Integrand size = 113, antiderivative size = 25 \[ \int \frac {-30+3 e^{2 x}+e^{9 x+x^2} \left (1-e^{2 x}-2 x\right )+\left (-6+2 e^{9 x+x^2}\right ) \log \left (-3+e^{9 x+x^2}\right )}{-15-3 e^{2 x} x+e^{9 x+x^2} \left (5+e^{2 x} x\right )+\left (-3+e^{9 x+x^2}\right ) \log \left (-3+e^{9 x+x^2}\right )} \, dx=\log \left (\frac {3}{x+e^{-2 x} \left (5+\log \left (-3+e^{x (9+x)}\right )\right )}\right ) \]
\[ \int \frac {-30+3 e^{2 x}+e^{9 x+x^2} \left (1-e^{2 x}-2 x\right )+\left (-6+2 e^{9 x+x^2}\right ) \log \left (-3+e^{9 x+x^2}\right )}{-15-3 e^{2 x} x+e^{9 x+x^2} \left (5+e^{2 x} x\right )+\left (-3+e^{9 x+x^2}\right ) \log \left (-3+e^{9 x+x^2}\right )} \, dx=\int \frac {-30+3 e^{2 x}+e^{9 x+x^2} \left (1-e^{2 x}-2 x\right )+\left (-6+2 e^{9 x+x^2}\right ) \log \left (-3+e^{9 x+x^2}\right )}{-15-3 e^{2 x} x+e^{9 x+x^2} \left (5+e^{2 x} x\right )+\left (-3+e^{9 x+x^2}\right ) \log \left (-3+e^{9 x+x^2}\right )} \, dx \]
Integrate[(-30 + 3*E^(2*x) + E^(9*x + x^2)*(1 - E^(2*x) - 2*x) + (-6 + 2*E ^(9*x + x^2))*Log[-3 + E^(9*x + x^2)])/(-15 - 3*E^(2*x)*x + E^(9*x + x^2)* (5 + E^(2*x)*x) + (-3 + E^(9*x + x^2))*Log[-3 + E^(9*x + x^2)]),x]
Integrate[(-30 + 3*E^(2*x) + E^(9*x + x^2)*(1 - E^(2*x) - 2*x) + (-6 + 2*E ^(9*x + x^2))*Log[-3 + E^(9*x + x^2)])/(-15 - 3*E^(2*x)*x + E^(9*x + x^2)* (5 + E^(2*x)*x) + (-3 + E^(9*x + x^2))*Log[-3 + E^(9*x + 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.
\(\displaystyle \int \frac {e^{x^2+9 x} \left (-2 x-e^{2 x}+1\right )+\left (2 e^{x^2+9 x}-6\right ) \log \left (e^{x^2+9 x}-3\right )+3 e^{2 x}-30}{e^{x^2+9 x} \left (e^{2 x} x+5\right )+\left (e^{x^2+9 x}-3\right ) \log \left (e^{x^2+9 x}-3\right )-3 e^{2 x} x-15} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-e^{x^2+9 x} \left (-2 x-e^{2 x}+1\right )-\left (2 e^{x^2+9 x}-6\right ) \log \left (e^{x^2+9 x}-3\right )-3 e^{2 x}+30}{\left (3-e^{x (x+9)}\right ) \left (e^{2 x} x+\log \left (e^{x (x+9)}-3\right )+5\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2 e^{x^2+9 x} x}{\left (3-e^{x (x+9)}\right ) \left (e^{2 x} x+\log \left (e^{x (x+9)}-3\right )+5\right )}+\frac {2 e^{x^2+9 x} \log \left (e^{x (x+9)}-3\right )}{\left (e^{x (x+9)}-3\right ) \left (e^{2 x} x+\log \left (e^{x (x+9)}-3\right )+5\right )}+\frac {e^{x^2+11 x}}{\left (3-e^{x (x+9)}\right ) \left (e^{2 x} x+\log \left (e^{x (x+9)}-3\right )+5\right )}+\frac {e^{x^2+9 x}}{\left (e^{x (x+9)}-3\right ) \left (e^{2 x} x+\log \left (e^{x (x+9)}-3\right )+5\right )}-\frac {6 \log \left (e^{x (x+9)}-3\right )}{\left (e^{x (x+9)}-3\right ) \left (e^{2 x} x+\log \left (e^{x (x+9)}-3\right )+5\right )}+\frac {3 e^{2 x}}{\left (e^{x (x+9)}-3\right ) \left (e^{2 x} x+\log \left (e^{x (x+9)}-3\right )+5\right )}-\frac {30}{\left (e^{x (x+9)}-3\right ) \left (e^{2 x} x+\log \left (e^{x (x+9)}-3\right )+5\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \int \frac {e^{x^2+11 x}}{\left (3-e^{x (x+9)}\right ) \left (e^{2 x} x+\log \left (-3+e^{x (x+9)}\right )+5\right )}dx+\int \frac {e^{x^2+9 x}}{\left (-3+e^{x (x+9)}\right ) \left (e^{2 x} x+\log \left (-3+e^{x (x+9)}\right )+5\right )}dx+2 \int \frac {e^{x^2+9 x} x}{\left (3-e^{x (x+9)}\right ) \left (e^{2 x} x+\log \left (-3+e^{x (x+9)}\right )+5\right )}dx+2 \int \frac {e^{x^2+9 x} \log \left (-3+e^{x (x+9)}\right )}{\left (-3+e^{x (x+9)}\right ) \left (e^{2 x} x+\log \left (-3+e^{x (x+9)}\right )+5\right )}dx-30 \int \frac {1}{\left (-3+e^{x (x+9)}\right ) \left (e^{2 x} x+\log \left (-3+e^{x (x+9)}\right )+5\right )}dx+3 \int \frac {e^{2 x}}{\left (-3+e^{x (x+9)}\right ) \left (e^{2 x} x+\log \left (-3+e^{x (x+9)}\right )+5\right )}dx-6 \int \frac {\log \left (-3+e^{x (x+9)}\right )}{\left (-3+e^{x (x+9)}\right ) \left (e^{2 x} x+\log \left (-3+e^{x (x+9)}\right )+5\right )}dx\) |
Int[(-30 + 3*E^(2*x) + E^(9*x + x^2)*(1 - E^(2*x) - 2*x) + (-6 + 2*E^(9*x + x^2))*Log[-3 + E^(9*x + x^2)])/(-15 - 3*E^(2*x)*x + E^(9*x + x^2)*(5 + E ^(2*x)*x) + (-3 + E^(9*x + x^2))*Log[-3 + E^(9*x + x^2)]),x]
3.2.90.3.1 Defintions of rubi rules used
Time = 0.64 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00
method | result | size |
risch | \(2 x -\ln \left (x \,{\mathrm e}^{2 x}+\ln \left ({\mathrm e}^{\left (x +9\right ) x}-3\right )+5\right )\) | \(25\) |
norman | \(2 x -\ln \left (x \,{\mathrm e}^{2 x}+\ln \left ({\mathrm e}^{x^{2}+9 x}-3\right )+5\right )\) | \(27\) |
parallelrisch | \(2 x -\ln \left (x \,{\mathrm e}^{2 x}+\ln \left ({\mathrm e}^{x^{2}+9 x}-3\right )+5\right )\) | \(27\) |
int(((2*exp(x^2+9*x)-6)*ln(exp(x^2+9*x)-3)+(-exp(2*x)+1-2*x)*exp(x^2+9*x)+ 3*exp(2*x)-30)/((exp(x^2+9*x)-3)*ln(exp(x^2+9*x)-3)+(x*exp(2*x)+5)*exp(x^2 +9*x)-3*x*exp(2*x)-15),x,method=_RETURNVERBOSE)
Time = 0.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {-30+3 e^{2 x}+e^{9 x+x^2} \left (1-e^{2 x}-2 x\right )+\left (-6+2 e^{9 x+x^2}\right ) \log \left (-3+e^{9 x+x^2}\right )}{-15-3 e^{2 x} x+e^{9 x+x^2} \left (5+e^{2 x} x\right )+\left (-3+e^{9 x+x^2}\right ) \log \left (-3+e^{9 x+x^2}\right )} \, dx=2 \, x - \log \left (x e^{\left (2 \, x\right )} + \log \left (e^{\left (x^{2} + 9 \, x\right )} - 3\right ) + 5\right ) \]
integrate(((2*exp(x^2+9*x)-6)*log(exp(x^2+9*x)-3)+(-exp(2*x)+1-2*x)*exp(x^ 2+9*x)+3*exp(2*x)-30)/((exp(x^2+9*x)-3)*log(exp(x^2+9*x)-3)+(x*exp(2*x)+5) *exp(x^2+9*x)-3*x*exp(2*x)-15),x, algorithm=\
Time = 0.35 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {-30+3 e^{2 x}+e^{9 x+x^2} \left (1-e^{2 x}-2 x\right )+\left (-6+2 e^{9 x+x^2}\right ) \log \left (-3+e^{9 x+x^2}\right )}{-15-3 e^{2 x} x+e^{9 x+x^2} \left (5+e^{2 x} x\right )+\left (-3+e^{9 x+x^2}\right ) \log \left (-3+e^{9 x+x^2}\right )} \, dx=2 x - \log {\left (x e^{2 x} + \log {\left (e^{x^{2} + 9 x} - 3 \right )} + 5 \right )} \]
integrate(((2*exp(x**2+9*x)-6)*ln(exp(x**2+9*x)-3)+(-exp(2*x)+1-2*x)*exp(x **2+9*x)+3*exp(2*x)-30)/((exp(x**2+9*x)-3)*ln(exp(x**2+9*x)-3)+(x*exp(2*x) +5)*exp(x**2+9*x)-3*x*exp(2*x)-15),x)
Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {-30+3 e^{2 x}+e^{9 x+x^2} \left (1-e^{2 x}-2 x\right )+\left (-6+2 e^{9 x+x^2}\right ) \log \left (-3+e^{9 x+x^2}\right )}{-15-3 e^{2 x} x+e^{9 x+x^2} \left (5+e^{2 x} x\right )+\left (-3+e^{9 x+x^2}\right ) \log \left (-3+e^{9 x+x^2}\right )} \, dx=2 \, x - \log \left (x e^{\left (2 \, x\right )} + \log \left (e^{\left (x^{2} + 9 \, x\right )} - 3\right ) + 5\right ) \]
integrate(((2*exp(x^2+9*x)-6)*log(exp(x^2+9*x)-3)+(-exp(2*x)+1-2*x)*exp(x^ 2+9*x)+3*exp(2*x)-30)/((exp(x^2+9*x)-3)*log(exp(x^2+9*x)-3)+(x*exp(2*x)+5) *exp(x^2+9*x)-3*x*exp(2*x)-15),x, algorithm=\
Time = 0.31 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {-30+3 e^{2 x}+e^{9 x+x^2} \left (1-e^{2 x}-2 x\right )+\left (-6+2 e^{9 x+x^2}\right ) \log \left (-3+e^{9 x+x^2}\right )}{-15-3 e^{2 x} x+e^{9 x+x^2} \left (5+e^{2 x} x\right )+\left (-3+e^{9 x+x^2}\right ) \log \left (-3+e^{9 x+x^2}\right )} \, dx=2 \, x - \log \left (x e^{\left (2 \, x\right )} + \log \left (e^{\left (x^{2} + 9 \, x\right )} - 3\right ) + 5\right ) \]
integrate(((2*exp(x^2+9*x)-6)*log(exp(x^2+9*x)-3)+(-exp(2*x)+1-2*x)*exp(x^ 2+9*x)+3*exp(2*x)-30)/((exp(x^2+9*x)-3)*log(exp(x^2+9*x)-3)+(x*exp(2*x)+5) *exp(x^2+9*x)-3*x*exp(2*x)-15),x, algorithm=\
Time = 0.68 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {-30+3 e^{2 x}+e^{9 x+x^2} \left (1-e^{2 x}-2 x\right )+\left (-6+2 e^{9 x+x^2}\right ) \log \left (-3+e^{9 x+x^2}\right )}{-15-3 e^{2 x} x+e^{9 x+x^2} \left (5+e^{2 x} x\right )+\left (-3+e^{9 x+x^2}\right ) \log \left (-3+e^{9 x+x^2}\right )} \, dx=2\,x-\ln \left (\ln \left ({\mathrm {e}}^{x\,\left (x+9\right )}-3\right )+x\,{\mathrm {e}}^{2\,x}+5\right ) \]