\(\int \frac {2^{-e^{e^x}} (2 x+\log (-3-3 x+\log (9)))^{e^{e^x}} (e^{e^x} (-9-6 x+2 \log (9))+e^{e^x} (e^x (-6 x-6 x^2+2 x \log (9))+e^x (-3-3 x+\log (9)) \log (-3-3 x+\log (9))) \log (\frac {1}{2} (2 x+\log (-3-3 x+\log (9)))))}{-6 x-6 x^2+2 x \log (9)+(-3-3 x+\log (9)) \log (-3-3 x+\log (9))} \, dx\) [10292]

Optimal result

Integrand size = 137, antiderivative size = 20 \[ \int \frac {2^{-e^{e^x}} (2 x+\log (-3-3 x+\log (9)))^{e^{e^x}} \left (e^{e^x} (-9-6 x+2 \log (9))+e^{e^x} \left (e^x \left (-6 x-6 x^2+2 x \log (9)\right )+e^x (-3-3 x+\log (9)) \log (-3-3 x+\log (9))\right ) \log \left (\frac {1}{2} (2 x+\log (-3-3 x+\log (9)))\right )\right )}{-6 x-6 x^2+2 x \log (9)+(-3-3 x+\log (9)) \log (-3-3 x+\log (9))} \, dx=\left (x+\frac {1}{2} \log (-3-3 x+\log (9))\right )^{e^{e^x}} \]

[Out]

exp(exp(exp(x))*ln(1/2*ln(2*ln(3)-3*x-3)+x))

Rubi [F]

\[ \int \frac {2^{-e^{e^x}} (2 x+\log (-3-3 x+\log (9)))^{e^{e^x}} \left (e^{e^x} (-9-6 x+2 \log (9))+e^{e^x} \left (e^x \left (-6 x-6 x^2+2 x \log (9)\right )+e^x (-3-3 x+\log (9)) \log (-3-3 x+\log (9))\right ) \log \left (\frac {1}{2} (2 x+\log (-3-3 x+\log (9)))\right )\right )}{-6 x-6 x^2+2 x \log (9)+(-3-3 x+\log (9)) \log (-3-3 x+\log (9))} \, dx=\int \frac {2^{-e^{e^x}} (2 x+\log (-3-3 x+\log (9)))^{e^{e^x}} \left (e^{e^x} (-9-6 x+2 \log (9))+e^{e^x} \left (e^x \left (-6 x-6 x^2+2 x \log (9)\right )+e^x (-3-3 x+\log (9)) \log (-3-3 x+\log (9))\right ) \log \left (\frac {1}{2} (2 x+\log (-3-3 x+\log (9)))\right )\right )}{-6 x-6 x^2+2 x \log (9)+(-3-3 x+\log (9)) \log (-3-3 x+\log (9))} \, dx \]

[In]

Int[((2*x + Log[-3 - 3*x + Log[9]])^E^E^x*(E^E^x*(-9 - 6*x + 2*Log[9]) + E^E^x*(E^x*(-6*x - 6*x^2 + 2*x*Log[9]
) + E^x*(-3 - 3*x + Log[9])*Log[-3 - 3*x + Log[9]])*Log[(2*x + Log[-3 - 3*x + Log[9]])/2]))/(2^E^E^x*(-6*x - 6
*x^2 + 2*x*Log[9] + (-3 - 3*x + Log[9])*Log[-3 - 3*x + Log[9]])),x]

[Out]

Defer[Int][2^(1 - E^E^x)*E^E^x*(2*x + Log[-3 - 3*x + Log[9]])^(-1 + E^E^x), x] + 3*Defer[Int][(E^E^x*(2*x + Lo
g[-3 - 3*x + Log[9]])^(-1 + E^E^x))/(2^E^E^x*(3 + 3*x - Log[9])), x] + Defer[Int][(E^(E^x + x)*(2*x + Log[-3 -
3*x + Log[9]])^E^E^x*Log[(2*x + Log[-3 - 3*x + Log[9]])/2])/2^E^E^x, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {2^{-e^{e^x}} (2 x+\log (-3-3 x+\log (9)))^{e^{e^x}} \left (e^{e^x} (-9-6 x+2 \log (9))+e^{e^x} \left (e^x \left (-6 x-6 x^2+2 x \log (9)\right )+e^x (-3-3 x+\log (9)) \log (-3-3 x+\log (9))\right ) \log \left (\frac {1}{2} (2 x+\log (-3-3 x+\log (9)))\right )\right )}{-6 x^2+x (-6+2 \log (9))+(-3-3 x+\log (9)) \log (-3-3 x+\log (9))} \, dx \\ & = \int \frac {2^{-e^{e^x}} e^{e^x} (2 x+\log (-3-3 x+\log (9)))^{-1+e^{e^x}} \left (6 x+9 \left (1-\frac {4 \log (3)}{9}\right )+e^x (3+3 x-\log (9)) (2 x+\log (-3-3 x+\log (9))) \log \left (x+\frac {1}{2} \log (-3-3 x+\log (9))\right )\right )}{3+3 x-\log (9)} \, dx \\ & = \int \left (\frac {2^{-e^{e^x}} e^{e^x} (9+6 x-4 \log (3)) (2 x+\log (-3-3 x+\log (9)))^{-1+e^{e^x}}}{3+3 x-\log (9)}+2^{-e^{e^x}} e^{e^x+x} (2 x+\log (-3-3 x+\log (9)))^{e^{e^x}} \log \left (\frac {1}{2} (2 x+\log (-3-3 x+\log (9)))\right )\right ) \, dx \\ & = \int \frac {2^{-e^{e^x}} e^{e^x} (9+6 x-4 \log (3)) (2 x+\log (-3-3 x+\log (9)))^{-1+e^{e^x}}}{3+3 x-\log (9)} \, dx+\int 2^{-e^{e^x}} e^{e^x+x} (2 x+\log (-3-3 x+\log (9)))^{e^{e^x}} \log \left (\frac {1}{2} (2 x+\log (-3-3 x+\log (9)))\right ) \, dx \\ & = \int \left (2^{1-e^{e^x}} e^{e^x} (2 x+\log (-3-3 x+\log (9)))^{-1+e^{e^x}}+\frac {3\ 2^{-e^{e^x}} e^{e^x} (2 x+\log (-3-3 x+\log (9)))^{-1+e^{e^x}}}{3+3 x-\log (9)}\right ) \, dx+\int 2^{-e^{e^x}} e^{e^x+x} (2 x+\log (-3-3 x+\log (9)))^{e^{e^x}} \log \left (\frac {1}{2} (2 x+\log (-3-3 x+\log (9)))\right ) \, dx \\ & = 3 \int \frac {2^{-e^{e^x}} e^{e^x} (2 x+\log (-3-3 x+\log (9)))^{-1+e^{e^x}}}{3+3 x-\log (9)} \, dx+\int 2^{1-e^{e^x}} e^{e^x} (2 x+\log (-3-3 x+\log (9)))^{-1+e^{e^x}} \, dx+\int 2^{-e^{e^x}} e^{e^x+x} (2 x+\log (-3-3 x+\log (9)))^{e^{e^x}} \log \left (\frac {1}{2} (2 x+\log (-3-3 x+\log (9)))\right ) \, dx \\ \end{align*}

Mathematica [F]

\[ \int \frac {2^{-e^{e^x}} (2 x+\log (-3-3 x+\log (9)))^{e^{e^x}} \left (e^{e^x} (-9-6 x+2 \log (9))+e^{e^x} \left (e^x \left (-6 x-6 x^2+2 x \log (9)\right )+e^x (-3-3 x+\log (9)) \log (-3-3 x+\log (9))\right ) \log \left (\frac {1}{2} (2 x+\log (-3-3 x+\log (9)))\right )\right )}{-6 x-6 x^2+2 x \log (9)+(-3-3 x+\log (9)) \log (-3-3 x+\log (9))} \, dx=\int \frac {2^{-e^{e^x}} (2 x+\log (-3-3 x+\log (9)))^{e^{e^x}} \left (e^{e^x} (-9-6 x+2 \log (9))+e^{e^x} \left (e^x \left (-6 x-6 x^2+2 x \log (9)\right )+e^x (-3-3 x+\log (9)) \log (-3-3 x+\log (9))\right ) \log \left (\frac {1}{2} (2 x+\log (-3-3 x+\log (9)))\right )\right )}{-6 x-6 x^2+2 x \log (9)+(-3-3 x+\log (9)) \log (-3-3 x+\log (9))} \, dx \]

[In]

Integrate[((2*x + Log[-3 - 3*x + Log[9]])^E^E^x*(E^E^x*(-9 - 6*x + 2*Log[9]) + E^E^x*(E^x*(-6*x - 6*x^2 + 2*x*
Log[9]) + E^x*(-3 - 3*x + Log[9])*Log[-3 - 3*x + Log[9]])*Log[(2*x + Log[-3 - 3*x + Log[9]])/2]))/(2^E^E^x*(-6
*x - 6*x^2 + 2*x*Log[9] + (-3 - 3*x + Log[9])*Log[-3 - 3*x + Log[9]])),x]

[Out]

Integrate[((2*x + Log[-3 - 3*x + Log[9]])^E^E^x*(E^E^x*(-9 - 6*x + 2*Log[9]) + E^E^x*(E^x*(-6*x - 6*x^2 + 2*x*
Log[9]) + E^x*(-3 - 3*x + Log[9])*Log[-3 - 3*x + Log[9]])*Log[(2*x + Log[-3 - 3*x + Log[9]])/2]))/(2^E^E^x*(-6
*x - 6*x^2 + 2*x*Log[9] + (-3 - 3*x + Log[9])*Log[-3 - 3*x + Log[9]])), x]

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95

\[\left (\frac {\ln \left (2 \ln \left (3\right )-3 x -3\right )}{2}+x \right )^{{\mathrm e}^{{\mathrm e}^{x}}}\]

[In]

int((((2*ln(3)-3*x-3)*exp(x)*ln(2*ln(3)-3*x-3)+(4*x*ln(3)-6*x^2-6*x)*exp(x))*exp(exp(x))*ln(1/2*ln(2*ln(3)-3*x
-3)+x)+(4*ln(3)-6*x-9)*exp(exp(x)))*exp(exp(exp(x))*ln(1/2*ln(2*ln(3)-3*x-3)+x))/((2*ln(3)-3*x-3)*ln(2*ln(3)-3
*x-3)+4*x*ln(3)-6*x^2-6*x),x)

[Out]

(1/2*ln(2*ln(3)-3*x-3)+x)^exp(exp(x))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {2^{-e^{e^x}} (2 x+\log (-3-3 x+\log (9)))^{e^{e^x}} \left (e^{e^x} (-9-6 x+2 \log (9))+e^{e^x} \left (e^x \left (-6 x-6 x^2+2 x \log (9)\right )+e^x (-3-3 x+\log (9)) \log (-3-3 x+\log (9))\right ) \log \left (\frac {1}{2} (2 x+\log (-3-3 x+\log (9)))\right )\right )}{-6 x-6 x^2+2 x \log (9)+(-3-3 x+\log (9)) \log (-3-3 x+\log (9))} \, dx={\left (x + \frac {1}{2} \, \log \left (-3 \, x + 2 \, \log \left (3\right ) - 3\right )\right )}^{e^{\left (e^{x}\right )}} \]

[In]

integrate((((2*log(3)-3*x-3)*exp(x)*log(2*log(3)-3*x-3)+(4*x*log(3)-6*x^2-6*x)*exp(x))*exp(exp(x))*log(1/2*log
(2*log(3)-3*x-3)+x)+(4*log(3)-6*x-9)*exp(exp(x)))*exp(exp(exp(x))*log(1/2*log(2*log(3)-3*x-3)+x))/((2*log(3)-3
*x-3)*log(2*log(3)-3*x-3)+4*x*log(3)-6*x^2-6*x),x, algorithm="fricas")

[Out]

(x + 1/2*log(-3*x + 2*log(3) - 3))^e^(e^x)

Sympy [F(-1)]

Timed out. \[ \int \frac {2^{-e^{e^x}} (2 x+\log (-3-3 x+\log (9)))^{e^{e^x}} \left (e^{e^x} (-9-6 x+2 \log (9))+e^{e^x} \left (e^x \left (-6 x-6 x^2+2 x \log (9)\right )+e^x (-3-3 x+\log (9)) \log (-3-3 x+\log (9))\right ) \log \left (\frac {1}{2} (2 x+\log (-3-3 x+\log (9)))\right )\right )}{-6 x-6 x^2+2 x \log (9)+(-3-3 x+\log (9)) \log (-3-3 x+\log (9))} \, dx=\text {Timed out} \]

[In]

integrate((((2*ln(3)-3*x-3)*exp(x)*ln(2*ln(3)-3*x-3)+(4*x*ln(3)-6*x**2-6*x)*exp(x))*exp(exp(x))*ln(1/2*ln(2*ln
(3)-3*x-3)+x)+(4*ln(3)-6*x-9)*exp(exp(x)))*exp(exp(exp(x))*ln(1/2*ln(2*ln(3)-3*x-3)+x))/((2*ln(3)-3*x-3)*ln(2*
ln(3)-3*x-3)+4*x*ln(3)-6*x**2-6*x),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.40 \[ \int \frac {2^{-e^{e^x}} (2 x+\log (-3-3 x+\log (9)))^{e^{e^x}} \left (e^{e^x} (-9-6 x+2 \log (9))+e^{e^x} \left (e^x \left (-6 x-6 x^2+2 x \log (9)\right )+e^x (-3-3 x+\log (9)) \log (-3-3 x+\log (9))\right ) \log \left (\frac {1}{2} (2 x+\log (-3-3 x+\log (9)))\right )\right )}{-6 x-6 x^2+2 x \log (9)+(-3-3 x+\log (9)) \log (-3-3 x+\log (9))} \, dx=e^{\left (-e^{\left (e^{x}\right )} \log \left (2\right ) + e^{\left (e^{x}\right )} \log \left (2 \, x + \log \left (-3 \, x + 2 \, \log \left (3\right ) - 3\right )\right )\right )} \]

[In]

integrate((((2*log(3)-3*x-3)*exp(x)*log(2*log(3)-3*x-3)+(4*x*log(3)-6*x^2-6*x)*exp(x))*exp(exp(x))*log(1/2*log
(2*log(3)-3*x-3)+x)+(4*log(3)-6*x-9)*exp(exp(x)))*exp(exp(exp(x))*log(1/2*log(2*log(3)-3*x-3)+x))/((2*log(3)-3
*x-3)*log(2*log(3)-3*x-3)+4*x*log(3)-6*x^2-6*x),x, algorithm="maxima")

[Out]

e^(-e^(e^x)*log(2) + e^(e^x)*log(2*x + log(-3*x + 2*log(3) - 3)))

Giac [F]

\[ \int \frac {2^{-e^{e^x}} (2 x+\log (-3-3 x+\log (9)))^{e^{e^x}} \left (e^{e^x} (-9-6 x+2 \log (9))+e^{e^x} \left (e^x \left (-6 x-6 x^2+2 x \log (9)\right )+e^x (-3-3 x+\log (9)) \log (-3-3 x+\log (9))\right ) \log \left (\frac {1}{2} (2 x+\log (-3-3 x+\log (9)))\right )\right )}{-6 x-6 x^2+2 x \log (9)+(-3-3 x+\log (9)) \log (-3-3 x+\log (9))} \, dx=\int { \frac {{\left ({\left ({\left (3 \, x - 2 \, \log \left (3\right ) + 3\right )} e^{x} \log \left (-3 \, x + 2 \, \log \left (3\right ) - 3\right ) + 2 \, {\left (3 \, x^{2} - 2 \, x \log \left (3\right ) + 3 \, x\right )} e^{x}\right )} e^{\left (e^{x}\right )} \log \left (x + \frac {1}{2} \, \log \left (-3 \, x + 2 \, \log \left (3\right ) - 3\right )\right ) + {\left (6 \, x - 4 \, \log \left (3\right ) + 9\right )} e^{\left (e^{x}\right )}\right )} {\left (x + \frac {1}{2} \, \log \left (-3 \, x + 2 \, \log \left (3\right ) - 3\right )\right )}^{e^{\left (e^{x}\right )}}}{6 \, x^{2} - 4 \, x \log \left (3\right ) + {\left (3 \, x - 2 \, \log \left (3\right ) + 3\right )} \log \left (-3 \, x + 2 \, \log \left (3\right ) - 3\right ) + 6 \, x} \,d x } \]

[In]

integrate((((2*log(3)-3*x-3)*exp(x)*log(2*log(3)-3*x-3)+(4*x*log(3)-6*x^2-6*x)*exp(x))*exp(exp(x))*log(1/2*log
(2*log(3)-3*x-3)+x)+(4*log(3)-6*x-9)*exp(exp(x)))*exp(exp(exp(x))*log(1/2*log(2*log(3)-3*x-3)+x))/((2*log(3)-3
*x-3)*log(2*log(3)-3*x-3)+4*x*log(3)-6*x^2-6*x),x, algorithm="giac")

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 18.13 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {2^{-e^{e^x}} (2 x+\log (-3-3 x+\log (9)))^{e^{e^x}} \left (e^{e^x} (-9-6 x+2 \log (9))+e^{e^x} \left (e^x \left (-6 x-6 x^2+2 x \log (9)\right )+e^x (-3-3 x+\log (9)) \log (-3-3 x+\log (9))\right ) \log \left (\frac {1}{2} (2 x+\log (-3-3 x+\log (9)))\right )\right )}{-6 x-6 x^2+2 x \log (9)+(-3-3 x+\log (9)) \log (-3-3 x+\log (9))} \, dx={\left (x+\frac {\ln \left (\ln \left (9\right )-3\,x-3\right )}{2}\right )}^{{\mathrm {e}}^{{\mathrm {e}}^x}} \]

[In]

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

[Out]

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