Integrand size = 153, antiderivative size = 32 \[ \int \frac {-6 e^{2+2 x} x^3+e^{\frac {e^{-2-2 x} \left (-4+x-x^2+\log (x)\right )}{x^2}} \left (27+21 x-6 x^2+3 e^{2+2 x} x^2+6 x^3+(-6-6 x) \log (x)\right )}{e^{2+2 x+\frac {2 e^{-2-2 x} \left (-4+x-x^2+\log (x)\right )}{x^2}} x^4-2 e^{2+2 x+\frac {e^{-2-2 x} \left (-4+x-x^2+\log (x)\right )}{x^2}} x^5+e^{2+2 x} x^6} \, dx=\frac {3}{x \left (-e^{e^{-2-2 x} \left (-1+\frac {-4+x+\log (x)}{x^2}\right )}+x\right )} \]
\[ \int \frac {-6 e^{2+2 x} x^3+e^{\frac {e^{-2-2 x} \left (-4+x-x^2+\log (x)\right )}{x^2}} \left (27+21 x-6 x^2+3 e^{2+2 x} x^2+6 x^3+(-6-6 x) \log (x)\right )}{e^{2+2 x+\frac {2 e^{-2-2 x} \left (-4+x-x^2+\log (x)\right )}{x^2}} x^4-2 e^{2+2 x+\frac {e^{-2-2 x} \left (-4+x-x^2+\log (x)\right )}{x^2}} x^5+e^{2+2 x} x^6} \, dx=\int \frac {-6 e^{2+2 x} x^3+e^{\frac {e^{-2-2 x} \left (-4+x-x^2+\log (x)\right )}{x^2}} \left (27+21 x-6 x^2+3 e^{2+2 x} x^2+6 x^3+(-6-6 x) \log (x)\right )}{e^{2+2 x+\frac {2 e^{-2-2 x} \left (-4+x-x^2+\log (x)\right )}{x^2}} x^4-2 e^{2+2 x+\frac {e^{-2-2 x} \left (-4+x-x^2+\log (x)\right )}{x^2}} x^5+e^{2+2 x} x^6} \, dx \]
Integrate[(-6*E^(2 + 2*x)*x^3 + E^((E^(-2 - 2*x)*(-4 + x - x^2 + Log[x]))/ x^2)*(27 + 21*x - 6*x^2 + 3*E^(2 + 2*x)*x^2 + 6*x^3 + (-6 - 6*x)*Log[x]))/ (E^(2 + 2*x + (2*E^(-2 - 2*x)*(-4 + x - x^2 + Log[x]))/x^2)*x^4 - 2*E^(2 + 2*x + (E^(-2 - 2*x)*(-4 + x - x^2 + Log[x]))/x^2)*x^5 + E^(2 + 2*x)*x^6), x]
Integrate[(-6*E^(2 + 2*x)*x^3 + E^((E^(-2 - 2*x)*(-4 + x - x^2 + Log[x]))/ x^2)*(27 + 21*x - 6*x^2 + 3*E^(2 + 2*x)*x^2 + 6*x^3 + (-6 - 6*x)*Log[x]))/ (E^(2 + 2*x + (2*E^(-2 - 2*x)*(-4 + x - x^2 + Log[x]))/x^2)*x^4 - 2*E^(2 + 2*x + (E^(-2 - 2*x)*(-4 + x - x^2 + Log[x]))/x^2)*x^5 + E^(2 + 2*x)*x^6), 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.
| \(\displaystyle \int \frac {e^{\frac {e^{-2 x-2} \left (-x^2+x+\log (x)-4\right )}{x^2}} \left (6 x^3+3 e^{2 x+2} x^2-6 x^2+21 x+(-6 x-6) \log (x)+27\right )-6 e^{2 x+2} x^3}{-2 x^5 \exp \left (\frac {e^{-2 x-2} \left (-x^2+x+\log (x)-4\right )}{x^2}+2 x+2\right )+x^4 \exp \left (\frac {2 e^{-2 x-2} \left (-x^2+x+\log (x)-4\right )}{x^2}+2 x+2\right )+e^{2 x+2} x^6} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {3 \exp \left (\frac {8 e^{-2 x-2}}{x^2}-2 x+2 e^{-2 x-2}-2\right ) \left (-2 x^3+e^{2 x+2} x^2+2 x^2-7 x+2 x \log (x)+2 \log (x)-9\right )}{x^3 \left (e^{\frac {4 e^{-2 x-2}}{x^2}+e^{-2 x-2}} x-e^{\frac {e^{-2 x-2}}{x}} x^{\frac {e^{-2 x-2}}{x^2}}\right )^2}-\frac {3 \exp \left (\frac {4 e^{-2 x-2}}{x^2}-2 x+e^{-2 x-2}-2\right ) \left (2 x^3+e^{2 x+2} x^2-2 x^2+7 x-2 x \log (x)-2 \log (x)+9\right )}{x^4 \left (e^{\frac {4 e^{-2 x-2}}{x^2}+e^{-2 x-2}} x-e^{\frac {e^{-2 x-2}}{x}} x^{\frac {e^{-2 x-2}}{x^2}}\right )}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (-\frac {3 \exp \left (\frac {8 e^{-2 x-2}}{x^2}-2 x+2 e^{-2 x-2}-2\right ) \left (-2 x^3+e^{2 x+2} x^2+2 x^2-7 x+2 x \log (x)+2 \log (x)-9\right )}{x^3 \left (e^{\frac {4 e^{-2 x-2}}{x^2}+e^{-2 x-2}} x-e^{\frac {e^{-2 x-2}}{x}} x^{\frac {e^{-2 x-2}}{x^2}}\right )^2}-\frac {3 \exp \left (\frac {4 e^{-2 x-2}}{x^2}-2 x+e^{-2 x-2}-2\right ) \left (2 x^3+e^{2 x+2} x^2-2 x^2+7 x-2 x \log (x)-2 \log (x)+9\right )}{x^4 \left (e^{\frac {4 e^{-2 x-2}}{x^2}+e^{-2 x-2}} x-e^{\frac {e^{-2 x-2}}{x}} x^{\frac {e^{-2 x-2}}{x^2}}\right )}\right )dx\) |
Int[(-6*E^(2 + 2*x)*x^3 + E^((E^(-2 - 2*x)*(-4 + x - x^2 + Log[x]))/x^2)*( 27 + 21*x - 6*x^2 + 3*E^(2 + 2*x)*x^2 + 6*x^3 + (-6 - 6*x)*Log[x]))/(E^(2 + 2*x + (2*E^(-2 - 2*x)*(-4 + x - x^2 + Log[x]))/x^2)*x^4 - 2*E^(2 + 2*x + (E^(-2 - 2*x)*(-4 + x - x^2 + Log[x]))/x^2)*x^5 + E^(2 + 2*x)*x^6),x]
3.10.61.3.1 Defintions of rubi rules used
Time = 17.29 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03
| method | result | size |
| risch | \(\frac {3}{x \left (x -{\mathrm e}^{\frac {\left (\ln \left (x \right )-x^{2}+x -4\right ) {\mathrm e}^{-2-2 x}}{x^{2}}}\right )}\) | \(33\) |
| parallelrisch | \(\frac {3}{x \left (x -{\mathrm e}^{\frac {\left (\ln \left (x \right )-x^{2}+x -4\right ) {\mathrm e}^{-2-2 x}}{x^{2}}}\right )}\) | \(35\) |
int((((-6*x-6)*ln(x)+3*x^2*exp(2+2*x)+6*x^3-6*x^2+21*x+27)*exp((ln(x)-x^2+ x-4)/x^2/exp(2+2*x))-6*x^3*exp(2+2*x))/(x^4*exp(2+2*x)*exp((ln(x)-x^2+x-4) /x^2/exp(2+2*x))^2-2*x^5*exp(2+2*x)*exp((ln(x)-x^2+x-4)/x^2/exp(2+2*x))+x^ 6*exp(2+2*x)),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (30) = 60\).
Time = 0.27 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.97 \[ \int \frac {-6 e^{2+2 x} x^3+e^{\frac {e^{-2-2 x} \left (-4+x-x^2+\log (x)\right )}{x^2}} \left (27+21 x-6 x^2+3 e^{2+2 x} x^2+6 x^3+(-6-6 x) \log (x)\right )}{e^{2+2 x+\frac {2 e^{-2-2 x} \left (-4+x-x^2+\log (x)\right )}{x^2}} x^4-2 e^{2+2 x+\frac {e^{-2-2 x} \left (-4+x-x^2+\log (x)\right )}{x^2}} x^5+e^{2+2 x} x^6} \, dx=\frac {3 \, e^{\left (2 \, x + 2\right )}}{x^{2} e^{\left (2 \, x + 2\right )} - x e^{\left (-\frac {{\left (x^{2} - 2 \, {\left (x^{3} + x^{2}\right )} e^{\left (2 \, x + 2\right )} - x - \log \left (x\right ) + 4\right )} e^{\left (-2 \, x - 2\right )}}{x^{2}}\right )}} \]
integrate((((-6*x-6)*log(x)+3*x^2*exp(2+2*x)+6*x^3-6*x^2+21*x+27)*exp((log (x)-x^2+x-4)/x^2/exp(2+2*x))-6*x^3*exp(2+2*x))/(x^4*exp(2+2*x)*exp((log(x) -x^2+x-4)/x^2/exp(2+2*x))^2-2*x^5*exp(2+2*x)*exp((log(x)-x^2+x-4)/x^2/exp( 2+2*x))+x^6*exp(2+2*x)),x, algorithm=\
3*e^(2*x + 2)/(x^2*e^(2*x + 2) - x*e^(-(x^2 - 2*(x^3 + x^2)*e^(2*x + 2) - x - log(x) + 4)*e^(-2*x - 2)/x^2))
Time = 0.30 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {-6 e^{2+2 x} x^3+e^{\frac {e^{-2-2 x} \left (-4+x-x^2+\log (x)\right )}{x^2}} \left (27+21 x-6 x^2+3 e^{2+2 x} x^2+6 x^3+(-6-6 x) \log (x)\right )}{e^{2+2 x+\frac {2 e^{-2-2 x} \left (-4+x-x^2+\log (x)\right )}{x^2}} x^4-2 e^{2+2 x+\frac {e^{-2-2 x} \left (-4+x-x^2+\log (x)\right )}{x^2}} x^5+e^{2+2 x} x^6} \, dx=- \frac {3}{- x^{2} + x e^{\frac {\left (- x^{2} + x + \log {\left (x \right )} - 4\right ) e^{- 2 x - 2}}{x^{2}}}} \]
integrate((((-6*x-6)*ln(x)+3*x**2*exp(2+2*x)+6*x**3-6*x**2+21*x+27)*exp((l n(x)-x**2+x-4)/x**2/exp(2+2*x))-6*x**3*exp(2+2*x))/(x**4*exp(2+2*x)*exp((l n(x)-x**2+x-4)/x**2/exp(2+2*x))**2-2*x**5*exp(2+2*x)*exp((ln(x)-x**2+x-4)/ x**2/exp(2+2*x))+x**6*exp(2+2*x)),x)
Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (30) = 60\).
Time = 0.38 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.31 \[ \int \frac {-6 e^{2+2 x} x^3+e^{\frac {e^{-2-2 x} \left (-4+x-x^2+\log (x)\right )}{x^2}} \left (27+21 x-6 x^2+3 e^{2+2 x} x^2+6 x^3+(-6-6 x) \log (x)\right )}{e^{2+2 x+\frac {2 e^{-2-2 x} \left (-4+x-x^2+\log (x)\right )}{x^2}} x^4-2 e^{2+2 x+\frac {e^{-2-2 x} \left (-4+x-x^2+\log (x)\right )}{x^2}} x^5+e^{2+2 x} x^6} \, dx=\frac {3 \, e^{\left (\frac {4 \, e^{\left (-2 \, x - 2\right )}}{x^{2}} + e^{\left (-2 \, x - 2\right )}\right )}}{x^{2} e^{\left (\frac {4 \, e^{\left (-2 \, x - 2\right )}}{x^{2}} + e^{\left (-2 \, x - 2\right )}\right )} - x e^{\left (\frac {e^{\left (-2 \, x - 2\right )}}{x} + \frac {e^{\left (-2 \, x - 2\right )} \log \left (x\right )}{x^{2}}\right )}} \]
integrate((((-6*x-6)*log(x)+3*x^2*exp(2+2*x)+6*x^3-6*x^2+21*x+27)*exp((log (x)-x^2+x-4)/x^2/exp(2+2*x))-6*x^3*exp(2+2*x))/(x^4*exp(2+2*x)*exp((log(x) -x^2+x-4)/x^2/exp(2+2*x))^2-2*x^5*exp(2+2*x)*exp((log(x)-x^2+x-4)/x^2/exp( 2+2*x))+x^6*exp(2+2*x)),x, algorithm=\
3*e^(4*e^(-2*x - 2)/x^2 + e^(-2*x - 2))/(x^2*e^(4*e^(-2*x - 2)/x^2 + e^(-2 *x - 2)) - x*e^(e^(-2*x - 2)/x + e^(-2*x - 2)*log(x)/x^2))
Leaf count of result is larger than twice the leaf count of optimal. 513 vs. \(2 (30) = 60\).
Time = 0.66 (sec) , antiderivative size = 513, normalized size of antiderivative = 16.03 \[ \int \frac {-6 e^{2+2 x} x^3+e^{\frac {e^{-2-2 x} \left (-4+x-x^2+\log (x)\right )}{x^2}} \left (27+21 x-6 x^2+3 e^{2+2 x} x^2+6 x^3+(-6-6 x) \log (x)\right )}{e^{2+2 x+\frac {2 e^{-2-2 x} \left (-4+x-x^2+\log (x)\right )}{x^2}} x^4-2 e^{2+2 x+\frac {e^{-2-2 x} \left (-4+x-x^2+\log (x)\right )}{x^2}} x^5+e^{2+2 x} x^6} \, dx=\frac {3 \, {\left (2 \, x^{3} - x^{2} e^{\left (2 \, x + 2\right )} - 2 \, x^{2} - 2 \, x \log \left (x\right ) + 7 \, x - 2 \, \log \left (x\right ) + 9\right )}}{2 \, x^{5} - x^{4} e^{\left (2 \, x + 2\right )} - 2 \, x^{4} e^{\left (-2 \, x + \frac {2 \, x^{3} - x^{2} e^{\left (-2 \, x - 2\right )} + 2 \, x^{2} + x e^{\left (-2 \, x - 2\right )} + e^{\left (-2 \, x - 2\right )} \log \left (x\right ) - 4 \, e^{\left (-2 \, x - 2\right )}}{x^{2}} - 2\right )} - 2 \, x^{4} + 2 \, x^{3} e^{\left (-2 \, x + \frac {2 \, x^{3} - x^{2} e^{\left (-2 \, x - 2\right )} + 2 \, x^{2} + x e^{\left (-2 \, x - 2\right )} + e^{\left (-2 \, x - 2\right )} \log \left (x\right ) - 4 \, e^{\left (-2 \, x - 2\right )}}{x^{2}} - 2\right )} + x^{3} e^{\left (\frac {2 \, x^{3} - x^{2} e^{\left (-2 \, x - 2\right )} + 2 \, x^{2} + x e^{\left (-2 \, x - 2\right )} + e^{\left (-2 \, x - 2\right )} \log \left (x\right ) - 4 \, e^{\left (-2 \, x - 2\right )}}{x^{2}}\right )} - 2 \, x^{3} \log \left (x\right ) + 2 \, x^{2} e^{\left (-2 \, x + \frac {2 \, x^{3} - x^{2} e^{\left (-2 \, x - 2\right )} + 2 \, x^{2} + x e^{\left (-2 \, x - 2\right )} + e^{\left (-2 \, x - 2\right )} \log \left (x\right ) - 4 \, e^{\left (-2 \, x - 2\right )}}{x^{2}} - 2\right )} \log \left (x\right ) + 7 \, x^{3} - 7 \, x^{2} e^{\left (-2 \, x + \frac {2 \, x^{3} - x^{2} e^{\left (-2 \, x - 2\right )} + 2 \, x^{2} + x e^{\left (-2 \, x - 2\right )} + e^{\left (-2 \, x - 2\right )} \log \left (x\right ) - 4 \, e^{\left (-2 \, x - 2\right )}}{x^{2}} - 2\right )} - 2 \, x^{2} \log \left (x\right ) + 2 \, x e^{\left (-2 \, x + \frac {2 \, x^{3} - x^{2} e^{\left (-2 \, x - 2\right )} + 2 \, x^{2} + x e^{\left (-2 \, x - 2\right )} + e^{\left (-2 \, x - 2\right )} \log \left (x\right ) - 4 \, e^{\left (-2 \, x - 2\right )}}{x^{2}} - 2\right )} \log \left (x\right ) + 9 \, x^{2} - 9 \, x e^{\left (-2 \, x + \frac {2 \, x^{3} - x^{2} e^{\left (-2 \, x - 2\right )} + 2 \, x^{2} + x e^{\left (-2 \, x - 2\right )} + e^{\left (-2 \, x - 2\right )} \log \left (x\right ) - 4 \, e^{\left (-2 \, x - 2\right )}}{x^{2}} - 2\right )}} \]
integrate((((-6*x-6)*log(x)+3*x^2*exp(2+2*x)+6*x^3-6*x^2+21*x+27)*exp((log (x)-x^2+x-4)/x^2/exp(2+2*x))-6*x^3*exp(2+2*x))/(x^4*exp(2+2*x)*exp((log(x) -x^2+x-4)/x^2/exp(2+2*x))^2-2*x^5*exp(2+2*x)*exp((log(x)-x^2+x-4)/x^2/exp( 2+2*x))+x^6*exp(2+2*x)),x, algorithm=\
3*(2*x^3 - x^2*e^(2*x + 2) - 2*x^2 - 2*x*log(x) + 7*x - 2*log(x) + 9)/(2*x ^5 - x^4*e^(2*x + 2) - 2*x^4*e^(-2*x + (2*x^3 - x^2*e^(-2*x - 2) + 2*x^2 + x*e^(-2*x - 2) + e^(-2*x - 2)*log(x) - 4*e^(-2*x - 2))/x^2 - 2) - 2*x^4 + 2*x^3*e^(-2*x + (2*x^3 - x^2*e^(-2*x - 2) + 2*x^2 + x*e^(-2*x - 2) + e^(- 2*x - 2)*log(x) - 4*e^(-2*x - 2))/x^2 - 2) + x^3*e^((2*x^3 - x^2*e^(-2*x - 2) + 2*x^2 + x*e^(-2*x - 2) + e^(-2*x - 2)*log(x) - 4*e^(-2*x - 2))/x^2) - 2*x^3*log(x) + 2*x^2*e^(-2*x + (2*x^3 - x^2*e^(-2*x - 2) + 2*x^2 + x*e^( -2*x - 2) + e^(-2*x - 2)*log(x) - 4*e^(-2*x - 2))/x^2 - 2)*log(x) + 7*x^3 - 7*x^2*e^(-2*x + (2*x^3 - x^2*e^(-2*x - 2) + 2*x^2 + x*e^(-2*x - 2) + e^( -2*x - 2)*log(x) - 4*e^(-2*x - 2))/x^2 - 2) - 2*x^2*log(x) + 2*x*e^(-2*x + (2*x^3 - x^2*e^(-2*x - 2) + 2*x^2 + x*e^(-2*x - 2) + e^(-2*x - 2)*log(x) - 4*e^(-2*x - 2))/x^2 - 2)*log(x) + 9*x^2 - 9*x*e^(-2*x + (2*x^3 - x^2*e^( -2*x - 2) + 2*x^2 + x*e^(-2*x - 2) + e^(-2*x - 2)*log(x) - 4*e^(-2*x - 2)) /x^2 - 2))
Time = 9.47 (sec) , antiderivative size = 214, normalized size of antiderivative = 6.69 \[ \int \frac {-6 e^{2+2 x} x^3+e^{\frac {e^{-2-2 x} \left (-4+x-x^2+\log (x)\right )}{x^2}} \left (27+21 x-6 x^2+3 e^{2+2 x} x^2+6 x^3+(-6-6 x) \log (x)\right )}{e^{2+2 x+\frac {2 e^{-2-2 x} \left (-4+x-x^2+\log (x)\right )}{x^2}} x^4-2 e^{2+2 x+\frac {e^{-2-2 x} \left (-4+x-x^2+\log (x)\right )}{x^2}} x^5+e^{2+2 x} x^6} \, dx=-\frac {x^4\,\left (21\,{\mathrm {e}}^{2\,x+2}-6\,{\mathrm {e}}^{2\,x+2}\,\ln \left (x\right )\right )+x^3\,\left (27\,{\mathrm {e}}^{2\,x+2}-6\,{\mathrm {e}}^{2\,x+2}\,\ln \left (x\right )\right )-x^5\,\left (6\,{\mathrm {e}}^{2\,x+2}+3\,{\mathrm {e}}^{4\,x+4}\right )+6\,x^6\,{\mathrm {e}}^{2\,x+2}}{\left (x-x^{\frac {{\mathrm {e}}^{-2\,x-2}}{x^2}}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{-2}}{x}-{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{-2}-\frac {4\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{-2}}{x^2}}\right )\,\left (2\,x^6\,{\mathrm {e}}^{2\,x+2}-7\,x^5\,{\mathrm {e}}^{2\,x+2}-9\,x^4\,{\mathrm {e}}^{2\,x+2}-2\,x^7\,{\mathrm {e}}^{2\,x+2}+x^6\,{\mathrm {e}}^{4\,x+4}+2\,x^4\,{\mathrm {e}}^{2\,x+2}\,\ln \left (x\right )+2\,x^5\,{\mathrm {e}}^{2\,x+2}\,\ln \left (x\right )\right )} \]
int((exp((exp(- 2*x - 2)*(x + log(x) - x^2 - 4))/x^2)*(21*x - log(x)*(6*x + 6) + 3*x^2*exp(2*x + 2) - 6*x^2 + 6*x^3 + 27) - 6*x^3*exp(2*x + 2))/(x^6 *exp(2*x + 2) - 2*x^5*exp((exp(- 2*x - 2)*(x + log(x) - x^2 - 4))/x^2)*exp (2*x + 2) + x^4*exp((2*exp(- 2*x - 2)*(x + log(x) - x^2 - 4))/x^2)*exp(2*x + 2)),x)
-(x^4*(21*exp(2*x + 2) - 6*exp(2*x + 2)*log(x)) + x^3*(27*exp(2*x + 2) - 6 *exp(2*x + 2)*log(x)) - x^5*(6*exp(2*x + 2) + 3*exp(4*x + 4)) + 6*x^6*exp( 2*x + 2))/((x - x^(exp(- 2*x - 2)/x^2)*exp((exp(-2*x)*exp(-2))/x - exp(-2* x)*exp(-2) - (4*exp(-2*x)*exp(-2))/x^2))*(2*x^6*exp(2*x + 2) - 7*x^5*exp(2 *x + 2) - 9*x^4*exp(2*x + 2) - 2*x^7*exp(2*x + 2) + x^6*exp(4*x + 4) + 2*x ^4*exp(2*x + 2)*log(x) + 2*x^5*exp(2*x + 2)*log(x)))