^4-256*x^3+80*x^ 2+32*x+16)*log(3*x)+45*x^10+333*x^9+912*x^8+1050*x^7+243*x^6-399*x^5-124*x ^4+156*x^3+60*x^2-16*x-16)/x,x, algorithm=\
9/16*x^10 + 9/2*x^9 + 105/8*x^8 + 15*x^7 + 1/16*x^6 - 10*x^5 + 3/2*x^4 + 1 5/2*x^3 + 1/16*(9*x^8 + 72*x^7 + 210*x^6 + 240*x^5 + x^4 - 184*x^3 - 72*x^ 2 + 32*x + 16)*log(3*x)^2 - x^2 + 1/8*(9*x^9 + 72*x^8 + 210*x^7 + 240*x^6 + x^5 - 172*x^4 - 24*x^3 + 76*x^2 - 16)*log(3*x) - 2*x
Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (24) = 48\).
Time = 0.16 (sec) , antiderivative size = 168, normalized size of antiderivative = 5.79 \[ \int \frac {-16-16 x+60 x^2+156 x^3-124 x^4-399 x^5+243 x^6+1050 x^7+912 x^8+333 x^9+45 x^{10}+\left (16+32 x+80 x^2-256 x^3-687 x^4+245 x^5+1650 x^6+1542 x^7+585 x^8+81 x^9\right ) \log (3 x)+\left (16 x-72 x^2-276 x^3+2 x^4+600 x^5+630 x^6+252 x^7+36 x^8\right ) \log ^2(3 x)}{8 x} \, dx=\frac {9 x^{10}}{16} + \frac {9 x^{9}}{2} + \frac {105 x^{8}}{8} + 15 x^{7} + \frac {x^{6}}{16} - 10 x^{5} + \frac {3 x^{4}}{2} + \frac {15 x^{3}}{2} - x^{2} - 2 x + \left (\frac {9 x^{9}}{8} + 9 x^{8} + \frac {105 x^{7}}{4} + 30 x^{6} + \frac {x^{5}}{8} - \frac {43 x^{4}}{2} - 3 x^{3} + \frac {19 x^{2}}{2}\right ) \log {\left (3 x \right )} + \left (\frac {9 x^{8}}{16} + \frac {9 x^{7}}{2} + \frac {105 x^{6}}{8} + 15 x^{5} + \frac {x^{4}}{16} - \frac {23 x^{3}}{2} - \frac {9 x^{2}}{2} + 2 x + 1\right ) \log {\left (3 x \right )}^{2} - 2 \log {\left (x \right )} \]
integrate(1/8*((36*x**8+252*x**7+630*x**6+600*x**5+2*x**4-276*x**3-72*x**2 +16*x)*ln(3*x)**2+(81*x**9+585*x**8+1542*x**7+1650*x**6+245*x**5-687*x**4- 256*x**3+80*x**2+32*x+16)*ln(3*x)+45*x**10+333*x**9+912*x**8+1050*x**7+243 *x**6-399*x**5-124*x**4+156*x**3+60*x**2-16*x-16)/x,x)
9*x**10/16 + 9*x**9/2 + 105*x**8/8 + 15*x**7 + x**6/16 - 10*x**5 + 3*x**4/ 2 + 15*x**3/2 - x**2 - 2*x + (9*x**9/8 + 9*x**8 + 105*x**7/4 + 30*x**6 + x **5/8 - 43*x**4/2 - 3*x**3 + 19*x**2/2)*log(3*x) + (9*x**8/16 + 9*x**7/2 + 105*x**6/8 + 15*x**5 + x**4/16 - 23*x**3/2 - 9*x**2/2 + 2*x + 1)*log(3*x) **2 - 2*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 302 vs. \(2 (27) = 54\).
Time = 0.18 (sec) , antiderivative size = 302, normalized size of antiderivative = 10.41 \[ \int \frac {-16-16 x+60 x^2+156 x^3-124 x^4-399 x^5+243 x^6+1050 x^7+912 x^8+333 x^9+45 x^{10}+\left (16+32 x+80 x^2-256 x^3-687 x^4+245 x^5+1650 x^6+1542 x^7+585 x^8+81 x^9\right ) \log (3 x)+\left (16 x-72 x^2-276 x^3+2 x^4+600 x^5+630 x^6+252 x^7+36 x^8\right ) \log ^2(3 x)}{8 x} \, dx=\frac {9}{16} \, x^{10} + \frac {9}{8} \, x^{9} \log \left (3 \, x\right ) + \frac {9}{512} \, {\left (32 \, \log \left (3 \, x\right )^{2} - 8 \, \log \left (3 \, x\right ) + 1\right )} x^{8} + \frac {9}{2} \, x^{9} + \frac {585}{64} \, x^{8} \log \left (3 \, x\right ) + \frac {9}{98} \, {\left (49 \, \log \left (3 \, x\right )^{2} - 14 \, \log \left (3 \, x\right ) + 2\right )} x^{7} + \frac {6711}{512} \, x^{8} + \frac {771}{28} \, x^{7} \log \left (3 \, x\right ) + \frac {35}{48} \, {\left (18 \, \log \left (3 \, x\right )^{2} - 6 \, \log \left (3 \, x\right ) + 1\right )} x^{6} + \frac {726}{49} \, x^{7} + \frac {275}{8} \, x^{6} \log \left (3 \, x\right ) + \frac {3}{5} \, {\left (25 \, \log \left (3 \, x\right )^{2} - 10 \, \log \left (3 \, x\right ) + 2\right )} x^{5} - \frac {2}{3} \, x^{6} + \frac {49}{8} \, x^{5} \log \left (3 \, x\right ) + \frac {1}{128} \, {\left (8 \, \log \left (3 \, x\right )^{2} - 4 \, \log \left (3 \, x\right ) + 1\right )} x^{4} - \frac {56}{5} \, x^{5} - \frac {687}{32} \, x^{4} \log \left (3 \, x\right ) - \frac {23}{18} \, {\left (9 \, \log \left (3 \, x\right )^{2} - 6 \, \log \left (3 \, x\right ) + 2\right )} x^{3} + \frac {191}{128} \, x^{4} - \frac {32}{3} \, x^{3} \log \left (3 \, x\right ) - \frac {9}{4} \, {\left (2 \, \log \left (3 \, x\right )^{2} - 2 \, \log \left (3 \, x\right ) + 1\right )} x^{2} + \frac {181}{18} \, x^{3} + 5 \, x^{2} \log \left (3 \, x\right ) + 2 \, {\left (\log \left (3 \, x\right )^{2} - 2 \, \log \left (3 \, x\right ) + 2\right )} x + \frac {5}{4} \, x^{2} + 4 \, x \log \left (3 \, x\right ) + \log \left (3 \, x\right )^{2} - 6 \, x - 2 \, \log \left (x\right ) \]
integrate(1/8*((36*x^8+252*x^7+630*x^6+600*x^5+2*x^4-276*x^3-72*x^2+16*x)* log(3*x)^2+(81*x^9+585*x^8+1542*x^7+1650*x^6+245*x^5-687*x^4-256*x^3+80*x^ 2+32*x+16)*log(3*x)+45*x^10+333*x^9+912*x^8+1050*x^7+243*x^6-399*x^5-124*x ^4+156*x^3+60*x^2-16*x-16)/x,x, algorithm=\
9/16*x^10 + 9/8*x^9*log(3*x) + 9/512*(32*log(3*x)^2 - 8*log(3*x) + 1)*x^8 + 9/2*x^9 + 585/64*x^8*log(3*x) + 9/98*(49*log(3*x)^2 - 14*log(3*x) + 2)*x ^7 + 6711/512*x^8 + 771/28*x^7*log(3*x) + 35/48*(18*log(3*x)^2 - 6*log(3*x ) + 1)*x^6 + 726/49*x^7 + 275/8*x^6*log(3*x) + 3/5*(25*log(3*x)^2 - 10*log (3*x) + 2)*x^5 - 2/3*x^6 + 49/8*x^5*log(3*x) + 1/128*(8*log(3*x)^2 - 4*log (3*x) + 1)*x^4 - 56/5*x^5 - 687/32*x^4*log(3*x) - 23/18*(9*log(3*x)^2 - 6* log(3*x) + 2)*x^3 + 191/128*x^4 - 32/3*x^3*log(3*x) - 9/4*(2*log(3*x)^2 - 2*log(3*x) + 1)*x^2 + 181/18*x^3 + 5*x^2*log(3*x) + 2*(log(3*x)^2 - 2*log( 3*x) + 2)*x + 5/4*x^2 + 4*x*log(3*x) + log(3*x)^2 - 6*x - 2*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 144 vs. \(2 (27) = 54\).
Time = 0.28 (sec) , antiderivative size = 144, normalized size of antiderivative = 4.97 \[ \int \frac {-16-16 x+60 x^2+156 x^3-124 x^4-399 x^5+243 x^6+1050 x^7+912 x^8+333 x^9+45 x^{10}+\left (16+32 x+80 x^2-256 x^3-687 x^4+245 x^5+1650 x^6+1542 x^7+585 x^8+81 x^9\right ) \log (3 x)+\left (16 x-72 x^2-276 x^3+2 x^4+600 x^5+630 x^6+252 x^7+36 x^8\right ) \log ^2(3 x)}{8 x} \, dx=\frac {9}{16} \, x^{10} + \frac {9}{2} \, x^{9} + \frac {105}{8} \, x^{8} + 15 \, x^{7} + \frac {1}{16} \, x^{6} - 10 \, x^{5} + \frac {3}{2} \, x^{4} + \frac {15}{2} \, x^{3} + \frac {1}{16} \, {\left (9 \, x^{8} + 72 \, x^{7} + 210 \, x^{6} + 240 \, x^{5} + x^{4} - 184 \, x^{3} - 72 \, x^{2} + 32 \, x + 16\right )} \log \left (3 \, x\right )^{2} - x^{2} + \frac {1}{8} \, {\left (9 \, x^{9} + 72 \, x^{8} + 210 \, x^{7} + 240 \, x^{6} + x^{5} - 172 \, x^{4} - 24 \, x^{3} + 76 \, x^{2}\right )} \log \left (3 \, x\right ) - 2 \, x - 2 \, \log \left (x\right ) \]
integrate(1/8*((36*x^8+252*x^7+630*x^6+600*x^5+2*x^4-276*x^3-72*x^2+16*x)* log(3*x)^2+(81*x^9+585*x^8+1542*x^7+1650*x^6+245*x^5-687*x^4-256*x^3+80*x^ 2+32*x+16)*log(3*x)+45*x^10+333*x^9+912*x^8+1050*x^7+243*x^6-399*x^5-124*x ^4+156*x^3+60*x^2-16*x-16)/x,x, algorithm=\
9/16*x^10 + 9/2*x^9 + 105/8*x^8 + 15*x^7 + 1/16*x^6 - 10*x^5 + 3/2*x^4 + 1 5/2*x^3 + 1/16*(9*x^8 + 72*x^7 + 210*x^6 + 240*x^5 + x^4 - 184*x^3 - 72*x^ 2 + 32*x + 16)*log(3*x)^2 - x^2 + 1/8*(9*x^9 + 72*x^8 + 210*x^7 + 240*x^6 + x^5 - 172*x^4 - 24*x^3 + 76*x^2)*log(3*x) - 2*x - 2*log(x)
Time = 9.10 (sec) , antiderivative size = 217, normalized size of antiderivative = 7.48 \[ \int \frac {-16-16 x+60 x^2+156 x^3-124 x^4-399 x^5+243 x^6+1050 x^7+912 x^8+333 x^9+45 x^{10}+\left (16+32 x+80 x^2-256 x^3-687 x^4+245 x^5+1650 x^6+1542 x^7+585 x^8+81 x^9\right ) \log (3 x)+\left (16 x-72 x^2-276 x^3+2 x^4+600 x^5+630 x^6+252 x^7+36 x^8\right ) \log ^2(3 x)}{8 x} \, dx=2\,x\,{\ln \left (3\,x\right )}^2-2\,\ln \left (x\right )-2\,x+\frac {19\,x^2\,\ln \left (3\,x\right )}{2}-3\,x^3\,\ln \left (3\,x\right )-\frac {43\,x^4\,\ln \left (3\,x\right )}{2}+\frac {x^5\,\ln \left (3\,x\right )}{8}+30\,x^6\,\ln \left (3\,x\right )+\frac {105\,x^7\,\ln \left (3\,x\right )}{4}+9\,x^8\,\ln \left (3\,x\right )+\frac {9\,x^9\,\ln \left (3\,x\right )}{8}+{\ln \left (3\,x\right )}^2-x^2+\frac {15\,x^3}{2}+\frac {3\,x^4}{2}-10\,x^5+\frac {x^6}{16}+15\,x^7+\frac {105\,x^8}{8}+\frac {9\,x^9}{2}+\frac {9\,x^{10}}{16}-\frac {9\,x^2\,{\ln \left (3\,x\right )}^2}{2}-\frac {23\,x^3\,{\ln \left (3\,x\right )}^2}{2}+\frac {x^4\,{\ln \left (3\,x\right )}^2}{16}+15\,x^5\,{\ln \left (3\,x\right )}^2+\frac {105\,x^6\,{\ln \left (3\,x\right )}^2}{8}+\frac {9\,x^7\,{\ln \left (3\,x\right )}^2}{2}+\frac {9\,x^8\,{\ln \left (3\,x\right )}^2}{16} \]
int(((log(3*x)*(32*x + 80*x^2 - 256*x^3 - 687*x^4 + 245*x^5 + 1650*x^6 + 1 542*x^7 + 585*x^8 + 81*x^9 + 16))/8 - 2*x + (log(3*x)^2*(16*x - 72*x^2 - 2 76*x^3 + 2*x^4 + 600*x^5 + 630*x^6 + 252*x^7 + 36*x^8))/8 + (15*x^2)/2 + ( 39*x^3)/2 - (31*x^4)/2 - (399*x^5)/8 + (243*x^6)/8 + (525*x^7)/4 + 114*x^8 + (333*x^9)/8 + (45*x^10)/8 - 2)/x,x)
2*x*log(3*x)^2 - 2*log(x) - 2*x + (19*x^2*log(3*x))/2 - 3*x^3*log(3*x) - ( 43*x^4*log(3*x))/2 + (x^5*log(3*x))/8 + 30*x^6*log(3*x) + (105*x^7*log(3*x ))/4 + 9*x^8*log(3*x) + (9*x^9*log(3*x))/8 + log(3*x)^2 - x^2 + (15*x^3)/2 + (3*x^4)/2 - 10*x^5 + x^6/16 + 15*x^7 + (105*x^8)/8 + (9*x^9)/2 + (9*x^1 0)/16 - (9*x^2*log(3*x)^2)/2 - (23*x^3*log(3*x)^2)/2 + (x^4*log(3*x)^2)/16 + 15*x^5*log(3*x)^2 + (105*x^6*log(3*x)^2)/8 + (9*x^7*log(3*x)^2)/2 + (9* x^8*log(3*x)^2)/16