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

Optimal. Leaf size=24 \[ 4+e^{25}+e^{x-\frac {4 x}{\log \left (\frac {e^{2 x}}{x^6}\right )}} \]

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Rubi [F]  time = 1.10, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.45, size = 19, normalized size = 0.79 \begin {gather*} e^{x-\frac {4 x}{\log \left (\frac {e^{2 x}}{x^6}\right )}} \end {gather*}

Antiderivative was successfully verified.

[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])

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fricas [A]  time = 0.71, size = 28, normalized size = 1.17 \begin {gather*} 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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [A]  time = 1.43, size = 17, normalized size = 0.71 \begin {gather*} e^{\left (x - \frac {4 \, x}{\log \left (\frac {e^{\left (2 \, x\right )}}{x^{6}}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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^(x - 4*x/log(e^(2*x)/x^6))

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maple [F(-1)]  time = 180.00, size = 0, normalized size = 0.00 hanged

Verification of antiderivative is not currently implemented for this CAS.

[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,method=_RETURNVERBOSE)

[Out]

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,method=_RETURNVERBOSE)

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maxima [A]  time = 0.50, size = 16, normalized size = 0.67 \begin {gather*} e^{\left (x - \frac {6 \, \log \relax (x)}{x - 3 \, \log \relax (x)} - 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 2.17, size = 39, normalized size = 1.62 \begin {gather*} {\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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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sympy [A]  time = 125.29, size = 26, normalized size = 1.08 \begin {gather*} e^{\frac {x \log {\left (\frac {e^{2 x}}{x^{6}} \right )} - 4 x}{\log {\left (\frac {e^{2 x}}{x^{6}} \right )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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