\(\int \frac {(-4+4 x^2-x^4+\log ^2(\frac {x^3}{3})+(4-2 x^2) \log (\log (x))-\log ^2(\log (x)))^x (-4+2 x^2+(-8 x^2+4 x^4) \log (x)-6 \log (x) \log (\frac {x^3}{3})+(2+4 x^2 \log (x)) \log (\log (x))+((4-4 x^2+x^4) \log (x)-\log (x) \log ^2(\frac {x^3}{3})+(-4+2 x^2) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))) \log (-4+4 x^2-x^4+\log ^2(\frac {x^3}{3})+(4-2 x^2) \log (\log (x))-\log ^2(\log (x))))}{(4-4 x^2+x^4) \log (x)-\log (x) \log ^2(\frac {x^3}{3})+(-4+2 x^2) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))} \, dx\) [2194]

Optimal result

Integrand size = 231, antiderivative size = 25 \[ \int \frac {\left (-4+4 x^2-x^4+\log ^2\left (\frac {x^3}{3}\right )+\left (4-2 x^2\right ) \log (\log (x))-\log ^2(\log (x))\right )^x \left (-4+2 x^2+\left (-8 x^2+4 x^4\right ) \log (x)-6 \log (x) \log \left (\frac {x^3}{3}\right )+\left (2+4 x^2 \log (x)\right ) \log (\log (x))+\left (\left (4-4 x^2+x^4\right ) \log (x)-\log (x) \log ^2\left (\frac {x^3}{3}\right )+\left (-4+2 x^2\right ) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))\right ) \log \left (-4+4 x^2-x^4+\log ^2\left (\frac {x^3}{3}\right )+\left (4-2 x^2\right ) \log (\log (x))-\log ^2(\log (x))\right )\right )}{\left (4-4 x^2+x^4\right ) \log (x)-\log (x) \log ^2\left (\frac {x^3}{3}\right )+\left (-4+2 x^2\right ) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))} \, dx=\left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^x \] Output:

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

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-4+4 x^2-x^4+\log ^2\left (\frac {x^3}{3}\right )+\left (4-2 x^2\right ) \log (\log (x))-\log ^2(\log (x))\right )^x \left (-4+2 x^2+\left (-8 x^2+4 x^4\right ) \log (x)-6 \log (x) \log \left (\frac {x^3}{3}\right )+\left (2+4 x^2 \log (x)\right ) \log (\log (x))+\left (\left (4-4 x^2+x^4\right ) \log (x)-\log (x) \log ^2\left (\frac {x^3}{3}\right )+\left (-4+2 x^2\right ) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))\right ) \log \left (-4+4 x^2-x^4+\log ^2\left (\frac {x^3}{3}\right )+\left (4-2 x^2\right ) \log (\log (x))-\log ^2(\log (x))\right )\right )}{\left (4-4 x^2+x^4\right ) \log (x)-\log (x) \log ^2\left (\frac {x^3}{3}\right )+\left (-4+2 x^2\right ) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))} \, dx=\left (\log ^2\left (\frac {x^3}{3}\right )-\left (-2+x^2+\log (\log (x))\right )^2\right )^x \] Input:

Integrate[((-4 + 4*x^2 - x^4 + Log[x^3/3]^2 + (4 - 2*x^2)*Log[Log[x]] - Lo 
g[Log[x]]^2)^x*(-4 + 2*x^2 + (-8*x^2 + 4*x^4)*Log[x] - 6*Log[x]*Log[x^3/3] 
 + (2 + 4*x^2*Log[x])*Log[Log[x]] + ((4 - 4*x^2 + x^4)*Log[x] - Log[x]*Log 
[x^3/3]^2 + (-4 + 2*x^2)*Log[x]*Log[Log[x]] + Log[x]*Log[Log[x]]^2)*Log[-4 
 + 4*x^2 - x^4 + Log[x^3/3]^2 + (4 - 2*x^2)*Log[Log[x]] - Log[Log[x]]^2])) 
/((4 - 4*x^2 + x^4)*Log[x] - Log[x]*Log[x^3/3]^2 + (-4 + 2*x^2)*Log[x]*Log 
[Log[x]] + Log[x]*Log[Log[x]]^2),x]
 

Output:

(Log[x^3/3]^2 - (-2 + x^2 + Log[Log[x]])^2)^x
 

Rubi [F]

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.

\(\displaystyle \int \frac {\left (-x^4+\log ^2\left (\frac {x^3}{3}\right )+4 x^2+\left (4-2 x^2\right ) \log (\log (x))-\log ^2(\log (x))-4\right )^x \left (-6 \log (x) \log \left (\frac {x^3}{3}\right )+2 x^2+\left (4 x^2 \log (x)+2\right ) \log (\log (x))+\left (4 x^4-8 x^2\right ) \log (x)+\left (-\log (x) \log ^2\left (\frac {x^3}{3}\right )+\left (2 x^2-4\right ) \log (x) \log (\log (x))+\left (x^4-4 x^2+4\right ) \log (x)+\log (x) \log ^2(\log (x))\right ) \log \left (-x^4+\log ^2\left (\frac {x^3}{3}\right )+4 x^2+\left (4-2 x^2\right ) \log (\log (x))-\log ^2(\log (x))-4\right )-4\right )}{-\log (x) \log ^2\left (\frac {x^3}{3}\right )+\left (2 x^2-4\right ) \log (x) \log (\log (x))+\left (x^4-4 x^2+4\right ) \log (x)+\log (x) \log ^2(\log (x))} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (\log ^2\left (\frac {x^3}{3}\right )-\left (x^2+\log (\log (x))-2\right )^2\right )^{x-1} \left (6 \log (x) \log \left (\frac {x^3}{3}\right )-2 x^2-4 \left (x^2-2\right ) x^2 \log (x)-\left (4 x^2 \log (x)+2\right ) \log (\log (x))-\log (x) \left (\left (x^2+\log (\log (x))-2\right )^2-\log ^2\left (\frac {x^3}{3}\right )\right ) \log \left (\log ^2\left (\frac {x^3}{3}\right )-\left (x^2+\log (\log (x))-2\right )^2\right )+4\right )}{\log (x)}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\left (-\log \left (\frac {x^3}{3}\right )-x^2-\log (\log (x))+2\right ) \left (-\log \left (x^3\right )+x^2+\log (\log (x))-2 \left (1-\frac {\log (3)}{2}\right )\right ) \left (\log ^2\left (\frac {x^3}{3}\right )-\left (x^2+\log (\log (x))-2\right )^2\right )^{x-1} \log \left (\log ^2\left (\frac {x^3}{3}\right )-\left (x^2+\log (\log (x))-2\right )^2\right )-\frac {2 \left (2 x^4 \log (x)-3 \log (x) \log \left (x^3\right )+x^2-4 x^2 \log (x)+2 x^2 \log (x) \log (\log (x))+\log (27) \log (x)+\log (\log (x))-2\right ) \left (\log ^2\left (\frac {x^3}{3}\right )-\left (x^2+\log (\log (x))-2\right )^2\right )^{x-1}}{\log (x)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -2 \log (27) \int \left (\log ^2\left (\frac {x^3}{3}\right )-\left (x^2+\log (\log (x))-2\right )^2\right )^{x-1}dx+8 \int x^2 \left (\log ^2\left (\frac {x^3}{3}\right )-\left (x^2+\log (\log (x))-2\right )^2\right )^{x-1}dx+4 \int \frac {\left (\log ^2\left (\frac {x^3}{3}\right )-\left (x^2+\log (\log (x))-2\right )^2\right )^{x-1}}{\log (x)}dx-2 \int \frac {x^2 \left (\log ^2\left (\frac {x^3}{3}\right )-\left (x^2+\log (\log (x))-2\right )^2\right )^{x-1}}{\log (x)}dx+6 \int \log \left (x^3\right ) \left (\log ^2\left (\frac {x^3}{3}\right )-\left (x^2+\log (\log (x))-2\right )^2\right )^{x-1}dx-4 \int x^2 \log (\log (x)) \left (\log ^2\left (\frac {x^3}{3}\right )-\left (x^2+\log (\log (x))-2\right )^2\right )^{x-1}dx-2 \int \frac {\log (\log (x)) \left (\log ^2\left (\frac {x^3}{3}\right )-\left (x^2+\log (\log (x))-2\right )^2\right )^{x-1}}{\log (x)}dx+\int \left (\log ^2\left (\frac {x^3}{3}\right )-\left (x^2+\log (\log (x))-2\right )^2\right )^x \log \left (\log ^2\left (\frac {x^3}{3}\right )-\left (x^2+\log (\log (x))-2\right )^2\right )dx-4 \int x^4 \left (\log ^2\left (\frac {x^3}{3}\right )-\left (x^2+\log (\log (x))-2\right )^2\right )^{x-1}dx\)

Input:

Int[((-4 + 4*x^2 - x^4 + Log[x^3/3]^2 + (4 - 2*x^2)*Log[Log[x]] - Log[Log[ 
x]]^2)^x*(-4 + 2*x^2 + (-8*x^2 + 4*x^4)*Log[x] - 6*Log[x]*Log[x^3/3] + (2 
+ 4*x^2*Log[x])*Log[Log[x]] + ((4 - 4*x^2 + x^4)*Log[x] - Log[x]*Log[x^3/3 
]^2 + (-4 + 2*x^2)*Log[x]*Log[Log[x]] + Log[x]*Log[Log[x]]^2)*Log[-4 + 4*x 
^2 - x^4 + Log[x^3/3]^2 + (4 - 2*x^2)*Log[Log[x]] - Log[Log[x]]^2]))/((4 - 
 4*x^2 + x^4)*Log[x] - Log[x]*Log[x^3/3]^2 + (-4 + 2*x^2)*Log[x]*Log[Log[x 
]] + Log[x]*Log[Log[x]]^2),x]
 

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.23 (sec) , antiderivative size = 113, normalized size of antiderivative = 4.52

\[\left (-\ln \left (\ln \left (x \right )\right )^{2}+\left (-2 x^{2}+4\right ) \ln \left (\ln \left (x \right )\right )+{\left (\ln \left (3\right )-3 \ln \left (x \right )+\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) {\left (-\operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x \right )\right )}^{2}}{2}+\frac {i \pi \,\operatorname {csgn}\left (i x^{3}\right ) \left (-\operatorname {csgn}\left (i x^{3}\right )+\operatorname {csgn}\left (i x^{2}\right )\right ) \left (-\operatorname {csgn}\left (i x^{3}\right )+\operatorname {csgn}\left (i x \right )\right )}{2}\right )}^{2}-x^{4}+4 x^{2}-4\right )^{x}\]

Input:

int(((ln(x)*ln(ln(x))^2+(2*x^2-4)*ln(x)*ln(ln(x))-ln(x)*ln(1/3*x^3)^2+(x^4 
-4*x^2+4)*ln(x))*ln(-ln(ln(x))^2+(-2*x^2+4)*ln(ln(x))+ln(1/3*x^3)^2-x^4+4* 
x^2-4)+(4*x^2*ln(x)+2)*ln(ln(x))-6*ln(x)*ln(1/3*x^3)+(4*x^4-8*x^2)*ln(x)+2 
*x^2-4)*exp(x*ln(-ln(ln(x))^2+(-2*x^2+4)*ln(ln(x))+ln(1/3*x^3)^2-x^4+4*x^2 
-4))/(ln(x)*ln(ln(x))^2+(2*x^2-4)*ln(x)*ln(ln(x))-ln(x)*ln(1/3*x^3)^2+(x^4 
-4*x^2+4)*ln(x)),x)
 

Output:

(-ln(ln(x))^2+(-2*x^2+4)*ln(ln(x))+(ln(3)-3*ln(x)+1/2*I*Pi*csgn(I*x^2)*(-c 
sgn(I*x^2)+csgn(I*x))^2+1/2*I*Pi*csgn(I*x^3)*(-csgn(I*x^3)+csgn(I*x^2))*(- 
csgn(I*x^3)+csgn(I*x)))^2-x^4+4*x^2-4)^x
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (23) = 46\).

Time = 0.08 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.88 \[ \int \frac {\left (-4+4 x^2-x^4+\log ^2\left (\frac {x^3}{3}\right )+\left (4-2 x^2\right ) \log (\log (x))-\log ^2(\log (x))\right )^x \left (-4+2 x^2+\left (-8 x^2+4 x^4\right ) \log (x)-6 \log (x) \log \left (\frac {x^3}{3}\right )+\left (2+4 x^2 \log (x)\right ) \log (\log (x))+\left (\left (4-4 x^2+x^4\right ) \log (x)-\log (x) \log ^2\left (\frac {x^3}{3}\right )+\left (-4+2 x^2\right ) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))\right ) \log \left (-4+4 x^2-x^4+\log ^2\left (\frac {x^3}{3}\right )+\left (4-2 x^2\right ) \log (\log (x))-\log ^2(\log (x))\right )\right )}{\left (4-4 x^2+x^4\right ) \log (x)-\log (x) \log ^2\left (\frac {x^3}{3}\right )+\left (-4+2 x^2\right ) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))} \, dx={\left (-x^{4} + 4 \, x^{2} + \log \left (3\right )^{2} - 6 \, \log \left (3\right ) \log \left (x\right ) + 9 \, \log \left (x\right )^{2} - 2 \, {\left (x^{2} - 2\right )} \log \left (\log \left (x\right )\right ) - \log \left (\log \left (x\right )\right )^{2} - 4\right )}^{x} \] Input:

integrate(((log(x)*log(log(x))^2+(2*x^2-4)*log(x)*log(log(x))-log(x)*log(1 
/3*x^3)^2+(x^4-4*x^2+4)*log(x))*log(-log(log(x))^2+(-2*x^2+4)*log(log(x))+ 
log(1/3*x^3)^2-x^4+4*x^2-4)+(4*x^2*log(x)+2)*log(log(x))-6*log(x)*log(1/3* 
x^3)+(4*x^4-8*x^2)*log(x)+2*x^2-4)*exp(x*log(-log(log(x))^2+(-2*x^2+4)*log 
(log(x))+log(1/3*x^3)^2-x^4+4*x^2-4))/(log(x)*log(log(x))^2+(2*x^2-4)*log( 
x)*log(log(x))-log(x)*log(1/3*x^3)^2+(x^4-4*x^2+4)*log(x)),x, algorithm="f 
ricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (-4+4 x^2-x^4+\log ^2\left (\frac {x^3}{3}\right )+\left (4-2 x^2\right ) \log (\log (x))-\log ^2(\log (x))\right )^x \left (-4+2 x^2+\left (-8 x^2+4 x^4\right ) \log (x)-6 \log (x) \log \left (\frac {x^3}{3}\right )+\left (2+4 x^2 \log (x)\right ) \log (\log (x))+\left (\left (4-4 x^2+x^4\right ) \log (x)-\log (x) \log ^2\left (\frac {x^3}{3}\right )+\left (-4+2 x^2\right ) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))\right ) \log \left (-4+4 x^2-x^4+\log ^2\left (\frac {x^3}{3}\right )+\left (4-2 x^2\right ) \log (\log (x))-\log ^2(\log (x))\right )\right )}{\left (4-4 x^2+x^4\right ) \log (x)-\log (x) \log ^2\left (\frac {x^3}{3}\right )+\left (-4+2 x^2\right ) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))} \, dx=\text {Timed out} \] Input:

integrate(((ln(x)*ln(ln(x))**2+(2*x**2-4)*ln(x)*ln(ln(x))-ln(x)*ln(1/3*x** 
3)**2+(x**4-4*x**2+4)*ln(x))*ln(-ln(ln(x))**2+(-2*x**2+4)*ln(ln(x))+ln(1/3 
*x**3)**2-x**4+4*x**2-4)+(4*x**2*ln(x)+2)*ln(ln(x))-6*ln(x)*ln(1/3*x**3)+( 
4*x**4-8*x**2)*ln(x)+2*x**2-4)*exp(x*ln(-ln(ln(x))**2+(-2*x**2+4)*ln(ln(x) 
)+ln(1/3*x**3)**2-x**4+4*x**2-4))/(ln(x)*ln(ln(x))**2+(2*x**2-4)*ln(x)*ln( 
ln(x))-ln(x)*ln(1/3*x**3)**2+(x**4-4*x**2+4)*ln(x)),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.76 \[ \int \frac {\left (-4+4 x^2-x^4+\log ^2\left (\frac {x^3}{3}\right )+\left (4-2 x^2\right ) \log (\log (x))-\log ^2(\log (x))\right )^x \left (-4+2 x^2+\left (-8 x^2+4 x^4\right ) \log (x)-6 \log (x) \log \left (\frac {x^3}{3}\right )+\left (2+4 x^2 \log (x)\right ) \log (\log (x))+\left (\left (4-4 x^2+x^4\right ) \log (x)-\log (x) \log ^2\left (\frac {x^3}{3}\right )+\left (-4+2 x^2\right ) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))\right ) \log \left (-4+4 x^2-x^4+\log ^2\left (\frac {x^3}{3}\right )+\left (4-2 x^2\right ) \log (\log (x))-\log ^2(\log (x))\right )\right )}{\left (4-4 x^2+x^4\right ) \log (x)-\log (x) \log ^2\left (\frac {x^3}{3}\right )+\left (-4+2 x^2\right ) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))} \, dx=e^{\left (x \log \left (x^{2} - \log \left (3\right ) + 3 \, \log \left (x\right ) + \log \left (\log \left (x\right )\right ) - 2\right ) + x \log \left (-x^{2} - \log \left (3\right ) + 3 \, \log \left (x\right ) - \log \left (\log \left (x\right )\right ) + 2\right )\right )} \] Input:

integrate(((log(x)*log(log(x))^2+(2*x^2-4)*log(x)*log(log(x))-log(x)*log(1 
/3*x^3)^2+(x^4-4*x^2+4)*log(x))*log(-log(log(x))^2+(-2*x^2+4)*log(log(x))+ 
log(1/3*x^3)^2-x^4+4*x^2-4)+(4*x^2*log(x)+2)*log(log(x))-6*log(x)*log(1/3* 
x^3)+(4*x^4-8*x^2)*log(x)+2*x^2-4)*exp(x*log(-log(log(x))^2+(-2*x^2+4)*log 
(log(x))+log(1/3*x^3)^2-x^4+4*x^2-4))/(log(x)*log(log(x))^2+(2*x^2-4)*log( 
x)*log(log(x))-log(x)*log(1/3*x^3)^2+(x^4-4*x^2+4)*log(x)),x, algorithm="m 
axima")
 

Output:

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

Giac [F]

\[ \int \frac {\left (-4+4 x^2-x^4+\log ^2\left (\frac {x^3}{3}\right )+\left (4-2 x^2\right ) \log (\log (x))-\log ^2(\log (x))\right )^x \left (-4+2 x^2+\left (-8 x^2+4 x^4\right ) \log (x)-6 \log (x) \log \left (\frac {x^3}{3}\right )+\left (2+4 x^2 \log (x)\right ) \log (\log (x))+\left (\left (4-4 x^2+x^4\right ) \log (x)-\log (x) \log ^2\left (\frac {x^3}{3}\right )+\left (-4+2 x^2\right ) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))\right ) \log \left (-4+4 x^2-x^4+\log ^2\left (\frac {x^3}{3}\right )+\left (4-2 x^2\right ) \log (\log (x))-\log ^2(\log (x))\right )\right )}{\left (4-4 x^2+x^4\right ) \log (x)-\log (x) \log ^2\left (\frac {x^3}{3}\right )+\left (-4+2 x^2\right ) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))} \, dx=\int { -\frac {{\left (2 \, x^{2} - {\left (\log \left (\frac {1}{3} \, x^{3}\right )^{2} \log \left (x\right ) - 2 \, {\left (x^{2} - 2\right )} \log \left (x\right ) \log \left (\log \left (x\right )\right ) - \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} - {\left (x^{4} - 4 \, x^{2} + 4\right )} \log \left (x\right )\right )} \log \left (-x^{4} + 4 \, x^{2} + \log \left (\frac {1}{3} \, x^{3}\right )^{2} - 2 \, {\left (x^{2} - 2\right )} \log \left (\log \left (x\right )\right ) - \log \left (\log \left (x\right )\right )^{2} - 4\right ) + 4 \, {\left (x^{4} - 2 \, x^{2}\right )} \log \left (x\right ) - 6 \, \log \left (\frac {1}{3} \, x^{3}\right ) \log \left (x\right ) + 2 \, {\left (2 \, x^{2} \log \left (x\right ) + 1\right )} \log \left (\log \left (x\right )\right ) - 4\right )} {\left (-x^{4} + 4 \, x^{2} + \log \left (\frac {1}{3} \, x^{3}\right )^{2} - 2 \, {\left (x^{2} - 2\right )} \log \left (\log \left (x\right )\right ) - \log \left (\log \left (x\right )\right )^{2} - 4\right )}^{x}}{\log \left (\frac {1}{3} \, x^{3}\right )^{2} \log \left (x\right ) - 2 \, {\left (x^{2} - 2\right )} \log \left (x\right ) \log \left (\log \left (x\right )\right ) - \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} - {\left (x^{4} - 4 \, x^{2} + 4\right )} \log \left (x\right )} \,d x } \] Input:

integrate(((log(x)*log(log(x))^2+(2*x^2-4)*log(x)*log(log(x))-log(x)*log(1 
/3*x^3)^2+(x^4-4*x^2+4)*log(x))*log(-log(log(x))^2+(-2*x^2+4)*log(log(x))+ 
log(1/3*x^3)^2-x^4+4*x^2-4)+(4*x^2*log(x)+2)*log(log(x))-6*log(x)*log(1/3* 
x^3)+(4*x^4-8*x^2)*log(x)+2*x^2-4)*exp(x*log(-log(log(x))^2+(-2*x^2+4)*log 
(log(x))+log(1/3*x^3)^2-x^4+4*x^2-4))/(log(x)*log(log(x))^2+(2*x^2-4)*log( 
x)*log(log(x))-log(x)*log(1/3*x^3)^2+(x^4-4*x^2+4)*log(x)),x, algorithm="g 
iac")
 

Output:

integrate(-(2*x^2 - (log(1/3*x^3)^2*log(x) - 2*(x^2 - 2)*log(x)*log(log(x) 
) - log(x)*log(log(x))^2 - (x^4 - 4*x^2 + 4)*log(x))*log(-x^4 + 4*x^2 + lo 
g(1/3*x^3)^2 - 2*(x^2 - 2)*log(log(x)) - log(log(x))^2 - 4) + 4*(x^4 - 2*x 
^2)*log(x) - 6*log(1/3*x^3)*log(x) + 2*(2*x^2*log(x) + 1)*log(log(x)) - 4) 
*(-x^4 + 4*x^2 + log(1/3*x^3)^2 - 2*(x^2 - 2)*log(log(x)) - log(log(x))^2 
- 4)^x/(log(1/3*x^3)^2*log(x) - 2*(x^2 - 2)*log(x)*log(log(x)) - log(x)*lo 
g(log(x))^2 - (x^4 - 4*x^2 + 4)*log(x)), x)
 

Mupad [B] (verification not implemented)

Time = 2.57 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.08 \[ \int \frac {\left (-4+4 x^2-x^4+\log ^2\left (\frac {x^3}{3}\right )+\left (4-2 x^2\right ) \log (\log (x))-\log ^2(\log (x))\right )^x \left (-4+2 x^2+\left (-8 x^2+4 x^4\right ) \log (x)-6 \log (x) \log \left (\frac {x^3}{3}\right )+\left (2+4 x^2 \log (x)\right ) \log (\log (x))+\left (\left (4-4 x^2+x^4\right ) \log (x)-\log (x) \log ^2\left (\frac {x^3}{3}\right )+\left (-4+2 x^2\right ) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))\right ) \log \left (-4+4 x^2-x^4+\log ^2\left (\frac {x^3}{3}\right )+\left (4-2 x^2\right ) \log (\log (x))-\log ^2(\log (x))\right )\right )}{\left (4-4 x^2+x^4\right ) \log (x)-\log (x) \log ^2\left (\frac {x^3}{3}\right )+\left (-4+2 x^2\right ) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))} \, dx={\left (-x^4-2\,x^2\,\ln \left (\ln \left (x\right )\right )+4\,x^2+{\ln \left (x^3\right )}^2-2\,\ln \left (3\right )\,\ln \left (x^3\right )-{\ln \left (\ln \left (x\right )\right )}^2+4\,\ln \left (\ln \left (x\right )\right )+{\ln \left (3\right )}^2-4\right )}^x \] Input:

int(-(exp(x*log(log(x^3/3)^2 - log(log(x))*(2*x^2 - 4) - log(log(x))^2 + 4 
*x^2 - x^4 - 4))*(log(x)*(8*x^2 - 4*x^4) - log(log(x))*(4*x^2*log(x) + 2) 
- log(log(x^3/3)^2 - log(log(x))*(2*x^2 - 4) - log(log(x))^2 + 4*x^2 - x^4 
 - 4)*(log(x)*(x^4 - 4*x^2 + 4) - log(x^3/3)^2*log(x) + log(log(x))^2*log( 
x) + log(log(x))*log(x)*(2*x^2 - 4)) - 2*x^2 + 6*log(x^3/3)*log(x) + 4))/( 
log(x)*(x^4 - 4*x^2 + 4) - log(x^3/3)^2*log(x) + log(log(x))^2*log(x) + lo 
g(log(x))*log(x)*(2*x^2 - 4)),x)
 

Output:

(4*log(log(x)) - 2*log(x^3)*log(3) - 2*x^2*log(log(x)) + log(x^3)^2 - log( 
log(x))^2 + log(3)^2 + 4*x^2 - x^4 - 4)^x
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.68 \[ \int \frac {\left (-4+4 x^2-x^4+\log ^2\left (\frac {x^3}{3}\right )+\left (4-2 x^2\right ) \log (\log (x))-\log ^2(\log (x))\right )^x \left (-4+2 x^2+\left (-8 x^2+4 x^4\right ) \log (x)-6 \log (x) \log \left (\frac {x^3}{3}\right )+\left (2+4 x^2 \log (x)\right ) \log (\log (x))+\left (\left (4-4 x^2+x^4\right ) \log (x)-\log (x) \log ^2\left (\frac {x^3}{3}\right )+\left (-4+2 x^2\right ) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))\right ) \log \left (-4+4 x^2-x^4+\log ^2\left (\frac {x^3}{3}\right )+\left (4-2 x^2\right ) \log (\log (x))-\log ^2(\log (x))\right )\right )}{\left (4-4 x^2+x^4\right ) \log (x)-\log (x) \log ^2\left (\frac {x^3}{3}\right )+\left (-4+2 x^2\right ) \log (x) \log (\log (x))+\log (x) \log ^2(\log (x))} \, dx=\left (-\mathrm {log}\left (\mathrm {log}\left (x \right )\right )^{2}-2 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x^{2}+4 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right )+\mathrm {log}\left (\frac {x^{3}}{3}\right )^{2}-x^{4}+4 x^{2}-4\right )^{x} \] Input:

int(((log(x)*log(log(x))^2+(2*x^2-4)*log(x)*log(log(x))-log(x)*log(1/3*x^3 
)^2+(x^4-4*x^2+4)*log(x))*log(-log(log(x))^2+(-2*x^2+4)*log(log(x))+log(1/ 
3*x^3)^2-x^4+4*x^2-4)+(4*x^2*log(x)+2)*log(log(x))-6*log(x)*log(1/3*x^3)+( 
4*x^4-8*x^2)*log(x)+2*x^2-4)*exp(x*log(-log(log(x))^2+(-2*x^2+4)*log(log(x 
))+log(1/3*x^3)^2-x^4+4*x^2-4))/(log(x)*log(log(x))^2+(2*x^2-4)*log(x)*log 
(log(x))-log(x)*log(1/3*x^3)^2+(x^4-4*x^2+4)*log(x)),x)
 

Output:

( - log(log(x))**2 - 2*log(log(x))*x**2 + 4*log(log(x)) + log(x**3/3)**2 - 
 x**4 + 4*x**2 - 4)**x