Integrand size = 133, antiderivative size = 31 \[ \int \frac {-216+216 x-54 x^2+e^x \left (36-144 x+99 x^2-18 x^3\right )+\left (-108+216 x-81 x^2\right ) \log (9)+\left (108-216 x+81 x^2\right ) \log (x)}{e^{3 x}-9 e^{2 x} \log (9)+27 e^x \log ^2(9)-27 \log ^3(9)+\left (9 e^{2 x}-54 e^x \log (9)+81 \log ^2(9)\right ) \log (x)+\left (27 e^x-81 \log (9)\right ) \log ^2(x)+27 \log ^3(x)} \, dx=\frac {\left (2 x-x^2\right )^2}{x \left (\frac {e^x}{3}-\log (9)+\log (x)\right )^2} \]
Time = 0.38 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.71 \[ \int \frac {-216+216 x-54 x^2+e^x \left (36-144 x+99 x^2-18 x^3\right )+\left (-108+216 x-81 x^2\right ) \log (9)+\left (108-216 x+81 x^2\right ) \log (x)}{e^{3 x}-9 e^{2 x} \log (9)+27 e^x \log ^2(9)-27 \log ^3(9)+\left (9 e^{2 x}-54 e^x \log (9)+81 \log ^2(9)\right ) \log (x)+\left (27 e^x-81 \log (9)\right ) \log ^2(x)+27 \log ^3(x)} \, dx=\frac {9 (-2+x)^2 x}{\left (e^x-3 \log (9)+3 \log (x)\right )^2} \]
Integrate[(-216 + 216*x - 54*x^2 + E^x*(36 - 144*x + 99*x^2 - 18*x^3) + (- 108 + 216*x - 81*x^2)*Log[9] + (108 - 216*x + 81*x^2)*Log[x])/(E^(3*x) - 9 *E^(2*x)*Log[9] + 27*E^x*Log[9]^2 - 27*Log[9]^3 + (9*E^(2*x) - 54*E^x*Log[ 9] + 81*Log[9]^2)*Log[x] + (27*E^x - 81*Log[9])*Log[x]^2 + 27*Log[x]^3),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.
| \(\displaystyle \int \frac {-54 x^2+\left (81 x^2-216 x+108\right ) \log (x)+\left (-81 x^2+216 x-108\right ) \log (9)+e^x \left (-18 x^3+99 x^2-144 x+36\right )+216 x-216}{e^{3 x}+27 \log ^3(x)+\left (27 e^x-81 \log (9)\right ) \log ^2(x)+\left (9 e^{2 x}-54 e^x \log (9)+81 \log ^2(9)\right ) \log (x)+27 e^x \log ^2(9)-9 e^{2 x} \log (9)-27 \log ^3(9)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {9 (2-x) \left (e^x \left (2 x^2-7 x+2\right )+6 x (1+\log (27))+(6-9 x) \log (x)-12 (1+\log (3))\right )}{\left (e^x+3 \log (x)-3 \log (9)\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 9 \int \frac {(2-x) \left (6 (1+\log (27)) x+e^x \left (2 x^2-7 x+2\right )+3 (2-3 x) \log (x)-12 (1+\log (3))\right )}{\left (3 \log (x)+e^x-3 \log (9)\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 9 \int \left (\frac {3 (2-x) \left (-2 \log (x) x^2+\log (81) x^2+4 \log (x) x+2 (1-\log (81)) x-4\right )}{\left (3 \log (x)+e^x-3 \log (9)\right )^3}-\frac {2 x^3-11 x^2+16 x-4}{\left (3 \log (x)+e^x-3 \log (9)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 9 \left (-3 \log (81) \int \frac {x^3}{\left (3 \log (x)+e^x-3 \log (9)\right )^3}dx+6 \int \frac {x^3 \log (x)}{\left (3 \log (x)+e^x-3 \log (9)\right )^3}dx-2 \int \frac {x^3}{\left (3 \log (x)+e^x-3 \log (9)\right )^2}dx+12 \log (81) \int \frac {x^2}{\left (3 \log (x)+e^x-3 \log (9)\right )^3}dx-6 \int \frac {x^2}{\left (3 \log (x)+e^x-3 \log (9)\right )^3}dx-24 \int \frac {x^2 \log (x)}{\left (3 \log (x)+e^x-3 \log (9)\right )^3}dx+11 \int \frac {x^2}{\left (3 \log (x)+e^x-3 \log (9)\right )^2}dx-24 \int \frac {1}{\left (3 \log (x)+e^x-3 \log (9)\right )^3}dx-12 \log (81) \int \frac {x}{\left (3 \log (x)+e^x-3 \log (9)\right )^3}dx+24 \int \frac {x}{\left (3 \log (x)+e^x-3 \log (9)\right )^3}dx+24 \int \frac {x \log (x)}{\left (3 \log (x)+e^x-3 \log (9)\right )^3}dx+4 \int \frac {1}{\left (3 \log (x)+e^x-3 \log (9)\right )^2}dx-16 \int \frac {x}{\left (3 \log (x)+e^x-3 \log (9)\right )^2}dx\right )\) |
Int[(-216 + 216*x - 54*x^2 + E^x*(36 - 144*x + 99*x^2 - 18*x^3) + (-108 + 216*x - 81*x^2)*Log[9] + (108 - 216*x + 81*x^2)*Log[x])/(E^(3*x) - 9*E^(2* x)*Log[9] + 27*E^x*Log[9]^2 - 27*Log[9]^3 + (9*E^(2*x) - 54*E^x*Log[9] + 8 1*Log[9]^2)*Log[x] + (27*E^x - 81*Log[9])*Log[x]^2 + 27*Log[x]^3),x]
3.1.22.3.1 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 = 0.39 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87
| method | result | size |
| risch | \(\frac {9 x \left (x^{2}-4 x +4\right )}{\left (6 \ln \left (3\right )-{\mathrm e}^{x}-3 \ln \left (x \right )\right )^{2}}\) | \(27\) |
| parallelrisch | \(\frac {9 x^{3}-36 x^{2}+36 x}{36 \ln \left (3\right )^{2}-36 \ln \left (3\right ) \ln \left (x \right )-12 \ln \left (3\right ) {\mathrm e}^{x}+9 \ln \left (x \right )^{2}+6 \,{\mathrm e}^{x} \ln \left (x \right )+{\mathrm e}^{2 x}}\) | \(53\) |
int(((81*x^2-216*x+108)*ln(x)+(-18*x^3+99*x^2-144*x+36)*exp(x)+2*(-81*x^2+ 216*x-108)*ln(3)-54*x^2+216*x-216)/(27*ln(x)^3+(27*exp(x)-162*ln(3))*ln(x) ^2+(9*exp(x)^2-108*ln(3)*exp(x)+324*ln(3)^2)*ln(x)+exp(x)^3-18*ln(3)*exp(x )^2+108*ln(3)^2*exp(x)-216*ln(3)^3),x,method=_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.68 \[ \int \frac {-216+216 x-54 x^2+e^x \left (36-144 x+99 x^2-18 x^3\right )+\left (-108+216 x-81 x^2\right ) \log (9)+\left (108-216 x+81 x^2\right ) \log (x)}{e^{3 x}-9 e^{2 x} \log (9)+27 e^x \log ^2(9)-27 \log ^3(9)+\left (9 e^{2 x}-54 e^x \log (9)+81 \log ^2(9)\right ) \log (x)+\left (27 e^x-81 \log (9)\right ) \log ^2(x)+27 \log ^3(x)} \, dx=-\frac {9 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )}}{12 \, e^{x} \log \left (3\right ) - 36 \, \log \left (3\right )^{2} - 6 \, {\left (e^{x} - 6 \, \log \left (3\right )\right )} \log \left (x\right ) - 9 \, \log \left (x\right )^{2} - e^{\left (2 \, x\right )}} \]
integrate(((81*x^2-216*x+108)*log(x)+(-18*x^3+99*x^2-144*x+36)*exp(x)+2*(- 81*x^2+216*x-108)*log(3)-54*x^2+216*x-216)/(27*log(x)^3+(27*exp(x)-162*log (3))*log(x)^2+(9*exp(x)^2-108*log(3)*exp(x)+324*log(3)^2)*log(x)+exp(x)^3- 18*log(3)*exp(x)^2+108*log(3)^2*exp(x)-216*log(3)^3),x, algorithm=\
-9*(x^3 - 4*x^2 + 4*x)/(12*e^x*log(3) - 36*log(3)^2 - 6*(e^x - 6*log(3))*l og(x) - 9*log(x)^2 - e^(2*x))
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (24) = 48\).
Time = 0.16 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.71 \[ \int \frac {-216+216 x-54 x^2+e^x \left (36-144 x+99 x^2-18 x^3\right )+\left (-108+216 x-81 x^2\right ) \log (9)+\left (108-216 x+81 x^2\right ) \log (x)}{e^{3 x}-9 e^{2 x} \log (9)+27 e^x \log ^2(9)-27 \log ^3(9)+\left (9 e^{2 x}-54 e^x \log (9)+81 \log ^2(9)\right ) \log (x)+\left (27 e^x-81 \log (9)\right ) \log ^2(x)+27 \log ^3(x)} \, dx=\frac {9 x^{3} - 36 x^{2} + 36 x}{\left (6 \log {\left (x \right )} - 12 \log {\left (3 \right )}\right ) e^{x} + e^{2 x} + 9 \log {\left (x \right )}^{2} - 36 \log {\left (3 \right )} \log {\left (x \right )} + 36 \log {\left (3 \right )}^{2}} \]
integrate(((81*x**2-216*x+108)*ln(x)+(-18*x**3+99*x**2-144*x+36)*exp(x)+2* (-81*x**2+216*x-108)*ln(3)-54*x**2+216*x-216)/(27*ln(x)**3+(27*exp(x)-162* ln(3))*ln(x)**2+(9*exp(x)**2-108*ln(3)*exp(x)+324*ln(3)**2)*ln(x)+exp(x)** 3-18*ln(3)*exp(x)**2+108*ln(3)**2*exp(x)-216*ln(3)**3),x)
(9*x**3 - 36*x**2 + 36*x)/((6*log(x) - 12*log(3))*exp(x) + exp(2*x) + 9*lo g(x)**2 - 36*log(3)*log(x) + 36*log(3)**2)
Time = 0.34 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.74 \[ \int \frac {-216+216 x-54 x^2+e^x \left (36-144 x+99 x^2-18 x^3\right )+\left (-108+216 x-81 x^2\right ) \log (9)+\left (108-216 x+81 x^2\right ) \log (x)}{e^{3 x}-9 e^{2 x} \log (9)+27 e^x \log ^2(9)-27 \log ^3(9)+\left (9 e^{2 x}-54 e^x \log (9)+81 \log ^2(9)\right ) \log (x)+\left (27 e^x-81 \log (9)\right ) \log ^2(x)+27 \log ^3(x)} \, dx=-\frac {9 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )}}{6 \, {\left (2 \, \log \left (3\right ) - \log \left (x\right )\right )} e^{x} - 36 \, \log \left (3\right )^{2} + 36 \, \log \left (3\right ) \log \left (x\right ) - 9 \, \log \left (x\right )^{2} - e^{\left (2 \, x\right )}} \]
integrate(((81*x^2-216*x+108)*log(x)+(-18*x^3+99*x^2-144*x+36)*exp(x)+2*(- 81*x^2+216*x-108)*log(3)-54*x^2+216*x-216)/(27*log(x)^3+(27*exp(x)-162*log (3))*log(x)^2+(9*exp(x)^2-108*log(3)*exp(x)+324*log(3)^2)*log(x)+exp(x)^3- 18*log(3)*exp(x)^2+108*log(3)^2*exp(x)-216*log(3)^3),x, algorithm=\
-9*(x^3 - 4*x^2 + 4*x)/(6*(2*log(3) - log(x))*e^x - 36*log(3)^2 + 36*log(3 )*log(x) - 9*log(x)^2 - e^(2*x))
Time = 0.34 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.71 \[ \int \frac {-216+216 x-54 x^2+e^x \left (36-144 x+99 x^2-18 x^3\right )+\left (-108+216 x-81 x^2\right ) \log (9)+\left (108-216 x+81 x^2\right ) \log (x)}{e^{3 x}-9 e^{2 x} \log (9)+27 e^x \log ^2(9)-27 \log ^3(9)+\left (9 e^{2 x}-54 e^x \log (9)+81 \log ^2(9)\right ) \log (x)+\left (27 e^x-81 \log (9)\right ) \log ^2(x)+27 \log ^3(x)} \, dx=-\frac {9 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )}}{12 \, e^{x} \log \left (3\right ) - 36 \, \log \left (3\right )^{2} - 6 \, e^{x} \log \left (x\right ) + 36 \, \log \left (3\right ) \log \left (x\right ) - 9 \, \log \left (x\right )^{2} - e^{\left (2 \, x\right )}} \]
integrate(((81*x^2-216*x+108)*log(x)+(-18*x^3+99*x^2-144*x+36)*exp(x)+2*(- 81*x^2+216*x-108)*log(3)-54*x^2+216*x-216)/(27*log(x)^3+(27*exp(x)-162*log (3))*log(x)^2+(9*exp(x)^2-108*log(3)*exp(x)+324*log(3)^2)*log(x)+exp(x)^3- 18*log(3)*exp(x)^2+108*log(3)^2*exp(x)-216*log(3)^3),x, algorithm=\
-9*(x^3 - 4*x^2 + 4*x)/(12*e^x*log(3) - 36*log(3)^2 - 6*e^x*log(x) + 36*lo g(3)*log(x) - 9*log(x)^2 - e^(2*x))
Timed out. \[ \int \frac {-216+216 x-54 x^2+e^x \left (36-144 x+99 x^2-18 x^3\right )+\left (-108+216 x-81 x^2\right ) \log (9)+\left (108-216 x+81 x^2\right ) \log (x)}{e^{3 x}-9 e^{2 x} \log (9)+27 e^x \log ^2(9)-27 \log ^3(9)+\left (9 e^{2 x}-54 e^x \log (9)+81 \log ^2(9)\right ) \log (x)+\left (27 e^x-81 \log (9)\right ) \log ^2(x)+27 \log ^3(x)} \, dx=\int -\frac {2\,\ln \left (3\right )\,\left (81\,x^2-216\,x+108\right )-216\,x-\ln \left (x\right )\,\left (81\,x^2-216\,x+108\right )+54\,x^2+{\mathrm {e}}^x\,\left (18\,x^3-99\,x^2+144\,x-36\right )+216}{27\,{\ln \left (x\right )}^3+\left (27\,{\mathrm {e}}^x-162\,\ln \left (3\right )\right )\,{\ln \left (x\right )}^2+\left (9\,{\mathrm {e}}^{2\,x}-108\,{\mathrm {e}}^x\,\ln \left (3\right )+324\,{\ln \left (3\right )}^2\right )\,\ln \left (x\right )+{\mathrm {e}}^{3\,x}-18\,{\mathrm {e}}^{2\,x}\,\ln \left (3\right )+108\,{\mathrm {e}}^x\,{\ln \left (3\right )}^2-216\,{\ln \left (3\right )}^3} \,d x \]
int(-(2*log(3)*(81*x^2 - 216*x + 108) - 216*x - log(x)*(81*x^2 - 216*x + 1 08) + 54*x^2 + exp(x)*(144*x - 99*x^2 + 18*x^3 - 36) + 216)/(exp(3*x) - 18 *exp(2*x)*log(3) + 108*exp(x)*log(3)^2 - log(x)^2*(162*log(3) - 27*exp(x)) + log(x)*(9*exp(2*x) - 108*exp(x)*log(3) + 324*log(3)^2) + 27*log(x)^3 - 216*log(3)^3),x)
int(-(2*log(3)*(81*x^2 - 216*x + 108) - 216*x - log(x)*(81*x^2 - 216*x + 1 08) + 54*x^2 + exp(x)*(144*x - 99*x^2 + 18*x^3 - 36) + 216)/(exp(3*x) - 18 *exp(2*x)*log(3) + 108*exp(x)*log(3)^2 - log(x)^2*(162*log(3) - 27*exp(x)) + log(x)*(9*exp(2*x) - 108*exp(x)*log(3) + 324*log(3)^2) + 27*log(x)^3 - 216*log(3)^3), x)