Integrand size = 251, antiderivative size = 29 \[ \int \frac {48+e^{8 x} (3-x)+80 x+40 x^2-5 x^4-x^5+e^{6 x} \left (24+4 x-4 x^2\right )+e^{4 x} \left (72+48 x-6 x^2-6 x^3\right )+e^{2 x} \left (96+112 x+24 x^2-12 x^3-4 x^4\right )+\left (-112 x-144 x^2-48 x^3+4 x^4+3 x^5+e^{8 x} \left (-25 x+8 x^2\right )+e^{6 x} \left (-164 x-24 x^2+24 x^3\right )+e^{4 x} \left (-384 x-228 x^2+30 x^3+24 x^4\right )+e^{2 x} \left (-368 x-368 x^2-60 x^3+32 x^4+8 x^5\right )\right ) \log (x)+\left (54 x-36 x^2+6 x^3\right ) \log ^2(x)}{\left (27 x-18 x^2+3 x^3\right ) \log ^2(x)} \, dx=2 (4+x)+\frac {\left (2+e^{2 x}+x\right )^4}{3 (-3+x) \log (x)} \]
Time = 1.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97 \[ \int \frac {48+e^{8 x} (3-x)+80 x+40 x^2-5 x^4-x^5+e^{6 x} \left (24+4 x-4 x^2\right )+e^{4 x} \left (72+48 x-6 x^2-6 x^3\right )+e^{2 x} \left (96+112 x+24 x^2-12 x^3-4 x^4\right )+\left (-112 x-144 x^2-48 x^3+4 x^4+3 x^5+e^{8 x} \left (-25 x+8 x^2\right )+e^{6 x} \left (-164 x-24 x^2+24 x^3\right )+e^{4 x} \left (-384 x-228 x^2+30 x^3+24 x^4\right )+e^{2 x} \left (-368 x-368 x^2-60 x^3+32 x^4+8 x^5\right )\right ) \log (x)+\left (54 x-36 x^2+6 x^3\right ) \log ^2(x)}{\left (27 x-18 x^2+3 x^3\right ) \log ^2(x)} \, dx=\frac {1}{3} \left (6 x+\frac {\left (2+e^{2 x}+x\right )^4}{(-3+x) \log (x)}\right ) \]
Integrate[(48 + E^(8*x)*(3 - x) + 80*x + 40*x^2 - 5*x^4 - x^5 + E^(6*x)*(2 4 + 4*x - 4*x^2) + E^(4*x)*(72 + 48*x - 6*x^2 - 6*x^3) + E^(2*x)*(96 + 112 *x + 24*x^2 - 12*x^3 - 4*x^4) + (-112*x - 144*x^2 - 48*x^3 + 4*x^4 + 3*x^5 + E^(8*x)*(-25*x + 8*x^2) + E^(6*x)*(-164*x - 24*x^2 + 24*x^3) + E^(4*x)* (-384*x - 228*x^2 + 30*x^3 + 24*x^4) + E^(2*x)*(-368*x - 368*x^2 - 60*x^3 + 32*x^4 + 8*x^5))*Log[x] + (54*x - 36*x^2 + 6*x^3)*Log[x]^2)/((27*x - 18* x^2 + 3*x^3)*Log[x]^2),x]
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.
| \(\displaystyle \int \frac {-x^5-5 x^4+40 x^2+e^{6 x} \left (-4 x^2+4 x+24\right )+e^{4 x} \left (-6 x^3-6 x^2+48 x+72\right )+\left (6 x^3-36 x^2+54 x\right ) \log ^2(x)+e^{2 x} \left (-4 x^4-12 x^3+24 x^2+112 x+96\right )+\left (3 x^5+4 x^4-48 x^3-144 x^2+e^{8 x} \left (8 x^2-25 x\right )+e^{6 x} \left (24 x^3-24 x^2-164 x\right )+e^{4 x} \left (24 x^4+30 x^3-228 x^2-384 x\right )+e^{2 x} \left (8 x^5+32 x^4-60 x^3-368 x^2-368 x\right )-112 x\right ) \log (x)+80 x+e^{8 x} (3-x)+48}{\left (3 x^3-18 x^2+27 x\right ) \log ^2(x)} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {-x^5-5 x^4+40 x^2+e^{6 x} \left (-4 x^2+4 x+24\right )+e^{4 x} \left (-6 x^3-6 x^2+48 x+72\right )+\left (6 x^3-36 x^2+54 x\right ) \log ^2(x)+e^{2 x} \left (-4 x^4-12 x^3+24 x^2+112 x+96\right )+\left (3 x^5+4 x^4-48 x^3-144 x^2+e^{8 x} \left (8 x^2-25 x\right )+e^{6 x} \left (24 x^3-24 x^2-164 x\right )+e^{4 x} \left (24 x^4+30 x^3-228 x^2-384 x\right )+e^{2 x} \left (8 x^5+32 x^4-60 x^3-368 x^2-368 x\right )-112 x\right ) \log (x)+80 x+e^{8 x} (3-x)+48}{x \left (3 x^2-18 x+27\right ) \log ^2(x)}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{3} \left (-\frac {\left (x+e^{2 x}+2\right )^4}{(x-3) x \log ^2(x)}+\frac {\left (3 x+e^{2 x} (8 x-25)-14\right ) \left (x+e^{2 x}+2\right )^3}{(x-3)^2 \log (x)}+6\right )dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \int \left (\frac {\left (x+e^{2 x}+2\right )^4}{(3-x) x \log ^2(x)}-\frac {\left (e^{2 x} (25-8 x)-3 x+14\right ) \left (x+e^{2 x}+2\right )^3}{(3-x)^2 \log (x)}+6\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} \left (3 \int \frac {x^4}{(x-3)^2 \log (x)}dx+4 \int \frac {x^3}{(x-3)^2 \log (x)}dx+8 \int \frac {e^{2 x} x^2}{\log (x)}dx-48 \int \frac {x^2}{(x-3)^2 \log (x)}dx-36 \int \frac {e^{2 x}}{\log ^2(x)}dx-6 \int \frac {e^{4 x}}{\log ^2(x)}dx-\frac {500}{3} \int \frac {e^{2 x}}{(x-3) \log ^2(x)}dx-50 \int \frac {e^{4 x}}{(x-3) \log ^2(x)}dx-\frac {20}{3} \int \frac {e^{6 x}}{(x-3) \log ^2(x)}dx-\frac {1}{3} \int \frac {e^{8 x}}{(x-3) \log ^2(x)}dx+\frac {32}{3} \int \frac {e^{2 x}}{x \log ^2(x)}dx+8 \int \frac {e^{4 x}}{x \log ^2(x)}dx+\frac {8}{3} \int \frac {e^{6 x}}{x \log ^2(x)}dx+\frac {1}{3} \int \frac {e^{8 x}}{x \log ^2(x)}dx-4 \int \frac {e^{2 x} x}{\log ^2(x)}dx-\int \frac {(x+2)^4}{(x-3) x \log ^2(x)}dx+348 \int \frac {e^{2 x}}{\log (x)}dx+174 \int \frac {e^{4 x}}{\log (x)}dx+24 \int \frac {e^{6 x}}{\log (x)}dx-112 \int \frac {1}{(x-3)^2 \log (x)}dx-500 \int \frac {e^{2 x}}{(x-3)^2 \log (x)}dx-150 \int \frac {e^{4 x}}{(x-3)^2 \log (x)}dx-20 \int \frac {e^{6 x}}{(x-3)^2 \log (x)}dx-\int \frac {e^{8 x}}{(x-3)^2 \log (x)}dx+1000 \int \frac {e^{2 x}}{(x-3) \log (x)}dx+600 \int \frac {e^{4 x}}{(x-3) \log (x)}dx+120 \int \frac {e^{6 x}}{(x-3) \log (x)}dx+8 \int \frac {e^{8 x}}{(x-3) \log (x)}dx+80 \int \frac {e^{2 x} x}{\log (x)}dx+24 \int \frac {e^{4 x} x}{\log (x)}dx-144 \int \frac {x}{(x-3)^2 \log (x)}dx+6 x\right )\) |
Int[(48 + E^(8*x)*(3 - x) + 80*x + 40*x^2 - 5*x^4 - x^5 + E^(6*x)*(24 + 4* x - 4*x^2) + E^(4*x)*(72 + 48*x - 6*x^2 - 6*x^3) + E^(2*x)*(96 + 112*x + 2 4*x^2 - 12*x^3 - 4*x^4) + (-112*x - 144*x^2 - 48*x^3 + 4*x^4 + 3*x^5 + E^( 8*x)*(-25*x + 8*x^2) + E^(6*x)*(-164*x - 24*x^2 + 24*x^3) + E^(4*x)*(-384* x - 228*x^2 + 30*x^3 + 24*x^4) + E^(2*x)*(-368*x - 368*x^2 - 60*x^3 + 32*x ^4 + 8*x^5))*Log[x] + (54*x - 36*x^2 + 6*x^3)*Log[x]^2)/((27*x - 18*x^2 + 3*x^3)*Log[x]^2),x]
3.8.63.3.1 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[(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[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(103\) vs. \(2(25)=50\).
Time = 19.96 (sec) , antiderivative size = 104, normalized size of antiderivative = 3.59
| method | result | size |
| risch | \(2 x +\frac {x^{4}+4 \,{\mathrm e}^{2 x} x^{3}+6 x^{2} {\mathrm e}^{4 x}+4 x \,{\mathrm e}^{6 x}+{\mathrm e}^{8 x}+8 x^{3}+24 \,{\mathrm e}^{2 x} x^{2}+24 x \,{\mathrm e}^{4 x}+8 \,{\mathrm e}^{6 x}+24 x^{2}+48 x \,{\mathrm e}^{2 x}+24 \,{\mathrm e}^{4 x}+32 x +32 \,{\mathrm e}^{2 x}+16}{3 \left (-3+x \right ) \ln \left (x \right )}\) | \(104\) |
| parallelrisch | \(\frac {3 x^{4}+12 \,{\mathrm e}^{2 x} x^{3}+18 x^{2} {\mathrm e}^{4 x}+12 x \,{\mathrm e}^{6 x}+3 \,{\mathrm e}^{8 x}+24 x^{3}+18 x^{2} \ln \left (x \right )+72 \,{\mathrm e}^{2 x} x^{2}+48+72 x \,{\mathrm e}^{4 x}+24 \,{\mathrm e}^{6 x}+72 x^{2}+144 x \,{\mathrm e}^{2 x}+72 \,{\mathrm e}^{4 x}+96 x -162 \ln \left (x \right )+96 \,{\mathrm e}^{2 x}}{9 \ln \left (x \right ) \left (-3+x \right )}\) | \(127\) |
int(((6*x^3-36*x^2+54*x)*ln(x)^2+((8*x^2-25*x)*exp(2*x)^4+(24*x^3-24*x^2-1 64*x)*exp(2*x)^3+(24*x^4+30*x^3-228*x^2-384*x)*exp(2*x)^2+(8*x^5+32*x^4-60 *x^3-368*x^2-368*x)*exp(2*x)+3*x^5+4*x^4-48*x^3-144*x^2-112*x)*ln(x)+(-x+3 )*exp(2*x)^4+(-4*x^2+4*x+24)*exp(2*x)^3+(-6*x^3-6*x^2+48*x+72)*exp(2*x)^2+ (-4*x^4-12*x^3+24*x^2+112*x+96)*exp(2*x)-x^5-5*x^4+40*x^2+80*x+48)/(3*x^3- 18*x^2+27*x)/ln(x)^2,x,method=_RETURNVERBOSE)
2*x+1/3*(x^4+4*exp(2*x)*x^3+6*x^2*exp(4*x)+4*x*exp(6*x)+exp(8*x)+8*x^3+24* exp(2*x)*x^2+24*x*exp(4*x)+8*exp(6*x)+24*x^2+48*x*exp(2*x)+24*exp(4*x)+32* x+32*exp(2*x)+16)/(-3+x)/ln(x)
Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (25) = 50\).
Time = 0.29 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.97 \[ \int \frac {48+e^{8 x} (3-x)+80 x+40 x^2-5 x^4-x^5+e^{6 x} \left (24+4 x-4 x^2\right )+e^{4 x} \left (72+48 x-6 x^2-6 x^3\right )+e^{2 x} \left (96+112 x+24 x^2-12 x^3-4 x^4\right )+\left (-112 x-144 x^2-48 x^3+4 x^4+3 x^5+e^{8 x} \left (-25 x+8 x^2\right )+e^{6 x} \left (-164 x-24 x^2+24 x^3\right )+e^{4 x} \left (-384 x-228 x^2+30 x^3+24 x^4\right )+e^{2 x} \left (-368 x-368 x^2-60 x^3+32 x^4+8 x^5\right )\right ) \log (x)+\left (54 x-36 x^2+6 x^3\right ) \log ^2(x)}{\left (27 x-18 x^2+3 x^3\right ) \log ^2(x)} \, dx=\frac {x^{4} + 8 \, x^{3} + 24 \, x^{2} + 4 \, {\left (x + 2\right )} e^{\left (6 \, x\right )} + 6 \, {\left (x^{2} + 4 \, x + 4\right )} e^{\left (4 \, x\right )} + 4 \, {\left (x^{3} + 6 \, x^{2} + 12 \, x + 8\right )} e^{\left (2 \, x\right )} + 6 \, {\left (x^{2} - 3 \, x\right )} \log \left (x\right ) + 32 \, x + e^{\left (8 \, x\right )} + 16}{3 \, {\left (x - 3\right )} \log \left (x\right )} \]
integrate(((6*x^3-36*x^2+54*x)*log(x)^2+((8*x^2-25*x)*exp(2*x)^4+(24*x^3-2 4*x^2-164*x)*exp(2*x)^3+(24*x^4+30*x^3-228*x^2-384*x)*exp(2*x)^2+(8*x^5+32 *x^4-60*x^3-368*x^2-368*x)*exp(2*x)+3*x^5+4*x^4-48*x^3-144*x^2-112*x)*log( x)+(-x+3)*exp(2*x)^4+(-4*x^2+4*x+24)*exp(2*x)^3+(-6*x^3-6*x^2+48*x+72)*exp (2*x)^2+(-4*x^4-12*x^3+24*x^2+112*x+96)*exp(2*x)-x^5-5*x^4+40*x^2+80*x+48) /(3*x^3-18*x^2+27*x)/log(x)^2,x, algorithm=\
1/3*(x^4 + 8*x^3 + 24*x^2 + 4*(x + 2)*e^(6*x) + 6*(x^2 + 4*x + 4)*e^(4*x) + 4*(x^3 + 6*x^2 + 12*x + 8)*e^(2*x) + 6*(x^2 - 3*x)*log(x) + 32*x + e^(8* x) + 16)/((x - 3)*log(x))
Leaf count of result is larger than twice the leaf count of optimal. 291 vs. \(2 (22) = 44\).
Time = 0.35 (sec) , antiderivative size = 291, normalized size of antiderivative = 10.03 \[ \int \frac {48+e^{8 x} (3-x)+80 x+40 x^2-5 x^4-x^5+e^{6 x} \left (24+4 x-4 x^2\right )+e^{4 x} \left (72+48 x-6 x^2-6 x^3\right )+e^{2 x} \left (96+112 x+24 x^2-12 x^3-4 x^4\right )+\left (-112 x-144 x^2-48 x^3+4 x^4+3 x^5+e^{8 x} \left (-25 x+8 x^2\right )+e^{6 x} \left (-164 x-24 x^2+24 x^3\right )+e^{4 x} \left (-384 x-228 x^2+30 x^3+24 x^4\right )+e^{2 x} \left (-368 x-368 x^2-60 x^3+32 x^4+8 x^5\right )\right ) \log (x)+\left (54 x-36 x^2+6 x^3\right ) \log ^2(x)}{\left (27 x-18 x^2+3 x^3\right ) \log ^2(x)} \, dx=2 x + \frac {\left (9 x^{3} \log {\left (x \right )}^{3} - 81 x^{2} \log {\left (x \right )}^{3} + 243 x \log {\left (x \right )}^{3} - 243 \log {\left (x \right )}^{3}\right ) e^{8 x} + \left (36 x^{4} \log {\left (x \right )}^{3} - 252 x^{3} \log {\left (x \right )}^{3} + 324 x^{2} \log {\left (x \right )}^{3} + 972 x \log {\left (x \right )}^{3} - 1944 \log {\left (x \right )}^{3}\right ) e^{6 x} + \left (54 x^{5} \log {\left (x \right )}^{3} - 270 x^{4} \log {\left (x \right )}^{3} - 270 x^{3} \log {\left (x \right )}^{3} + 2430 x^{2} \log {\left (x \right )}^{3} - 5832 \log {\left (x \right )}^{3}\right ) e^{4 x} + \left (36 x^{6} \log {\left (x \right )}^{3} - 108 x^{5} \log {\left (x \right )}^{3} - 540 x^{4} \log {\left (x \right )}^{3} + 1260 x^{3} \log {\left (x \right )}^{3} + 3240 x^{2} \log {\left (x \right )}^{3} - 3888 x \log {\left (x \right )}^{3} - 7776 \log {\left (x \right )}^{3}\right ) e^{2 x}}{27 x^{4} \log {\left (x \right )}^{4} - 324 x^{3} \log {\left (x \right )}^{4} + 1458 x^{2} \log {\left (x \right )}^{4} - 2916 x \log {\left (x \right )}^{4} + 2187 \log {\left (x \right )}^{4}} + \frac {x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 16}{\left (3 x - 9\right ) \log {\left (x \right )}} \]
integrate(((6*x**3-36*x**2+54*x)*ln(x)**2+((8*x**2-25*x)*exp(2*x)**4+(24*x **3-24*x**2-164*x)*exp(2*x)**3+(24*x**4+30*x**3-228*x**2-384*x)*exp(2*x)** 2+(8*x**5+32*x**4-60*x**3-368*x**2-368*x)*exp(2*x)+3*x**5+4*x**4-48*x**3-1 44*x**2-112*x)*ln(x)+(-x+3)*exp(2*x)**4+(-4*x**2+4*x+24)*exp(2*x)**3+(-6*x **3-6*x**2+48*x+72)*exp(2*x)**2+(-4*x**4-12*x**3+24*x**2+112*x+96)*exp(2*x )-x**5-5*x**4+40*x**2+80*x+48)/(3*x**3-18*x**2+27*x)/ln(x)**2,x)
2*x + ((9*x**3*log(x)**3 - 81*x**2*log(x)**3 + 243*x*log(x)**3 - 243*log(x )**3)*exp(8*x) + (36*x**4*log(x)**3 - 252*x**3*log(x)**3 + 324*x**2*log(x) **3 + 972*x*log(x)**3 - 1944*log(x)**3)*exp(6*x) + (54*x**5*log(x)**3 - 27 0*x**4*log(x)**3 - 270*x**3*log(x)**3 + 2430*x**2*log(x)**3 - 5832*log(x)* *3)*exp(4*x) + (36*x**6*log(x)**3 - 108*x**5*log(x)**3 - 540*x**4*log(x)** 3 + 1260*x**3*log(x)**3 + 3240*x**2*log(x)**3 - 3888*x*log(x)**3 - 7776*lo g(x)**3)*exp(2*x))/(27*x**4*log(x)**4 - 324*x**3*log(x)**4 + 1458*x**2*log (x)**4 - 2916*x*log(x)**4 + 2187*log(x)**4) + (x**4 + 8*x**3 + 24*x**2 + 3 2*x + 16)/((3*x - 9)*log(x))
Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (25) = 50\).
Time = 0.25 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.97 \[ \int \frac {48+e^{8 x} (3-x)+80 x+40 x^2-5 x^4-x^5+e^{6 x} \left (24+4 x-4 x^2\right )+e^{4 x} \left (72+48 x-6 x^2-6 x^3\right )+e^{2 x} \left (96+112 x+24 x^2-12 x^3-4 x^4\right )+\left (-112 x-144 x^2-48 x^3+4 x^4+3 x^5+e^{8 x} \left (-25 x+8 x^2\right )+e^{6 x} \left (-164 x-24 x^2+24 x^3\right )+e^{4 x} \left (-384 x-228 x^2+30 x^3+24 x^4\right )+e^{2 x} \left (-368 x-368 x^2-60 x^3+32 x^4+8 x^5\right )\right ) \log (x)+\left (54 x-36 x^2+6 x^3\right ) \log ^2(x)}{\left (27 x-18 x^2+3 x^3\right ) \log ^2(x)} \, dx=\frac {x^{4} + 8 \, x^{3} + 24 \, x^{2} + 4 \, {\left (x + 2\right )} e^{\left (6 \, x\right )} + 6 \, {\left (x^{2} + 4 \, x + 4\right )} e^{\left (4 \, x\right )} + 4 \, {\left (x^{3} + 6 \, x^{2} + 12 \, x + 8\right )} e^{\left (2 \, x\right )} + 6 \, {\left (x^{2} - 3 \, x\right )} \log \left (x\right ) + 32 \, x + e^{\left (8 \, x\right )} + 16}{3 \, {\left (x - 3\right )} \log \left (x\right )} \]
integrate(((6*x^3-36*x^2+54*x)*log(x)^2+((8*x^2-25*x)*exp(2*x)^4+(24*x^3-2 4*x^2-164*x)*exp(2*x)^3+(24*x^4+30*x^3-228*x^2-384*x)*exp(2*x)^2+(8*x^5+32 *x^4-60*x^3-368*x^2-368*x)*exp(2*x)+3*x^5+4*x^4-48*x^3-144*x^2-112*x)*log( x)+(-x+3)*exp(2*x)^4+(-4*x^2+4*x+24)*exp(2*x)^3+(-6*x^3-6*x^2+48*x+72)*exp (2*x)^2+(-4*x^4-12*x^3+24*x^2+112*x+96)*exp(2*x)-x^5-5*x^4+40*x^2+80*x+48) /(3*x^3-18*x^2+27*x)/log(x)^2,x, algorithm=\
1/3*(x^4 + 8*x^3 + 24*x^2 + 4*(x + 2)*e^(6*x) + 6*(x^2 + 4*x + 4)*e^(4*x) + 4*(x^3 + 6*x^2 + 12*x + 8)*e^(2*x) + 6*(x^2 - 3*x)*log(x) + 32*x + e^(8* x) + 16)/((x - 3)*log(x))
Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (25) = 50\).
Time = 0.27 (sec) , antiderivative size = 113, normalized size of antiderivative = 3.90 \[ \int \frac {48+e^{8 x} (3-x)+80 x+40 x^2-5 x^4-x^5+e^{6 x} \left (24+4 x-4 x^2\right )+e^{4 x} \left (72+48 x-6 x^2-6 x^3\right )+e^{2 x} \left (96+112 x+24 x^2-12 x^3-4 x^4\right )+\left (-112 x-144 x^2-48 x^3+4 x^4+3 x^5+e^{8 x} \left (-25 x+8 x^2\right )+e^{6 x} \left (-164 x-24 x^2+24 x^3\right )+e^{4 x} \left (-384 x-228 x^2+30 x^3+24 x^4\right )+e^{2 x} \left (-368 x-368 x^2-60 x^3+32 x^4+8 x^5\right )\right ) \log (x)+\left (54 x-36 x^2+6 x^3\right ) \log ^2(x)}{\left (27 x-18 x^2+3 x^3\right ) \log ^2(x)} \, dx=\frac {x^{4} + 4 \, x^{3} e^{\left (2 \, x\right )} + 8 \, x^{3} + 6 \, x^{2} e^{\left (4 \, x\right )} + 24 \, x^{2} e^{\left (2 \, x\right )} + 6 \, x^{2} \log \left (x\right ) + 24 \, x^{2} + 4 \, x e^{\left (6 \, x\right )} + 24 \, x e^{\left (4 \, x\right )} + 48 \, x e^{\left (2 \, x\right )} - 18 \, x \log \left (x\right ) + 32 \, x + e^{\left (8 \, x\right )} + 8 \, e^{\left (6 \, x\right )} + 24 \, e^{\left (4 \, x\right )} + 32 \, e^{\left (2 \, x\right )} + 16}{3 \, {\left (x \log \left (x\right ) - 3 \, \log \left (x\right )\right )}} \]
integrate(((6*x^3-36*x^2+54*x)*log(x)^2+((8*x^2-25*x)*exp(2*x)^4+(24*x^3-2 4*x^2-164*x)*exp(2*x)^3+(24*x^4+30*x^3-228*x^2-384*x)*exp(2*x)^2+(8*x^5+32 *x^4-60*x^3-368*x^2-368*x)*exp(2*x)+3*x^5+4*x^4-48*x^3-144*x^2-112*x)*log( x)+(-x+3)*exp(2*x)^4+(-4*x^2+4*x+24)*exp(2*x)^3+(-6*x^3-6*x^2+48*x+72)*exp (2*x)^2+(-4*x^4-12*x^3+24*x^2+112*x+96)*exp(2*x)-x^5-5*x^4+40*x^2+80*x+48) /(3*x^3-18*x^2+27*x)/log(x)^2,x, algorithm=\
1/3*(x^4 + 4*x^3*e^(2*x) + 8*x^3 + 6*x^2*e^(4*x) + 24*x^2*e^(2*x) + 6*x^2* log(x) + 24*x^2 + 4*x*e^(6*x) + 24*x*e^(4*x) + 48*x*e^(2*x) - 18*x*log(x) + 32*x + e^(8*x) + 8*e^(6*x) + 24*e^(4*x) + 32*e^(2*x) + 16)/(x*log(x) - 3 *log(x))
Time = 9.27 (sec) , antiderivative size = 217, normalized size of antiderivative = 7.48 \[ \int \frac {48+e^{8 x} (3-x)+80 x+40 x^2-5 x^4-x^5+e^{6 x} \left (24+4 x-4 x^2\right )+e^{4 x} \left (72+48 x-6 x^2-6 x^3\right )+e^{2 x} \left (96+112 x+24 x^2-12 x^3-4 x^4\right )+\left (-112 x-144 x^2-48 x^3+4 x^4+3 x^5+e^{8 x} \left (-25 x+8 x^2\right )+e^{6 x} \left (-164 x-24 x^2+24 x^3\right )+e^{4 x} \left (-384 x-228 x^2+30 x^3+24 x^4\right )+e^{2 x} \left (-368 x-368 x^2-60 x^3+32 x^4+8 x^5\right )\right ) \log (x)+\left (54 x-36 x^2+6 x^3\right ) \log ^2(x)}{\left (27 x-18 x^2+3 x^3\right ) \log ^2(x)} \, dx=21\,x-\frac {625\,x}{3\,\left (x^2-6\,x+9\right )}+\frac {\frac {{\left (x+{\mathrm {e}}^{2\,x}+2\right )}^4}{3\,\left (x-3\right )}-\frac {x\,\ln \left (x\right )\,{\left (x+{\mathrm {e}}^{2\,x}+2\right )}^3\,\left (3\,x-25\,{\mathrm {e}}^{2\,x}+8\,x\,{\mathrm {e}}^{2\,x}-14\right )}{3\,{\left (x-3\right )}^2}}{\ln \left (x\right )}+\frac {22\,x^2}{3}+x^3-\frac {{\mathrm {e}}^{2\,x}\,\left (-\frac {8\,x^5}{3}-\frac {32\,x^4}{3}+20\,x^3+\frac {368\,x^2}{3}+\frac {368\,x}{3}\right )}{x^2-6\,x+9}-\frac {{\mathrm {e}}^{8\,x}\,\left (\frac {25\,x}{3}-\frac {8\,x^2}{3}\right )}{x^2-6\,x+9}-\frac {{\mathrm {e}}^{6\,x}\,\left (-8\,x^3+8\,x^2+\frac {164\,x}{3}\right )}{x^2-6\,x+9}-\frac {{\mathrm {e}}^{4\,x}\,\left (-8\,x^4-10\,x^3+76\,x^2+128\,x\right )}{x^2-6\,x+9} \]
int((80*x + exp(6*x)*(4*x - 4*x^2 + 24) + exp(4*x)*(48*x - 6*x^2 - 6*x^3 + 72) + log(x)^2*(54*x - 36*x^2 + 6*x^3) - exp(8*x)*(x - 3) + exp(2*x)*(112 *x + 24*x^2 - 12*x^3 - 4*x^4 + 96) - log(x)*(112*x + exp(8*x)*(25*x - 8*x^ 2) + exp(6*x)*(164*x + 24*x^2 - 24*x^3) + exp(4*x)*(384*x + 228*x^2 - 30*x ^3 - 24*x^4) + exp(2*x)*(368*x + 368*x^2 + 60*x^3 - 32*x^4 - 8*x^5) + 144* x^2 + 48*x^3 - 4*x^4 - 3*x^5) + 40*x^2 - 5*x^4 - x^5 + 48)/(log(x)^2*(27*x - 18*x^2 + 3*x^3)),x)
21*x - (625*x)/(3*(x^2 - 6*x + 9)) + ((x + exp(2*x) + 2)^4/(3*(x - 3)) - ( x*log(x)*(x + exp(2*x) + 2)^3*(3*x - 25*exp(2*x) + 8*x*exp(2*x) - 14))/(3* (x - 3)^2))/log(x) + (22*x^2)/3 + x^3 - (exp(2*x)*((368*x)/3 + (368*x^2)/3 + 20*x^3 - (32*x^4)/3 - (8*x^5)/3))/(x^2 - 6*x + 9) - (exp(8*x)*((25*x)/3 - (8*x^2)/3))/(x^2 - 6*x + 9) - (exp(6*x)*((164*x)/3 + 8*x^2 - 8*x^3))/(x ^2 - 6*x + 9) - (exp(4*x)*(128*x + 76*x^2 - 10*x^3 - 8*x^4))/(x^2 - 6*x + 9)