Integrand size = 197, antiderivative size = 31 \[ \int \frac {e^{3 x} (8-72 x)+e^{4 x} \left (36 x-108 x^2+36 x^3\right )+\left (24 e^{3 x} x+e^{4 x} \left (-24 x+72 x^2-24 x^3\right )\right ) \log (x)+e^{4 x} \left (4 x-12 x^2+4 x^3\right ) \log ^2(x)}{16 x+e^x \left (48 x^2-24 x^3\right )+e^{2 x} \left (36 x^3-36 x^4+9 x^5\right )+\left (e^x \left (-16 x^2+8 x^3\right )+e^{2 x} \left (-24 x^3+24 x^4-6 x^5\right )\right ) \log (x)+e^{2 x} \left (4 x^3-4 x^4+x^5\right ) \log ^2(x)} \, dx=\frac {2 e^{2 x}}{x \left (-2+x+\frac {4 e^{-x}}{x (-3+\log (x))}\right )} \]
Time = 0.13 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13 \[ \int \frac {e^{3 x} (8-72 x)+e^{4 x} \left (36 x-108 x^2+36 x^3\right )+\left (24 e^{3 x} x+e^{4 x} \left (-24 x+72 x^2-24 x^3\right )\right ) \log (x)+e^{4 x} \left (4 x-12 x^2+4 x^3\right ) \log ^2(x)}{16 x+e^x \left (48 x^2-24 x^3\right )+e^{2 x} \left (36 x^3-36 x^4+9 x^5\right )+\left (e^x \left (-16 x^2+8 x^3\right )+e^{2 x} \left (-24 x^3+24 x^4-6 x^5\right )\right ) \log (x)+e^{2 x} \left (4 x^3-4 x^4+x^5\right ) \log ^2(x)} \, dx=\frac {4 e^{3 x} (-3+\log (x))}{8-6 e^x (-2+x) x+2 e^x (-2+x) x \log (x)} \]
Integrate[(E^(3*x)*(8 - 72*x) + E^(4*x)*(36*x - 108*x^2 + 36*x^3) + (24*E^ (3*x)*x + E^(4*x)*(-24*x + 72*x^2 - 24*x^3))*Log[x] + E^(4*x)*(4*x - 12*x^ 2 + 4*x^3)*Log[x]^2)/(16*x + E^x*(48*x^2 - 24*x^3) + E^(2*x)*(36*x^3 - 36* x^4 + 9*x^5) + (E^x*(-16*x^2 + 8*x^3) + E^(2*x)*(-24*x^3 + 24*x^4 - 6*x^5) )*Log[x] + E^(2*x)*(4*x^3 - 4*x^4 + x^5)*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.
| \(\displaystyle \int \frac {e^{4 x} \left (36 x^3-108 x^2+36 x\right )+e^{4 x} \left (4 x^3-12 x^2+4 x\right ) \log ^2(x)+\left (e^{4 x} \left (-24 x^3+72 x^2-24 x\right )+24 e^{3 x} x\right ) \log (x)+e^{3 x} (8-72 x)}{e^x \left (48 x^2-24 x^3\right )+e^{2 x} \left (9 x^5-36 x^4+36 x^3\right )+e^{2 x} \left (x^5-4 x^4+4 x^3\right ) \log ^2(x)+\left (e^x \left (8 x^3-16 x^2\right )+e^{2 x} \left (-6 x^5+24 x^4-24 x^3\right )\right ) \log (x)+16 x} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {4 e^{3 x} \left (9 e^x x^3-27 e^x x^2+e^x \left (x^2-3 x+1\right ) x \log ^2(x)-6 \left (e^x \left (x^2-3 x+1\right )-1\right ) x \log (x)+9 \left (e^x-2\right ) x+2\right )}{x \left (-3 e^x (x-2) x+e^x (x-2) x \log (x)+4\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 \int \frac {e^{3 x} \left (9 e^x x^3-27 e^x x^2+e^x \left (x^2-3 x+1\right ) \log ^2(x) x-9 \left (2-e^x\right ) x+6 \left (1-e^x \left (x^2-3 x+1\right )\right ) \log (x) x+2\right )}{x \left (3 e^x (2-x) x-e^x (2-x) \log (x) x+4\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 4 \int \left (\frac {e^{3 x} \left (x^2-3 x+1\right ) (\log (x)-3)}{(x-2) x \left (-3 e^x x^2+e^x \log (x) x^2+6 e^x x-2 e^x \log (x) x+4\right )}+\frac {2 e^{3 x} \left (\log (x) x^2-3 x^2+x-2 \log (x)+4\right )}{(x-2) x \left (-3 e^x x^2+e^x \log (x) x^2+6 e^x x-2 e^x \log (x) x+4\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle 4 \int \left (\frac {e^{3 x} \left (x^2-3 x+1\right ) (\log (x)-3)}{(x-2) x \left (-3 e^x x^2+e^x \log (x) x^2+6 e^x x-2 e^x \log (x) x+4\right )}+\frac {2 e^{3 x} \left (\log (x) x^2-3 x^2+x-2 \log (x)+4\right )}{(x-2) x \left (-3 e^x x^2+e^x \log (x) x^2+6 e^x x-2 e^x \log (x) x+4\right )^2}\right )dx\) |
Int[(E^(3*x)*(8 - 72*x) + E^(4*x)*(36*x - 108*x^2 + 36*x^3) + (24*E^(3*x)* x + E^(4*x)*(-24*x + 72*x^2 - 24*x^3))*Log[x] + E^(4*x)*(4*x - 12*x^2 + 4* x^3)*Log[x]^2)/(16*x + E^x*(48*x^2 - 24*x^3) + E^(2*x)*(36*x^3 - 36*x^4 + 9*x^5) + (E^x*(-16*x^2 + 8*x^3) + E^(2*x)*(-24*x^3 + 24*x^4 - 6*x^5))*Log[ x] + E^(2*x)*(4*x^3 - 4*x^4 + x^5)*Log[x]^2),x]
3.10.40.3.1 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_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.80 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35
| method | result | size |
| risch | \(\frac {2 \left (\ln \left (x \right )-3\right ) {\mathrm e}^{3 x}}{x^{2} {\mathrm e}^{x} \ln \left (x \right )-2 x \,{\mathrm e}^{x} \ln \left (x \right )-3 \,{\mathrm e}^{x} x^{2}+6 \,{\mathrm e}^{x} x +4}\) | \(42\) |
| parallelrisch | \(\frac {2 \,{\mathrm e}^{3 x} \ln \left (x \right )-6 \,{\mathrm e}^{3 x}}{x^{2} {\mathrm e}^{x} \ln \left (x \right )-2 x \,{\mathrm e}^{x} \ln \left (x \right )-3 \,{\mathrm e}^{x} x^{2}+6 \,{\mathrm e}^{x} x +4}\) | \(48\) |
int(((4*x^3-12*x^2+4*x)*exp(x)^4*ln(x)^2+((-24*x^3+72*x^2-24*x)*exp(x)^4+2 4*x*exp(x)^3)*ln(x)+(36*x^3-108*x^2+36*x)*exp(x)^4+(-72*x+8)*exp(x)^3)/((x ^5-4*x^4+4*x^3)*exp(x)^2*ln(x)^2+((-6*x^5+24*x^4-24*x^3)*exp(x)^2+(8*x^3-1 6*x^2)*exp(x))*ln(x)+(9*x^5-36*x^4+36*x^3)*exp(x)^2+(-24*x^3+48*x^2)*exp(x )+16*x),x,method=_RETURNVERBOSE)
Time = 0.28 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.39 \[ \int \frac {e^{3 x} (8-72 x)+e^{4 x} \left (36 x-108 x^2+36 x^3\right )+\left (24 e^{3 x} x+e^{4 x} \left (-24 x+72 x^2-24 x^3\right )\right ) \log (x)+e^{4 x} \left (4 x-12 x^2+4 x^3\right ) \log ^2(x)}{16 x+e^x \left (48 x^2-24 x^3\right )+e^{2 x} \left (36 x^3-36 x^4+9 x^5\right )+\left (e^x \left (-16 x^2+8 x^3\right )+e^{2 x} \left (-24 x^3+24 x^4-6 x^5\right )\right ) \log (x)+e^{2 x} \left (4 x^3-4 x^4+x^5\right ) \log ^2(x)} \, dx=\frac {2 \, {\left (e^{\left (3 \, x\right )} \log \left (x\right ) - 3 \, e^{\left (3 \, x\right )}\right )}}{{\left (x^{2} - 2 \, x\right )} e^{x} \log \left (x\right ) - 3 \, {\left (x^{2} - 2 \, x\right )} e^{x} + 4} \]
integrate(((4*x^3-12*x^2+4*x)*exp(x)^4*log(x)^2+((-24*x^3+72*x^2-24*x)*exp (x)^4+24*x*exp(x)^3)*log(x)+(36*x^3-108*x^2+36*x)*exp(x)^4+(-72*x+8)*exp(x )^3)/((x^5-4*x^4+4*x^3)*exp(x)^2*log(x)^2+((-6*x^5+24*x^4-24*x^3)*exp(x)^2 +(8*x^3-16*x^2)*exp(x))*log(x)+(9*x^5-36*x^4+36*x^3)*exp(x)^2+(-24*x^3+48* x^2)*exp(x)+16*x),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 447 vs. \(2 (26) = 52\).
Time = 0.61 (sec) , antiderivative size = 447, normalized size of antiderivative = 14.42 \[ \int \frac {e^{3 x} (8-72 x)+e^{4 x} \left (36 x-108 x^2+36 x^3\right )+\left (24 e^{3 x} x+e^{4 x} \left (-24 x+72 x^2-24 x^3\right )\right ) \log (x)+e^{4 x} \left (4 x-12 x^2+4 x^3\right ) \log ^2(x)}{16 x+e^x \left (48 x^2-24 x^3\right )+e^{2 x} \left (36 x^3-36 x^4+9 x^5\right )+\left (e^x \left (-16 x^2+8 x^3\right )+e^{2 x} \left (-24 x^3+24 x^4-6 x^5\right )\right ) \log (x)+e^{2 x} \left (4 x^3-4 x^4+x^5\right ) \log ^2(x)} \, dx=\frac {\left (- 8 x^{2} + 16 x\right ) e^{x} + \left (2 x^{4} \log {\left (x \right )} - 6 x^{4} - 8 x^{3} \log {\left (x \right )} + 24 x^{3} + 8 x^{2} \log {\left (x \right )} - 24 x^{2}\right ) e^{2 x}}{x^{6} \log {\left (x \right )} - 3 x^{6} - 6 x^{5} \log {\left (x \right )} + 18 x^{5} + 12 x^{4} \log {\left (x \right )} - 36 x^{4} - 8 x^{3} \log {\left (x \right )} + 24 x^{3}} - \frac {128}{4 x^{6} \log {\left (x \right )}^{2} - 24 x^{6} \log {\left (x \right )} + 36 x^{6} - 24 x^{5} \log {\left (x \right )}^{2} + 144 x^{5} \log {\left (x \right )} - 216 x^{5} + 48 x^{4} \log {\left (x \right )}^{2} - 288 x^{4} \log {\left (x \right )} + 432 x^{4} - 32 x^{3} \log {\left (x \right )}^{2} + 192 x^{3} \log {\left (x \right )} - 288 x^{3} + \left (x^{8} \log {\left (x \right )}^{3} - 9 x^{8} \log {\left (x \right )}^{2} + 27 x^{8} \log {\left (x \right )} - 27 x^{8} - 8 x^{7} \log {\left (x \right )}^{3} + 72 x^{7} \log {\left (x \right )}^{2} - 216 x^{7} \log {\left (x \right )} + 216 x^{7} + 24 x^{6} \log {\left (x \right )}^{3} - 216 x^{6} \log {\left (x \right )}^{2} + 648 x^{6} \log {\left (x \right )} - 648 x^{6} - 32 x^{5} \log {\left (x \right )}^{3} + 288 x^{5} \log {\left (x \right )}^{2} - 864 x^{5} \log {\left (x \right )} + 864 x^{5} + 16 x^{4} \log {\left (x \right )}^{3} - 144 x^{4} \log {\left (x \right )}^{2} + 432 x^{4} \log {\left (x \right )} - 432 x^{4}\right ) e^{x}} + \frac {32}{9 x^{6} - 54 x^{5} + 108 x^{4} - 72 x^{3} + \left (- 6 x^{6} + 36 x^{5} - 72 x^{4} + 48 x^{3}\right ) \log {\left (x \right )} + \left (x^{6} - 6 x^{5} + 12 x^{4} - 8 x^{3}\right ) \log {\left (x \right )}^{2}} \]
integrate(((4*x**3-12*x**2+4*x)*exp(x)**4*ln(x)**2+((-24*x**3+72*x**2-24*x )*exp(x)**4+24*x*exp(x)**3)*ln(x)+(36*x**3-108*x**2+36*x)*exp(x)**4+(-72*x +8)*exp(x)**3)/((x**5-4*x**4+4*x**3)*exp(x)**2*ln(x)**2+((-6*x**5+24*x**4- 24*x**3)*exp(x)**2+(8*x**3-16*x**2)*exp(x))*ln(x)+(9*x**5-36*x**4+36*x**3) *exp(x)**2+(-24*x**3+48*x**2)*exp(x)+16*x),x)
((-8*x**2 + 16*x)*exp(x) + (2*x**4*log(x) - 6*x**4 - 8*x**3*log(x) + 24*x* *3 + 8*x**2*log(x) - 24*x**2)*exp(2*x))/(x**6*log(x) - 3*x**6 - 6*x**5*log (x) + 18*x**5 + 12*x**4*log(x) - 36*x**4 - 8*x**3*log(x) + 24*x**3) - 128/ (4*x**6*log(x)**2 - 24*x**6*log(x) + 36*x**6 - 24*x**5*log(x)**2 + 144*x** 5*log(x) - 216*x**5 + 48*x**4*log(x)**2 - 288*x**4*log(x) + 432*x**4 - 32* x**3*log(x)**2 + 192*x**3*log(x) - 288*x**3 + (x**8*log(x)**3 - 9*x**8*log (x)**2 + 27*x**8*log(x) - 27*x**8 - 8*x**7*log(x)**3 + 72*x**7*log(x)**2 - 216*x**7*log(x) + 216*x**7 + 24*x**6*log(x)**3 - 216*x**6*log(x)**2 + 648 *x**6*log(x) - 648*x**6 - 32*x**5*log(x)**3 + 288*x**5*log(x)**2 - 864*x** 5*log(x) + 864*x**5 + 16*x**4*log(x)**3 - 144*x**4*log(x)**2 + 432*x**4*lo g(x) - 432*x**4)*exp(x)) + 32/(9*x**6 - 54*x**5 + 108*x**4 - 72*x**3 + (-6 *x**6 + 36*x**5 - 72*x**4 + 48*x**3)*log(x) + (x**6 - 6*x**5 + 12*x**4 - 8 *x**3)*log(x)**2)
Time = 0.29 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.19 \[ \int \frac {e^{3 x} (8-72 x)+e^{4 x} \left (36 x-108 x^2+36 x^3\right )+\left (24 e^{3 x} x+e^{4 x} \left (-24 x+72 x^2-24 x^3\right )\right ) \log (x)+e^{4 x} \left (4 x-12 x^2+4 x^3\right ) \log ^2(x)}{16 x+e^x \left (48 x^2-24 x^3\right )+e^{2 x} \left (36 x^3-36 x^4+9 x^5\right )+\left (e^x \left (-16 x^2+8 x^3\right )+e^{2 x} \left (-24 x^3+24 x^4-6 x^5\right )\right ) \log (x)+e^{2 x} \left (4 x^3-4 x^4+x^5\right ) \log ^2(x)} \, dx=-\frac {2 \, {\left (\log \left (x\right ) - 3\right )} e^{\left (3 \, x\right )}}{{\left (3 \, x^{2} - {\left (x^{2} - 2 \, x\right )} \log \left (x\right ) - 6 \, x\right )} e^{x} - 4} \]
integrate(((4*x^3-12*x^2+4*x)*exp(x)^4*log(x)^2+((-24*x^3+72*x^2-24*x)*exp (x)^4+24*x*exp(x)^3)*log(x)+(36*x^3-108*x^2+36*x)*exp(x)^4+(-72*x+8)*exp(x )^3)/((x^5-4*x^4+4*x^3)*exp(x)^2*log(x)^2+((-6*x^5+24*x^4-24*x^3)*exp(x)^2 +(8*x^3-16*x^2)*exp(x))*log(x)+(9*x^5-36*x^4+36*x^3)*exp(x)^2+(-24*x^3+48* x^2)*exp(x)+16*x),x, algorithm=\
Time = 0.33 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.52 \[ \int \frac {e^{3 x} (8-72 x)+e^{4 x} \left (36 x-108 x^2+36 x^3\right )+\left (24 e^{3 x} x+e^{4 x} \left (-24 x+72 x^2-24 x^3\right )\right ) \log (x)+e^{4 x} \left (4 x-12 x^2+4 x^3\right ) \log ^2(x)}{16 x+e^x \left (48 x^2-24 x^3\right )+e^{2 x} \left (36 x^3-36 x^4+9 x^5\right )+\left (e^x \left (-16 x^2+8 x^3\right )+e^{2 x} \left (-24 x^3+24 x^4-6 x^5\right )\right ) \log (x)+e^{2 x} \left (4 x^3-4 x^4+x^5\right ) \log ^2(x)} \, dx=\frac {2 \, {\left (e^{\left (3 \, x\right )} \log \left (x\right ) - 3 \, e^{\left (3 \, x\right )}\right )}}{x^{2} e^{x} \log \left (x\right ) - 3 \, x^{2} e^{x} - 2 \, x e^{x} \log \left (x\right ) + 6 \, x e^{x} + 4} \]
integrate(((4*x^3-12*x^2+4*x)*exp(x)^4*log(x)^2+((-24*x^3+72*x^2-24*x)*exp (x)^4+24*x*exp(x)^3)*log(x)+(36*x^3-108*x^2+36*x)*exp(x)^4+(-72*x+8)*exp(x )^3)/((x^5-4*x^4+4*x^3)*exp(x)^2*log(x)^2+((-6*x^5+24*x^4-24*x^3)*exp(x)^2 +(8*x^3-16*x^2)*exp(x))*log(x)+(9*x^5-36*x^4+36*x^3)*exp(x)^2+(-24*x^3+48* x^2)*exp(x)+16*x),x, algorithm=\
Timed out. \[ \int \frac {e^{3 x} (8-72 x)+e^{4 x} \left (36 x-108 x^2+36 x^3\right )+\left (24 e^{3 x} x+e^{4 x} \left (-24 x+72 x^2-24 x^3\right )\right ) \log (x)+e^{4 x} \left (4 x-12 x^2+4 x^3\right ) \log ^2(x)}{16 x+e^x \left (48 x^2-24 x^3\right )+e^{2 x} \left (36 x^3-36 x^4+9 x^5\right )+\left (e^x \left (-16 x^2+8 x^3\right )+e^{2 x} \left (-24 x^3+24 x^4-6 x^5\right )\right ) \log (x)+e^{2 x} \left (4 x^3-4 x^4+x^5\right ) \log ^2(x)} \, dx=-\int -\frac {{\mathrm {e}}^{4\,x}\,\left (4\,x^3-12\,x^2+4\,x\right )\,{\ln \left (x\right )}^2+\left (24\,x\,{\mathrm {e}}^{3\,x}-{\mathrm {e}}^{4\,x}\,\left (24\,x^3-72\,x^2+24\,x\right )\right )\,\ln \left (x\right )+{\mathrm {e}}^{4\,x}\,\left (36\,x^3-108\,x^2+36\,x\right )-{\mathrm {e}}^{3\,x}\,\left (72\,x-8\right )}{{\mathrm {e}}^{2\,x}\,\left (x^5-4\,x^4+4\,x^3\right )\,{\ln \left (x\right )}^2+\left (-{\mathrm {e}}^x\,\left (16\,x^2-8\,x^3\right )-{\mathrm {e}}^{2\,x}\,\left (6\,x^5-24\,x^4+24\,x^3\right )\right )\,\ln \left (x\right )+16\,x+{\mathrm {e}}^x\,\left (48\,x^2-24\,x^3\right )+{\mathrm {e}}^{2\,x}\,\left (9\,x^5-36\,x^4+36\,x^3\right )} \,d x \]
int((exp(4*x)*(36*x - 108*x^2 + 36*x^3) - exp(3*x)*(72*x - 8) + log(x)*(24 *x*exp(3*x) - exp(4*x)*(24*x - 72*x^2 + 24*x^3)) + exp(4*x)*log(x)^2*(4*x - 12*x^2 + 4*x^3))/(16*x + exp(x)*(48*x^2 - 24*x^3) - log(x)*(exp(x)*(16*x ^2 - 8*x^3) + exp(2*x)*(24*x^3 - 24*x^4 + 6*x^5)) + exp(2*x)*(36*x^3 - 36* x^4 + 9*x^5) + exp(2*x)*log(x)^2*(4*x^3 - 4*x^4 + x^5)),x)
-int(-(exp(4*x)*(36*x - 108*x^2 + 36*x^3) - exp(3*x)*(72*x - 8) + log(x)*( 24*x*exp(3*x) - exp(4*x)*(24*x - 72*x^2 + 24*x^3)) + exp(4*x)*log(x)^2*(4* x - 12*x^2 + 4*x^3))/(16*x + exp(x)*(48*x^2 - 24*x^3) - log(x)*(exp(x)*(16 *x^2 - 8*x^3) + exp(2*x)*(24*x^3 - 24*x^4 + 6*x^5)) + exp(2*x)*(36*x^3 - 3 6*x^4 + 9*x^5) + exp(2*x)*log(x)^2*(4*x^3 - 4*x^4 + x^5)), x)