\(\int \frac {e^{\frac {-4 x+x \log (\frac {e^{2 x}}{x^6})}{\log (\frac {e^{2 x}}{x^6})}} (-24+8 x-4 \log (\frac {e^{2 x}}{x^6})+\log ^2(\frac {e^{2 x}}{x^6}))}{\log ^2(\frac {e^{2 x}}{x^6})} \, dx\) [3440]

Optimal result

Integrand size = 73, antiderivative size = 24 \[ \int \frac {e^{\frac {-4 x+x \log \left (\frac {e^{2 x}}{x^6}\right )}{\log \left (\frac {e^{2 x}}{x^6}\right )}} \left (-24+8 x-4 \log \left (\frac {e^{2 x}}{x^6}\right )+\log ^2\left (\frac {e^{2 x}}{x^6}\right )\right )}{\log ^2\left (\frac {e^{2 x}}{x^6}\right )} \, dx=4+e^{25}+e^{x-\frac {4 x}{\log \left (\frac {e^{2 x}}{x^6}\right )}} \]

[Out]

4+exp(25)+exp(x-4*x/ln(exp(x)^2/x^6))

Rubi [F]

\[ \int \frac {e^{\frac {-4 x+x \log \left (\frac {e^{2 x}}{x^6}\right )}{\log \left (\frac {e^{2 x}}{x^6}\right )}} \left (-24+8 x-4 \log \left (\frac {e^{2 x}}{x^6}\right )+\log ^2\left (\frac {e^{2 x}}{x^6}\right )\right )}{\log ^2\left (\frac {e^{2 x}}{x^6}\right )} \, dx=\int \frac {e^{\frac {-4 x+x \log \left (\frac {e^{2 x}}{x^6}\right )}{\log \left (\frac {e^{2 x}}{x^6}\right )}} \left (-24+8 x-4 \log \left (\frac {e^{2 x}}{x^6}\right )+\log ^2\left (\frac {e^{2 x}}{x^6}\right )\right )}{\log ^2\left (\frac {e^{2 x}}{x^6}\right )} \, dx \]

[In]

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))/L
og[E^(2*x)/x^6]^2,x]

[Out]

Defer[Int][E^(x - (4*x)/Log[E^(2*x)/x^6]), x] - 24*Defer[Int][E^(x - (4*x)/Log[E^(2*x)/x^6])/Log[E^(2*x)/x^6]^
2, x] + 8*Defer[Int][(E^(x - (4*x)/Log[E^(2*x)/x^6])*x)/Log[E^(2*x)/x^6]^2, x] - 4*Defer[Int][E^(x - (4*x)/Log
[E^(2*x)/x^6])/Log[E^(2*x)/x^6], x]

Rubi steps \begin{align*} \text {integral}& = \int \left (e^{\frac {-4 x+x \log \left (\frac {e^{2 x}}{x^6}\right )}{\log \left (\frac {e^{2 x}}{x^6}\right )}}+\frac {8 e^{\frac {-4 x+x \log \left (\frac {e^{2 x}}{x^6}\right )}{\log \left (\frac {e^{2 x}}{x^6}\right )}} (-3+x)}{\log ^2\left (\frac {e^{2 x}}{x^6}\right )}-\frac {4 e^{\frac {-4 x+x \log \left (\frac {e^{2 x}}{x^6}\right )}{\log \left (\frac {e^{2 x}}{x^6}\right )}}}{\log \left (\frac {e^{2 x}}{x^6}\right )}\right ) \, dx \\ & = -\left (4 \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 )}}}{\log \left (\frac {e^{2 x}}{x^6}\right )} \, dx\right )+8 \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 )}} (-3+x)}{\log ^2\left (\frac {e^{2 x}}{x^6}\right )} \, dx+\int e^{\frac {-4 x+x \log \left (\frac {e^{2 x}}{x^6}\right )}{\log \left (\frac {e^{2 x}}{x^6}\right )}} \, dx \\ & = -\left (4 \int \frac {e^{x-\frac {4 x}{\log \left (\frac {e^{2 x}}{x^6}\right )}}}{\log \left (\frac {e^{2 x}}{x^6}\right )} \, dx\right )+8 \int \frac {e^{x-\frac {4 x}{\log \left (\frac {e^{2 x}}{x^6}\right )}} (-3+x)}{\log ^2\left (\frac {e^{2 x}}{x^6}\right )} \, dx+\int e^{x-\frac {4 x}{\log \left (\frac {e^{2 x}}{x^6}\right )}} \, dx \\ & = -\left (4 \int \frac {e^{x-\frac {4 x}{\log \left (\frac {e^{2 x}}{x^6}\right )}}}{\log \left (\frac {e^{2 x}}{x^6}\right )} \, dx\right )+8 \int \left (-\frac {3 e^{x-\frac {4 x}{\log \left (\frac {e^{2 x}}{x^6}\right )}}}{\log ^2\left (\frac {e^{2 x}}{x^6}\right )}+\frac {e^{x-\frac {4 x}{\log \left (\frac {e^{2 x}}{x^6}\right )}} x}{\log ^2\left (\frac {e^{2 x}}{x^6}\right )}\right ) \, dx+\int e^{x-\frac {4 x}{\log \left (\frac {e^{2 x}}{x^6}\right )}} \, dx \\ & = -\left (4 \int \frac {e^{x-\frac {4 x}{\log \left (\frac {e^{2 x}}{x^6}\right )}}}{\log \left (\frac {e^{2 x}}{x^6}\right )} \, dx\right )+8 \int \frac {e^{x-\frac {4 x}{\log \left (\frac {e^{2 x}}{x^6}\right )}} x}{\log ^2\left (\frac {e^{2 x}}{x^6}\right )} \, dx-24 \int \frac {e^{x-\frac {4 x}{\log \left (\frac {e^{2 x}}{x^6}\right )}}}{\log ^2\left (\frac {e^{2 x}}{x^6}\right )} \, dx+\int e^{x-\frac {4 x}{\log \left (\frac {e^{2 x}}{x^6}\right )}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.62 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {e^{\frac {-4 x+x \log \left (\frac {e^{2 x}}{x^6}\right )}{\log \left (\frac {e^{2 x}}{x^6}\right )}} \left (-24+8 x-4 \log \left (\frac {e^{2 x}}{x^6}\right )+\log ^2\left (\frac {e^{2 x}}{x^6}\right )\right )}{\log ^2\left (\frac {e^{2 x}}{x^6}\right )} \, dx=e^{x-\frac {4 x}{\log \left (\frac {e^{2 x}}{x^6}\right )}} \]

[In]

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]

[Out]

E^(x - (4*x)/Log[E^(2*x)/x^6])

Maple [A] (verified)

Time = 4.75 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21

\[{\mathrm e}^{\frac {x \ln \left (\frac {{\mathrm e}^{2 x}}{x^{6}}\right )-4 x}{\ln \left (\frac {{\mathrm e}^{2 x}}{x^{6}}\right )}}\]

[In]

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)

[Out]

exp((x*ln(exp(x)^2/x^6)-4*x)/ln(exp(x)^2/x^6))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int \frac {e^{\frac {-4 x+x \log \left (\frac {e^{2 x}}{x^6}\right )}{\log \left (\frac {e^{2 x}}{x^6}\right )}} \left (-24+8 x-4 \log \left (\frac {e^{2 x}}{x^6}\right )+\log ^2\left (\frac {e^{2 x}}{x^6}\right )\right )}{\log ^2\left (\frac {e^{2 x}}{x^6}\right )} \, dx=e^{\left (\frac {x \log \left (\frac {e^{\left (2 \, x\right )}}{x^{6}}\right ) - 4 \, x}{\log \left (\frac {e^{\left (2 \, x\right )}}{x^{6}}\right )}\right )} \]

[In]

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))/lo
g(exp(x)^2/x^6)^2,x, algorithm="fricas")

[Out]

e^((x*log(e^(2*x)/x^6) - 4*x)/log(e^(2*x)/x^6))

Sympy [A] (verification not implemented)

Time = 110.64 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {e^{\frac {-4 x+x \log \left (\frac {e^{2 x}}{x^6}\right )}{\log \left (\frac {e^{2 x}}{x^6}\right )}} \left (-24+8 x-4 \log \left (\frac {e^{2 x}}{x^6}\right )+\log ^2\left (\frac {e^{2 x}}{x^6}\right )\right )}{\log ^2\left (\frac {e^{2 x}}{x^6}\right )} \, dx=e^{\frac {x \log {\left (\frac {e^{2 x}}{x^{6}} \right )} - 4 x}{\log {\left (\frac {e^{2 x}}{x^{6}} \right )}}} \]

[In]

integrate((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)

[Out]

exp((x*log(exp(2*x)/x**6) - 4*x)/log(exp(2*x)/x**6))

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.67 \[ \int \frac {e^{\frac {-4 x+x \log \left (\frac {e^{2 x}}{x^6}\right )}{\log \left (\frac {e^{2 x}}{x^6}\right )}} \left (-24+8 x-4 \log \left (\frac {e^{2 x}}{x^6}\right )+\log ^2\left (\frac {e^{2 x}}{x^6}\right )\right )}{\log ^2\left (\frac {e^{2 x}}{x^6}\right )} \, dx=e^{\left (x - \frac {6 \, \log \left (x\right )}{x - 3 \, \log \left (x\right )} - 2\right )} \]

[In]

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))/lo
g(exp(x)^2/x^6)^2,x, algorithm="maxima")

[Out]

e^(x - 6*log(x)/(x - 3*log(x)) - 2)

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.25 \[ \int \frac {e^{\frac {-4 x+x \log \left (\frac {e^{2 x}}{x^6}\right )}{\log \left (\frac {e^{2 x}}{x^6}\right )}} \left (-24+8 x-4 \log \left (\frac {e^{2 x}}{x^6}\right )+\log ^2\left (\frac {e^{2 x}}{x^6}\right )\right )}{\log ^2\left (\frac {e^{2 x}}{x^6}\right )} \, dx=e^{\left (\frac {2 \, x^{2} - x \log \left (x^{6}\right ) - 4 \, x}{2 \, x - \log \left (x^{6}\right )}\right )} \]

[In]

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))/lo
g(exp(x)^2/x^6)^2,x, algorithm="giac")

[Out]

e^((2*x^2 - x*log(x^6) - 4*x)/(2*x - log(x^6)))

Mupad [B] (verification not implemented)

Time = 8.53 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62 \[ \int \frac {e^{\frac {-4 x+x \log \left (\frac {e^{2 x}}{x^6}\right )}{\log \left (\frac {e^{2 x}}{x^6}\right )}} \left (-24+8 x-4 \log \left (\frac {e^{2 x}}{x^6}\right )+\log ^2\left (\frac {e^{2 x}}{x^6}\right )\right )}{\log ^2\left (\frac {e^{2 x}}{x^6}\right )} \, dx={\mathrm {e}}^{-\frac {4\,x-2\,x^2}{2\,x+\ln \left (\frac {1}{x^6}\right )}}\,{\left (\frac {1}{x^6}\right )}^{\frac {x}{2\,x+\ln \left (\frac {1}{x^6}\right )}} \]

[In]

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)

[Out]

exp(-(4*x - 2*x^2)/(2*x + log(1/x^6)))*(1/x^6)^(x/(2*x + log(1/x^6)))