Integrand size = 73, antiderivative size = 24 \[ \int \frac {e^{\frac {-4 x+x \log \left (\frac {e^{2 x}}{x^6}\right )}{\log \left (\frac {e^{2 x}}{x^6}\right )}} \left (-24+8 x-4 \log \left (\frac {e^{2 x}}{x^6}\right )+\log ^2\left (\frac {e^{2 x}}{x^6}\right )\right )}{\log ^2\left (\frac {e^{2 x}}{x^6}\right )} \, dx=4+e^{25}+e^{x-\frac {4 x}{\log \left (\frac {e^{2 x}}{x^6}\right )}} \] Output:
4+exp(25)+exp(x-4*x/ln(exp(x)^2/x^6))
Time = 0.40 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {e^{\frac {-4 x+x \log \left (\frac {e^{2 x}}{x^6}\right )}{\log \left (\frac {e^{2 x}}{x^6}\right )}} \left (-24+8 x-4 \log \left (\frac {e^{2 x}}{x^6}\right )+\log ^2\left (\frac {e^{2 x}}{x^6}\right )\right )}{\log ^2\left (\frac {e^{2 x}}{x^6}\right )} \, dx=e^{x-\frac {4 x}{\log \left (\frac {e^{2 x}}{x^6}\right )}} \] Input:
Integrate[(E^((-4*x + x*Log[E^(2*x)/x^6])/Log[E^(2*x)/x^6])*(-24 + 8*x - 4 *Log[E^(2*x)/x^6] + Log[E^(2*x)/x^6]^2))/Log[E^(2*x)/x^6]^2,x]
Output:
E^(x - (4*x)/Log[E^(2*x)/x^6])
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.
\(\displaystyle \int \frac {e^{\frac {x \log \left (\frac {e^{2 x}}{x^6}\right )-4 x}{\log \left (\frac {e^{2 x}}{x^6}\right )}} \left (\log ^2\left (\frac {e^{2 x}}{x^6}\right )-4 \log \left (\frac {e^{2 x}}{x^6}\right )+8 x-24\right )}{\log ^2\left (\frac {e^{2 x}}{x^6}\right )} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {8 (x-3) e^{\frac {x \log \left (\frac {e^{2 x}}{x^6}\right )-4 x}{\log \left (\frac {e^{2 x}}{x^6}\right )}}}{\log ^2\left (\frac {e^{2 x}}{x^6}\right )}+e^{\frac {x \log \left (\frac {e^{2 x}}{x^6}\right )-4 x}{\log \left (\frac {e^{2 x}}{x^6}\right )}}-\frac {4 e^{\frac {x \log \left (\frac {e^{2 x}}{x^6}\right )-4 x}{\log \left (\frac {e^{2 x}}{x^6}\right )}}}{\log \left (\frac {e^{2 x}}{x^6}\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -24 \int \frac {e^{\frac {x \log \left (\frac {e^{2 x}}{x^6}\right )-4 x}{\log \left (\frac {e^{2 x}}{x^6}\right )}}}{\log ^2\left (\frac {e^{2 x}}{x^6}\right )}dx+8 \int \frac {e^{\frac {x \log \left (\frac {e^{2 x}}{x^6}\right )-4 x}{\log \left (\frac {e^{2 x}}{x^6}\right )}} x}{\log ^2\left (\frac {e^{2 x}}{x^6}\right )}dx+\int e^{\frac {x \log \left (\frac {e^{2 x}}{x^6}\right )-4 x}{\log \left (\frac {e^{2 x}}{x^6}\right )}}dx-4 \int \frac {e^{\frac {x \log \left (\frac {e^{2 x}}{x^6}\right )-4 x}{\log \left (\frac {e^{2 x}}{x^6}\right )}}}{\log \left (\frac {e^{2 x}}{x^6}\right )}dx\) |
Input:
Int[(E^((-4*x + x*Log[E^(2*x)/x^6])/Log[E^(2*x)/x^6])*(-24 + 8*x - 4*Log[E ^(2*x)/x^6] + Log[E^(2*x)/x^6]^2))/Log[E^(2*x)/x^6]^2,x]
Output:
$Aborted
Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21
\[{\mathrm e}^{\frac {x \ln \left (\frac {{\mathrm e}^{2 x}}{x^{6}}\right )-4 x}{\ln \left (\frac {{\mathrm e}^{2 x}}{x^{6}}\right )}}\]
Input:
int((ln(exp(x)^2/x^6)^2-4*ln(exp(x)^2/x^6)+8*x-24)*exp((x*ln(exp(x)^2/x^6) -4*x)/ln(exp(x)^2/x^6))/ln(exp(x)^2/x^6)^2,x)
Output:
exp((x*ln(exp(x)^2/x^6)-4*x)/ln(exp(x)^2/x^6))
Time = 0.08 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int \frac {e^{\frac {-4 x+x \log \left (\frac {e^{2 x}}{x^6}\right )}{\log \left (\frac {e^{2 x}}{x^6}\right )}} \left (-24+8 x-4 \log \left (\frac {e^{2 x}}{x^6}\right )+\log ^2\left (\frac {e^{2 x}}{x^6}\right )\right )}{\log ^2\left (\frac {e^{2 x}}{x^6}\right )} \, dx=e^{\left (\frac {x \log \left (\frac {e^{\left (2 \, x\right )}}{x^{6}}\right ) - 4 \, x}{\log \left (\frac {e^{\left (2 \, x\right )}}{x^{6}}\right )}\right )} \] Input:
integrate((log(exp(x)^2/x^6)^2-4*log(exp(x)^2/x^6)+8*x-24)*exp((x*log(exp( x)^2/x^6)-4*x)/log(exp(x)^2/x^6))/log(exp(x)^2/x^6)^2,x, algorithm="fricas ")
Output:
e^((x*log(e^(2*x)/x^6) - 4*x)/log(e^(2*x)/x^6))
Time = 99.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {e^{\frac {-4 x+x \log \left (\frac {e^{2 x}}{x^6}\right )}{\log \left (\frac {e^{2 x}}{x^6}\right )}} \left (-24+8 x-4 \log \left (\frac {e^{2 x}}{x^6}\right )+\log ^2\left (\frac {e^{2 x}}{x^6}\right )\right )}{\log ^2\left (\frac {e^{2 x}}{x^6}\right )} \, dx=e^{\frac {x \log {\left (\frac {e^{2 x}}{x^{6}} \right )} - 4 x}{\log {\left (\frac {e^{2 x}}{x^{6}} \right )}}} \] Input:
integrate((ln(exp(x)**2/x**6)**2-4*ln(exp(x)**2/x**6)+8*x-24)*exp((x*ln(ex p(x)**2/x**6)-4*x)/ln(exp(x)**2/x**6))/ln(exp(x)**2/x**6)**2,x)
Output:
exp((x*log(exp(2*x)/x**6) - 4*x)/log(exp(2*x)/x**6))
Time = 0.10 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.67 \[ \int \frac {e^{\frac {-4 x+x \log \left (\frac {e^{2 x}}{x^6}\right )}{\log \left (\frac {e^{2 x}}{x^6}\right )}} \left (-24+8 x-4 \log \left (\frac {e^{2 x}}{x^6}\right )+\log ^2\left (\frac {e^{2 x}}{x^6}\right )\right )}{\log ^2\left (\frac {e^{2 x}}{x^6}\right )} \, dx=e^{\left (x - \frac {6 \, \log \left (x\right )}{x - 3 \, \log \left (x\right )} - 2\right )} \] Input:
integrate((log(exp(x)^2/x^6)^2-4*log(exp(x)^2/x^6)+8*x-24)*exp((x*log(exp( x)^2/x^6)-4*x)/log(exp(x)^2/x^6))/log(exp(x)^2/x^6)^2,x, algorithm="maxima ")
Output:
e^(x - 6*log(x)/(x - 3*log(x)) - 2)
Time = 0.18 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.25 \[ \int \frac {e^{\frac {-4 x+x \log \left (\frac {e^{2 x}}{x^6}\right )}{\log \left (\frac {e^{2 x}}{x^6}\right )}} \left (-24+8 x-4 \log \left (\frac {e^{2 x}}{x^6}\right )+\log ^2\left (\frac {e^{2 x}}{x^6}\right )\right )}{\log ^2\left (\frac {e^{2 x}}{x^6}\right )} \, dx=e^{\left (\frac {2 \, x^{2} - x \log \left (x^{6}\right ) - 4 \, x}{2 \, x - \log \left (x^{6}\right )}\right )} \] Input:
integrate((log(exp(x)^2/x^6)^2-4*log(exp(x)^2/x^6)+8*x-24)*exp((x*log(exp( x)^2/x^6)-4*x)/log(exp(x)^2/x^6))/log(exp(x)^2/x^6)^2,x, algorithm="giac")
Output:
e^((2*x^2 - x*log(x^6) - 4*x)/(2*x - log(x^6)))
Time = 2.51 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62 \[ \int \frac {e^{\frac {-4 x+x \log \left (\frac {e^{2 x}}{x^6}\right )}{\log \left (\frac {e^{2 x}}{x^6}\right )}} \left (-24+8 x-4 \log \left (\frac {e^{2 x}}{x^6}\right )+\log ^2\left (\frac {e^{2 x}}{x^6}\right )\right )}{\log ^2\left (\frac {e^{2 x}}{x^6}\right )} \, dx={\mathrm {e}}^{-\frac {4\,x-2\,x^2}{2\,x+\ln \left (\frac {1}{x^6}\right )}}\,{\left (\frac {1}{x^6}\right )}^{\frac {x}{2\,x+\ln \left (\frac {1}{x^6}\right )}} \] Input:
int((exp(-(4*x - x*log(exp(2*x)/x^6))/log(exp(2*x)/x^6))*(8*x - 4*log(exp( 2*x)/x^6) + log(exp(2*x)/x^6)^2 - 24))/log(exp(2*x)/x^6)^2,x)
Output:
exp(-(4*x - 2*x^2)/(2*x + log(1/x^6)))*(1/x^6)^(x/(2*x + log(1/x^6)))
Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {e^{\frac {-4 x+x \log \left (\frac {e^{2 x}}{x^6}\right )}{\log \left (\frac {e^{2 x}}{x^6}\right )}} \left (-24+8 x-4 \log \left (\frac {e^{2 x}}{x^6}\right )+\log ^2\left (\frac {e^{2 x}}{x^6}\right )\right )}{\log ^2\left (\frac {e^{2 x}}{x^6}\right )} \, dx=\frac {e^{x}}{e^{\frac {4 x}{\mathrm {log}\left (\frac {e^{2 x}}{x^{6}}\right )}}} \] Input:
int((log(exp(x)^2/x^6)^2-4*log(exp(x)^2/x^6)+8*x-24)*exp((x*log(exp(x)^2/x ^6)-4*x)/log(exp(x)^2/x^6))/log(exp(x)^2/x^6)^2,x)
Output:
e**x/e**((4*x)/log(e**(2*x)/x**6))