Integrand size = 95, antiderivative size = 21 \[ \int \frac {e^{\frac {-12-6 x-x^2+(4+x) \log \left (4 x^2\right )}{-3-x+\log \left (4 x^2\right )}} \left (4+6 x+x^2+(-5-2 x) \log \left (4 x^2\right )+\log ^2\left (4 x^2\right )\right )}{9+6 x+x^2+(-6-2 x) \log \left (4 x^2\right )+\log ^2\left (4 x^2\right )} \, dx=e^{4+x-\frac {x}{3+x-\log \left (4 x^2\right )}} \] Output:
exp(4+x-x/(x-ln(4*x^2)+3))
Time = 5.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {e^{\frac {-12-6 x-x^2+(4+x) \log \left (4 x^2\right )}{-3-x+\log \left (4 x^2\right )}} \left (4+6 x+x^2+(-5-2 x) \log \left (4 x^2\right )+\log ^2\left (4 x^2\right )\right )}{9+6 x+x^2+(-6-2 x) \log \left (4 x^2\right )+\log ^2\left (4 x^2\right )} \, dx=e^{4+x+\frac {x}{-3-x+\log \left (4 x^2\right )}} \] Input:
Integrate[(E^((-12 - 6*x - x^2 + (4 + x)*Log[4*x^2])/(-3 - x + Log[4*x^2]) )*(4 + 6*x + x^2 + (-5 - 2*x)*Log[4*x^2] + Log[4*x^2]^2))/(9 + 6*x + x^2 + (-6 - 2*x)*Log[4*x^2] + Log[4*x^2]^2),x]
Output:
E^(4 + x + x/(-3 - x + Log[4*x^2]))
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.
\(\displaystyle \int \frac {\left (x^2+\log ^2\left (4 x^2\right )+(-2 x-5) \log \left (4 x^2\right )+6 x+4\right ) \exp \left (\frac {-x^2+(x+4) \log \left (4 x^2\right )-6 x-12}{\log \left (4 x^2\right )-x-3}\right )}{x^2+\log ^2\left (4 x^2\right )+(-2 x-6) \log \left (4 x^2\right )+6 x+9} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {\left (x^2+\log ^2\left (4 x^2\right )+(-2 x-5) \log \left (4 x^2\right )+6 x+4\right ) \exp \left (\frac {-x^2+(x+4) \log \left (4 x^2\right )-6 x-12}{\log \left (4 x^2\right )-x-3}\right )}{\left (-\log \left (4 x^2\right )+x+3\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {(x-2) \exp \left (\frac {-x^2+(x+4) \log \left (4 x^2\right )-6 x-12}{\log \left (4 x^2\right )-x-3}\right )}{\left (-\log \left (4 x^2\right )+x+3\right )^2}+\exp \left (\frac {-x^2+(x+4) \log \left (4 x^2\right )-6 x-12}{\log \left (4 x^2\right )-x-3}\right )+\frac {\exp \left (\frac {-x^2+(x+4) \log \left (4 x^2\right )-6 x-12}{\log \left (4 x^2\right )-x-3}\right )}{\log \left (4 x^2\right )-x-3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \int \exp \left (\frac {-x^2-6 x+(x+4) \log \left (4 x^2\right )-12}{-x+\log \left (4 x^2\right )-3}\right )dx-2 \int \frac {\exp \left (\frac {-x^2-6 x+(x+4) \log \left (4 x^2\right )-12}{-x+\log \left (4 x^2\right )-3}\right )}{\left (x-\log \left (4 x^2\right )+3\right )^2}dx+\int \frac {\exp \left (\frac {-x^2-6 x+(x+4) \log \left (4 x^2\right )-12}{-x+\log \left (4 x^2\right )-3}\right ) x}{\left (x-\log \left (4 x^2\right )+3\right )^2}dx+\int \frac {\exp \left (\frac {-x^2-6 x+(x+4) \log \left (4 x^2\right )-12}{-x+\log \left (4 x^2\right )-3}\right )}{-x+\log \left (4 x^2\right )-3}dx\) |
Input:
Int[(E^((-12 - 6*x - x^2 + (4 + x)*Log[4*x^2])/(-3 - x + Log[4*x^2]))*(4 + 6*x + x^2 + (-5 - 2*x)*Log[4*x^2] + Log[4*x^2]^2))/(9 + 6*x + x^2 + (-6 - 2*x)*Log[4*x^2] + Log[4*x^2]^2),x]
Output:
$Aborted
Time = 0.60 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.71
method | result | size |
parallelrisch | \({\mathrm e}^{\frac {\left (4+x \right ) \ln \left (4 x^{2}\right )-x^{2}-6 x -12}{\ln \left (4 x^{2}\right )-3-x}}\) | \(36\) |
risch | \({\mathrm e}^{\frac {-\ln \left (4 x^{2}\right ) x +x^{2}-4 \ln \left (4 x^{2}\right )+6 x +12}{x -\ln \left (4 x^{2}\right )+3}}\) | \(41\) |
norman | \(\frac {x \,{\mathrm e}^{\frac {\left (4+x \right ) \ln \left (4 x^{2}\right )-x^{2}-6 x -12}{\ln \left (4 x^{2}\right )-3-x}}-\ln \left (4 x^{2}\right ) {\mathrm e}^{\frac {\left (4+x \right ) \ln \left (4 x^{2}\right )-x^{2}-6 x -12}{\ln \left (4 x^{2}\right )-3-x}}+3 \,{\mathrm e}^{\frac {\left (4+x \right ) \ln \left (4 x^{2}\right )-x^{2}-6 x -12}{\ln \left (4 x^{2}\right )-3-x}}}{x -\ln \left (4 x^{2}\right )+3}\) | \(133\) |
default | \(\frac {x \,{\mathrm e}^{\frac {\left (4+x \right ) \ln \left (4 x^{2}\right )-x^{2}-6 x -12}{\ln \left (4 x^{2}\right )-3-x}}+\left (-\ln \left (4 x^{2}\right )+2 \ln \left (x \right )+3\right ) {\mathrm e}^{\frac {\left (4+x \right ) \ln \left (4 x^{2}\right )-x^{2}-6 x -12}{\ln \left (4 x^{2}\right )-3-x}}-2 \ln \left (x \right ) {\mathrm e}^{\frac {\left (4+x \right ) \ln \left (4 x^{2}\right )-x^{2}-6 x -12}{\ln \left (4 x^{2}\right )-3-x}}}{x -\ln \left (4 x^{2}\right )+3}\) | \(142\) |
Input:
int((ln(4*x^2)^2+(-2*x-5)*ln(4*x^2)+x^2+6*x+4)*exp(((4+x)*ln(4*x^2)-x^2-6* x-12)/(ln(4*x^2)-3-x))/(ln(4*x^2)^2+(-2*x-6)*ln(4*x^2)+x^2+6*x+9),x,method =_RETURNVERBOSE)
Output:
exp(((4+x)*ln(4*x^2)-x^2-6*x-12)/(ln(4*x^2)-3-x))
Time = 0.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.62 \[ \int \frac {e^{\frac {-12-6 x-x^2+(4+x) \log \left (4 x^2\right )}{-3-x+\log \left (4 x^2\right )}} \left (4+6 x+x^2+(-5-2 x) \log \left (4 x^2\right )+\log ^2\left (4 x^2\right )\right )}{9+6 x+x^2+(-6-2 x) \log \left (4 x^2\right )+\log ^2\left (4 x^2\right )} \, dx=e^{\left (\frac {x^{2} - {\left (x + 4\right )} \log \left (4 \, x^{2}\right ) + 6 \, x + 12}{x - \log \left (4 \, x^{2}\right ) + 3}\right )} \] Input:
integrate((log(4*x^2)^2+(-2*x-5)*log(4*x^2)+x^2+6*x+4)*exp(((4+x)*log(4*x^ 2)-x^2-6*x-12)/(log(4*x^2)-3-x))/(log(4*x^2)^2+(-2*x-6)*log(4*x^2)+x^2+6*x +9),x, algorithm="fricas")
Output:
e^((x^2 - (x + 4)*log(4*x^2) + 6*x + 12)/(x - log(4*x^2) + 3))
Time = 0.34 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.38 \[ \int \frac {e^{\frac {-12-6 x-x^2+(4+x) \log \left (4 x^2\right )}{-3-x+\log \left (4 x^2\right )}} \left (4+6 x+x^2+(-5-2 x) \log \left (4 x^2\right )+\log ^2\left (4 x^2\right )\right )}{9+6 x+x^2+(-6-2 x) \log \left (4 x^2\right )+\log ^2\left (4 x^2\right )} \, dx=e^{\frac {- x^{2} - 6 x + \left (x + 4\right ) \log {\left (4 x^{2} \right )} - 12}{- x + \log {\left (4 x^{2} \right )} - 3}} \] Input:
integrate((ln(4*x**2)**2+(-2*x-5)*ln(4*x**2)+x**2+6*x+4)*exp(((4+x)*ln(4*x **2)-x**2-6*x-12)/(ln(4*x**2)-3-x))/(ln(4*x**2)**2+(-2*x-6)*ln(4*x**2)+x** 2+6*x+9),x)
Output:
exp((-x**2 - 6*x + (x + 4)*log(4*x**2) - 12)/(-x + log(4*x**2) - 3))
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (20) = 40\).
Time = 0.38 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.52 \[ \int \frac {e^{\frac {-12-6 x-x^2+(4+x) \log \left (4 x^2\right )}{-3-x+\log \left (4 x^2\right )}} \left (4+6 x+x^2+(-5-2 x) \log \left (4 x^2\right )+\log ^2\left (4 x^2\right )\right )}{9+6 x+x^2+(-6-2 x) \log \left (4 x^2\right )+\log ^2\left (4 x^2\right )} \, dx=e^{\left (x - \frac {2 \, \log \left (2\right )}{x - 2 \, \log \left (2\right ) - 2 \, \log \left (x\right ) + 3} - \frac {2 \, \log \left (x\right )}{x - 2 \, \log \left (2\right ) - 2 \, \log \left (x\right ) + 3} + \frac {3}{x - 2 \, \log \left (2\right ) - 2 \, \log \left (x\right ) + 3} + 3\right )} \] Input:
integrate((log(4*x^2)^2+(-2*x-5)*log(4*x^2)+x^2+6*x+4)*exp(((4+x)*log(4*x^ 2)-x^2-6*x-12)/(log(4*x^2)-3-x))/(log(4*x^2)^2+(-2*x-6)*log(4*x^2)+x^2+6*x +9),x, algorithm="maxima")
Output:
e^(x - 2*log(2)/(x - 2*log(2) - 2*log(x) + 3) - 2*log(x)/(x - 2*log(2) - 2 *log(x) + 3) + 3/(x - 2*log(2) - 2*log(x) + 3) + 3)
Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (20) = 40\).
Time = 0.47 (sec) , antiderivative size = 93, normalized size of antiderivative = 4.43 \[ \int \frac {e^{\frac {-12-6 x-x^2+(4+x) \log \left (4 x^2\right )}{-3-x+\log \left (4 x^2\right )}} \left (4+6 x+x^2+(-5-2 x) \log \left (4 x^2\right )+\log ^2\left (4 x^2\right )\right )}{9+6 x+x^2+(-6-2 x) \log \left (4 x^2\right )+\log ^2\left (4 x^2\right )} \, dx=e^{\left (\frac {x^{2}}{x - \log \left (4 \, x^{2}\right ) + 3} - \frac {x \log \left (4 \, x^{2}\right )}{x - \log \left (4 \, x^{2}\right ) + 3} + \frac {6 \, x}{x - \log \left (4 \, x^{2}\right ) + 3} - \frac {4 \, \log \left (4 \, x^{2}\right )}{x - \log \left (4 \, x^{2}\right ) + 3} + \frac {12}{x - \log \left (4 \, x^{2}\right ) + 3}\right )} \] Input:
integrate((log(4*x^2)^2+(-2*x-5)*log(4*x^2)+x^2+6*x+4)*exp(((4+x)*log(4*x^ 2)-x^2-6*x-12)/(log(4*x^2)-3-x))/(log(4*x^2)^2+(-2*x-6)*log(4*x^2)+x^2+6*x +9),x, algorithm="giac")
Output:
e^(x^2/(x - log(4*x^2) + 3) - x*log(4*x^2)/(x - log(4*x^2) + 3) + 6*x/(x - log(4*x^2) + 3) - 4*log(4*x^2)/(x - log(4*x^2) + 3) + 12/(x - log(4*x^2) + 3))
Time = 2.98 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.95 \[ \int \frac {e^{\frac {-12-6 x-x^2+(4+x) \log \left (4 x^2\right )}{-3-x+\log \left (4 x^2\right )}} \left (4+6 x+x^2+(-5-2 x) \log \left (4 x^2\right )+\log ^2\left (4 x^2\right )\right )}{9+6 x+x^2+(-6-2 x) \log \left (4 x^2\right )+\log ^2\left (4 x^2\right )} \, dx={\mathrm {e}}^{\frac {x^2}{x-\ln \left (x^2\right )-2\,\ln \left (2\right )+3}}\,{\mathrm {e}}^{\frac {12}{x-\ln \left (x^2\right )-2\,\ln \left (2\right )+3}}\,{\mathrm {e}}^{\frac {6\,x}{x-\ln \left (x^2\right )-2\,\ln \left (2\right )+3}}\,{\left (\frac {1}{4\,x^2}\right )}^{\frac {x+4}{x-\ln \left (x^2\right )-2\,\ln \left (2\right )+3}} \] Input:
int((exp((6*x + x^2 - log(4*x^2)*(x + 4) + 12)/(x - log(4*x^2) + 3))*(6*x + log(4*x^2)^2 - log(4*x^2)*(2*x + 5) + x^2 + 4))/(6*x + log(4*x^2)^2 - lo g(4*x^2)*(2*x + 6) + x^2 + 9),x)
Output:
exp(x^2/(x - log(x^2) - 2*log(2) + 3))*exp(12/(x - log(x^2) - 2*log(2) + 3 ))*exp((6*x)/(x - log(x^2) - 2*log(2) + 3))*(1/(4*x^2))^((x + 4)/(x - log( x^2) - 2*log(2) + 3))
Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.76 \[ \int \frac {e^{\frac {-12-6 x-x^2+(4+x) \log \left (4 x^2\right )}{-3-x+\log \left (4 x^2\right )}} \left (4+6 x+x^2+(-5-2 x) \log \left (4 x^2\right )+\log ^2\left (4 x^2\right )\right )}{9+6 x+x^2+(-6-2 x) \log \left (4 x^2\right )+\log ^2\left (4 x^2\right )} \, dx=e^{\frac {\mathrm {log}\left (4 x^{2}\right ) x -x^{2}-2 x}{\mathrm {log}\left (4 x^{2}\right )-x -3}} e^{4} \] Input:
int((log(4*x^2)^2+(-2*x-5)*log(4*x^2)+x^2+6*x+4)*exp(((4+x)*log(4*x^2)-x^2 -6*x-12)/(log(4*x^2)-3-x))/(log(4*x^2)^2+(-2*x-6)*log(4*x^2)+x^2+6*x+9),x)
Output:
e**((log(4*x**2)*x - x**2 - 2*x)/(log(4*x**2) - x - 3))*e**4