Integrand size = 92, antiderivative size = 23 \[ \int \frac {e^{9 e^{\frac {2 \left (-4 x-x^2\right )}{6-3 x+3 \log (x)}}+\frac {2 \left (-4 x-x^2\right )}{6-3 x+3 \log (x)}} \left (-24-18 x+6 x^2+(-24-12 x) \log (x)\right )}{4-4 x+x^2+(4-2 x) \log (x)+\log ^2(x)} \, dx=e^{9 e^{\frac {2 x (4+x)}{3 (-2+x-\log (x))}}} \] Output:
exp(9*exp((4+x)*x/(3*x-3*ln(x)-6))^2)
Time = 3.84 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {e^{9 e^{\frac {2 \left (-4 x-x^2\right )}{6-3 x+3 \log (x)}}+\frac {2 \left (-4 x-x^2\right )}{6-3 x+3 \log (x)}} \left (-24-18 x+6 x^2+(-24-12 x) \log (x)\right )}{4-4 x+x^2+(4-2 x) \log (x)+\log ^2(x)} \, dx=e^{9 e^{-\frac {2 x (4+x)}{3 (2-x+\log (x))}}} \] Input:
Integrate[(E^(9*E^((2*(-4*x - x^2))/(6 - 3*x + 3*Log[x])) + (2*(-4*x - x^2 ))/(6 - 3*x + 3*Log[x]))*(-24 - 18*x + 6*x^2 + (-24 - 12*x)*Log[x]))/(4 - 4*x + x^2 + (4 - 2*x)*Log[x] + Log[x]^2),x]
Output:
E^(9/E^((2*x*(4 + x))/(3*(2 - x + Log[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.
| \(\displaystyle \int \frac {\left (6 x^2-18 x+(-12 x-24) \log (x)-24\right ) \exp \left (\frac {2 \left (-x^2-4 x\right )}{-3 x+3 \log (x)+6}+9 e^{\frac {2 \left (-x^2-4 x\right )}{-3 x+3 \log (x)+6}}\right )}{x^2-4 x+\log ^2(x)+(4-2 x) \log (x)+4} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {6 \left (x^2-3 x-2 x \log (x)-4 \log (x)-4\right ) \exp \left (\frac {2 \left (-x^2-4 x\right )}{-3 x+3 \log (x)+6}+9 e^{\frac {2 \left (-x^2-4 x\right )}{-3 x+3 \log (x)+6}}\right )}{(-x+\log (x)+2)^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 6 \int -\frac {\exp \left (9 e^{-\frac {2 \left (x^2+4 x\right )}{3 (-x+\log (x)+2)}}-\frac {2 \left (x^2+4 x\right )}{3 (-x+\log (x)+2)}\right ) \left (-x^2+2 \log (x) x+3 x+4 \log (x)+4\right )}{(-x+\log (x)+2)^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -6 \int \frac {\exp \left (9 e^{-\frac {2 \left (x^2+4 x\right )}{3 (-x+\log (x)+2)}}-\frac {2 \left (x^2+4 x\right )}{3 (-x+\log (x)+2)}\right ) \left (-x^2+2 \log (x) x+3 x+4 \log (x)+4\right )}{(-x+\log (x)+2)^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -6 \int \left (\frac {\exp \left (9 e^{-\frac {2 \left (x^2+4 x\right )}{3 (-x+\log (x)+2)}}-\frac {2 \left (x^2+4 x\right )}{3 (-x+\log (x)+2)}\right ) \left (x^2+3 x-4\right )}{(x-\log (x)-2)^2}-\frac {2 \exp \left (9 e^{-\frac {2 \left (x^2+4 x\right )}{3 (-x+\log (x)+2)}}-\frac {2 \left (x^2+4 x\right )}{3 (-x+\log (x)+2)}\right ) (x+2)}{x-\log (x)-2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -6 \left (-4 \int \frac {\exp \left (9 e^{-\frac {2 \left (x^2+4 x\right )}{3 (-x+\log (x)+2)}}-\frac {2 \left (x^2+4 x\right )}{3 (-x+\log (x)+2)}\right )}{(x-\log (x)-2)^2}dx+3 \int \frac {\exp \left (9 e^{-\frac {2 \left (x^2+4 x\right )}{3 (-x+\log (x)+2)}}-\frac {2 \left (x^2+4 x\right )}{3 (-x+\log (x)+2)}\right ) x}{(x-\log (x)-2)^2}dx+\int \frac {\exp \left (9 e^{-\frac {2 \left (x^2+4 x\right )}{3 (-x+\log (x)+2)}}-\frac {2 \left (x^2+4 x\right )}{3 (-x+\log (x)+2)}\right ) x^2}{(x-\log (x)-2)^2}dx-4 \int \frac {\exp \left (9 e^{-\frac {2 \left (x^2+4 x\right )}{3 (-x+\log (x)+2)}}-\frac {2 \left (x^2+4 x\right )}{3 (-x+\log (x)+2)}\right )}{x-\log (x)-2}dx-2 \int \frac {\exp \left (9 e^{-\frac {2 \left (x^2+4 x\right )}{3 (-x+\log (x)+2)}}-\frac {2 \left (x^2+4 x\right )}{3 (-x+\log (x)+2)}\right ) x}{x-\log (x)-2}dx\right )\) |
Input:
Int[(E^(9*E^((2*(-4*x - x^2))/(6 - 3*x + 3*Log[x])) + (2*(-4*x - x^2))/(6 - 3*x + 3*Log[x]))*(-24 - 18*x + 6*x^2 + (-24 - 12*x)*Log[x]))/(4 - 4*x + x^2 + (4 - 2*x)*Log[x] + Log[x]^2),x]
Output:
$Aborted
Time = 19.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87
| method | result | size |
| risch | \({\mathrm e}^{9 \,{\mathrm e}^{-\frac {2 \left (4+x \right ) x}{3 \left (\ln \left (x \right )-x +2\right )}}}\) | \(20\) |
Input:
int(((-12*x-24)*ln(x)+6*x^2-18*x-24)*exp((-x^2-4*x)/(3*ln(x)-3*x+6))^2*exp (9*exp((-x^2-4*x)/(3*ln(x)-3*x+6))^2)/(ln(x)^2+(4-2*x)*ln(x)+x^2-4*x+4),x, method=_RETURNVERBOSE)
Output:
exp(9*exp(-2/3*(4+x)*x/(ln(x)-x+2)))
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (19) = 38\).
Time = 0.10 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.96 \[ \int \frac {e^{9 e^{\frac {2 \left (-4 x-x^2\right )}{6-3 x+3 \log (x)}}+\frac {2 \left (-4 x-x^2\right )}{6-3 x+3 \log (x)}} \left (-24-18 x+6 x^2+(-24-12 x) \log (x)\right )}{4-4 x+x^2+(4-2 x) \log (x)+\log ^2(x)} \, dx=e^{\left (\frac {2 \, x^{2} + 27 \, {\left (x - \log \left (x\right ) - 2\right )} e^{\left (\frac {2 \, {\left (x^{2} + 4 \, x\right )}}{3 \, {\left (x - \log \left (x\right ) - 2\right )}}\right )} + 8 \, x}{3 \, {\left (x - \log \left (x\right ) - 2\right )}} - \frac {2 \, {\left (x^{2} + 4 \, x\right )}}{3 \, {\left (x - \log \left (x\right ) - 2\right )}}\right )} \] Input:
integrate(((-12*x-24)*log(x)+6*x^2-18*x-24)*exp((-x^2-4*x)/(3*log(x)-3*x+6 ))^2*exp(9*exp((-x^2-4*x)/(3*log(x)-3*x+6))^2)/(log(x)^2+(4-2*x)*log(x)+x^ 2-4*x+4),x, algorithm="fricas")
Output:
e^(1/3*(2*x^2 + 27*(x - log(x) - 2)*e^(2/3*(x^2 + 4*x)/(x - log(x) - 2)) + 8*x)/(x - log(x) - 2) - 2/3*(x^2 + 4*x)/(x - log(x) - 2))
Exception generated. \[ \int \frac {e^{9 e^{\frac {2 \left (-4 x-x^2\right )}{6-3 x+3 \log (x)}}+\frac {2 \left (-4 x-x^2\right )}{6-3 x+3 \log (x)}} \left (-24-18 x+6 x^2+(-24-12 x) \log (x)\right )}{4-4 x+x^2+(4-2 x) \log (x)+\log ^2(x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(((-12*x-24)*ln(x)+6*x**2-18*x-24)*exp((-x**2-4*x)/(3*ln(x)-3*x+6 ))**2*exp(9*exp((-x**2-4*x)/(3*ln(x)-3*x+6))**2)/(ln(x)**2+(4-2*x)*ln(x)+x **2-4*x+4),x)
Output:
Exception raised: TypeError >> '>' not supported between instances of 'Pol y' and 'int'
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (19) = 38\).
Time = 0.46 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.22 \[ \int \frac {e^{9 e^{\frac {2 \left (-4 x-x^2\right )}{6-3 x+3 \log (x)}}+\frac {2 \left (-4 x-x^2\right )}{6-3 x+3 \log (x)}} \left (-24-18 x+6 x^2+(-24-12 x) \log (x)\right )}{4-4 x+x^2+(4-2 x) \log (x)+\log ^2(x)} \, dx=e^{\left (9 \, x^{\frac {2}{3}} e^{\left (\frac {2}{3} \, x + \frac {2 \, \log \left (x\right )^{2}}{3 \, {\left (x - \log \left (x\right ) - 2\right )}} + \frac {16 \, \log \left (x\right )}{3 \, {\left (x - \log \left (x\right ) - 2\right )}} + \frac {8}{x - \log \left (x\right ) - 2} + 4\right )}\right )} \] Input:
integrate(((-12*x-24)*log(x)+6*x^2-18*x-24)*exp((-x^2-4*x)/(3*log(x)-3*x+6 ))^2*exp(9*exp((-x^2-4*x)/(3*log(x)-3*x+6))^2)/(log(x)^2+(4-2*x)*log(x)+x^ 2-4*x+4),x, algorithm="maxima")
Output:
e^(9*x^(2/3)*e^(2/3*x + 2/3*log(x)^2/(x - log(x) - 2) + 16/3*log(x)/(x - l og(x) - 2) + 8/(x - log(x) - 2) + 4))
\[ \int \frac {e^{9 e^{\frac {2 \left (-4 x-x^2\right )}{6-3 x+3 \log (x)}}+\frac {2 \left (-4 x-x^2\right )}{6-3 x+3 \log (x)}} \left (-24-18 x+6 x^2+(-24-12 x) \log (x)\right )}{4-4 x+x^2+(4-2 x) \log (x)+\log ^2(x)} \, dx=\int { \frac {6 \, {\left (x^{2} - 2 \, {\left (x + 2\right )} \log \left (x\right ) - 3 \, x - 4\right )} e^{\left (\frac {2 \, {\left (x^{2} + 4 \, x\right )}}{3 \, {\left (x - \log \left (x\right ) - 2\right )}} + 9 \, e^{\left (\frac {2 \, {\left (x^{2} + 4 \, x\right )}}{3 \, {\left (x - \log \left (x\right ) - 2\right )}}\right )}\right )}}{x^{2} - 2 \, {\left (x - 2\right )} \log \left (x\right ) + \log \left (x\right )^{2} - 4 \, x + 4} \,d x } \] Input:
integrate(((-12*x-24)*log(x)+6*x^2-18*x-24)*exp((-x^2-4*x)/(3*log(x)-3*x+6 ))^2*exp(9*exp((-x^2-4*x)/(3*log(x)-3*x+6))^2)/(log(x)^2+(4-2*x)*log(x)+x^ 2-4*x+4),x, algorithm="giac")
Output:
integrate(6*(x^2 - 2*(x + 2)*log(x) - 3*x - 4)*e^(2/3*(x^2 + 4*x)/(x - log (x) - 2) + 9*e^(2/3*(x^2 + 4*x)/(x - log(x) - 2)))/(x^2 - 2*(x - 2)*log(x) + log(x)^2 - 4*x + 4), x)
Time = 7.76 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {e^{9 e^{\frac {2 \left (-4 x-x^2\right )}{6-3 x+3 \log (x)}}+\frac {2 \left (-4 x-x^2\right )}{6-3 x+3 \log (x)}} \left (-24-18 x+6 x^2+(-24-12 x) \log (x)\right )}{4-4 x+x^2+(4-2 x) \log (x)+\log ^2(x)} \, dx={\mathrm {e}}^{9\,{\mathrm {e}}^{-\frac {2\,x^2+8\,x}{3\,\ln \left (x\right )-3\,x+6}}} \] Input:
int(-(exp(9*exp(-(2*(4*x + x^2))/(3*log(x) - 3*x + 6)))*exp(-(2*(4*x + x^2 ))/(3*log(x) - 3*x + 6))*(18*x + log(x)*(12*x + 24) - 6*x^2 + 24))/(log(x) ^2 - 4*x - log(x)*(2*x - 4) + x^2 + 4),x)
Output:
exp(9*exp(-(8*x + 2*x^2)/(3*log(x) - 3*x + 6)))
\[ \int \frac {e^{9 e^{\frac {2 \left (-4 x-x^2\right )}{6-3 x+3 \log (x)}}+\frac {2 \left (-4 x-x^2\right )}{6-3 x+3 \log (x)}} \left (-24-18 x+6 x^2+(-24-12 x) \log (x)\right )}{4-4 x+x^2+(4-2 x) \log (x)+\log ^2(x)} \, dx =\text {Too large to display} \] Input:
int(((-12*x-24)*log(x)+6*x^2-18*x-24)*exp((-x^2-4*x)/(3*log(x)-3*x+6))^2*e xp(9*exp((-x^2-4*x)/(3*log(x)-3*x+6))^2)/(log(x)^2+(4-2*x)*log(x)+x^2-4*x+ 4),x)
Output:
6*( - 4*int(e**(9/e**((2*x**2 + 8*x)/(3*log(x) - 3*x + 6)))/(e**((2*x**2 + 8*x)/(3*log(x) - 3*x + 6))*log(x)**2 - 2*e**((2*x**2 + 8*x)/(3*log(x) - 3 *x + 6))*log(x)*x + 4*e**((2*x**2 + 8*x)/(3*log(x) - 3*x + 6))*log(x) + e* *((2*x**2 + 8*x)/(3*log(x) - 3*x + 6))*x**2 - 4*e**((2*x**2 + 8*x)/(3*log( x) - 3*x + 6))*x + 4*e**((2*x**2 + 8*x)/(3*log(x) - 3*x + 6))),x) + int((e **(9/e**((2*x**2 + 8*x)/(3*log(x) - 3*x + 6)))*x**2)/(e**((2*x**2 + 8*x)/( 3*log(x) - 3*x + 6))*log(x)**2 - 2*e**((2*x**2 + 8*x)/(3*log(x) - 3*x + 6) )*log(x)*x + 4*e**((2*x**2 + 8*x)/(3*log(x) - 3*x + 6))*log(x) + e**((2*x* *2 + 8*x)/(3*log(x) - 3*x + 6))*x**2 - 4*e**((2*x**2 + 8*x)/(3*log(x) - 3* x + 6))*x + 4*e**((2*x**2 + 8*x)/(3*log(x) - 3*x + 6))),x) - 2*int((e**(9/ e**((2*x**2 + 8*x)/(3*log(x) - 3*x + 6)))*log(x)*x)/(e**((2*x**2 + 8*x)/(3 *log(x) - 3*x + 6))*log(x)**2 - 2*e**((2*x**2 + 8*x)/(3*log(x) - 3*x + 6)) *log(x)*x + 4*e**((2*x**2 + 8*x)/(3*log(x) - 3*x + 6))*log(x) + e**((2*x** 2 + 8*x)/(3*log(x) - 3*x + 6))*x**2 - 4*e**((2*x**2 + 8*x)/(3*log(x) - 3*x + 6))*x + 4*e**((2*x**2 + 8*x)/(3*log(x) - 3*x + 6))),x) - 4*int((e**(9/e **((2*x**2 + 8*x)/(3*log(x) - 3*x + 6)))*log(x))/(e**((2*x**2 + 8*x)/(3*lo g(x) - 3*x + 6))*log(x)**2 - 2*e**((2*x**2 + 8*x)/(3*log(x) - 3*x + 6))*lo g(x)*x + 4*e**((2*x**2 + 8*x)/(3*log(x) - 3*x + 6))*log(x) + e**((2*x**2 + 8*x)/(3*log(x) - 3*x + 6))*x**2 - 4*e**((2*x**2 + 8*x)/(3*log(x) - 3*x + 6))*x + 4*e**((2*x**2 + 8*x)/(3*log(x) - 3*x + 6))),x) - 3*int((e**(9/e...