Integrand size = 160, antiderivative size = 30 \[ \int \frac {3 x^2+x^3+e^{x+x^2} \left (-6 x+6 x^2+21 x^3+6 x^4\right )+\left (6 x+2 x^2+e^{x+x^2} \left (9 x+21 x^2+6 x^3\right )\right ) \log \left (\frac {2}{e^{5/3} (3+x)}\right )+(3+x) \log ^2\left (\frac {2}{e^{5/3} (3+x)}\right )}{3 x^3+x^4+\left (6 x^2+2 x^3\right ) \log \left (\frac {2}{e^{5/3} (3+x)}\right )+\left (3 x+x^2\right ) \log ^2\left (\frac {2}{e^{5/3} (3+x)}\right )} \, dx=2+\log (x)+\frac {3 e^{x+x^2}}{x+\log \left (\frac {2}{e^{5/3} (3+x)}\right )} \] Output:
2+3*exp(x^2+x)/(x+ln(2/(3+x)/exp(5/3)))+ln(x)
Time = 0.14 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {3 x^2+x^3+e^{x+x^2} \left (-6 x+6 x^2+21 x^3+6 x^4\right )+\left (6 x+2 x^2+e^{x+x^2} \left (9 x+21 x^2+6 x^3\right )\right ) \log \left (\frac {2}{e^{5/3} (3+x)}\right )+(3+x) \log ^2\left (\frac {2}{e^{5/3} (3+x)}\right )}{3 x^3+x^4+\left (6 x^2+2 x^3\right ) \log \left (\frac {2}{e^{5/3} (3+x)}\right )+\left (3 x+x^2\right ) \log ^2\left (\frac {2}{e^{5/3} (3+x)}\right )} \, dx=\log (x)+\frac {9 e^{x+x^2}}{-5+3 x+3 \log (2)+3 \log \left (\frac {1}{3+x}\right )} \] Input:
Integrate[(3*x^2 + x^3 + E^(x + x^2)*(-6*x + 6*x^2 + 21*x^3 + 6*x^4) + (6* x + 2*x^2 + E^(x + x^2)*(9*x + 21*x^2 + 6*x^3))*Log[2/(E^(5/3)*(3 + x))] + (3 + x)*Log[2/(E^(5/3)*(3 + x))]^2)/(3*x^3 + x^4 + (6*x^2 + 2*x^3)*Log[2/ (E^(5/3)*(3 + x))] + (3*x + x^2)*Log[2/(E^(5/3)*(3 + x))]^2),x]
Output:
Log[x] + (9*E^(x + x^2))/(-5 + 3*x + 3*Log[2] + 3*Log[(3 + x)^(-1)])
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.
| \(\displaystyle \int \frac {x^3+3 x^2+\left (2 x^2+e^{x^2+x} \left (6 x^3+21 x^2+9 x\right )+6 x\right ) \log \left (\frac {2}{e^{5/3} (x+3)}\right )+e^{x^2+x} \left (6 x^4+21 x^3+6 x^2-6 x\right )+(x+3) \log ^2\left (\frac {2}{e^{5/3} (x+3)}\right )}{x^4+3 x^3+\left (x^2+3 x\right ) \log ^2\left (\frac {2}{e^{5/3} (x+3)}\right )+\left (2 x^3+6 x^2\right ) \log \left (\frac {2}{e^{5/3} (x+3)}\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {9 \left (x^3+3 x^2+\left (2 x^2+e^{x^2+x} \left (6 x^3+21 x^2+9 x\right )+6 x\right ) \log \left (\frac {2}{e^{5/3} (x+3)}\right )+e^{x^2+x} \left (6 x^4+21 x^3+6 x^2-6 x\right )+(x+3) \log ^2\left (\frac {2}{e^{5/3} (x+3)}\right )\right )}{x (x+3) \left (3 x+3 \log \left (\frac {1}{x+3}\right )-5 \left (1-\frac {3 \log (2)}{5}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 9 \int \frac {x^3+3 x^2+(x+3) \log ^2\left (\frac {2}{e^{5/3} (x+3)}\right )-3 e^{x^2+x} \left (-2 x^4-7 x^3-2 x^2+2 x\right )+\left (2 x^2+6 x+3 e^{x^2+x} \left (2 x^3+7 x^2+3 x\right )\right ) \log \left (\frac {2}{e^{5/3} (x+3)}\right )}{x (x+3) \left (-3 x-3 \log \left (\frac {1}{x+3}\right )-\log (8)+5\right )^2}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle 9 \int \frac {x^3+3 x^2+(x+3) \log ^2\left (\frac {2}{e^{5/3} (x+3)}\right )-3 e^{x^2+x} \left (-2 x^4-7 x^3-2 x^2+2 x\right )+\left (2 x^2+6 x+3 e^{x^2+x} \left (2 x^3+7 x^2+3 x\right )\right ) \log \left (\frac {2}{e^{5/3} (x+3)}\right )}{x (x+3) \left (3 x+3 \log \left (\frac {1}{x+3}\right )-5 \left (1-\frac {3 \log (2)}{5}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 9 \int \left (\frac {x^2}{(x+3) \left (3 x+3 \log \left (\frac {1}{x+3}\right )-5 \left (1-\frac {3 \log (2)}{5}\right )\right )^2}+\frac {2 \left (3 \log \left (\frac {1}{x+3}\right )-5 \left (1-\frac {3 \log (2)}{5}\right )\right ) x}{3 (x+3) \left (3 x+3 \log \left (\frac {1}{x+3}\right )-5 \left (1-\frac {3 \log (2)}{5}\right )\right )^2}+\frac {3 x}{(x+3) \left (3 x+3 \log \left (\frac {1}{x+3}\right )-5 \left (1-\frac {3 \log (2)}{5}\right )\right )^2}+\frac {2 \left (3 \log \left (\frac {2}{x+3}\right )-5\right )}{(x+3) \left (3 x+3 \log \left (\frac {1}{x+3}\right )-5 \left (1-\frac {3 \log (2)}{5}\right )\right )^2}+\frac {e^{x^2+x} \left (6 x^3+6 \log \left (\frac {2}{x+3}\right ) x^2+11 x^2+21 \log \left (\frac {2}{x+3}\right ) x-29 x+9 \log \left (\frac {1}{x+3}\right )-21 \left (1-\frac {3 \log (2)}{7}\right )\right )}{(x+3) \left (3 x+3 \log \left (\frac {1}{x+3}\right )-5 \left (1-\frac {3 \log (2)}{5}\right )\right )^2}+\frac {\left (5 \left (1-\frac {3 \log (2)}{5}\right )-3 \log \left (\frac {1}{x+3}\right )\right )^2}{9 \left (3 x+3 \log \left (\frac {1}{x+3}\right )-5 \left (1-\frac {3 \log (2)}{5}\right )\right )^2 x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 9 \left (-17 \int \frac {e^{x^2+x}}{\left (3 x+3 \log \left (\frac {1}{x+3}\right )-5 \left (1-\frac {3 \log (2)}{5}\right )\right )^2}dx+162 \int \frac {e^{x^2+x}}{(-x-3) \left (3 x+3 \log \left (\frac {1}{x+3}\right )-5 \left (1-\frac {3 \log (2)}{5}\right )\right )^2}dx-7 \int \frac {e^{x^2+x} x}{\left (3 x+3 \log \left (\frac {1}{x+3}\right )-5 \left (1-\frac {3 \log (2)}{5}\right )\right )^2}dx+6 \int \frac {e^{x^2+x} x^2}{\left (3 x+3 \log \left (\frac {1}{x+3}\right )-5 \left (1-\frac {3 \log (2)}{5}\right )\right )^2}dx-(21-\log (512)) \int \frac {e^{x^2+x}}{(x+3) \left (3 x+3 \log \left (\frac {1}{x+3}\right )-5 \left (1-\frac {3 \log (2)}{5}\right )\right )^2}dx+3 (14-\log (8)) \int \frac {e^{x^2+x}}{(x+3) \left (3 x+3 \log \left (\frac {1}{x+3}\right )-5 \left (1-\frac {3 \log (2)}{5}\right )\right )^2}dx+186 \int \frac {e^{x^2+x}}{(x+3) \left (3 x+3 \log \left (\frac {1}{x+3}\right )-5 \left (1-\frac {3 \log (2)}{5}\right )\right )^2}dx+3 \int \frac {e^{x^2+x}}{(x+3) \left (3 x+3 \log \left (\frac {1}{x+3}\right )-5 \left (1-\frac {3 \log (2)}{5}\right )\right )}dx+3 \int \frac {e^{x^2+x} \log \left (\frac {2}{x+3}\right )}{\left (3 x+3 \log \left (\frac {1}{x+3}\right )-5 \left (1-\frac {3 \log (2)}{5}\right )\right )^2}dx+9 \int \frac {e^{x^2+x} \log \left (\frac {2}{x+3}\right )}{(-x-3) \left (3 x+3 \log \left (\frac {1}{x+3}\right )-5 \left (1-\frac {3 \log (2)}{5}\right )\right )^2}dx+6 \int \frac {e^{x^2+x} x \log \left (\frac {2}{x+3}\right )}{\left (3 x+3 \log \left (\frac {1}{x+3}\right )-5 \left (1-\frac {3 \log (2)}{5}\right )\right )^2}dx+\frac {2}{3} \int \frac {1}{-3 x-3 \log \left (\frac {1}{x+3}\right )+5 \left (1-\frac {3 \log (2)}{5}\right )}dx+6 \int \frac {1}{\left (3 x+3 \log \left (\frac {1}{x+3}\right )-5 \left (1-\frac {3 \log (2)}{5}\right )\right )^2}dx+37 \int \frac {1}{(-x-3) \left (3 x+3 \log \left (\frac {1}{x+3}\right )-5 \left (1-\frac {3 \log (2)}{5}\right )\right )^2}dx+9 \int \frac {1}{(x+3) \left (3 x+3 \log \left (\frac {1}{x+3}\right )-5 \left (1-\frac {3 \log (2)}{5}\right )\right )^2}dx+\frac {2}{3} \int \frac {1}{3 x+3 \log \left (\frac {1}{x+3}\right )-5 \left (1-\frac {3 \log (2)}{5}\right )}dx+2 \int \frac {1}{(-x-3) \left (3 x+3 \log \left (\frac {1}{x+3}\right )-5 \left (1-\frac {3 \log (2)}{5}\right )\right )}dx+6 \int \frac {\log \left (\frac {2}{x+3}\right )}{(x+3) \left (3 x+3 \log \left (\frac {1}{x+3}\right )-5 \left (1-\frac {3 \log (2)}{5}\right )\right )^2}dx+\frac {\log (x)}{9}\right )\) |
Input:
Int[(3*x^2 + x^3 + E^(x + x^2)*(-6*x + 6*x^2 + 21*x^3 + 6*x^4) + (6*x + 2* x^2 + E^(x + x^2)*(9*x + 21*x^2 + 6*x^3))*Log[2/(E^(5/3)*(3 + x))] + (3 + x)*Log[2/(E^(5/3)*(3 + x))]^2)/(3*x^3 + x^4 + (6*x^2 + 2*x^3)*Log[2/(E^(5/ 3)*(3 + x))] + (3*x + x^2)*Log[2/(E^(5/3)*(3 + x))]^2),x]
Output:
$Aborted
Result contains complex when optimal does not.
Time = 40.78 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13
| method | result | size |
| risch | \(\ln \left (x \right )+\frac {18 i {\mathrm e}^{\left (1+x \right ) x}}{6 i \ln \left (2\right )+6 i x -6 i \ln \left (3+x \right )-10 i}\) | \(34\) |
| parallelrisch | \(\frac {x \ln \left (x \right )+\ln \left (x \right ) \ln \left (\frac {2 \,{\mathrm e}^{-\frac {5}{3}}}{3+x}\right )+3 \,{\mathrm e}^{x^{2}+x}}{x +\ln \left (\frac {2 \,{\mathrm e}^{-\frac {5}{3}}}{3+x}\right )}\) | \(46\) |
Input:
int(((3+x)*ln(2/(3+x)/exp(5/3))^2+((6*x^3+21*x^2+9*x)*exp(x^2+x)+2*x^2+6*x )*ln(2/(3+x)/exp(5/3))+(6*x^4+21*x^3+6*x^2-6*x)*exp(x^2+x)+x^3+3*x^2)/((x^ 2+3*x)*ln(2/(3+x)/exp(5/3))^2+(2*x^3+6*x^2)*ln(2/(3+x)/exp(5/3))+x^4+3*x^3 ),x,method=_RETURNVERBOSE)
Output:
ln(x)+18*I*exp((1+x)*x)/(6*I*ln(2)+6*I*x-6*I*ln(3+x)-10*I)
Time = 0.10 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.30 \[ \int \frac {3 x^2+x^3+e^{x+x^2} \left (-6 x+6 x^2+21 x^3+6 x^4\right )+\left (6 x+2 x^2+e^{x+x^2} \left (9 x+21 x^2+6 x^3\right )\right ) \log \left (\frac {2}{e^{5/3} (3+x)}\right )+(3+x) \log ^2\left (\frac {2}{e^{5/3} (3+x)}\right )}{3 x^3+x^4+\left (6 x^2+2 x^3\right ) \log \left (\frac {2}{e^{5/3} (3+x)}\right )+\left (3 x+x^2\right ) \log ^2\left (\frac {2}{e^{5/3} (3+x)}\right )} \, dx=\frac {{\left (x + \log \left (\frac {2 \, e^{\left (-\frac {5}{3}\right )}}{x + 3}\right )\right )} \log \left (x\right ) + 3 \, e^{\left (x^{2} + x\right )}}{x + \log \left (\frac {2 \, e^{\left (-\frac {5}{3}\right )}}{x + 3}\right )} \] Input:
integrate(((3+x)*log(2/(3+x)/exp(5/3))^2+((6*x^3+21*x^2+9*x)*exp(x^2+x)+2* x^2+6*x)*log(2/(3+x)/exp(5/3))+(6*x^4+21*x^3+6*x^2-6*x)*exp(x^2+x)+x^3+3*x ^2)/((x^2+3*x)*log(2/(3+x)/exp(5/3))^2+(2*x^3+6*x^2)*log(2/(3+x)/exp(5/3)) +x^4+3*x^3),x, algorithm="fricas")
Output:
((x + log(2*e^(-5/3)/(x + 3)))*log(x) + 3*e^(x^2 + x))/(x + log(2*e^(-5/3) /(x + 3)))
Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {3 x^2+x^3+e^{x+x^2} \left (-6 x+6 x^2+21 x^3+6 x^4\right )+\left (6 x+2 x^2+e^{x+x^2} \left (9 x+21 x^2+6 x^3\right )\right ) \log \left (\frac {2}{e^{5/3} (3+x)}\right )+(3+x) \log ^2\left (\frac {2}{e^{5/3} (3+x)}\right )}{3 x^3+x^4+\left (6 x^2+2 x^3\right ) \log \left (\frac {2}{e^{5/3} (3+x)}\right )+\left (3 x+x^2\right ) \log ^2\left (\frac {2}{e^{5/3} (3+x)}\right )} \, dx=\log {\left (x \right )} + \frac {3 e^{x^{2} + x}}{x + \log {\left (\frac {2}{\left (x + 3\right ) e^{\frac {5}{3}}} \right )}} \] Input:
integrate(((3+x)*ln(2/(3+x)/exp(5/3))**2+((6*x**3+21*x**2+9*x)*exp(x**2+x) +2*x**2+6*x)*ln(2/(3+x)/exp(5/3))+(6*x**4+21*x**3+6*x**2-6*x)*exp(x**2+x)+ x**3+3*x**2)/((x**2+3*x)*ln(2/(3+x)/exp(5/3))**2+(2*x**3+6*x**2)*ln(2/(3+x )/exp(5/3))+x**4+3*x**3),x)
Output:
log(x) + 3*exp(x**2 + x)/(x + log(2*exp(-5/3)/(x + 3)))
Time = 0.17 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93 \[ \int \frac {3 x^2+x^3+e^{x+x^2} \left (-6 x+6 x^2+21 x^3+6 x^4\right )+\left (6 x+2 x^2+e^{x+x^2} \left (9 x+21 x^2+6 x^3\right )\right ) \log \left (\frac {2}{e^{5/3} (3+x)}\right )+(3+x) \log ^2\left (\frac {2}{e^{5/3} (3+x)}\right )}{3 x^3+x^4+\left (6 x^2+2 x^3\right ) \log \left (\frac {2}{e^{5/3} (3+x)}\right )+\left (3 x+x^2\right ) \log ^2\left (\frac {2}{e^{5/3} (3+x)}\right )} \, dx=\frac {9 \, e^{\left (x^{2} + x\right )}}{3 \, x + 3 \, \log \left (2\right ) - 3 \, \log \left (x + 3\right ) - 5} + \log \left (x\right ) \] Input:
integrate(((3+x)*log(2/(3+x)/exp(5/3))^2+((6*x^3+21*x^2+9*x)*exp(x^2+x)+2* x^2+6*x)*log(2/(3+x)/exp(5/3))+(6*x^4+21*x^3+6*x^2-6*x)*exp(x^2+x)+x^3+3*x ^2)/((x^2+3*x)*log(2/(3+x)/exp(5/3))^2+(2*x^3+6*x^2)*log(2/(3+x)/exp(5/3)) +x^4+3*x^3),x, algorithm="maxima")
Output:
9*e^(x^2 + x)/(3*x + 3*log(2) - 3*log(x + 3) - 5) + log(x)
Time = 0.20 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.60 \[ \int \frac {3 x^2+x^3+e^{x+x^2} \left (-6 x+6 x^2+21 x^3+6 x^4\right )+\left (6 x+2 x^2+e^{x+x^2} \left (9 x+21 x^2+6 x^3\right )\right ) \log \left (\frac {2}{e^{5/3} (3+x)}\right )+(3+x) \log ^2\left (\frac {2}{e^{5/3} (3+x)}\right )}{3 x^3+x^4+\left (6 x^2+2 x^3\right ) \log \left (\frac {2}{e^{5/3} (3+x)}\right )+\left (3 x+x^2\right ) \log ^2\left (\frac {2}{e^{5/3} (3+x)}\right )} \, dx=\frac {3 \, x \log \left (x\right ) + 3 \, \log \left (x\right ) \log \left (\frac {2}{x + 3}\right ) + 9 \, e^{\left (x^{2} + x\right )} - 5 \, \log \left (x\right )}{3 \, x + 3 \, \log \left (\frac {2}{x + 3}\right ) - 5} \] Input:
integrate(((3+x)*log(2/(3+x)/exp(5/3))^2+((6*x^3+21*x^2+9*x)*exp(x^2+x)+2* x^2+6*x)*log(2/(3+x)/exp(5/3))+(6*x^4+21*x^3+6*x^2-6*x)*exp(x^2+x)+x^3+3*x ^2)/((x^2+3*x)*log(2/(3+x)/exp(5/3))^2+(2*x^3+6*x^2)*log(2/(3+x)/exp(5/3)) +x^4+3*x^3),x, algorithm="giac")
Output:
(3*x*log(x) + 3*log(x)*log(2/(x + 3)) + 9*e^(x^2 + x) - 5*log(x))/(3*x + 3 *log(2/(x + 3)) - 5)
Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37 \[ \int \frac {3 x^2+x^3+e^{x+x^2} \left (-6 x+6 x^2+21 x^3+6 x^4\right )+\left (6 x+2 x^2+e^{x+x^2} \left (9 x+21 x^2+6 x^3\right )\right ) \log \left (\frac {2}{e^{5/3} (3+x)}\right )+(3+x) \log ^2\left (\frac {2}{e^{5/3} (3+x)}\right )}{3 x^3+x^4+\left (6 x^2+2 x^3\right ) \log \left (\frac {2}{e^{5/3} (3+x)}\right )+\left (3 x+x^2\right ) \log ^2\left (\frac {2}{e^{5/3} (3+x)}\right )} \, dx=\frac {3\,{\mathrm {e}}^{x^2+x}+\ln \left (\frac {2\,{\mathrm {e}}^{-\frac {5}{3}}}{x+3}\right )\,\ln \left (x\right )+x\,\ln \left (x\right )}{x+\ln \left (\frac {2\,{\mathrm {e}}^{-\frac {5}{3}}}{x+3}\right )} \] Input:
int((log((2*exp(-5/3))/(x + 3))*(6*x + exp(x + x^2)*(9*x + 21*x^2 + 6*x^3) + 2*x^2) + exp(x + x^2)*(6*x^2 - 6*x + 21*x^3 + 6*x^4) + 3*x^2 + x^3 + lo g((2*exp(-5/3))/(x + 3))^2*(x + 3))/(log((2*exp(-5/3))/(x + 3))^2*(3*x + x ^2) + log((2*exp(-5/3))/(x + 3))*(6*x^2 + 2*x^3) + 3*x^3 + x^4),x)
Output:
(3*exp(x + x^2) + log((2*exp(-5/3))/(x + 3))*log(x) + x*log(x))/(x + log(( 2*exp(-5/3))/(x + 3)))
Time = 0.46 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.80 \[ \int \frac {3 x^2+x^3+e^{x+x^2} \left (-6 x+6 x^2+21 x^3+6 x^4\right )+\left (6 x+2 x^2+e^{x+x^2} \left (9 x+21 x^2+6 x^3\right )\right ) \log \left (\frac {2}{e^{5/3} (3+x)}\right )+(3+x) \log ^2\left (\frac {2}{e^{5/3} (3+x)}\right )}{3 x^3+x^4+\left (6 x^2+2 x^3\right ) \log \left (\frac {2}{e^{5/3} (3+x)}\right )+\left (3 x+x^2\right ) \log ^2\left (\frac {2}{e^{5/3} (3+x)}\right )} \, dx=\frac {3 e^{x^{2}+x}+\mathrm {log}\left (\frac {2}{e^{\frac {5}{3}} x +3 e^{\frac {5}{3}}}\right ) \mathrm {log}\left (x \right )+\mathrm {log}\left (x \right ) x}{\mathrm {log}\left (\frac {2}{e^{\frac {5}{3}} x +3 e^{\frac {5}{3}}}\right )+x} \] Input:
int(((3+x)*log(2/(3+x)/exp(5/3))^2+((6*x^3+21*x^2+9*x)*exp(x^2+x)+2*x^2+6* x)*log(2/(3+x)/exp(5/3))+(6*x^4+21*x^3+6*x^2-6*x)*exp(x^2+x)+x^3+3*x^2)/(( x^2+3*x)*log(2/(3+x)/exp(5/3))^2+(2*x^3+6*x^2)*log(2/(3+x)/exp(5/3))+x^4+3 *x^3),x)
Output:
(3*e**(x**2 + x) + log(2/(e**(2/3)*e*x + 3*e**(2/3)*e))*log(x) + log(x)*x) /(log(2/(e**(2/3)*e*x + 3*e**(2/3)*e)) + x)