Integrand size = 308, antiderivative size = 34 \[ \int \frac {\left (648 x^4+162 x^6+e^x \left (864 x^5+216 x^7\right )+e^{2 x} \left (432 x^6+108 x^8\right )+e^{3 x} \left (96 x^7+24 x^9\right )+e^{4 x} \left (8 x^8+2 x^{10}\right )\right ) \log (x)+\left (-216 x^2+1242 x^4+162 x^6+e^x \left (-144 x^3+2124 x^5+432 x^6+324 x^7+108 x^8\right )+e^{2 x} \left (-24 x^4+1290 x^6+432 x^7+216 x^8+108 x^9\right )+e^{3 x} \left (336 x^7+144 x^8+60 x^9+36 x^{10}\right )+e^{4 x} \left (32 x^8+16 x^9+6 x^{10}+4 x^{11}\right )\right ) \log ^2(x)+\left (16-140 x^2+e^x \left (-144 x^3-48 x^4-12 x^5-12 x^6\right )+e^{2 x} \left (-32 x^4-16 x^5-4 x^6-4 x^7\right )\right ) \log ^3(x)-2 x^2 \log ^4(x)}{144 x+72 x^3+9 x^5} \, dx=\frac {\log ^2(x) \left (-x^2 \left (3+e^x x\right )^2+\log (x)\right )^2}{9 \left (4+x^2\right )} \]
Time = 0.30 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {\left (648 x^4+162 x^6+e^x \left (864 x^5+216 x^7\right )+e^{2 x} \left (432 x^6+108 x^8\right )+e^{3 x} \left (96 x^7+24 x^9\right )+e^{4 x} \left (8 x^8+2 x^{10}\right )\right ) \log (x)+\left (-216 x^2+1242 x^4+162 x^6+e^x \left (-144 x^3+2124 x^5+432 x^6+324 x^7+108 x^8\right )+e^{2 x} \left (-24 x^4+1290 x^6+432 x^7+216 x^8+108 x^9\right )+e^{3 x} \left (336 x^7+144 x^8+60 x^9+36 x^{10}\right )+e^{4 x} \left (32 x^8+16 x^9+6 x^{10}+4 x^{11}\right )\right ) \log ^2(x)+\left (16-140 x^2+e^x \left (-144 x^3-48 x^4-12 x^5-12 x^6\right )+e^{2 x} \left (-32 x^4-16 x^5-4 x^6-4 x^7\right )\right ) \log ^3(x)-2 x^2 \log ^4(x)}{144 x+72 x^3+9 x^5} \, dx=\frac {\log ^2(x) \left (-x^2 \left (3+e^x x\right )^2+\log (x)\right )^2}{9 \left (4+x^2\right )} \]
Integrate[((648*x^4 + 162*x^6 + E^x*(864*x^5 + 216*x^7) + E^(2*x)*(432*x^6 + 108*x^8) + E^(3*x)*(96*x^7 + 24*x^9) + E^(4*x)*(8*x^8 + 2*x^10))*Log[x] + (-216*x^2 + 1242*x^4 + 162*x^6 + E^x*(-144*x^3 + 2124*x^5 + 432*x^6 + 3 24*x^7 + 108*x^8) + E^(2*x)*(-24*x^4 + 1290*x^6 + 432*x^7 + 216*x^8 + 108* x^9) + E^(3*x)*(336*x^7 + 144*x^8 + 60*x^9 + 36*x^10) + E^(4*x)*(32*x^8 + 16*x^9 + 6*x^10 + 4*x^11))*Log[x]^2 + (16 - 140*x^2 + E^x*(-144*x^3 - 48*x ^4 - 12*x^5 - 12*x^6) + E^(2*x)*(-32*x^4 - 16*x^5 - 4*x^6 - 4*x^7))*Log[x] ^3 - 2*x^2*Log[x]^4)/(144*x + 72*x^3 + 9*x^5),x]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 5.55 (sec) , antiderivative size = 350, normalized size of antiderivative = 10.29, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {2026, 1380, 27, 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.
| \(\displaystyle \int \frac {-2 x^2 \log ^4(x)+\left (-140 x^2+e^{2 x} \left (-4 x^7-4 x^6-16 x^5-32 x^4\right )+e^x \left (-12 x^6-12 x^5-48 x^4-144 x^3\right )+16\right ) \log ^3(x)+\left (162 x^6+648 x^4+e^{4 x} \left (2 x^{10}+8 x^8\right )+e^{3 x} \left (24 x^9+96 x^7\right )+e^{2 x} \left (108 x^8+432 x^6\right )+e^x \left (216 x^7+864 x^5\right )\right ) \log (x)+\left (162 x^6+1242 x^4-216 x^2+e^{4 x} \left (4 x^{11}+6 x^{10}+16 x^9+32 x^8\right )+e^{3 x} \left (36 x^{10}+60 x^9+144 x^8+336 x^7\right )+e^{2 x} \left (108 x^9+216 x^8+432 x^7+1290 x^6-24 x^4\right )+e^x \left (108 x^8+324 x^7+432 x^6+2124 x^5-144 x^3\right )\right ) \log ^2(x)}{9 x^5+72 x^3+144 x} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {-2 x^2 \log ^4(x)+\left (-140 x^2+e^{2 x} \left (-4 x^7-4 x^6-16 x^5-32 x^4\right )+e^x \left (-12 x^6-12 x^5-48 x^4-144 x^3\right )+16\right ) \log ^3(x)+\left (162 x^6+648 x^4+e^{4 x} \left (2 x^{10}+8 x^8\right )+e^{3 x} \left (24 x^9+96 x^7\right )+e^{2 x} \left (108 x^8+432 x^6\right )+e^x \left (216 x^7+864 x^5\right )\right ) \log (x)+\left (162 x^6+1242 x^4-216 x^2+e^{4 x} \left (4 x^{11}+6 x^{10}+16 x^9+32 x^8\right )+e^{3 x} \left (36 x^{10}+60 x^9+144 x^8+336 x^7\right )+e^{2 x} \left (108 x^9+216 x^8+432 x^7+1290 x^6-24 x^4\right )+e^x \left (108 x^8+324 x^7+432 x^6+2124 x^5-144 x^3\right )\right ) \log ^2(x)}{x \left (9 x^4+72 x^2+144\right )}dx\) |
| \(\Big \downarrow \) 1380 |
| \(\displaystyle 9 \int \frac {2 \left (-x^2 \log ^4(x)+2 \left (-35 x^2-3 e^x \left (x^6+x^5+4 x^4+12 x^3\right )-e^{2 x} \left (x^7+x^6+4 x^5+8 x^4\right )+4\right ) \log ^3(x)-\left (-81 x^6-621 x^4+108 x^2+18 e^x \left (-3 x^8-9 x^7-12 x^6-59 x^5+4 x^3\right )+3 e^{2 x} \left (-18 x^9-36 x^8-72 x^7-215 x^6+4 x^4\right )-6 e^{3 x} \left (3 x^{10}+5 x^9+12 x^8+28 x^7\right )-e^{4 x} \left (2 x^{11}+3 x^{10}+8 x^9+16 x^8\right )\right ) \log ^2(x)+\left (81 x^6+324 x^4+108 e^x \left (x^7+4 x^5\right )+54 e^{2 x} \left (x^8+4 x^6\right )+12 e^{3 x} \left (x^9+4 x^7\right )+e^{4 x} \left (x^{10}+4 x^8\right )\right ) \log (x)\right )}{81 x \left (x^2+4\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{9} \int \frac {-x^2 \log ^4(x)+2 \left (-35 x^2-3 e^x \left (x^6+x^5+4 x^4+12 x^3\right )-e^{2 x} \left (x^7+x^6+4 x^5+8 x^4\right )+4\right ) \log ^3(x)-\left (-81 x^6-621 x^4+108 x^2+18 e^x \left (-3 x^8-9 x^7-12 x^6-59 x^5+4 x^3\right )+3 e^{2 x} \left (-18 x^9-36 x^8-72 x^7-215 x^6+4 x^4\right )-6 e^{3 x} \left (3 x^{10}+5 x^9+12 x^8+28 x^7\right )-e^{4 x} \left (2 x^{11}+3 x^{10}+8 x^9+16 x^8\right )\right ) \log ^2(x)+\left (81 x^6+324 x^4+108 e^x \left (x^7+4 x^5\right )+54 e^{2 x} \left (x^8+4 x^6\right )+12 e^{3 x} \left (x^9+4 x^7\right )+e^{4 x} \left (x^{10}+4 x^8\right )\right ) \log (x)}{x \left (x^2+4\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2}{9} \int \left (\frac {e^{4 x} \log (x) \left (2 \log (x) x^3+3 \log (x) x^2+x^2+8 \log (x) x+16 \log (x)+4\right ) x^7}{\left (x^2+4\right )^2}+\frac {6 e^{3 x} \log (x) \left (3 \log (x) x^3+5 \log (x) x^2+2 x^2+12 \log (x) x+28 \log (x)+8\right ) x^6}{\left (x^2+4\right )^2}+\frac {81 \log ^2(x) x^5}{\left (x^2+4\right )^2}+\frac {81 \log (x) x^5}{\left (x^2+4\right )^2}+\frac {621 \log ^2(x) x^3}{\left (x^2+4\right )^2}+\frac {324 \log (x) x^3}{\left (x^2+4\right )^2}+\frac {e^{2 x} \log (x) \left (54 \log (x) x^5+108 \log (x) x^4+54 x^4-2 \log ^2(x) x^3+216 \log (x) x^3-2 \log ^2(x) x^2+645 \log (x) x^2+216 x^2-8 \log ^2(x) x-16 \log ^2(x)-12 \log (x)\right ) x^3}{\left (x^2+4\right )^2}+\frac {6 e^x \log (x) \left (9 \log (x) x^5+27 \log (x) x^4+18 x^4-\log ^2(x) x^3+36 \log (x) x^3-\log ^2(x) x^2+177 \log (x) x^2+72 x^2-4 \log ^2(x) x-12 \log ^2(x)-12 \log (x)\right ) x^2}{\left (x^2+4\right )^2}-\frac {\log ^4(x) x}{\left (x^2+4\right )^2}-\frac {70 \log ^3(x) x}{\left (x^2+4\right )^2}-\frac {108 \log ^2(x) x}{\left (x^2+4\right )^2}+\frac {8 \log ^3(x)}{\left (x^2+4\right )^2 x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2}{9} \left (-81 \operatorname {PolyLog}\left (2,-\frac {4}{x^2}\right )-81 \operatorname {PolyLog}\left (2,-\frac {x^2}{4}\right )-\frac {27}{4} \operatorname {PolyLog}\left (3,-\frac {4}{x^2}\right )+\frac {27}{4} \operatorname {PolyLog}\left (3,-\frac {x^2}{4}\right )-\frac {27}{2} \operatorname {PolyLog}\left (2,-\frac {4}{x^2}\right ) \log (x)-\frac {27}{2} \operatorname {PolyLog}\left (2,-\frac {x^2}{4}\right ) \log (x)+\frac {\log ^4(x)}{2 \left (x^2+4\right )}+\frac {36 \log ^3(x)}{x^2+4}+\frac {81}{2} x^2 \log ^2(x)+\frac {27}{2} \log \left (\frac {4}{x^2}+1\right ) \log ^2(x)+\frac {648 \log ^2(x)}{x^2+4}-\frac {27}{2} \log ^2(x) \log \left (\frac {x^2}{4}+1\right )+162 \log \left (\frac {4}{x^2}+1\right ) \log (x)-162 \log (x) \log \left (\frac {x^2}{4}+1\right )+\frac {e^{4 x} x^7 \log (x) \left (x^3 \log (x)+4 x \log (x)\right )}{2 \left (x^2+4\right )^2}+\frac {6 e^{3 x} x^6 \log (x) \left (x^3 \log (x)+4 x \log (x)\right )}{\left (x^2+4\right )^2}+\frac {e^{2 x} x^3 \log (x) \left (27 x^5 \log (x)-x^3 \log ^2(x)+108 x^3 \log (x)-4 x \log ^2(x)\right )}{\left (x^2+4\right )^2}+\frac {6 e^x x^2 \log (x) \left (9 x^5 \log (x)-x^3 \log ^2(x)+36 x^3 \log (x)-4 x \log ^2(x)\right )}{\left (x^2+4\right )^2}\right )\) |
Int[((648*x^4 + 162*x^6 + E^x*(864*x^5 + 216*x^7) + E^(2*x)*(432*x^6 + 108 *x^8) + E^(3*x)*(96*x^7 + 24*x^9) + E^(4*x)*(8*x^8 + 2*x^10))*Log[x] + (-2 16*x^2 + 1242*x^4 + 162*x^6 + E^x*(-144*x^3 + 2124*x^5 + 432*x^6 + 324*x^7 + 108*x^8) + E^(2*x)*(-24*x^4 + 1290*x^6 + 432*x^7 + 216*x^8 + 108*x^9) + E^(3*x)*(336*x^7 + 144*x^8 + 60*x^9 + 36*x^10) + E^(4*x)*(32*x^8 + 16*x^9 + 6*x^10 + 4*x^11))*Log[x]^2 + (16 - 140*x^2 + E^x*(-144*x^3 - 48*x^4 - 1 2*x^5 - 12*x^6) + E^(2*x)*(-32*x^4 - 16*x^5 - 4*x^6 - 4*x^7))*Log[x]^3 - 2 *x^2*Log[x]^4)/(144*x + 72*x^3 + 9*x^5),x]
(2*(162*Log[1 + 4/x^2]*Log[x] + (81*x^2*Log[x]^2)/2 + (648*Log[x]^2)/(4 + x^2) + (27*Log[1 + 4/x^2]*Log[x]^2)/2 + (36*Log[x]^3)/(4 + x^2) + Log[x]^4 /(2*(4 + x^2)) + (6*E^(3*x)*x^6*Log[x]*(4*x*Log[x] + x^3*Log[x]))/(4 + x^2 )^2 + (E^(4*x)*x^7*Log[x]*(4*x*Log[x] + x^3*Log[x]))/(2*(4 + x^2)^2) + (6* E^x*x^2*Log[x]*(36*x^3*Log[x] + 9*x^5*Log[x] - 4*x*Log[x]^2 - x^3*Log[x]^2 ))/(4 + x^2)^2 + (E^(2*x)*x^3*Log[x]*(108*x^3*Log[x] + 27*x^5*Log[x] - 4*x *Log[x]^2 - x^3*Log[x]^2))/(4 + x^2)^2 - 162*Log[x]*Log[1 + x^2/4] - (27*L og[x]^2*Log[1 + x^2/4])/2 - 81*PolyLog[2, -4/x^2] - (27*Log[x]*PolyLog[2, -4/x^2])/2 - 81*PolyLog[2, -1/4*x^2] - (27*Log[x]*PolyLog[2, -1/4*x^2])/2 - (27*PolyLog[3, -4/x^2])/4 + (27*PolyLog[3, -1/4*x^2])/4))/9
3.13.25.3.1 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_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S imp[1/c^p Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
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. \(94\) vs. \(2(31)=62\).
Time = 4.44 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.79
| method | result | size |
| risch | \(\frac {\ln \left (x \right )^{4}}{9 x^{2}+36}-\frac {2 x^{2} \left ({\mathrm e}^{2 x} x^{2}+6 \,{\mathrm e}^{x} x +9\right ) \ln \left (x \right )^{3}}{9 \left (x^{2}+4\right )}+\frac {\left (x^{4} {\mathrm e}^{4 x}+12 x^{3} {\mathrm e}^{3 x}+54 \,{\mathrm e}^{2 x} x^{2}+108 \,{\mathrm e}^{x} x +81\right ) x^{4} \ln \left (x \right )^{2}}{9 x^{2}+36}\) | \(95\) |
| parallelrisch | \(-\frac {-4 \ln \left (x \right )^{2} {\mathrm e}^{4 x} x^{8}-48 \ln \left (x \right )^{2} {\mathrm e}^{3 x} x^{7}-216 \ln \left (x \right )^{2} {\mathrm e}^{2 x} x^{6}+8 \ln \left (x \right )^{3} {\mathrm e}^{2 x} x^{4}-432 \ln \left (x \right )^{2} {\mathrm e}^{x} x^{5}+48 \ln \left (x \right )^{3} {\mathrm e}^{x} x^{3}-324 x^{4} \ln \left (x \right )^{2}+72 x^{2} \ln \left (x \right )^{3}-4 \ln \left (x \right )^{4}}{36 \left (x^{2}+4\right )}\) | \(109\) |
int((-2*x^2*ln(x)^4+((-4*x^7-4*x^6-16*x^5-32*x^4)*exp(x)^2+(-12*x^6-12*x^5 -48*x^4-144*x^3)*exp(x)-140*x^2+16)*ln(x)^3+((4*x^11+6*x^10+16*x^9+32*x^8) *exp(x)^4+(36*x^10+60*x^9+144*x^8+336*x^7)*exp(x)^3+(108*x^9+216*x^8+432*x ^7+1290*x^6-24*x^4)*exp(x)^2+(108*x^8+324*x^7+432*x^6+2124*x^5-144*x^3)*ex p(x)+162*x^6+1242*x^4-216*x^2)*ln(x)^2+((2*x^10+8*x^8)*exp(x)^4+(24*x^9+96 *x^7)*exp(x)^3+(108*x^8+432*x^6)*exp(x)^2+(216*x^7+864*x^5)*exp(x)+162*x^6 +648*x^4)*ln(x))/(9*x^5+72*x^3+144*x),x,method=_RETURNVERBOSE)
1/9/(x^2+4)*ln(x)^4-2/9*x^2*(exp(x)^2*x^2+6*exp(x)*x+9)/(x^2+4)*ln(x)^3+1/ 9*(x^4*exp(x)^4+12*x^3*exp(x)^3+54*exp(x)^2*x^2+108*exp(x)*x+81)*x^4/(x^2+ 4)*ln(x)^2
Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (32) = 64\).
Time = 0.27 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.59 \[ \int \frac {\left (648 x^4+162 x^6+e^x \left (864 x^5+216 x^7\right )+e^{2 x} \left (432 x^6+108 x^8\right )+e^{3 x} \left (96 x^7+24 x^9\right )+e^{4 x} \left (8 x^8+2 x^{10}\right )\right ) \log (x)+\left (-216 x^2+1242 x^4+162 x^6+e^x \left (-144 x^3+2124 x^5+432 x^6+324 x^7+108 x^8\right )+e^{2 x} \left (-24 x^4+1290 x^6+432 x^7+216 x^8+108 x^9\right )+e^{3 x} \left (336 x^7+144 x^8+60 x^9+36 x^{10}\right )+e^{4 x} \left (32 x^8+16 x^9+6 x^{10}+4 x^{11}\right )\right ) \log ^2(x)+\left (16-140 x^2+e^x \left (-144 x^3-48 x^4-12 x^5-12 x^6\right )+e^{2 x} \left (-32 x^4-16 x^5-4 x^6-4 x^7\right )\right ) \log ^3(x)-2 x^2 \log ^4(x)}{144 x+72 x^3+9 x^5} \, dx=-\frac {2 \, {\left (x^{4} e^{\left (2 \, x\right )} + 6 \, x^{3} e^{x} + 9 \, x^{2}\right )} \log \left (x\right )^{3} - \log \left (x\right )^{4} - {\left (x^{8} e^{\left (4 \, x\right )} + 12 \, x^{7} e^{\left (3 \, x\right )} + 54 \, x^{6} e^{\left (2 \, x\right )} + 108 \, x^{5} e^{x} + 81 \, x^{4}\right )} \log \left (x\right )^{2}}{9 \, {\left (x^{2} + 4\right )}} \]
integrate((-2*x^2*log(x)^4+((-4*x^7-4*x^6-16*x^5-32*x^4)*exp(x)^2+(-12*x^6 -12*x^5-48*x^4-144*x^3)*exp(x)-140*x^2+16)*log(x)^3+((4*x^11+6*x^10+16*x^9 +32*x^8)*exp(x)^4+(36*x^10+60*x^9+144*x^8+336*x^7)*exp(x)^3+(108*x^9+216*x ^8+432*x^7+1290*x^6-24*x^4)*exp(x)^2+(108*x^8+324*x^7+432*x^6+2124*x^5-144 *x^3)*exp(x)+162*x^6+1242*x^4-216*x^2)*log(x)^2+((2*x^10+8*x^8)*exp(x)^4+( 24*x^9+96*x^7)*exp(x)^3+(108*x^8+432*x^6)*exp(x)^2+(216*x^7+864*x^5)*exp(x )+162*x^6+648*x^4)*log(x))/(9*x^5+72*x^3+144*x),x, algorithm=\
-1/9*(2*(x^4*e^(2*x) + 6*x^3*e^x + 9*x^2)*log(x)^3 - log(x)^4 - (x^8*e^(4* x) + 12*x^7*e^(3*x) + 54*x^6*e^(2*x) + 108*x^5*e^x + 81*x^4)*log(x)^2)/(x^ 2 + 4)
Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (27) = 54\).
Time = 0.43 (sec) , antiderivative size = 326, normalized size of antiderivative = 9.59 \[ \int \frac {\left (648 x^4+162 x^6+e^x \left (864 x^5+216 x^7\right )+e^{2 x} \left (432 x^6+108 x^8\right )+e^{3 x} \left (96 x^7+24 x^9\right )+e^{4 x} \left (8 x^8+2 x^{10}\right )\right ) \log (x)+\left (-216 x^2+1242 x^4+162 x^6+e^x \left (-144 x^3+2124 x^5+432 x^6+324 x^7+108 x^8\right )+e^{2 x} \left (-24 x^4+1290 x^6+432 x^7+216 x^8+108 x^9\right )+e^{3 x} \left (336 x^7+144 x^8+60 x^9+36 x^{10}\right )+e^{4 x} \left (32 x^8+16 x^9+6 x^{10}+4 x^{11}\right )\right ) \log ^2(x)+\left (16-140 x^2+e^x \left (-144 x^3-48 x^4-12 x^5-12 x^6\right )+e^{2 x} \left (-32 x^4-16 x^5-4 x^6-4 x^7\right )\right ) \log ^3(x)-2 x^2 \log ^4(x)}{144 x+72 x^3+9 x^5} \, dx=\frac {9 x^{4} \log {\left (x \right )}^{2}}{x^{2} + 4} - \frac {2 x^{2} \log {\left (x \right )}^{3}}{x^{2} + 4} + \frac {\left (972 x^{13} \log {\left (x \right )}^{2} + 11664 x^{11} \log {\left (x \right )}^{2} + 46656 x^{9} \log {\left (x \right )}^{2} + 62208 x^{7} \log {\left (x \right )}^{2}\right ) e^{3 x} + \left (81 x^{14} \log {\left (x \right )}^{2} + 972 x^{12} \log {\left (x \right )}^{2} + 3888 x^{10} \log {\left (x \right )}^{2} + 5184 x^{8} \log {\left (x \right )}^{2}\right ) e^{4 x} + \left (8748 x^{11} \log {\left (x \right )}^{2} - 972 x^{9} \log {\left (x \right )}^{3} + 104976 x^{9} \log {\left (x \right )}^{2} - 11664 x^{7} \log {\left (x \right )}^{3} + 419904 x^{7} \log {\left (x \right )}^{2} - 46656 x^{5} \log {\left (x \right )}^{3} + 559872 x^{5} \log {\left (x \right )}^{2} - 62208 x^{3} \log {\left (x \right )}^{3}\right ) e^{x} + \left (4374 x^{12} \log {\left (x \right )}^{2} - 162 x^{10} \log {\left (x \right )}^{3} + 52488 x^{10} \log {\left (x \right )}^{2} - 1944 x^{8} \log {\left (x \right )}^{3} + 209952 x^{8} \log {\left (x \right )}^{2} - 7776 x^{6} \log {\left (x \right )}^{3} + 279936 x^{6} \log {\left (x \right )}^{2} - 10368 x^{4} \log {\left (x \right )}^{3}\right ) e^{2 x}}{729 x^{8} + 11664 x^{6} + 69984 x^{4} + 186624 x^{2} + 186624} + \frac {\log {\left (x \right )}^{4}}{9 x^{2} + 36} \]
integrate((-2*x**2*ln(x)**4+((-4*x**7-4*x**6-16*x**5-32*x**4)*exp(x)**2+(- 12*x**6-12*x**5-48*x**4-144*x**3)*exp(x)-140*x**2+16)*ln(x)**3+((4*x**11+6 *x**10+16*x**9+32*x**8)*exp(x)**4+(36*x**10+60*x**9+144*x**8+336*x**7)*exp (x)**3+(108*x**9+216*x**8+432*x**7+1290*x**6-24*x**4)*exp(x)**2+(108*x**8+ 324*x**7+432*x**6+2124*x**5-144*x**3)*exp(x)+162*x**6+1242*x**4-216*x**2)* ln(x)**2+((2*x**10+8*x**8)*exp(x)**4+(24*x**9+96*x**7)*exp(x)**3+(108*x**8 +432*x**6)*exp(x)**2+(216*x**7+864*x**5)*exp(x)+162*x**6+648*x**4)*ln(x))/ (9*x**5+72*x**3+144*x),x)
9*x**4*log(x)**2/(x**2 + 4) - 2*x**2*log(x)**3/(x**2 + 4) + ((972*x**13*lo g(x)**2 + 11664*x**11*log(x)**2 + 46656*x**9*log(x)**2 + 62208*x**7*log(x) **2)*exp(3*x) + (81*x**14*log(x)**2 + 972*x**12*log(x)**2 + 3888*x**10*log (x)**2 + 5184*x**8*log(x)**2)*exp(4*x) + (8748*x**11*log(x)**2 - 972*x**9* log(x)**3 + 104976*x**9*log(x)**2 - 11664*x**7*log(x)**3 + 419904*x**7*log (x)**2 - 46656*x**5*log(x)**3 + 559872*x**5*log(x)**2 - 62208*x**3*log(x)* *3)*exp(x) + (4374*x**12*log(x)**2 - 162*x**10*log(x)**3 + 52488*x**10*log (x)**2 - 1944*x**8*log(x)**3 + 209952*x**8*log(x)**2 - 7776*x**6*log(x)**3 + 279936*x**6*log(x)**2 - 10368*x**4*log(x)**3)*exp(2*x))/(729*x**8 + 116 64*x**6 + 69984*x**4 + 186624*x**2 + 186624) + log(x)**4/(9*x**2 + 36)
Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (32) = 64\).
Time = 0.26 (sec) , antiderivative size = 105, normalized size of antiderivative = 3.09 \[ \int \frac {\left (648 x^4+162 x^6+e^x \left (864 x^5+216 x^7\right )+e^{2 x} \left (432 x^6+108 x^8\right )+e^{3 x} \left (96 x^7+24 x^9\right )+e^{4 x} \left (8 x^8+2 x^{10}\right )\right ) \log (x)+\left (-216 x^2+1242 x^4+162 x^6+e^x \left (-144 x^3+2124 x^5+432 x^6+324 x^7+108 x^8\right )+e^{2 x} \left (-24 x^4+1290 x^6+432 x^7+216 x^8+108 x^9\right )+e^{3 x} \left (336 x^7+144 x^8+60 x^9+36 x^{10}\right )+e^{4 x} \left (32 x^8+16 x^9+6 x^{10}+4 x^{11}\right )\right ) \log ^2(x)+\left (16-140 x^2+e^x \left (-144 x^3-48 x^4-12 x^5-12 x^6\right )+e^{2 x} \left (-32 x^4-16 x^5-4 x^6-4 x^7\right )\right ) \log ^3(x)-2 x^2 \log ^4(x)}{144 x+72 x^3+9 x^5} \, dx=\frac {x^{8} e^{\left (4 \, x\right )} \log \left (x\right )^{2} + 12 \, x^{7} e^{\left (3 \, x\right )} \log \left (x\right )^{2} + 81 \, x^{4} \log \left (x\right )^{2} - 18 \, x^{2} \log \left (x\right )^{3} + \log \left (x\right )^{4} + 2 \, {\left (27 \, x^{6} \log \left (x\right )^{2} - x^{4} \log \left (x\right )^{3}\right )} e^{\left (2 \, x\right )} + 12 \, {\left (9 \, x^{5} \log \left (x\right )^{2} - x^{3} \log \left (x\right )^{3}\right )} e^{x}}{9 \, {\left (x^{2} + 4\right )}} \]
integrate((-2*x^2*log(x)^4+((-4*x^7-4*x^6-16*x^5-32*x^4)*exp(x)^2+(-12*x^6 -12*x^5-48*x^4-144*x^3)*exp(x)-140*x^2+16)*log(x)^3+((4*x^11+6*x^10+16*x^9 +32*x^8)*exp(x)^4+(36*x^10+60*x^9+144*x^8+336*x^7)*exp(x)^3+(108*x^9+216*x ^8+432*x^7+1290*x^6-24*x^4)*exp(x)^2+(108*x^8+324*x^7+432*x^6+2124*x^5-144 *x^3)*exp(x)+162*x^6+1242*x^4-216*x^2)*log(x)^2+((2*x^10+8*x^8)*exp(x)^4+( 24*x^9+96*x^7)*exp(x)^3+(108*x^8+432*x^6)*exp(x)^2+(216*x^7+864*x^5)*exp(x )+162*x^6+648*x^4)*log(x))/(9*x^5+72*x^3+144*x),x, algorithm=\
1/9*(x^8*e^(4*x)*log(x)^2 + 12*x^7*e^(3*x)*log(x)^2 + 81*x^4*log(x)^2 - 18 *x^2*log(x)^3 + log(x)^4 + 2*(27*x^6*log(x)^2 - x^4*log(x)^3)*e^(2*x) + 12 *(9*x^5*log(x)^2 - x^3*log(x)^3)*e^x)/(x^2 + 4)
\[ \int \frac {\left (648 x^4+162 x^6+e^x \left (864 x^5+216 x^7\right )+e^{2 x} \left (432 x^6+108 x^8\right )+e^{3 x} \left (96 x^7+24 x^9\right )+e^{4 x} \left (8 x^8+2 x^{10}\right )\right ) \log (x)+\left (-216 x^2+1242 x^4+162 x^6+e^x \left (-144 x^3+2124 x^5+432 x^6+324 x^7+108 x^8\right )+e^{2 x} \left (-24 x^4+1290 x^6+432 x^7+216 x^8+108 x^9\right )+e^{3 x} \left (336 x^7+144 x^8+60 x^9+36 x^{10}\right )+e^{4 x} \left (32 x^8+16 x^9+6 x^{10}+4 x^{11}\right )\right ) \log ^2(x)+\left (16-140 x^2+e^x \left (-144 x^3-48 x^4-12 x^5-12 x^6\right )+e^{2 x} \left (-32 x^4-16 x^5-4 x^6-4 x^7\right )\right ) \log ^3(x)-2 x^2 \log ^4(x)}{144 x+72 x^3+9 x^5} \, dx=\int { -\frac {2 \, {\left (x^{2} \log \left (x\right )^{4} + 2 \, {\left (35 \, x^{2} + {\left (x^{7} + x^{6} + 4 \, x^{5} + 8 \, x^{4}\right )} e^{\left (2 \, x\right )} + 3 \, {\left (x^{6} + x^{5} + 4 \, x^{4} + 12 \, x^{3}\right )} e^{x} - 4\right )} \log \left (x\right )^{3} - {\left (81 \, x^{6} + 621 \, x^{4} - 108 \, x^{2} + {\left (2 \, x^{11} + 3 \, x^{10} + 8 \, x^{9} + 16 \, x^{8}\right )} e^{\left (4 \, x\right )} + 6 \, {\left (3 \, x^{10} + 5 \, x^{9} + 12 \, x^{8} + 28 \, x^{7}\right )} e^{\left (3 \, x\right )} + 3 \, {\left (18 \, x^{9} + 36 \, x^{8} + 72 \, x^{7} + 215 \, x^{6} - 4 \, x^{4}\right )} e^{\left (2 \, x\right )} + 18 \, {\left (3 \, x^{8} + 9 \, x^{7} + 12 \, x^{6} + 59 \, x^{5} - 4 \, x^{3}\right )} e^{x}\right )} \log \left (x\right )^{2} - {\left (81 \, x^{6} + 324 \, x^{4} + {\left (x^{10} + 4 \, x^{8}\right )} e^{\left (4 \, x\right )} + 12 \, {\left (x^{9} + 4 \, x^{7}\right )} e^{\left (3 \, x\right )} + 54 \, {\left (x^{8} + 4 \, x^{6}\right )} e^{\left (2 \, x\right )} + 108 \, {\left (x^{7} + 4 \, x^{5}\right )} e^{x}\right )} \log \left (x\right )\right )}}{9 \, {\left (x^{5} + 8 \, x^{3} + 16 \, x\right )}} \,d x } \]
integrate((-2*x^2*log(x)^4+((-4*x^7-4*x^6-16*x^5-32*x^4)*exp(x)^2+(-12*x^6 -12*x^5-48*x^4-144*x^3)*exp(x)-140*x^2+16)*log(x)^3+((4*x^11+6*x^10+16*x^9 +32*x^8)*exp(x)^4+(36*x^10+60*x^9+144*x^8+336*x^7)*exp(x)^3+(108*x^9+216*x ^8+432*x^7+1290*x^6-24*x^4)*exp(x)^2+(108*x^8+324*x^7+432*x^6+2124*x^5-144 *x^3)*exp(x)+162*x^6+1242*x^4-216*x^2)*log(x)^2+((2*x^10+8*x^8)*exp(x)^4+( 24*x^9+96*x^7)*exp(x)^3+(108*x^8+432*x^6)*exp(x)^2+(216*x^7+864*x^5)*exp(x )+162*x^6+648*x^4)*log(x))/(9*x^5+72*x^3+144*x),x, algorithm=\
integrate(-2/9*(x^2*log(x)^4 + 2*(35*x^2 + (x^7 + x^6 + 4*x^5 + 8*x^4)*e^( 2*x) + 3*(x^6 + x^5 + 4*x^4 + 12*x^3)*e^x - 4)*log(x)^3 - (81*x^6 + 621*x^ 4 - 108*x^2 + (2*x^11 + 3*x^10 + 8*x^9 + 16*x^8)*e^(4*x) + 6*(3*x^10 + 5*x ^9 + 12*x^8 + 28*x^7)*e^(3*x) + 3*(18*x^9 + 36*x^8 + 72*x^7 + 215*x^6 - 4* x^4)*e^(2*x) + 18*(3*x^8 + 9*x^7 + 12*x^6 + 59*x^5 - 4*x^3)*e^x)*log(x)^2 - (81*x^6 + 324*x^4 + (x^10 + 4*x^8)*e^(4*x) + 12*(x^9 + 4*x^7)*e^(3*x) + 54*(x^8 + 4*x^6)*e^(2*x) + 108*(x^7 + 4*x^5)*e^x)*log(x))/(x^5 + 8*x^3 + 1 6*x), x)
Time = 8.79 (sec) , antiderivative size = 173, normalized size of antiderivative = 5.09 \[ \int \frac {\left (648 x^4+162 x^6+e^x \left (864 x^5+216 x^7\right )+e^{2 x} \left (432 x^6+108 x^8\right )+e^{3 x} \left (96 x^7+24 x^9\right )+e^{4 x} \left (8 x^8+2 x^{10}\right )\right ) \log (x)+\left (-216 x^2+1242 x^4+162 x^6+e^x \left (-144 x^3+2124 x^5+432 x^6+324 x^7+108 x^8\right )+e^{2 x} \left (-24 x^4+1290 x^6+432 x^7+216 x^8+108 x^9\right )+e^{3 x} \left (336 x^7+144 x^8+60 x^9+36 x^{10}\right )+e^{4 x} \left (32 x^8+16 x^9+6 x^{10}+4 x^{11}\right )\right ) \log ^2(x)+\left (16-140 x^2+e^x \left (-144 x^3-48 x^4-12 x^5-12 x^6\right )+e^{2 x} \left (-32 x^4-16 x^5-4 x^6-4 x^7\right )\right ) \log ^3(x)-2 x^2 \log ^4(x)}{144 x+72 x^3+9 x^5} \, dx=\frac {8\,{\ln \left (x\right )}^3}{x^2+4}+\frac {{\ln \left (x\right )}^4}{9\,\left (x^2+4\right )}-2\,{\ln \left (x\right )}^3+\frac {9\,x^4\,{\ln \left (x\right )}^2}{x^2+4}-\frac {2\,x^4\,{\mathrm {e}}^{2\,x}\,{\ln \left (x\right )}^3}{9\,\left (x^2+4\right )}+\frac {6\,x^6\,{\mathrm {e}}^{2\,x}\,{\ln \left (x\right )}^2}{x^2+4}+\frac {4\,x^7\,{\mathrm {e}}^{3\,x}\,{\ln \left (x\right )}^2}{3\,\left (x^2+4\right )}+\frac {x^8\,{\mathrm {e}}^{4\,x}\,{\ln \left (x\right )}^2}{9\,\left (x^2+4\right )}-\frac {4\,x^3\,{\mathrm {e}}^x\,{\ln \left (x\right )}^3}{3\,\left (x^2+4\right )}+\frac {12\,x^5\,{\mathrm {e}}^x\,{\ln \left (x\right )}^2}{x^2+4} \]
int((log(x)^2*(exp(2*x)*(1290*x^6 - 24*x^4 + 432*x^7 + 216*x^8 + 108*x^9) + exp(x)*(2124*x^5 - 144*x^3 + 432*x^6 + 324*x^7 + 108*x^8) + exp(4*x)*(32 *x^8 + 16*x^9 + 6*x^10 + 4*x^11) + exp(3*x)*(336*x^7 + 144*x^8 + 60*x^9 + 36*x^10) - 216*x^2 + 1242*x^4 + 162*x^6) + log(x)*(exp(x)*(864*x^5 + 216*x ^7) + exp(4*x)*(8*x^8 + 2*x^10) + exp(3*x)*(96*x^7 + 24*x^9) + exp(2*x)*(4 32*x^6 + 108*x^8) + 648*x^4 + 162*x^6) - 2*x^2*log(x)^4 - log(x)^3*(exp(x) *(144*x^3 + 48*x^4 + 12*x^5 + 12*x^6) + exp(2*x)*(32*x^4 + 16*x^5 + 4*x^6 + 4*x^7) + 140*x^2 - 16))/(144*x + 72*x^3 + 9*x^5),x)
(8*log(x)^3)/(x^2 + 4) + log(x)^4/(9*(x^2 + 4)) - 2*log(x)^3 + (9*x^4*log( x)^2)/(x^2 + 4) - (2*x^4*exp(2*x)*log(x)^3)/(9*(x^2 + 4)) + (6*x^6*exp(2*x )*log(x)^2)/(x^2 + 4) + (4*x^7*exp(3*x)*log(x)^2)/(3*(x^2 + 4)) + (x^8*exp (4*x)*log(x)^2)/(9*(x^2 + 4)) - (4*x^3*exp(x)*log(x)^3)/(3*(x^2 + 4)) + (1 2*x^5*exp(x)*log(x)^2)/(x^2 + 4)