3.101.34 \(\int \frac {e^{9 e^{\frac {2 (-4 x-x^2)}{6-3 x+3 \log (x)}}+\frac {2 (-4 x-x^2)}{6-3 x+3 \log (x)}} (-24-18 x+6 x^2+(-24-12 x) \log (x))}{4-4 x+x^2+(4-2 x) \log (x)+\log ^2(x)} \, dx\)

Optimal. Leaf size=23 \[ e^{9 e^{\frac {2 x (4+x)}{3 (-2+x-\log (x))}}} \]

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Rubi [F]  time = 15.83, 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 {\exp \left (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)}\right ) \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 \end {gather*}

Verification is not applicable to the result.

[In]

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]

[Out]

24*Defer[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]))/(-2 + x
- Log[x])^2, x] - 18*Defer[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]))*x)/(-2 + x - Log[x])^2, x] - 6*Defer[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]))*x^2)/(-2 + x - Log[x])^2, x] + 24*Defer[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]))/(-2 + x - Log[x]), x] + 12*Defer[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]))*x)/(-2 + x - Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {6 \exp \left (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)}\right ) \left (-4-3 x+x^2-4 \log (x)-2 x \log (x)\right )}{(2-x+\log (x))^2} \, dx\\ &=6 \int \frac {\exp \left (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)}\right ) \left (-4-3 x+x^2-4 \log (x)-2 x \log (x)\right )}{(2-x+\log (x))^2} \, dx\\ &=6 \int \left (\frac {\exp \left (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)}\right ) \left (4-3 x-x^2\right )}{(-2+x-\log (x))^2}+\frac {2 \exp \left (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)}\right ) (2+x)}{-2+x-\log (x)}\right ) \, dx\\ &=6 \int \frac {\exp \left (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)}\right ) \left (4-3 x-x^2\right )}{(-2+x-\log (x))^2} \, dx+12 \int \frac {\exp \left (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)}\right ) (2+x)}{-2+x-\log (x)} \, dx\\ &=6 \int \left (\frac {4 \exp \left (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)}\right )}{(-2+x-\log (x))^2}-\frac {3 \exp \left (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)}\right ) x}{(-2+x-\log (x))^2}-\frac {\exp \left (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)}\right ) x^2}{(-2+x-\log (x))^2}\right ) \, dx+12 \int \left (\frac {2 \exp \left (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)}\right )}{-2+x-\log (x)}+\frac {\exp \left (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)}\right ) x}{-2+x-\log (x)}\right ) \, dx\\ &=-\left (6 \int \frac {\exp \left (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)}\right ) x^2}{(-2+x-\log (x))^2} \, dx\right )+12 \int \frac {\exp \left (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)}\right ) x}{-2+x-\log (x)} \, dx-18 \int \frac {\exp \left (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)}\right ) x}{(-2+x-\log (x))^2} \, dx+24 \int \frac {\exp \left (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)}\right )}{(-2+x-\log (x))^2} \, dx+24 \int \frac {\exp \left (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)}\right )}{-2+x-\log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 4.98, size = 23, normalized size = 1.00 \begin {gather*} e^{9 e^{-\frac {2 x (4+x)}{3 (2-x+\log (x))}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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]

[Out]

E^(9/E^((2*x*(4 + x))/(3*(2 - x + Log[x]))))

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fricas [B]  time = 1.16, size = 68, normalized size = 2.96 \begin {gather*} e^{\left (\frac {2 \, x^{2} + 27 \, {\left (x - \log \relax (x) - 2\right )} e^{\left (\frac {2 \, {\left (x^{2} + 4 \, x\right )}}{3 \, {\left (x - \log \relax (x) - 2\right )}}\right )} + 8 \, x}{3 \, {\left (x - \log \relax (x) - 2\right )}} - \frac {2 \, {\left (x^{2} + 4 \, x\right )}}{3 \, {\left (x - \log \relax (x) - 2\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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")

[Out]

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))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {6 \, {\left (x^{2} - 2 \, {\left (x + 2\right )} \log \relax (x) - 3 \, x - 4\right )} e^{\left (\frac {2 \, {\left (x^{2} + 4 \, x\right )}}{3 \, {\left (x - \log \relax (x) - 2\right )}} + 9 \, e^{\left (\frac {2 \, {\left (x^{2} + 4 \, x\right )}}{3 \, {\left (x - \log \relax (x) - 2\right )}}\right )}\right )}}{x^{2} - 2 \, {\left (x - 2\right )} \log \relax (x) + \log \relax (x)^{2} - 4 \, x + 4}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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")

[Out]

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)

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maple [A]  time = 0.04, size = 20, normalized size = 0.87

method	result	size
risch	\({\mathrm e}^{9 \,{\mathrm e}^{-\frac {2 \left (4+x \right ) x}{3 \left (2+\ln \relax (x )-x \right )}}}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

exp(9*exp(-2/3*(4+x)*x/(2+ln(x)-x)))

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maxima [B]  time = 0.78, size = 51, normalized size = 2.22 \begin {gather*} e^{\left (9 \, x^{\frac {2}{3}} e^{\left (\frac {2}{3} \, x + \frac {2 \, \log \relax (x)^{2}}{3 \, {\left (x - \log \relax (x) - 2\right )}} + \frac {16 \, \log \relax (x)}{3 \, {\left (x - \log \relax (x) - 2\right )}} + \frac {8}{x - \log \relax (x) - 2} + 4\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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")

[Out]

e^(9*x^(2/3)*e^(2/3*x + 2/3*log(x)^2/(x - log(x) - 2) + 16/3*log(x)/(x - log(x) - 2) + 8/(x - log(x) - 2) + 4)
)

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mupad [B]  time = 7.59, size = 26, normalized size = 1.13 \begin {gather*} {\mathrm {e}}^{9\,{\mathrm {e}}^{-\frac {2\,x^2+8\,x}{3\,\ln \relax (x)-3\,x+6}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

exp(9*exp(-(8*x + 2*x^2)/(3*log(x) - 3*x + 6)))

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sympy [A]  time = 2.51, size = 24, normalized size = 1.04 \begin {gather*} e^{9 e^{\frac {2 \left (- x^{2} - 4 x\right )}{- 3 x + 3 \log {\relax (x )} + 6}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

exp(9*exp(2*(-x**2 - 4*x)/(-3*x + 3*log(x) + 6)))

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