Integrand size = 117, antiderivative size = 31 \[ \int \frac {e^{\frac {6}{-9-5 x+3 \log (x)}} \left (-36-39 x-60 x^2-25 x^3+\left (54 x+30 x^2\right ) \log (x)-9 x \log ^2(x)\right )}{324 x+684 x^2+541 x^3+190 x^4+25 x^5+\left (-216 x-336 x^2-174 x^3-30 x^4\right ) \log (x)+\left (36 x+36 x^2+9 x^3\right ) \log ^2(x)} \, dx=4-e+\frac {e^{\frac {2}{-4-2 x+\frac {3+x}{3}+\log (x)}}}{2+x} \]
Time = 0.73 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.68 \[ \int \frac {e^{\frac {6}{-9-5 x+3 \log (x)}} \left (-36-39 x-60 x^2-25 x^3+\left (54 x+30 x^2\right ) \log (x)-9 x \log ^2(x)\right )}{324 x+684 x^2+541 x^3+190 x^4+25 x^5+\left (-216 x-336 x^2-174 x^3-30 x^4\right ) \log (x)+\left (36 x+36 x^2+9 x^3\right ) \log ^2(x)} \, dx=\frac {e^{\frac {6}{-9-5 x+3 \log (x)}}}{2+x} \]
Integrate[(E^(6/(-9 - 5*x + 3*Log[x]))*(-36 - 39*x - 60*x^2 - 25*x^3 + (54 *x + 30*x^2)*Log[x] - 9*x*Log[x]^2))/(324*x + 684*x^2 + 541*x^3 + 190*x^4 + 25*x^5 + (-216*x - 336*x^2 - 174*x^3 - 30*x^4)*Log[x] + (36*x + 36*x^2 + 9*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.
\(\displaystyle \int \frac {e^{\frac {6}{-5 x+3 \log (x)-9}} \left (-25 x^3-60 x^2+\left (30 x^2+54 x\right ) \log (x)-39 x-9 x \log ^2(x)-36\right )}{25 x^5+190 x^4+541 x^3+684 x^2+\left (9 x^3+36 x^2+36 x\right ) \log ^2(x)+\left (-30 x^4-174 x^3-336 x^2-216 x\right ) \log (x)+324 x} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {e^{-\frac {6}{5 x-3 \log (x)+9}} \left (-25 x^3-60 x^2-39 x-9 x \log ^2(x)+6 (5 x+9) x \log (x)-36\right )}{x (x+2)^2 (5 x-3 \log (x)+9)^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {6 (5 x-3) e^{-\frac {6}{5 x-3 \log (x)+9}}}{x (x+2) (5 x-3 \log (x)+9)^2}-\frac {e^{-\frac {6}{5 x-3 \log (x)+9}}}{(x+2)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\int \frac {e^{-\frac {6}{5 x-3 \log (x)+9}}}{(x+2)^2}dx-9 \int \frac {e^{-\frac {6}{5 x-3 \log (x)+9}}}{x (5 x-3 \log (x)+9)^2}dx+39 \int \frac {e^{-\frac {6}{5 x-3 \log (x)+9}}}{(x+2) (5 x-3 \log (x)+9)^2}dx\) |
Int[(E^(6/(-9 - 5*x + 3*Log[x]))*(-36 - 39*x - 60*x^2 - 25*x^3 + (54*x + 3 0*x^2)*Log[x] - 9*x*Log[x]^2))/(324*x + 684*x^2 + 541*x^3 + 190*x^4 + 25*x ^5 + (-216*x - 336*x^2 - 174*x^3 - 30*x^4)*Log[x] + (36*x + 36*x^2 + 9*x^3 )*Log[x]^2),x]
3.14.75.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.50 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.68
method | result | size |
risch | \(\frac {{\mathrm e}^{\frac {6}{3 \ln \left (x \right )-5 x -9}}}{2+x}\) | \(21\) |
parallelrisch | \(\frac {{\mathrm e}^{\frac {6}{3 \ln \left (x \right )-5 x -9}}}{2+x}\) | \(21\) |
int((-9*x*ln(x)^2+(30*x^2+54*x)*ln(x)-25*x^3-60*x^2-39*x-36)*exp(6/(3*ln(x )-5*x-9))/((9*x^3+36*x^2+36*x)*ln(x)^2+(-30*x^4-174*x^3-336*x^2-216*x)*ln( x)+25*x^5+190*x^4+541*x^3+684*x^2+324*x),x,method=_RETURNVERBOSE)
Time = 0.23 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65 \[ \int \frac {e^{\frac {6}{-9-5 x+3 \log (x)}} \left (-36-39 x-60 x^2-25 x^3+\left (54 x+30 x^2\right ) \log (x)-9 x \log ^2(x)\right )}{324 x+684 x^2+541 x^3+190 x^4+25 x^5+\left (-216 x-336 x^2-174 x^3-30 x^4\right ) \log (x)+\left (36 x+36 x^2+9 x^3\right ) \log ^2(x)} \, dx=\frac {e^{\left (-\frac {6}{5 \, x - 3 \, \log \left (x\right ) + 9}\right )}}{x + 2} \]
integrate((-9*x*log(x)^2+(30*x^2+54*x)*log(x)-25*x^3-60*x^2-39*x-36)*exp(6 /(3*log(x)-5*x-9))/((9*x^3+36*x^2+36*x)*log(x)^2+(-30*x^4-174*x^3-336*x^2- 216*x)*log(x)+25*x^5+190*x^4+541*x^3+684*x^2+324*x),x, algorithm=\
Time = 0.21 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.48 \[ \int \frac {e^{\frac {6}{-9-5 x+3 \log (x)}} \left (-36-39 x-60 x^2-25 x^3+\left (54 x+30 x^2\right ) \log (x)-9 x \log ^2(x)\right )}{324 x+684 x^2+541 x^3+190 x^4+25 x^5+\left (-216 x-336 x^2-174 x^3-30 x^4\right ) \log (x)+\left (36 x+36 x^2+9 x^3\right ) \log ^2(x)} \, dx=\frac {e^{\frac {6}{- 5 x + 3 \log {\left (x \right )} - 9}}}{x + 2} \]
integrate((-9*x*ln(x)**2+(30*x**2+54*x)*ln(x)-25*x**3-60*x**2-39*x-36)*exp (6/(3*ln(x)-5*x-9))/((9*x**3+36*x**2+36*x)*ln(x)**2+(-30*x**4-174*x**3-336 *x**2-216*x)*ln(x)+25*x**5+190*x**4+541*x**3+684*x**2+324*x),x)
\[ \int \frac {e^{\frac {6}{-9-5 x+3 \log (x)}} \left (-36-39 x-60 x^2-25 x^3+\left (54 x+30 x^2\right ) \log (x)-9 x \log ^2(x)\right )}{324 x+684 x^2+541 x^3+190 x^4+25 x^5+\left (-216 x-336 x^2-174 x^3-30 x^4\right ) \log (x)+\left (36 x+36 x^2+9 x^3\right ) \log ^2(x)} \, dx=\int { -\frac {{\left (25 \, x^{3} + 9 \, x \log \left (x\right )^{2} + 60 \, x^{2} - 6 \, {\left (5 \, x^{2} + 9 \, x\right )} \log \left (x\right ) + 39 \, x + 36\right )} e^{\left (-\frac {6}{5 \, x - 3 \, \log \left (x\right ) + 9}\right )}}{25 \, x^{5} + 190 \, x^{4} + 541 \, x^{3} + 9 \, {\left (x^{3} + 4 \, x^{2} + 4 \, x\right )} \log \left (x\right )^{2} + 684 \, x^{2} - 6 \, {\left (5 \, x^{4} + 29 \, x^{3} + 56 \, x^{2} + 36 \, x\right )} \log \left (x\right ) + 324 \, x} \,d x } \]
integrate((-9*x*log(x)^2+(30*x^2+54*x)*log(x)-25*x^3-60*x^2-39*x-36)*exp(6 /(3*log(x)-5*x-9))/((9*x^3+36*x^2+36*x)*log(x)^2+(-30*x^4-174*x^3-336*x^2- 216*x)*log(x)+25*x^5+190*x^4+541*x^3+684*x^2+324*x),x, algorithm=\
-25/6*x^3*e^(-6/(5*x - 3*log(x) + 9))/(5*x^3 + 17*x^2 + 8*x - 12) - 10*x^2 *e^(-6/(5*x - 3*log(x) + 9))/(5*x^3 + 17*x^2 + 8*x - 12) - 13/2*x*e^(-6/(5 *x - 3*log(x) + 9))/(5*x^3 + 17*x^2 + 8*x - 12) - 6*e^(-6/(5*x - 3*log(x) + 9))/(5*x^3 + 17*x^2 + 8*x - 12) - integrate(-3*(2*(5*x + 9)*log(x) - 3*l og(x)^2)*e^(-6/(5*x - 3*log(x) + 9))/(25*x^4 + 190*x^3 + 9*(x^2 + 4*x + 4) *log(x)^2 + 541*x^2 - 6*(5*x^3 + 29*x^2 + 56*x + 36)*log(x) + 684*x + 324) , x)
Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int \frac {e^{\frac {6}{-9-5 x+3 \log (x)}} \left (-36-39 x-60 x^2-25 x^3+\left (54 x+30 x^2\right ) \log (x)-9 x \log ^2(x)\right )}{324 x+684 x^2+541 x^3+190 x^4+25 x^5+\left (-216 x-336 x^2-174 x^3-30 x^4\right ) \log (x)+\left (36 x+36 x^2+9 x^3\right ) \log ^2(x)} \, dx=\frac {e^{\left (\frac {2 \, {\left (5 \, x - 3 \, \log \left (x\right )\right )}}{3 \, {\left (5 \, x - 3 \, \log \left (x\right ) + 9\right )}} - \frac {2}{3}\right )}}{x + 2} \]
integrate((-9*x*log(x)^2+(30*x^2+54*x)*log(x)-25*x^3-60*x^2-39*x-36)*exp(6 /(3*log(x)-5*x-9))/((9*x^3+36*x^2+36*x)*log(x)^2+(-30*x^4-174*x^3-336*x^2- 216*x)*log(x)+25*x^5+190*x^4+541*x^3+684*x^2+324*x),x, algorithm=\
Time = 15.48 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65 \[ \int \frac {e^{\frac {6}{-9-5 x+3 \log (x)}} \left (-36-39 x-60 x^2-25 x^3+\left (54 x+30 x^2\right ) \log (x)-9 x \log ^2(x)\right )}{324 x+684 x^2+541 x^3+190 x^4+25 x^5+\left (-216 x-336 x^2-174 x^3-30 x^4\right ) \log (x)+\left (36 x+36 x^2+9 x^3\right ) \log ^2(x)} \, dx=\frac {{\mathrm {e}}^{-\frac {6}{5\,x-3\,\ln \left (x\right )+9}}}{x+2} \]