Integrand size = 164, antiderivative size = 35 \[ \int \frac {e^{2 x} (9-5 x)+e^x \left (54-84 x+30 x^2\right )+\left (-5 e^{2 x} x+e^x \left (78 x-84 x^2+30 x^3\right )\right ) \log (x)+e^x \left (162 x-180 x^2+50 x^3\right ) \log ^2(x)}{36 x+e^{2 x} x-72 x^2+36 x^3+e^x \left (12 x-12 x^2\right )+\left (216 x-336 x^2+120 x^3+e^x \left (36 x-20 x^2\right )\right ) \log (x)+\left (324 x-360 x^2+100 x^3\right ) \log ^2(x)} \, dx=\frac {e^x}{2+\frac {e^x-3 (-2+2 x)}{(4+5 (1-x)) \log (x)}} \]
Time = 5.09 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \frac {e^{2 x} (9-5 x)+e^x \left (54-84 x+30 x^2\right )+\left (-5 e^{2 x} x+e^x \left (78 x-84 x^2+30 x^3\right )\right ) \log (x)+e^x \left (162 x-180 x^2+50 x^3\right ) \log ^2(x)}{36 x+e^{2 x} x-72 x^2+36 x^3+e^x \left (12 x-12 x^2\right )+\left (216 x-336 x^2+120 x^3+e^x \left (36 x-20 x^2\right )\right ) \log (x)+\left (324 x-360 x^2+100 x^3\right ) \log ^2(x)} \, dx=-\frac {e^x (-9+5 x) \log (x)}{6+e^x-6 x-2 (-9+5 x) \log (x)} \]
Integrate[(E^(2*x)*(9 - 5*x) + E^x*(54 - 84*x + 30*x^2) + (-5*E^(2*x)*x + E^x*(78*x - 84*x^2 + 30*x^3))*Log[x] + E^x*(162*x - 180*x^2 + 50*x^3)*Log[ x]^2)/(36*x + E^(2*x)*x - 72*x^2 + 36*x^3 + E^x*(12*x - 12*x^2) + (216*x - 336*x^2 + 120*x^3 + E^x*(36*x - 20*x^2))*Log[x] + (324*x - 360*x^2 + 100* 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.
| \(\displaystyle \int \frac {e^x \left (30 x^2-84 x+54\right )+e^x \left (50 x^3-180 x^2+162 x\right ) \log ^2(x)+\left (e^x \left (30 x^3-84 x^2+78 x\right )-5 e^{2 x} x\right ) \log (x)+e^{2 x} (9-5 x)}{36 x^3-72 x^2+e^x \left (12 x-12 x^2\right )+\left (100 x^3-360 x^2+324 x\right ) \log ^2(x)+\left (120 x^3-336 x^2+e^x \left (36 x-20 x^2\right )+216 x\right ) \log (x)+e^{2 x} x+36 x} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^x \left (x \left (30 x^2-84 x-5 e^x+78\right ) \log (x)-\left (-6 x+e^x+6\right ) (5 x-9)+2 (9-5 x)^2 x \log ^2(x)\right )}{x \left (-6 x+e^x-2 (5 x-9) \log (x)+6\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 e^x (5 x-9) \log (x) \left (3 x^2+5 x^2 \log (x)-11 x-14 x \log (x)+9\right )}{x \left (6 x-e^x+10 x \log (x)-18 \log (x)-6\right )^2}+\frac {e^x (5 x+5 x \log (x)-9)}{x \left (6 x-e^x+10 x \log (x)-18 \log (x)-6\right )}\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 e^x (5 x-9) \log (x) \left (3 x^2+5 x^2 \log (x)-11 x-14 x \log (x)+9\right )}{x \left (-6 x+e^x-10 x \log (x)+18 \log (x)+6\right )^2}-\frac {e^x (5 x+5 x \log (x)-9)}{x \left (-6 x+e^x-10 x \log (x)+18 \log (x)+6\right )}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {2 e^x (5 x-9) \log (x) \left (3 x^2+5 x^2 \log (x)-11 x-14 x \log (x)+9\right )}{x \left (-6 x+e^x-10 x \log (x)+18 \log (x)+6\right )^2}-\frac {e^x (5 x+5 x \log (x)-9)}{x \left (-6 x+e^x-10 x \log (x)+18 \log (x)+6\right )}\right )dx\) |
Int[(E^(2*x)*(9 - 5*x) + E^x*(54 - 84*x + 30*x^2) + (-5*E^(2*x)*x + E^x*(7 8*x - 84*x^2 + 30*x^3))*Log[x] + E^x*(162*x - 180*x^2 + 50*x^3)*Log[x]^2)/ (36*x + E^(2*x)*x - 72*x^2 + 36*x^3 + E^x*(12*x - 12*x^2) + (216*x - 336*x ^2 + 120*x^3 + E^x*(36*x - 20*x^2))*Log[x] + (324*x - 360*x^2 + 100*x^3)*L og[x]^2),x]
3.24.54.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.34 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.03
| method | result | size |
| parallelrisch | \(\frac {5 x \,{\mathrm e}^{x} \ln \left (x \right )-9 \,{\mathrm e}^{x} \ln \left (x \right )}{10 x \ln \left (x \right )-18 \ln \left (x \right )-{\mathrm e}^{x}+6 x -6}\) | \(36\) |
| risch | \(\frac {{\mathrm e}^{x}}{2}-\frac {\left (6 x -{\mathrm e}^{x}-6\right ) {\mathrm e}^{x}}{2 \left (10 x \ln \left (x \right )-18 \ln \left (x \right )-{\mathrm e}^{x}+6 x -6\right )}\) | \(39\) |
int(((50*x^3-180*x^2+162*x)*exp(x)*ln(x)^2+(-5*x*exp(x)^2+(30*x^3-84*x^2+7 8*x)*exp(x))*ln(x)+(-5*x+9)*exp(x)^2+(30*x^2-84*x+54)*exp(x))/((100*x^3-36 0*x^2+324*x)*ln(x)^2+((-20*x^2+36*x)*exp(x)+120*x^3-336*x^2+216*x)*ln(x)+x *exp(x)^2+(-12*x^2+12*x)*exp(x)+36*x^3-72*x^2+36*x),x,method=_RETURNVERBOS E)
Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.86 \[ \int \frac {e^{2 x} (9-5 x)+e^x \left (54-84 x+30 x^2\right )+\left (-5 e^{2 x} x+e^x \left (78 x-84 x^2+30 x^3\right )\right ) \log (x)+e^x \left (162 x-180 x^2+50 x^3\right ) \log ^2(x)}{36 x+e^{2 x} x-72 x^2+36 x^3+e^x \left (12 x-12 x^2\right )+\left (216 x-336 x^2+120 x^3+e^x \left (36 x-20 x^2\right )\right ) \log (x)+\left (324 x-360 x^2+100 x^3\right ) \log ^2(x)} \, dx=\frac {{\left (5 \, x - 9\right )} e^{x} \log \left (x\right )}{2 \, {\left (5 \, x - 9\right )} \log \left (x\right ) + 6 \, x - e^{x} - 6} \]
integrate(((50*x^3-180*x^2+162*x)*exp(x)*log(x)^2+(-5*x*exp(x)^2+(30*x^3-8 4*x^2+78*x)*exp(x))*log(x)+(-5*x+9)*exp(x)^2+(30*x^2-84*x+54)*exp(x))/((10 0*x^3-360*x^2+324*x)*log(x)^2+((-20*x^2+36*x)*exp(x)+120*x^3-336*x^2+216*x )*log(x)+x*exp(x)^2+(-12*x^2+12*x)*exp(x)+36*x^3-72*x^2+36*x),x, algorithm =\
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (20) = 40\).
Time = 0.13 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.17 \[ \int \frac {e^{2 x} (9-5 x)+e^x \left (54-84 x+30 x^2\right )+\left (-5 e^{2 x} x+e^x \left (78 x-84 x^2+30 x^3\right )\right ) \log (x)+e^x \left (162 x-180 x^2+50 x^3\right ) \log ^2(x)}{36 x+e^{2 x} x-72 x^2+36 x^3+e^x \left (12 x-12 x^2\right )+\left (216 x-336 x^2+120 x^3+e^x \left (36 x-20 x^2\right )\right ) \log (x)+\left (324 x-360 x^2+100 x^3\right ) \log ^2(x)} \, dx=- 5 x \log {\left (x \right )} + 9 \log {\left (x \right )} + \frac {- 50 x^{2} \log {\left (x \right )}^{2} - 30 x^{2} \log {\left (x \right )} + 180 x \log {\left (x \right )}^{2} + 84 x \log {\left (x \right )} - 162 \log {\left (x \right )}^{2} - 54 \log {\left (x \right )}}{- 10 x \log {\left (x \right )} - 6 x + e^{x} + 18 \log {\left (x \right )} + 6} \]
integrate(((50*x**3-180*x**2+162*x)*exp(x)*ln(x)**2+(-5*x*exp(x)**2+(30*x* *3-84*x**2+78*x)*exp(x))*ln(x)+(-5*x+9)*exp(x)**2+(30*x**2-84*x+54)*exp(x) )/((100*x**3-360*x**2+324*x)*ln(x)**2+((-20*x**2+36*x)*exp(x)+120*x**3-336 *x**2+216*x)*ln(x)+x*exp(x)**2+(-12*x**2+12*x)*exp(x)+36*x**3-72*x**2+36*x ),x)
-5*x*log(x) + 9*log(x) + (-50*x**2*log(x)**2 - 30*x**2*log(x) + 180*x*log( x)**2 + 84*x*log(x) - 162*log(x)**2 - 54*log(x))/(-10*x*log(x) - 6*x + exp (x) + 18*log(x) + 6)
Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.86 \[ \int \frac {e^{2 x} (9-5 x)+e^x \left (54-84 x+30 x^2\right )+\left (-5 e^{2 x} x+e^x \left (78 x-84 x^2+30 x^3\right )\right ) \log (x)+e^x \left (162 x-180 x^2+50 x^3\right ) \log ^2(x)}{36 x+e^{2 x} x-72 x^2+36 x^3+e^x \left (12 x-12 x^2\right )+\left (216 x-336 x^2+120 x^3+e^x \left (36 x-20 x^2\right )\right ) \log (x)+\left (324 x-360 x^2+100 x^3\right ) \log ^2(x)} \, dx=\frac {{\left (5 \, x - 9\right )} e^{x} \log \left (x\right )}{2 \, {\left (5 \, x - 9\right )} \log \left (x\right ) + 6 \, x - e^{x} - 6} \]
integrate(((50*x^3-180*x^2+162*x)*exp(x)*log(x)^2+(-5*x*exp(x)^2+(30*x^3-8 4*x^2+78*x)*exp(x))*log(x)+(-5*x+9)*exp(x)^2+(30*x^2-84*x+54)*exp(x))/((10 0*x^3-360*x^2+324*x)*log(x)^2+((-20*x^2+36*x)*exp(x)+120*x^3-336*x^2+216*x )*log(x)+x*exp(x)^2+(-12*x^2+12*x)*exp(x)+36*x^3-72*x^2+36*x),x, algorithm =\
Time = 0.34 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {e^{2 x} (9-5 x)+e^x \left (54-84 x+30 x^2\right )+\left (-5 e^{2 x} x+e^x \left (78 x-84 x^2+30 x^3\right )\right ) \log (x)+e^x \left (162 x-180 x^2+50 x^3\right ) \log ^2(x)}{36 x+e^{2 x} x-72 x^2+36 x^3+e^x \left (12 x-12 x^2\right )+\left (216 x-336 x^2+120 x^3+e^x \left (36 x-20 x^2\right )\right ) \log (x)+\left (324 x-360 x^2+100 x^3\right ) \log ^2(x)} \, dx=\frac {5 \, x e^{x} \log \left (x\right ) - 9 \, e^{x} \log \left (x\right )}{10 \, x \log \left (x\right ) + 6 \, x - e^{x} - 18 \, \log \left (x\right ) - 6} \]
integrate(((50*x^3-180*x^2+162*x)*exp(x)*log(x)^2+(-5*x*exp(x)^2+(30*x^3-8 4*x^2+78*x)*exp(x))*log(x)+(-5*x+9)*exp(x)^2+(30*x^2-84*x+54)*exp(x))/((10 0*x^3-360*x^2+324*x)*log(x)^2+((-20*x^2+36*x)*exp(x)+120*x^3-336*x^2+216*x )*log(x)+x*exp(x)^2+(-12*x^2+12*x)*exp(x)+36*x^3-72*x^2+36*x),x, algorithm =\
Timed out. \[ \int \frac {e^{2 x} (9-5 x)+e^x \left (54-84 x+30 x^2\right )+\left (-5 e^{2 x} x+e^x \left (78 x-84 x^2+30 x^3\right )\right ) \log (x)+e^x \left (162 x-180 x^2+50 x^3\right ) \log ^2(x)}{36 x+e^{2 x} x-72 x^2+36 x^3+e^x \left (12 x-12 x^2\right )+\left (216 x-336 x^2+120 x^3+e^x \left (36 x-20 x^2\right )\right ) \log (x)+\left (324 x-360 x^2+100 x^3\right ) \log ^2(x)} \, dx=\int -\frac {-{\mathrm {e}}^x\,\left (50\,x^3-180\,x^2+162\,x\right )\,{\ln \left (x\right )}^2+\left (5\,x\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^x\,\left (30\,x^3-84\,x^2+78\,x\right )\right )\,\ln \left (x\right )-{\mathrm {e}}^x\,\left (30\,x^2-84\,x+54\right )+{\mathrm {e}}^{2\,x}\,\left (5\,x-9\right )}{36\,x+x\,{\mathrm {e}}^{2\,x}+{\ln \left (x\right )}^2\,\left (100\,x^3-360\,x^2+324\,x\right )+\ln \left (x\right )\,\left (216\,x+{\mathrm {e}}^x\,\left (36\,x-20\,x^2\right )-336\,x^2+120\,x^3\right )+{\mathrm {e}}^x\,\left (12\,x-12\,x^2\right )-72\,x^2+36\,x^3} \,d x \]
int(-(log(x)*(5*x*exp(2*x) - exp(x)*(78*x - 84*x^2 + 30*x^3)) - exp(x)*(30 *x^2 - 84*x + 54) + exp(2*x)*(5*x - 9) - exp(x)*log(x)^2*(162*x - 180*x^2 + 50*x^3))/(36*x + x*exp(2*x) + log(x)^2*(324*x - 360*x^2 + 100*x^3) + log (x)*(216*x + exp(x)*(36*x - 20*x^2) - 336*x^2 + 120*x^3) + exp(x)*(12*x - 12*x^2) - 72*x^2 + 36*x^3),x)
int(-(log(x)*(5*x*exp(2*x) - exp(x)*(78*x - 84*x^2 + 30*x^3)) - exp(x)*(30 *x^2 - 84*x + 54) + exp(2*x)*(5*x - 9) - exp(x)*log(x)^2*(162*x - 180*x^2 + 50*x^3))/(36*x + x*exp(2*x) + log(x)^2*(324*x - 360*x^2 + 100*x^3) + log (x)*(216*x + exp(x)*(36*x - 20*x^2) - 336*x^2 + 120*x^3) + exp(x)*(12*x - 12*x^2) - 72*x^2 + 36*x^3), x)