Integrand size = 109, antiderivative size = 27 \[ \int \frac {3+e^{2 x} (3+6 x)+e^x (9+9 x)+\left (4 x-2 e^{2 x} x\right ) \log (x)+\left (6 x+2 e^{2 x} x^2+e^x \left (6 x+6 x^2\right )\right ) \log ^2(x)+\left (x^2+e^x \left (x^2+x^3\right )\right ) \log ^4(x)}{9+6 x \log ^2(x)+x^2 \log ^4(x)} \, dx=5+x+x \left (e^x+\frac {-2+e^{2 x}}{3+x \log ^2(x)}\right ) \]
Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {3+e^{2 x} (3+6 x)+e^x (9+9 x)+\left (4 x-2 e^{2 x} x\right ) \log (x)+\left (6 x+2 e^{2 x} x^2+e^x \left (6 x+6 x^2\right )\right ) \log ^2(x)+\left (x^2+e^x \left (x^2+x^3\right )\right ) \log ^4(x)}{9+6 x \log ^2(x)+x^2 \log ^4(x)} \, dx=x+e^x x+\frac {\left (-2+e^{2 x}\right ) x}{3+x \log ^2(x)} \]
Integrate[(3 + E^(2*x)*(3 + 6*x) + E^x*(9 + 9*x) + (4*x - 2*E^(2*x)*x)*Log [x] + (6*x + 2*E^(2*x)*x^2 + E^x*(6*x + 6*x^2))*Log[x]^2 + (x^2 + E^x*(x^2 + x^3))*Log[x]^4)/(9 + 6*x*Log[x]^2 + x^2*Log[x]^4),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.
\(\displaystyle \int \frac {\left (2 e^{2 x} x^2+e^x \left (6 x^2+6 x\right )+6 x\right ) \log ^2(x)+\left (x^2+e^x \left (x^3+x^2\right )\right ) \log ^4(x)+e^{2 x} (6 x+3)+e^x (9 x+9)+\left (4 x-2 e^{2 x} x\right ) \log (x)+3}{x^2 \log ^4(x)+6 x \log ^2(x)+9} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {\left (2 e^{2 x} x^2+e^x \left (6 x^2+6 x\right )+6 x\right ) \log ^2(x)+\left (x^2+e^x \left (x^3+x^2\right )\right ) \log ^4(x)+e^{2 x} (6 x+3)+e^x (9 x+9)+\left (4 x-2 e^{2 x} x\right ) \log (x)+3}{\left (x \log ^2(x)+3\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {e^{2 x} \left (2 x^2 \log ^2(x)+6 x-2 x \log (x)+3\right )}{\left (x \log ^2(x)+3\right )^2}+\frac {x^2 \log ^4(x)}{\left (x \log ^2(x)+3\right )^2}+e^x (x+1)+\frac {6 x \log ^2(x)}{\left (x \log ^2(x)+3\right )^2}+\frac {4 x \log (x)}{\left (x \log ^2(x)+3\right )^2}+\frac {3}{\left (x \log ^2(x)+3\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -6 \int \frac {1}{\left (x \log ^2(x)+3\right )^2}dx+4 \int \frac {x \log (x)}{\left (x \log ^2(x)+3\right )^2}dx+\frac {e^{2 x} \left (x^2 \log ^2(x)+3 x\right )}{\left (x \log ^2(x)+3\right )^2}+x-e^x+e^x (x+1)\) |
Int[(3 + E^(2*x)*(3 + 6*x) + E^x*(9 + 9*x) + (4*x - 2*E^(2*x)*x)*Log[x] + (6*x + 2*E^(2*x)*x^2 + E^x*(6*x + 6*x^2))*Log[x]^2 + (x^2 + E^x*(x^2 + x^3 ))*Log[x]^4)/(9 + 6*x*Log[x]^2 + x^2*Log[x]^4),x]
3.2.41.3.1 Defintions of rubi rules used
Time = 1.21 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93
method | result | size |
risch | \({\mathrm e}^{x} x +x +\frac {x \left ({\mathrm e}^{2 x}-2\right )}{x \ln \left (x \right )^{2}+3}\) | \(25\) |
parallelrisch | \(-\frac {-x^{2} {\mathrm e}^{x} \ln \left (x \right )^{2}-x^{2} \ln \left (x \right )^{2}-x \,{\mathrm e}^{2 x}-3 \,{\mathrm e}^{x} x -x}{x \ln \left (x \right )^{2}+3}\) | \(49\) |
int((((x^3+x^2)*exp(x)+x^2)*ln(x)^4+(2*exp(x)^2*x^2+(6*x^2+6*x)*exp(x)+6*x )*ln(x)^2+(-2*x*exp(x)^2+4*x)*ln(x)+(6*x+3)*exp(x)^2+(9*x+9)*exp(x)+3)/(x^ 2*ln(x)^4+6*x*ln(x)^2+9),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int \frac {3+e^{2 x} (3+6 x)+e^x (9+9 x)+\left (4 x-2 e^{2 x} x\right ) \log (x)+\left (6 x+2 e^{2 x} x^2+e^x \left (6 x+6 x^2\right )\right ) \log ^2(x)+\left (x^2+e^x \left (x^2+x^3\right )\right ) \log ^4(x)}{9+6 x \log ^2(x)+x^2 \log ^4(x)} \, dx=\frac {{\left (x^{2} e^{x} + x^{2}\right )} \log \left (x\right )^{2} + x e^{\left (2 \, x\right )} + 3 \, x e^{x} + x}{x \log \left (x\right )^{2} + 3} \]
integrate((((x^3+x^2)*exp(x)+x^2)*log(x)^4+(2*exp(x)^2*x^2+(6*x^2+6*x)*exp (x)+6*x)*log(x)^2+(-2*x*exp(x)^2+4*x)*log(x)+(6*x+3)*exp(x)^2+(9*x+9)*exp( x)+3)/(x^2*log(x)^4+6*x*log(x)^2+9),x, algorithm=\
Time = 0.15 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \[ \int \frac {3+e^{2 x} (3+6 x)+e^x (9+9 x)+\left (4 x-2 e^{2 x} x\right ) \log (x)+\left (6 x+2 e^{2 x} x^2+e^x \left (6 x+6 x^2\right )\right ) \log ^2(x)+\left (x^2+e^x \left (x^2+x^3\right )\right ) \log ^4(x)}{9+6 x \log ^2(x)+x^2 \log ^4(x)} \, dx=x - \frac {2 x}{x \log {\left (x \right )}^{2} + 3} + \frac {x e^{2 x} + \left (x^{2} \log {\left (x \right )}^{2} + 3 x\right ) e^{x}}{x \log {\left (x \right )}^{2} + 3} \]
integrate((((x**3+x**2)*exp(x)+x**2)*ln(x)**4+(2*exp(x)**2*x**2+(6*x**2+6* x)*exp(x)+6*x)*ln(x)**2+(-2*x*exp(x)**2+4*x)*ln(x)+(6*x+3)*exp(x)**2+(9*x+ 9)*exp(x)+3)/(x**2*ln(x)**4+6*x*ln(x)**2+9),x)
Time = 0.22 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \[ \int \frac {3+e^{2 x} (3+6 x)+e^x (9+9 x)+\left (4 x-2 e^{2 x} x\right ) \log (x)+\left (6 x+2 e^{2 x} x^2+e^x \left (6 x+6 x^2\right )\right ) \log ^2(x)+\left (x^2+e^x \left (x^2+x^3\right )\right ) \log ^4(x)}{9+6 x \log ^2(x)+x^2 \log ^4(x)} \, dx=\frac {x^{2} \log \left (x\right )^{2} + x e^{\left (2 \, x\right )} + {\left (x^{2} \log \left (x\right )^{2} + 3 \, x\right )} e^{x} + x}{x \log \left (x\right )^{2} + 3} \]
integrate((((x^3+x^2)*exp(x)+x^2)*log(x)^4+(2*exp(x)^2*x^2+(6*x^2+6*x)*exp (x)+6*x)*log(x)^2+(-2*x*exp(x)^2+4*x)*log(x)+(6*x+3)*exp(x)^2+(9*x+9)*exp( x)+3)/(x^2*log(x)^4+6*x*log(x)^2+9),x, algorithm=\
Time = 0.34 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \[ \int \frac {3+e^{2 x} (3+6 x)+e^x (9+9 x)+\left (4 x-2 e^{2 x} x\right ) \log (x)+\left (6 x+2 e^{2 x} x^2+e^x \left (6 x+6 x^2\right )\right ) \log ^2(x)+\left (x^2+e^x \left (x^2+x^3\right )\right ) \log ^4(x)}{9+6 x \log ^2(x)+x^2 \log ^4(x)} \, dx=\frac {x^{2} e^{x} \log \left (x\right )^{2} + x^{2} \log \left (x\right )^{2} + x e^{\left (2 \, x\right )} + 3 \, x e^{x} + x}{x \log \left (x\right )^{2} + 3} \]
integrate((((x^3+x^2)*exp(x)+x^2)*log(x)^4+(2*exp(x)^2*x^2+(6*x^2+6*x)*exp (x)+6*x)*log(x)^2+(-2*x*exp(x)^2+4*x)*log(x)+(6*x+3)*exp(x)^2+(9*x+9)*exp( x)+3)/(x^2*log(x)^4+6*x*log(x)^2+9),x, algorithm=\
Time = 11.72 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {3+e^{2 x} (3+6 x)+e^x (9+9 x)+\left (4 x-2 e^{2 x} x\right ) \log (x)+\left (6 x+2 e^{2 x} x^2+e^x \left (6 x+6 x^2\right )\right ) \log ^2(x)+\left (x^2+e^x \left (x^2+x^3\right )\right ) \log ^4(x)}{9+6 x \log ^2(x)+x^2 \log ^4(x)} \, dx=\frac {x\,\left ({\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^x+x\,{\ln \left (x\right )}^2+x\,{\mathrm {e}}^x\,{\ln \left (x\right )}^2+1\right )}{x\,{\ln \left (x\right )}^2+3} \]