Integrand size = 187, antiderivative size = 26 \[ \int \frac {4^{\frac {-16-8 x-x^2}{-2-2 x+2 \log \left (\log \left (x^2\right )\right )}} \left (\left (32+48 x+18 x^2+2 x^3\right ) \log (4)+\left (2+8 x+10 x^2+4 x^3+\left (-8 x-6 x^2+3 x^3+x^4\right ) \log (4)\right ) \log \left (x^2\right )+\left (-4-12 x-8 x^2+\left (-8 x-10 x^2-2 x^3\right ) \log (4)\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+(2+4 x) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )\right )}{\left (2+4 x+2 x^2\right ) \log \left (x^2\right )+(-4-4 x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+2 \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )} \, dx=2^{\frac {(4+x)^2}{1+x-\log \left (\log \left (x^2\right )\right )}} \left (x+x^2\right ) \]
\[ \int \frac {4^{\frac {-16-8 x-x^2}{-2-2 x+2 \log \left (\log \left (x^2\right )\right )}} \left (\left (32+48 x+18 x^2+2 x^3\right ) \log (4)+\left (2+8 x+10 x^2+4 x^3+\left (-8 x-6 x^2+3 x^3+x^4\right ) \log (4)\right ) \log \left (x^2\right )+\left (-4-12 x-8 x^2+\left (-8 x-10 x^2-2 x^3\right ) \log (4)\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+(2+4 x) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )\right )}{\left (2+4 x+2 x^2\right ) \log \left (x^2\right )+(-4-4 x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+2 \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )} \, dx=\int \frac {4^{\frac {-16-8 x-x^2}{-2-2 x+2 \log \left (\log \left (x^2\right )\right )}} \left (\left (32+48 x+18 x^2+2 x^3\right ) \log (4)+\left (2+8 x+10 x^2+4 x^3+\left (-8 x-6 x^2+3 x^3+x^4\right ) \log (4)\right ) \log \left (x^2\right )+\left (-4-12 x-8 x^2+\left (-8 x-10 x^2-2 x^3\right ) \log (4)\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+(2+4 x) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )\right )}{\left (2+4 x+2 x^2\right ) \log \left (x^2\right )+(-4-4 x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+2 \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )} \, dx \]
Integrate[(4^((-16 - 8*x - x^2)/(-2 - 2*x + 2*Log[Log[x^2]]))*((32 + 48*x + 18*x^2 + 2*x^3)*Log[4] + (2 + 8*x + 10*x^2 + 4*x^3 + (-8*x - 6*x^2 + 3*x ^3 + x^4)*Log[4])*Log[x^2] + (-4 - 12*x - 8*x^2 + (-8*x - 10*x^2 - 2*x^3)* Log[4])*Log[x^2]*Log[Log[x^2]] + (2 + 4*x)*Log[x^2]*Log[Log[x^2]]^2))/((2 + 4*x + 2*x^2)*Log[x^2] + (-4 - 4*x)*Log[x^2]*Log[Log[x^2]] + 2*Log[x^2]*L og[Log[x^2]]^2),x]
Integrate[(4^((-16 - 8*x - x^2)/(-2 - 2*x + 2*Log[Log[x^2]]))*((32 + 48*x + 18*x^2 + 2*x^3)*Log[4] + (2 + 8*x + 10*x^2 + 4*x^3 + (-8*x - 6*x^2 + 3*x ^3 + x^4)*Log[4])*Log[x^2] + (-4 - 12*x - 8*x^2 + (-8*x - 10*x^2 - 2*x^3)* Log[4])*Log[x^2]*Log[Log[x^2]] + (2 + 4*x)*Log[x^2]*Log[Log[x^2]]^2))/((2 + 4*x + 2*x^2)*Log[x^2] + (-4 - 4*x)*Log[x^2]*Log[Log[x^2]] + 2*Log[x^2]*L og[Log[x^2]]^2), x]
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.
| \(\displaystyle \int \frac {4^{\frac {-x^2-8 x-16}{2 \log \left (\log \left (x^2\right )\right )-2 x-2}} \left ((4 x+2) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )+\left (-8 x^2+\left (-2 x^3-10 x^2-8 x\right ) \log (4)-12 x-4\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+\left (2 x^3+18 x^2+48 x+32\right ) \log (4)+\left (4 x^3+10 x^2+\left (x^4+3 x^3-6 x^2-8 x\right ) \log (4)+8 x+2\right ) \log \left (x^2\right )\right )}{2 \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )+(-4 x-4) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+\left (2 x^2+4 x+2\right ) \log \left (x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {2^{-\frac {-x^2-8 x-16}{-\log \left (\log \left (x^2\right )\right )+x+1}-1} \left ((4 x+2) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )+\left (-8 x^2+\left (-2 x^3-10 x^2-8 x\right ) \log (4)-12 x-4\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+\left (2 x^3+18 x^2+48 x+32\right ) \log (4)+\left (4 x^3+10 x^2+\left (x^4+3 x^3-6 x^2-8 x\right ) \log (4)+8 x+2\right ) \log \left (x^2\right )\right )}{\log \left (x^2\right ) \left (-\log \left (\log \left (x^2\right )\right )+x+1\right )^2}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {2^{\frac {x^2+\log \left (\log \left (x^2\right )\right )+7 x+15}{-\log \left (\log \left (x^2\right )\right )+x+1}} \left ((4 x+2) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )+\left (-8 x^2+\left (-2 x^3-10 x^2-8 x\right ) \log (4)-12 x-4\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+\left (2 x^3+18 x^2+48 x+32\right ) \log (4)+\left (4 x^3+10 x^2+\left (x^4+3 x^3-6 x^2-8 x\right ) \log (4)+8 x+2\right ) \log \left (x^2\right )\right )}{\log \left (x^2\right ) \left (-\log \left (\log \left (x^2\right )\right )+x+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {(x+1) (x+4)^2 \log (4) 2^{\frac {x^2+\log \left (\log \left (x^2\right )\right )+7 x+15}{-\log \left (\log \left (x^2\right )\right )+x+1}} \left (x \log \left (x^2\right )-2\right )}{\log \left (x^2\right ) \left (-\log \left (\log \left (x^2\right )\right )+x+1\right )^2}+(2 x+1) 2^{\frac {x^2+\log \left (\log \left (x^2\right )\right )+7 x+15}{-\log \left (\log \left (x^2\right )\right )+x+1}+1}+\frac {x \left (x^2+5 x+4\right ) \log (4) 2^{\frac {x^2+\log \left (\log \left (x^2\right )\right )+7 x+15}{-\log \left (\log \left (x^2\right )\right )+x+1}+1}}{-\log \left (\log \left (x^2\right )\right )+x+1}\right )dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \left (-\frac {(x+1) (x+4)^2 \log (4) 2^{\frac {x^2+\log \left (\log \left (x^2\right )\right )+7 x+15}{-\log \left (\log \left (x^2\right )\right )+x+1}} \left (x \log \left (x^2\right )-2\right )}{\log \left (x^2\right ) \left (-\log \left (\log \left (x^2\right )\right )+x+1\right )^2}+(2 x+1) 2^{\frac {(x+4)^2}{-\log \left (\log \left (x^2\right )\right )+x+1}}+\frac {x \left (x^2+5 x+4\right ) \log (4) 2^{\frac {(x+4)^2}{-\log \left (\log \left (x^2\right )\right )+x+1}}}{-\log \left (\log \left (x^2\right )\right )+x+1}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \int 2^{\frac {(x+4)^2}{x-\log \left (\log \left (x^2\right )\right )+1}}dx+\int 2^{\frac {(x+4)^2}{x-\log \left (\log \left (x^2\right )\right )+1}+1} xdx-\log (4) \int \frac {2^{\frac {x^2+11 x-3 \log \left (\log \left (x^2\right )\right )+19}{x-\log \left (\log \left (x^2\right )\right )+1}} x}{\left (x-\log \left (\log \left (x^2\right )\right )+1\right )^2}dx-3 \log (4) \int \frac {2^{\frac {x^2+10 x-2 \log \left (\log \left (x^2\right )\right )+18}{x-\log \left (\log \left (x^2\right )\right )+1}} x^2}{\left (x-\log \left (\log \left (x^2\right )\right )+1\right )^2}dx+\log (4) \int \frac {2^{\frac {x^2+12 x-4 \log \left (\log \left (x^2\right )\right )+20}{x-\log \left (\log \left (x^2\right )\right )+1}}}{\log \left (x^2\right ) \left (x-\log \left (\log \left (x^2\right )\right )+1\right )^2}dx+3 \log (4) \int \frac {2^{\frac {x^2+11 x-3 \log \left (\log \left (x^2\right )\right )+19}{x-\log \left (\log \left (x^2\right )\right )+1}} x}{\log \left (x^2\right ) \left (x-\log \left (\log \left (x^2\right )\right )+1\right )^2}dx+9 \log (4) \int \frac {2^{\frac {(x+4)^2}{x-\log \left (\log \left (x^2\right )\right )+1}} x^2}{\log \left (x^2\right ) \left (x-\log \left (\log \left (x^2\right )\right )+1\right )^2}dx+\log (4) \int \frac {2^{\frac {(x+4)^2}{x-\log \left (\log \left (x^2\right )\right )+1}+2} x}{x-\log \left (\log \left (x^2\right )\right )+1}dx+5 \log (4) \int \frac {2^{\frac {(x+4)^2}{x-\log \left (\log \left (x^2\right )\right )+1}} x^2}{x-\log \left (\log \left (x^2\right )\right )+1}dx-\log (4) \int \frac {2^{\frac {x^2+7 x+\log \left (\log \left (x^2\right )\right )+15}{x-\log \left (\log \left (x^2\right )\right )+1}} x^4}{\left (x-\log \left (\log \left (x^2\right )\right )+1\right )^2}dx-9 \log (4) \int \frac {2^{\frac {x^2+7 x+\log \left (\log \left (x^2\right )\right )+15}{x-\log \left (\log \left (x^2\right )\right )+1}} x^3}{\left (x-\log \left (\log \left (x^2\right )\right )+1\right )^2}dx+\log (4) \int \frac {2^{\frac {(x+4)^2}{x-\log \left (\log \left (x^2\right )\right )+1}} x^3}{\log \left (x^2\right ) \left (x-\log \left (\log \left (x^2\right )\right )+1\right )^2}dx+\log (4) \int \frac {2^{\frac {(x+4)^2}{x-\log \left (\log \left (x^2\right )\right )+1}} x^3}{x-\log \left (\log \left (x^2\right )\right )+1}dx\) |
Int[(4^((-16 - 8*x - x^2)/(-2 - 2*x + 2*Log[Log[x^2]]))*((32 + 48*x + 18*x ^2 + 2*x^3)*Log[4] + (2 + 8*x + 10*x^2 + 4*x^3 + (-8*x - 6*x^2 + 3*x^3 + x ^4)*Log[4])*Log[x^2] + (-4 - 12*x - 8*x^2 + (-8*x - 10*x^2 - 2*x^3)*Log[4] )*Log[x^2]*Log[Log[x^2]] + (2 + 4*x)*Log[x^2]*Log[Log[x^2]]^2))/((2 + 4*x + 2*x^2)*Log[x^2] + (-4 - 4*x)*Log[x^2]*Log[Log[x^2]] + 2*Log[x^2]*Log[Log [x^2]]^2),x]
3.9.35.3.1 Defintions of rubi rules used
Leaf count of result is larger than twice the leaf count of optimal. \(57\) vs. \(2(27)=54\).
Time = 46.11 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.23
| method | result | size |
| parallelrisch | \({\mathrm e}^{-\frac {\left (x^{2}+8 x +16\right ) \ln \left (2\right )}{\ln \left (\ln \left (x^{2}\right )\right )-x -1}} x^{2}+{\mathrm e}^{-\frac {\left (x^{2}+8 x +16\right ) \ln \left (2\right )}{\ln \left (\ln \left (x^{2}\right )\right )-x -1}} x\) | \(58\) |
int(((4*x+2)*ln(x^2)*ln(ln(x^2))^2+(2*(-2*x^3-10*x^2-8*x)*ln(2)-8*x^2-12*x -4)*ln(x^2)*ln(ln(x^2))+(2*(x^4+3*x^3-6*x^2-8*x)*ln(2)+4*x^3+10*x^2+8*x+2) *ln(x^2)+2*(2*x^3+18*x^2+48*x+32)*ln(2))*exp(2*(-x^2-8*x-16)*ln(2)/(2*ln(l n(x^2))-2*x-2))/(2*ln(x^2)*ln(ln(x^2))^2+(-4-4*x)*ln(x^2)*ln(ln(x^2))+(2*x ^2+4*x+2)*ln(x^2)),x,method=_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {4^{\frac {-16-8 x-x^2}{-2-2 x+2 \log \left (\log \left (x^2\right )\right )}} \left (\left (32+48 x+18 x^2+2 x^3\right ) \log (4)+\left (2+8 x+10 x^2+4 x^3+\left (-8 x-6 x^2+3 x^3+x^4\right ) \log (4)\right ) \log \left (x^2\right )+\left (-4-12 x-8 x^2+\left (-8 x-10 x^2-2 x^3\right ) \log (4)\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+(2+4 x) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )\right )}{\left (2+4 x+2 x^2\right ) \log \left (x^2\right )+(-4-4 x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+2 \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )} \, dx={\left (x^{2} + x\right )} 2^{\frac {x^{2} + 8 \, x + 16}{x - \log \left (\log \left (x^{2}\right )\right ) + 1}} \]
integrate(((4*x+2)*log(x^2)*log(log(x^2))^2+(2*(-2*x^3-10*x^2-8*x)*log(2)- 8*x^2-12*x-4)*log(x^2)*log(log(x^2))+(2*(x^4+3*x^3-6*x^2-8*x)*log(2)+4*x^3 +10*x^2+8*x+2)*log(x^2)+2*(2*x^3+18*x^2+48*x+32)*log(2))*exp(2*(-x^2-8*x-1 6)*log(2)/(2*log(log(x^2))-2*x-2))/(2*log(x^2)*log(log(x^2))^2+(-4-4*x)*lo g(x^2)*log(log(x^2))+(2*x^2+4*x+2)*log(x^2)),x, algorithm=\
Exception generated. \[ \int \frac {4^{\frac {-16-8 x-x^2}{-2-2 x+2 \log \left (\log \left (x^2\right )\right )}} \left (\left (32+48 x+18 x^2+2 x^3\right ) \log (4)+\left (2+8 x+10 x^2+4 x^3+\left (-8 x-6 x^2+3 x^3+x^4\right ) \log (4)\right ) \log \left (x^2\right )+\left (-4-12 x-8 x^2+\left (-8 x-10 x^2-2 x^3\right ) \log (4)\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+(2+4 x) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )\right )}{\left (2+4 x+2 x^2\right ) \log \left (x^2\right )+(-4-4 x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+2 \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )} \, dx=\text {Exception raised: TypeError} \]
integrate(((4*x+2)*ln(x**2)*ln(ln(x**2))**2+(2*(-2*x**3-10*x**2-8*x)*ln(2) -8*x**2-12*x-4)*ln(x**2)*ln(ln(x**2))+(2*(x**4+3*x**3-6*x**2-8*x)*ln(2)+4* x**3+10*x**2+8*x+2)*ln(x**2)+2*(2*x**3+18*x**2+48*x+32)*ln(2))*exp(2*(-x** 2-8*x-16)*ln(2)/(2*ln(ln(x**2))-2*x-2))/(2*ln(x**2)*ln(ln(x**2))**2+(-4-4* x)*ln(x**2)*ln(ln(x**2))+(2*x**2+4*x+2)*ln(x**2)),x)
Exception generated. \[ \int \frac {4^{\frac {-16-8 x-x^2}{-2-2 x+2 \log \left (\log \left (x^2\right )\right )}} \left (\left (32+48 x+18 x^2+2 x^3\right ) \log (4)+\left (2+8 x+10 x^2+4 x^3+\left (-8 x-6 x^2+3 x^3+x^4\right ) \log (4)\right ) \log \left (x^2\right )+\left (-4-12 x-8 x^2+\left (-8 x-10 x^2-2 x^3\right ) \log (4)\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+(2+4 x) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )\right )}{\left (2+4 x+2 x^2\right ) \log \left (x^2\right )+(-4-4 x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+2 \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )} \, dx=\text {Exception raised: RuntimeError} \]
integrate(((4*x+2)*log(x^2)*log(log(x^2))^2+(2*(-2*x^3-10*x^2-8*x)*log(2)- 8*x^2-12*x-4)*log(x^2)*log(log(x^2))+(2*(x^4+3*x^3-6*x^2-8*x)*log(2)+4*x^3 +10*x^2+8*x+2)*log(x^2)+2*(2*x^3+18*x^2+48*x+32)*log(2))*exp(2*(-x^2-8*x-1 6)*log(2)/(2*log(log(x^2))-2*x-2))/(2*log(x^2)*log(log(x^2))^2+(-4-4*x)*lo g(x^2)*log(log(x^2))+(2*x^2+4*x+2)*log(x^2)),x, algorithm=\
Exception raised: RuntimeError >> ECL says: In function CAR, the value of the first argument is 0which is not of the expected type LIST
\[ \int \frac {4^{\frac {-16-8 x-x^2}{-2-2 x+2 \log \left (\log \left (x^2\right )\right )}} \left (\left (32+48 x+18 x^2+2 x^3\right ) \log (4)+\left (2+8 x+10 x^2+4 x^3+\left (-8 x-6 x^2+3 x^3+x^4\right ) \log (4)\right ) \log \left (x^2\right )+\left (-4-12 x-8 x^2+\left (-8 x-10 x^2-2 x^3\right ) \log (4)\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+(2+4 x) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )\right )}{\left (2+4 x+2 x^2\right ) \log \left (x^2\right )+(-4-4 x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+2 \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )} \, dx=\int { -\frac {{\left ({\left (2 \, x + 1\right )} \log \left (x^{2}\right ) \log \left (\log \left (x^{2}\right )\right )^{2} - 2 \, {\left (2 \, x^{2} + {\left (x^{3} + 5 \, x^{2} + 4 \, x\right )} \log \left (2\right ) + 3 \, x + 1\right )} \log \left (x^{2}\right ) \log \left (\log \left (x^{2}\right )\right ) + 2 \, {\left (x^{3} + 9 \, x^{2} + 24 \, x + 16\right )} \log \left (2\right ) + {\left (2 \, x^{3} + 5 \, x^{2} + {\left (x^{4} + 3 \, x^{3} - 6 \, x^{2} - 8 \, x\right )} \log \left (2\right ) + 4 \, x + 1\right )} \log \left (x^{2}\right )\right )} 2^{\frac {x^{2} + 8 \, x + 16}{x - \log \left (\log \left (x^{2}\right )\right ) + 1}}}{2 \, {\left (x + 1\right )} \log \left (x^{2}\right ) \log \left (\log \left (x^{2}\right )\right ) - \log \left (x^{2}\right ) \log \left (\log \left (x^{2}\right )\right )^{2} - {\left (x^{2} + 2 \, x + 1\right )} \log \left (x^{2}\right )} \,d x } \]
integrate(((4*x+2)*log(x^2)*log(log(x^2))^2+(2*(-2*x^3-10*x^2-8*x)*log(2)- 8*x^2-12*x-4)*log(x^2)*log(log(x^2))+(2*(x^4+3*x^3-6*x^2-8*x)*log(2)+4*x^3 +10*x^2+8*x+2)*log(x^2)+2*(2*x^3+18*x^2+48*x+32)*log(2))*exp(2*(-x^2-8*x-1 6)*log(2)/(2*log(log(x^2))-2*x-2))/(2*log(x^2)*log(log(x^2))^2+(-4-4*x)*lo g(x^2)*log(log(x^2))+(2*x^2+4*x+2)*log(x^2)),x, algorithm=\
integrate(-((2*x + 1)*log(x^2)*log(log(x^2))^2 - 2*(2*x^2 + (x^3 + 5*x^2 + 4*x)*log(2) + 3*x + 1)*log(x^2)*log(log(x^2)) + 2*(x^3 + 9*x^2 + 24*x + 1 6)*log(2) + (2*x^3 + 5*x^2 + (x^4 + 3*x^3 - 6*x^2 - 8*x)*log(2) + 4*x + 1) *log(x^2))*2^((x^2 + 8*x + 16)/(x - log(log(x^2)) + 1))/(2*(x + 1)*log(x^2 )*log(log(x^2)) - log(x^2)*log(log(x^2))^2 - (x^2 + 2*x + 1)*log(x^2)), x)
Timed out. \[ \int \frac {4^{\frac {-16-8 x-x^2}{-2-2 x+2 \log \left (\log \left (x^2\right )\right )}} \left (\left (32+48 x+18 x^2+2 x^3\right ) \log (4)+\left (2+8 x+10 x^2+4 x^3+\left (-8 x-6 x^2+3 x^3+x^4\right ) \log (4)\right ) \log \left (x^2\right )+\left (-4-12 x-8 x^2+\left (-8 x-10 x^2-2 x^3\right ) \log (4)\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+(2+4 x) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )\right )}{\left (2+4 x+2 x^2\right ) \log \left (x^2\right )+(-4-4 x) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+2 \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )} \, dx=\int \frac {{\mathrm {e}}^{\frac {2\,\ln \left (2\right )\,\left (x^2+8\,x+16\right )}{2\,x-2\,\ln \left (\ln \left (x^2\right )\right )+2}}\,\left (\ln \left (x^2\right )\,\left (4\,x+2\right )\,{\ln \left (\ln \left (x^2\right )\right )}^2-\ln \left (x^2\right )\,\left (12\,x+2\,\ln \left (2\right )\,\left (2\,x^3+10\,x^2+8\,x\right )+8\,x^2+4\right )\,\ln \left (\ln \left (x^2\right )\right )+2\,\ln \left (2\right )\,\left (2\,x^3+18\,x^2+48\,x+32\right )+\ln \left (x^2\right )\,\left (8\,x-2\,\ln \left (2\right )\,\left (-x^4-3\,x^3+6\,x^2+8\,x\right )+10\,x^2+4\,x^3+2\right )\right )}{2\,\ln \left (x^2\right )\,{\ln \left (\ln \left (x^2\right )\right )}^2-\ln \left (x^2\right )\,\left (4\,x+4\right )\,\ln \left (\ln \left (x^2\right )\right )+\ln \left (x^2\right )\,\left (2\,x^2+4\,x+2\right )} \,d x \]
int((exp((2*log(2)*(8*x + x^2 + 16))/(2*x - 2*log(log(x^2)) + 2))*(2*log(2 )*(48*x + 18*x^2 + 2*x^3 + 32) + log(x^2)*(8*x - 2*log(2)*(8*x + 6*x^2 - 3 *x^3 - x^4) + 10*x^2 + 4*x^3 + 2) - log(x^2)*log(log(x^2))*(12*x + 2*log(2 )*(8*x + 10*x^2 + 2*x^3) + 8*x^2 + 4) + log(x^2)*log(log(x^2))^2*(4*x + 2) ))/(log(x^2)*(4*x + 2*x^2 + 2) + 2*log(x^2)*log(log(x^2))^2 - log(x^2)*log (log(x^2))*(4*x + 4)),x)
int((exp((2*log(2)*(8*x + x^2 + 16))/(2*x - 2*log(log(x^2)) + 2))*(2*log(2 )*(48*x + 18*x^2 + 2*x^3 + 32) + log(x^2)*(8*x - 2*log(2)*(8*x + 6*x^2 - 3 *x^3 - x^4) + 10*x^2 + 4*x^3 + 2) - log(x^2)*log(log(x^2))*(12*x + 2*log(2 )*(8*x + 10*x^2 + 2*x^3) + 8*x^2 + 4) + log(x^2)*log(log(x^2))^2*(4*x + 2) ))/(log(x^2)*(4*x + 2*x^2 + 2) + 2*log(x^2)*log(log(x^2))^2 - log(x^2)*log (log(x^2))*(4*x + 4)), x)