\(\int \frac {-8 e^{2 x}+e^{4 x} (4 e^{2 x}+e^x (1-2 x)-4 x)+e^x (-2+4 x)+(-2+e^{4 x}+e^{e^x} (-2 e^x+e^{5 x})) \log ^2(2-e^{4 x})+(-8 e^{2 x}+e^{4 x} (4 e^{2 x}+e^x (1-x))+e^x (-2+2 x)) \log (x)+(-2 e^{2 x}+e^{6 x}) \log ^2(x)+e^{e^x} (-8 e^{3 x}+4 e^{7 x}+(-8 e^{3 x}+4 e^{7 x}) \log (x)+(-2 e^{3 x}+e^{7 x}) \log ^2(x))+\log (2-e^{4 x}) (-2-8 e^x+e^{4 x} (1+4 e^x)+(-4 e^x+2 e^{5 x}) \log (x)+e^{e^x} (-8 e^{2 x}+4 e^{6 x}+(-4 e^{2 x}+2 e^{6 x}) \log (x)))}{-8 e^{2 x}+4 e^{6 x}+(-2+e^{4 x}) \log ^2(2-e^{4 x})+(-8 e^{2 x}+4 e^{6 x}) \log (x)+(-2 e^{2 x}+e^{6 x}) \log ^2(x)+\log (2-e^{4 x}) (-8 e^x+4 e^{5 x}+(-4 e^x+2 e^{5 x}) \log (x))} \, dx\) [1817]

Optimal result

Integrand size = 398, antiderivative size = 30 \[ \int \frac {-8 e^{2 x}+e^{4 x} \left (4 e^{2 x}+e^x (1-2 x)-4 x\right )+e^x (-2+4 x)+\left (-2+e^{4 x}+e^{e^x} \left (-2 e^x+e^{5 x}\right )\right ) \log ^2\left (2-e^{4 x}\right )+\left (-8 e^{2 x}+e^{4 x} \left (4 e^{2 x}+e^x (1-x)\right )+e^x (-2+2 x)\right ) \log (x)+\left (-2 e^{2 x}+e^{6 x}\right ) \log ^2(x)+e^{e^x} \left (-8 e^{3 x}+4 e^{7 x}+\left (-8 e^{3 x}+4 e^{7 x}\right ) \log (x)+\left (-2 e^{3 x}+e^{7 x}\right ) \log ^2(x)\right )+\log \left (2-e^{4 x}\right ) \left (-2-8 e^x+e^{4 x} \left (1+4 e^x\right )+\left (-4 e^x+2 e^{5 x}\right ) \log (x)+e^{e^x} \left (-8 e^{2 x}+4 e^{6 x}+\left (-4 e^{2 x}+2 e^{6 x}\right ) \log (x)\right )\right )}{-8 e^{2 x}+4 e^{6 x}+\left (-2+e^{4 x}\right ) \log ^2\left (2-e^{4 x}\right )+\left (-8 e^{2 x}+4 e^{6 x}\right ) \log (x)+\left (-2 e^{2 x}+e^{6 x}\right ) \log ^2(x)+\log \left (2-e^{4 x}\right ) \left (-8 e^x+4 e^{5 x}+\left (-4 e^x+2 e^{5 x}\right ) \log (x)\right )} \, dx=e^{e^x}+x+\frac {x}{\log \left (2-e^{4 x}\right )+e^x (2+\log (x))} \] Output:

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

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {-8 e^{2 x}+e^{4 x} \left (4 e^{2 x}+e^x (1-2 x)-4 x\right )+e^x (-2+4 x)+\left (-2+e^{4 x}+e^{e^x} \left (-2 e^x+e^{5 x}\right )\right ) \log ^2\left (2-e^{4 x}\right )+\left (-8 e^{2 x}+e^{4 x} \left (4 e^{2 x}+e^x (1-x)\right )+e^x (-2+2 x)\right ) \log (x)+\left (-2 e^{2 x}+e^{6 x}\right ) \log ^2(x)+e^{e^x} \left (-8 e^{3 x}+4 e^{7 x}+\left (-8 e^{3 x}+4 e^{7 x}\right ) \log (x)+\left (-2 e^{3 x}+e^{7 x}\right ) \log ^2(x)\right )+\log \left (2-e^{4 x}\right ) \left (-2-8 e^x+e^{4 x} \left (1+4 e^x\right )+\left (-4 e^x+2 e^{5 x}\right ) \log (x)+e^{e^x} \left (-8 e^{2 x}+4 e^{6 x}+\left (-4 e^{2 x}+2 e^{6 x}\right ) \log (x)\right )\right )}{-8 e^{2 x}+4 e^{6 x}+\left (-2+e^{4 x}\right ) \log ^2\left (2-e^{4 x}\right )+\left (-8 e^{2 x}+4 e^{6 x}\right ) \log (x)+\left (-2 e^{2 x}+e^{6 x}\right ) \log ^2(x)+\log \left (2-e^{4 x}\right ) \left (-8 e^x+4 e^{5 x}+\left (-4 e^x+2 e^{5 x}\right ) \log (x)\right )} \, dx=e^{e^x}+x+\frac {x}{\log \left (2-e^{4 x}\right )+e^x (2+\log (x))} \] Input:

Integrate[(-8*E^(2*x) + E^(4*x)*(4*E^(2*x) + E^x*(1 - 2*x) - 4*x) + E^x*(- 
2 + 4*x) + (-2 + E^(4*x) + E^E^x*(-2*E^x + E^(5*x)))*Log[2 - E^(4*x)]^2 + 
(-8*E^(2*x) + E^(4*x)*(4*E^(2*x) + E^x*(1 - x)) + E^x*(-2 + 2*x))*Log[x] + 
 (-2*E^(2*x) + E^(6*x))*Log[x]^2 + E^E^x*(-8*E^(3*x) + 4*E^(7*x) + (-8*E^( 
3*x) + 4*E^(7*x))*Log[x] + (-2*E^(3*x) + E^(7*x))*Log[x]^2) + Log[2 - E^(4 
*x)]*(-2 - 8*E^x + E^(4*x)*(1 + 4*E^x) + (-4*E^x + 2*E^(5*x))*Log[x] + E^E 
^x*(-8*E^(2*x) + 4*E^(6*x) + (-4*E^(2*x) + 2*E^(6*x))*Log[x])))/(-8*E^(2*x 
) + 4*E^(6*x) + (-2 + E^(4*x))*Log[2 - E^(4*x)]^2 + (-8*E^(2*x) + 4*E^(6*x 
))*Log[x] + (-2*E^(2*x) + E^(6*x))*Log[x]^2 + Log[2 - E^(4*x)]*(-8*E^x + 4 
*E^(5*x) + (-4*E^x + 2*E^(5*x))*Log[x])),x]
 

Output:

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

\(\displaystyle \int \frac {-8 e^{2 x}+e^{4 x} \left (e^x (1-2 x)+4 e^{2 x}-4 x\right )+e^x (4 x-2)+\left (e^{e^x} \left (e^{5 x}-2 e^x\right )+e^{4 x}-2\right ) \log ^2\left (2-e^{4 x}\right )+\left (e^{6 x}-2 e^{2 x}\right ) \log ^2(x)+e^{e^x} \left (-8 e^{3 x}+4 e^{7 x}+\left (e^{7 x}-2 e^{3 x}\right ) \log ^2(x)+\left (4 e^{7 x}-8 e^{3 x}\right ) \log (x)\right )+\left (e^{4 x} \left (4 e^x+1\right )-8 e^x+\left (2 e^{5 x}-4 e^x\right ) \log (x)+e^{e^x} \left (-8 e^{2 x}+4 e^{6 x}+\left (2 e^{6 x}-4 e^{2 x}\right ) \log (x)\right )-2\right ) \log \left (2-e^{4 x}\right )+\left (e^{4 x} \left (e^x (1-x)+4 e^{2 x}\right )-8 e^{2 x}+e^x (2 x-2)\right ) \log (x)}{-8 e^{2 x}+4 e^{6 x}+\left (e^{4 x}-2\right ) \log ^2\left (2-e^{4 x}\right )+\left (e^{6 x}-2 e^{2 x}\right ) \log ^2(x)+\left (-8 e^x+4 e^{5 x}+\left (2 e^{5 x}-4 e^x\right ) \log (x)\right ) \log \left (2-e^{4 x}\right )+\left (4 e^{6 x}-8 e^{2 x}\right ) \log (x)} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-\left (\left (e^{4 x}-2\right ) \left (e^{x+e^x}+1\right ) \log ^2\left (2-e^{4 x}\right )\right )-e^x \left (-8 e^x+4 e^{5 x}-8 e^{2 x+e^x}+4 e^{6 x+e^x}+e^{4 x} (1-2 x)-4 e^{3 x} x+4 x+e^x \left (e^{4 x}-2\right ) \left (e^{x+e^x}+1\right ) \log ^2(x)+\left (e^{4 x}-2\right ) \left (-x+4 e^x+4 e^{2 x+e^x}+1\right ) \log (x)-2\right )-\left (e^{4 x}-2\right ) \left (4 e^x+4 e^{2 x+e^x}+2 e^x \left (e^{x+e^x}+1\right ) \log (x)+1\right ) \log \left (2-e^{4 x}\right )}{\left (2-e^{4 x}\right ) \left (\log \left (2-e^{4 x}\right )+e^x (\log (x)+2)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\log \left (2-e^{4 x}\right ) \left (-2 x \log ^5(x)-20 x \log ^4(x)-2 \log ^4(x)-80 x \log ^3(x)-16 \log ^3(x)-160 x \log ^2(x)-48 \log ^2(x)+x \log ^4\left (2-e^{4 x}\right ) \log (x)-4 x \log ^3\left (2-e^{4 x}\right ) \log (x)-160 x \log (x)-64 \log (x)+2 x \log ^4\left (2-e^{4 x}\right )+\log ^4\left (2-e^{4 x}\right )-8 x \log ^3\left (2-e^{4 x}\right )-64 x-32\right )}{(\log (x)+2) \left (\log \left (2-e^{4 x}\right )+2 e^x+e^x \log (x)\right )^2 \left (\log ^4\left (2-e^{4 x}\right )-2 \log ^4(x)-16 \log ^3(x)-48 \log ^2(x)-64 \log (x)-32\right )}+e^{x+e^x}+\frac {8 x \left (-\log ^6\left (2-e^{4 x}\right )+4 e^x \log ^5\left (2-e^{4 x}\right )+2 e^x \log (x) \log ^5\left (2-e^{4 x}\right )-12 e^{2 x} \log ^4\left (2-e^{4 x}\right )-3 e^{2 x} \log ^2(x) \log ^4\left (2-e^{4 x}\right )-12 e^{2 x} \log (x) \log ^4\left (2-e^{4 x}\right )+32 e^{3 x} \log ^3\left (2-e^{4 x}\right )+4 e^{3 x} \log ^3(x) \log ^3\left (2-e^{4 x}\right )+24 e^{3 x} \log ^2(x) \log ^3\left (2-e^{4 x}\right )+48 e^{3 x} \log (x) \log ^3\left (2-e^{4 x}\right )-6 \log ^4(x) \log ^2\left (2-e^{4 x}\right )-48 \log ^3(x) \log ^2\left (2-e^{4 x}\right )-144 \log ^2(x) \log ^2\left (2-e^{4 x}\right )-192 \log (x) \log ^2\left (2-e^{4 x}\right )-96 \log ^2\left (2-e^{4 x}\right )+128 e^x \log \left (2-e^{4 x}\right )+4 e^x \log ^5(x) \log \left (2-e^{4 x}\right )+40 e^x \log ^4(x) \log \left (2-e^{4 x}\right )+160 e^x \log ^3(x) \log \left (2-e^{4 x}\right )+320 e^x \log ^2(x) \log \left (2-e^{4 x}\right )+320 e^x \log (x) \log \left (2-e^{4 x}\right )-128 e^{2 x}-2 e^{2 x} \log ^6(x)-24 e^{2 x} \log ^5(x)-120 e^{2 x} \log ^4(x)-320 e^{2 x} \log ^3(x)-480 e^{2 x} \log ^2(x)-384 e^{2 x} \log (x)\right )}{\left (-2+e^{4 x}\right ) \left (\log ^4\left (2-e^{4 x}\right )-2 \log ^4(x)-16 \log ^3(x)-48 \log ^2(x)-64 \log (x)-32\right )^2}+\frac {-4 x \log ^9(x)+4 \log ^9(x)-72 x \log ^8(x)+68 \log ^8(x)-576 x \log ^7(x)+512 \log ^7(x)-2688 x \log ^6(x)+2240 \log ^6(x)+4 x \log ^4\left (2-e^{4 x}\right ) \log ^5(x)-4 \log ^4\left (2-e^{4 x}\right ) \log ^5(x)-32 x \log ^3\left (2-e^{4 x}\right ) \log ^5(x)-8064 x \log ^5(x)+6272 \log ^5(x)+40 x \log ^4\left (2-e^{4 x}\right ) \log ^4(x)-36 \log ^4\left (2-e^{4 x}\right ) \log ^4(x)-320 x \log ^3\left (2-e^{4 x}\right ) \log ^4(x)-16128 x \log ^4(x)+11648 \log ^4(x)+160 x \log ^4\left (2-e^{4 x}\right ) \log ^3(x)-128 \log ^4\left (2-e^{4 x}\right ) \log ^3(x)-1280 x \log ^3\left (2-e^{4 x}\right ) \log ^3(x)-21504 x \log ^3(x)+14336 \log ^3(x)+320 x \log ^4\left (2-e^{4 x}\right ) \log ^2(x)-224 \log ^4\left (2-e^{4 x}\right ) \log ^2(x)-2560 x \log ^3\left (2-e^{4 x}\right ) \log ^2(x)-18432 x \log ^2(x)+11264 \log ^2(x)-x \log ^8\left (2-e^{4 x}\right ) \log (x)+\log ^8\left (2-e^{4 x}\right ) \log (x)+320 x \log ^4\left (2-e^{4 x}\right ) \log (x)-192 \log ^4\left (2-e^{4 x}\right ) \log (x)-2560 x \log ^3\left (2-e^{4 x}\right ) \log (x)-9216 x \log (x)+5120 \log (x)-2 x \log ^8\left (2-e^{4 x}\right )+\log ^8\left (2-e^{4 x}\right )+128 x \log ^4\left (2-e^{4 x}\right )-64 \log ^4\left (2-e^{4 x}\right )-1024 x \log ^3\left (2-e^{4 x}\right )-2048 x+1024}{(\log (x)+2) \left (\log \left (2-e^{4 x}\right )+2 e^x+e^x \log (x)\right ) \left (-\log ^4\left (2-e^{4 x}\right )+2 \log ^4(x)+16 \log ^3(x)+48 \log ^2(x)+64 \log (x)+32\right )^2}+1\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-\left (\left (e^{4 x}-2\right ) \left (e^{x+e^x}+1\right ) \log ^2\left (2-e^{4 x}\right )\right )-e^x \left (-8 e^x+4 e^{5 x}-8 e^{2 x+e^x}+4 e^{6 x+e^x}+e^{4 x} (1-2 x)-4 e^{3 x} x+4 x+e^x \left (e^{4 x}-2\right ) \left (e^{x+e^x}+1\right ) \log ^2(x)+\left (e^{4 x}-2\right ) \left (-x+4 e^x+4 e^{2 x+e^x}+1\right ) \log (x)-2\right )-\left (e^{4 x}-2\right ) \left (4 e^x+4 e^{2 x+e^x}+2 e^x \left (e^{x+e^x}+1\right ) \log (x)+1\right ) \log \left (2-e^{4 x}\right )}{\left (2-e^{4 x}\right ) \left (\log \left (2-e^{4 x}\right )+e^x (\log (x)+2)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\log \left (2-e^{4 x}\right ) \left (-2 x \log ^5(x)-20 x \log ^4(x)-2 \log ^4(x)-80 x \log ^3(x)-16 \log ^3(x)-160 x \log ^2(x)-48 \log ^2(x)+x \log ^4\left (2-e^{4 x}\right ) \log (x)-4 x \log ^3\left (2-e^{4 x}\right ) \log (x)-160 x \log (x)-64 \log (x)+2 x \log ^4\left (2-e^{4 x}\right )+\log ^4\left (2-e^{4 x}\right )-8 x \log ^3\left (2-e^{4 x}\right )-64 x-32\right )}{(\log (x)+2) \left (\log \left (2-e^{4 x}\right )+2 e^x+e^x \log (x)\right )^2 \left (\log ^4\left (2-e^{4 x}\right )-2 \log ^4(x)-16 \log ^3(x)-48 \log ^2(x)-64 \log (x)-32\right )}+e^{x+e^x}+\frac {8 x \left (-\log ^6\left (2-e^{4 x}\right )+4 e^x \log ^5\left (2-e^{4 x}\right )+2 e^x \log (x) \log ^5\left (2-e^{4 x}\right )-12 e^{2 x} \log ^4\left (2-e^{4 x}\right )-3 e^{2 x} \log ^2(x) \log ^4\left (2-e^{4 x}\right )-12 e^{2 x} \log (x) \log ^4\left (2-e^{4 x}\right )+32 e^{3 x} \log ^3\left (2-e^{4 x}\right )+4 e^{3 x} \log ^3(x) \log ^3\left (2-e^{4 x}\right )+24 e^{3 x} \log ^2(x) \log ^3\left (2-e^{4 x}\right )+48 e^{3 x} \log (x) \log ^3\left (2-e^{4 x}\right )-6 \log ^4(x) \log ^2\left (2-e^{4 x}\right )-48 \log ^3(x) \log ^2\left (2-e^{4 x}\right )-144 \log ^2(x) \log ^2\left (2-e^{4 x}\right )-192 \log (x) \log ^2\left (2-e^{4 x}\right )-96 \log ^2\left (2-e^{4 x}\right )+128 e^x \log \left (2-e^{4 x}\right )+4 e^x \log ^5(x) \log \left (2-e^{4 x}\right )+40 e^x \log ^4(x) \log \left (2-e^{4 x}\right )+160 e^x \log ^3(x) \log \left (2-e^{4 x}\right )+320 e^x \log ^2(x) \log \left (2-e^{4 x}\right )+320 e^x \log (x) \log \left (2-e^{4 x}\right )-128 e^{2 x}-2 e^{2 x} \log ^6(x)-24 e^{2 x} \log ^5(x)-120 e^{2 x} \log ^4(x)-320 e^{2 x} \log ^3(x)-480 e^{2 x} \log ^2(x)-384 e^{2 x} \log (x)\right )}{\left (-2+e^{4 x}\right ) \left (\log ^4\left (2-e^{4 x}\right )-2 \log ^4(x)-16 \log ^3(x)-48 \log ^2(x)-64 \log (x)-32\right )^2}+\frac {-4 x \log ^9(x)+4 \log ^9(x)-72 x \log ^8(x)+68 \log ^8(x)-576 x \log ^7(x)+512 \log ^7(x)-2688 x \log ^6(x)+2240 \log ^6(x)+4 x \log ^4\left (2-e^{4 x}\right ) \log ^5(x)-4 \log ^4\left (2-e^{4 x}\right ) \log ^5(x)-32 x \log ^3\left (2-e^{4 x}\right ) \log ^5(x)-8064 x \log ^5(x)+6272 \log ^5(x)+40 x \log ^4\left (2-e^{4 x}\right ) \log ^4(x)-36 \log ^4\left (2-e^{4 x}\right ) \log ^4(x)-320 x \log ^3\left (2-e^{4 x}\right ) \log ^4(x)-16128 x \log ^4(x)+11648 \log ^4(x)+160 x \log ^4\left (2-e^{4 x}\right ) \log ^3(x)-128 \log ^4\left (2-e^{4 x}\right ) \log ^3(x)-1280 x \log ^3\left (2-e^{4 x}\right ) \log ^3(x)-21504 x \log ^3(x)+14336 \log ^3(x)+320 x \log ^4\left (2-e^{4 x}\right ) \log ^2(x)-224 \log ^4\left (2-e^{4 x}\right ) \log ^2(x)-2560 x \log ^3\left (2-e^{4 x}\right ) \log ^2(x)-18432 x \log ^2(x)+11264 \log ^2(x)-x \log ^8\left (2-e^{4 x}\right ) \log (x)+\log ^8\left (2-e^{4 x}\right ) \log (x)+320 x \log ^4\left (2-e^{4 x}\right ) \log (x)-192 \log ^4\left (2-e^{4 x}\right ) \log (x)-2560 x \log ^3\left (2-e^{4 x}\right ) \log (x)-9216 x \log (x)+5120 \log (x)-2 x \log ^8\left (2-e^{4 x}\right )+\log ^8\left (2-e^{4 x}\right )+128 x \log ^4\left (2-e^{4 x}\right )-64 \log ^4\left (2-e^{4 x}\right )-1024 x \log ^3\left (2-e^{4 x}\right )-2048 x+1024}{(\log (x)+2) \left (\log \left (2-e^{4 x}\right )+2 e^x+e^x \log (x)\right ) \left (-\log ^4\left (2-e^{4 x}\right )+2 \log ^4(x)+16 \log ^3(x)+48 \log ^2(x)+64 \log (x)+32\right )^2}+1\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-\left (\left (e^{4 x}-2\right ) \left (e^{x+e^x}+1\right ) \log ^2\left (2-e^{4 x}\right )\right )-e^x \left (-8 e^x+4 e^{5 x}-8 e^{2 x+e^x}+4 e^{6 x+e^x}+e^{4 x} (1-2 x)-4 e^{3 x} x+4 x+e^x \left (e^{4 x}-2\right ) \left (e^{x+e^x}+1\right ) \log ^2(x)+\left (e^{4 x}-2\right ) \left (-x+4 e^x+4 e^{2 x+e^x}+1\right ) \log (x)-2\right )-\left (e^{4 x}-2\right ) \left (4 e^x+4 e^{2 x+e^x}+2 e^x \left (e^{x+e^x}+1\right ) \log (x)+1\right ) \log \left (2-e^{4 x}\right )}{\left (2-e^{4 x}\right ) \left (\log \left (2-e^{4 x}\right )+e^x (\log (x)+2)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\log \left (2-e^{4 x}\right ) \left (-2 x \log ^5(x)-20 x \log ^4(x)-2 \log ^4(x)-80 x \log ^3(x)-16 \log ^3(x)-160 x \log ^2(x)-48 \log ^2(x)+x \log ^4\left (2-e^{4 x}\right ) \log (x)-4 x \log ^3\left (2-e^{4 x}\right ) \log (x)-160 x \log (x)-64 \log (x)+2 x \log ^4\left (2-e^{4 x}\right )+\log ^4\left (2-e^{4 x}\right )-8 x \log ^3\left (2-e^{4 x}\right )-64 x-32\right )}{(\log (x)+2) \left (\log \left (2-e^{4 x}\right )+2 e^x+e^x \log (x)\right )^2 \left (\log ^4\left (2-e^{4 x}\right )-2 \log ^4(x)-16 \log ^3(x)-48 \log ^2(x)-64 \log (x)-32\right )}+e^{x+e^x}+\frac {8 x \left (-\log ^6\left (2-e^{4 x}\right )+4 e^x \log ^5\left (2-e^{4 x}\right )+2 e^x \log (x) \log ^5\left (2-e^{4 x}\right )-12 e^{2 x} \log ^4\left (2-e^{4 x}\right )-3 e^{2 x} \log ^2(x) \log ^4\left (2-e^{4 x}\right )-12 e^{2 x} \log (x) \log ^4\left (2-e^{4 x}\right )+32 e^{3 x} \log ^3\left (2-e^{4 x}\right )+4 e^{3 x} \log ^3(x) \log ^3\left (2-e^{4 x}\right )+24 e^{3 x} \log ^2(x) \log ^3\left (2-e^{4 x}\right )+48 e^{3 x} \log (x) \log ^3\left (2-e^{4 x}\right )-6 \log ^4(x) \log ^2\left (2-e^{4 x}\right )-48 \log ^3(x) \log ^2\left (2-e^{4 x}\right )-144 \log ^2(x) \log ^2\left (2-e^{4 x}\right )-192 \log (x) \log ^2\left (2-e^{4 x}\right )-96 \log ^2\left (2-e^{4 x}\right )+128 e^x \log \left (2-e^{4 x}\right )+4 e^x \log ^5(x) \log \left (2-e^{4 x}\right )+40 e^x \log ^4(x) \log \left (2-e^{4 x}\right )+160 e^x \log ^3(x) \log \left (2-e^{4 x}\right )+320 e^x \log ^2(x) \log \left (2-e^{4 x}\right )+320 e^x \log (x) \log \left (2-e^{4 x}\right )-128 e^{2 x}-2 e^{2 x} \log ^6(x)-24 e^{2 x} \log ^5(x)-120 e^{2 x} \log ^4(x)-320 e^{2 x} \log ^3(x)-480 e^{2 x} \log ^2(x)-384 e^{2 x} \log (x)\right )}{\left (-2+e^{4 x}\right ) \left (\log ^4\left (2-e^{4 x}\right )-2 \log ^4(x)-16 \log ^3(x)-48 \log ^2(x)-64 \log (x)-32\right )^2}+\frac {-4 x \log ^9(x)+4 \log ^9(x)-72 x \log ^8(x)+68 \log ^8(x)-576 x \log ^7(x)+512 \log ^7(x)-2688 x \log ^6(x)+2240 \log ^6(x)+4 x \log ^4\left (2-e^{4 x}\right ) \log ^5(x)-4 \log ^4\left (2-e^{4 x}\right ) \log ^5(x)-32 x \log ^3\left (2-e^{4 x}\right ) \log ^5(x)-8064 x \log ^5(x)+6272 \log ^5(x)+40 x \log ^4\left (2-e^{4 x}\right ) \log ^4(x)-36 \log ^4\left (2-e^{4 x}\right ) \log ^4(x)-320 x \log ^3\left (2-e^{4 x}\right ) \log ^4(x)-16128 x \log ^4(x)+11648 \log ^4(x)+160 x \log ^4\left (2-e^{4 x}\right ) \log ^3(x)-128 \log ^4\left (2-e^{4 x}\right ) \log ^3(x)-1280 x \log ^3\left (2-e^{4 x}\right ) \log ^3(x)-21504 x \log ^3(x)+14336 \log ^3(x)+320 x \log ^4\left (2-e^{4 x}\right ) \log ^2(x)-224 \log ^4\left (2-e^{4 x}\right ) \log ^2(x)-2560 x \log ^3\left (2-e^{4 x}\right ) \log ^2(x)-18432 x \log ^2(x)+11264 \log ^2(x)-x \log ^8\left (2-e^{4 x}\right ) \log (x)+\log ^8\left (2-e^{4 x}\right ) \log (x)+320 x \log ^4\left (2-e^{4 x}\right ) \log (x)-192 \log ^4\left (2-e^{4 x}\right ) \log (x)-2560 x \log ^3\left (2-e^{4 x}\right ) \log (x)-9216 x \log (x)+5120 \log (x)-2 x \log ^8\left (2-e^{4 x}\right )+\log ^8\left (2-e^{4 x}\right )+128 x \log ^4\left (2-e^{4 x}\right )-64 \log ^4\left (2-e^{4 x}\right )-1024 x \log ^3\left (2-e^{4 x}\right )-2048 x+1024}{(\log (x)+2) \left (\log \left (2-e^{4 x}\right )+2 e^x+e^x \log (x)\right ) \left (-\log ^4\left (2-e^{4 x}\right )+2 \log ^4(x)+16 \log ^3(x)+48 \log ^2(x)+64 \log (x)+32\right )^2}+1\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-\left (\left (e^{4 x}-2\right ) \left (e^{x+e^x}+1\right ) \log ^2\left (2-e^{4 x}\right )\right )-e^x \left (-8 e^x+4 e^{5 x}-8 e^{2 x+e^x}+4 e^{6 x+e^x}+e^{4 x} (1-2 x)-4 e^{3 x} x+4 x+e^x \left (e^{4 x}-2\right ) \left (e^{x+e^x}+1\right ) \log ^2(x)+\left (e^{4 x}-2\right ) \left (-x+4 e^x+4 e^{2 x+e^x}+1\right ) \log (x)-2\right )-\left (e^{4 x}-2\right ) \left (4 e^x+4 e^{2 x+e^x}+2 e^x \left (e^{x+e^x}+1\right ) \log (x)+1\right ) \log \left (2-e^{4 x}\right )}{\left (2-e^{4 x}\right ) \left (\log \left (2-e^{4 x}\right )+e^x (\log (x)+2)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\log \left (2-e^{4 x}\right ) \left (-2 x \log ^5(x)-20 x \log ^4(x)-2 \log ^4(x)-80 x \log ^3(x)-16 \log ^3(x)-160 x \log ^2(x)-48 \log ^2(x)+x \log ^4\left (2-e^{4 x}\right ) \log (x)-4 x \log ^3\left (2-e^{4 x}\right ) \log (x)-160 x \log (x)-64 \log (x)+2 x \log ^4\left (2-e^{4 x}\right )+\log ^4\left (2-e^{4 x}\right )-8 x \log ^3\left (2-e^{4 x}\right )-64 x-32\right )}{(\log (x)+2) \left (\log \left (2-e^{4 x}\right )+2 e^x+e^x \log (x)\right )^2 \left (\log ^4\left (2-e^{4 x}\right )-2 \log ^4(x)-16 \log ^3(x)-48 \log ^2(x)-64 \log (x)-32\right )}+e^{x+e^x}+\frac {8 x \left (-\log ^6\left (2-e^{4 x}\right )+4 e^x \log ^5\left (2-e^{4 x}\right )+2 e^x \log (x) \log ^5\left (2-e^{4 x}\right )-12 e^{2 x} \log ^4\left (2-e^{4 x}\right )-3 e^{2 x} \log ^2(x) \log ^4\left (2-e^{4 x}\right )-12 e^{2 x} \log (x) \log ^4\left (2-e^{4 x}\right )+32 e^{3 x} \log ^3\left (2-e^{4 x}\right )+4 e^{3 x} \log ^3(x) \log ^3\left (2-e^{4 x}\right )+24 e^{3 x} \log ^2(x) \log ^3\left (2-e^{4 x}\right )+48 e^{3 x} \log (x) \log ^3\left (2-e^{4 x}\right )-6 \log ^4(x) \log ^2\left (2-e^{4 x}\right )-48 \log ^3(x) \log ^2\left (2-e^{4 x}\right )-144 \log ^2(x) \log ^2\left (2-e^{4 x}\right )-192 \log (x) \log ^2\left (2-e^{4 x}\right )-96 \log ^2\left (2-e^{4 x}\right )+128 e^x \log \left (2-e^{4 x}\right )+4 e^x \log ^5(x) \log \left (2-e^{4 x}\right )+40 e^x \log ^4(x) \log \left (2-e^{4 x}\right )+160 e^x \log ^3(x) \log \left (2-e^{4 x}\right )+320 e^x \log ^2(x) \log \left (2-e^{4 x}\right )+320 e^x \log (x) \log \left (2-e^{4 x}\right )-128 e^{2 x}-2 e^{2 x} \log ^6(x)-24 e^{2 x} \log ^5(x)-120 e^{2 x} \log ^4(x)-320 e^{2 x} \log ^3(x)-480 e^{2 x} \log ^2(x)-384 e^{2 x} \log (x)\right )}{\left (-2+e^{4 x}\right ) \left (\log ^4\left (2-e^{4 x}\right )-2 \log ^4(x)-16 \log ^3(x)-48 \log ^2(x)-64 \log (x)-32\right )^2}+\frac {-4 x \log ^9(x)+4 \log ^9(x)-72 x \log ^8(x)+68 \log ^8(x)-576 x \log ^7(x)+512 \log ^7(x)-2688 x \log ^6(x)+2240 \log ^6(x)+4 x \log ^4\left (2-e^{4 x}\right ) \log ^5(x)-4 \log ^4\left (2-e^{4 x}\right ) \log ^5(x)-32 x \log ^3\left (2-e^{4 x}\right ) \log ^5(x)-8064 x \log ^5(x)+6272 \log ^5(x)+40 x \log ^4\left (2-e^{4 x}\right ) \log ^4(x)-36 \log ^4\left (2-e^{4 x}\right ) \log ^4(x)-320 x \log ^3\left (2-e^{4 x}\right ) \log ^4(x)-16128 x \log ^4(x)+11648 \log ^4(x)+160 x \log ^4\left (2-e^{4 x}\right ) \log ^3(x)-128 \log ^4\left (2-e^{4 x}\right ) \log ^3(x)-1280 x \log ^3\left (2-e^{4 x}\right ) \log ^3(x)-21504 x \log ^3(x)+14336 \log ^3(x)+320 x \log ^4\left (2-e^{4 x}\right ) \log ^2(x)-224 \log ^4\left (2-e^{4 x}\right ) \log ^2(x)-2560 x \log ^3\left (2-e^{4 x}\right ) \log ^2(x)-18432 x \log ^2(x)+11264 \log ^2(x)-x \log ^8\left (2-e^{4 x}\right ) \log (x)+\log ^8\left (2-e^{4 x}\right ) \log (x)+320 x \log ^4\left (2-e^{4 x}\right ) \log (x)-192 \log ^4\left (2-e^{4 x}\right ) \log (x)-2560 x \log ^3\left (2-e^{4 x}\right ) \log (x)-9216 x \log (x)+5120 \log (x)-2 x \log ^8\left (2-e^{4 x}\right )+\log ^8\left (2-e^{4 x}\right )+128 x \log ^4\left (2-e^{4 x}\right )-64 \log ^4\left (2-e^{4 x}\right )-1024 x \log ^3\left (2-e^{4 x}\right )-2048 x+1024}{(\log (x)+2) \left (\log \left (2-e^{4 x}\right )+2 e^x+e^x \log (x)\right ) \left (-\log ^4\left (2-e^{4 x}\right )+2 \log ^4(x)+16 \log ^3(x)+48 \log ^2(x)+64 \log (x)+32\right )^2}+1\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-\left (\left (e^{4 x}-2\right ) \left (e^{x+e^x}+1\right ) \log ^2\left (2-e^{4 x}\right )\right )-e^x \left (-8 e^x+4 e^{5 x}-8 e^{2 x+e^x}+4 e^{6 x+e^x}+e^{4 x} (1-2 x)-4 e^{3 x} x+4 x+e^x \left (e^{4 x}-2\right ) \left (e^{x+e^x}+1\right ) \log ^2(x)+\left (e^{4 x}-2\right ) \left (-x+4 e^x+4 e^{2 x+e^x}+1\right ) \log (x)-2\right )-\left (e^{4 x}-2\right ) \left (4 e^x+4 e^{2 x+e^x}+2 e^x \left (e^{x+e^x}+1\right ) \log (x)+1\right ) \log \left (2-e^{4 x}\right )}{\left (2-e^{4 x}\right ) \left (\log \left (2-e^{4 x}\right )+e^x (\log (x)+2)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\log \left (2-e^{4 x}\right ) \left (-2 x \log ^5(x)-20 x \log ^4(x)-2 \log ^4(x)-80 x \log ^3(x)-16 \log ^3(x)-160 x \log ^2(x)-48 \log ^2(x)+x \log ^4\left (2-e^{4 x}\right ) \log (x)-4 x \log ^3\left (2-e^{4 x}\right ) \log (x)-160 x \log (x)-64 \log (x)+2 x \log ^4\left (2-e^{4 x}\right )+\log ^4\left (2-e^{4 x}\right )-8 x \log ^3\left (2-e^{4 x}\right )-64 x-32\right )}{(\log (x)+2) \left (\log \left (2-e^{4 x}\right )+2 e^x+e^x \log (x)\right )^2 \left (\log ^4\left (2-e^{4 x}\right )-2 \log ^4(x)-16 \log ^3(x)-48 \log ^2(x)-64 \log (x)-32\right )}+e^{x+e^x}+\frac {8 x \left (-\log ^6\left (2-e^{4 x}\right )+4 e^x \log ^5\left (2-e^{4 x}\right )+2 e^x \log (x) \log ^5\left (2-e^{4 x}\right )-12 e^{2 x} \log ^4\left (2-e^{4 x}\right )-3 e^{2 x} \log ^2(x) \log ^4\left (2-e^{4 x}\right )-12 e^{2 x} \log (x) \log ^4\left (2-e^{4 x}\right )+32 e^{3 x} \log ^3\left (2-e^{4 x}\right )+4 e^{3 x} \log ^3(x) \log ^3\left (2-e^{4 x}\right )+24 e^{3 x} \log ^2(x) \log ^3\left (2-e^{4 x}\right )+48 e^{3 x} \log (x) \log ^3\left (2-e^{4 x}\right )-6 \log ^4(x) \log ^2\left (2-e^{4 x}\right )-48 \log ^3(x) \log ^2\left (2-e^{4 x}\right )-144 \log ^2(x) \log ^2\left (2-e^{4 x}\right )-192 \log (x) \log ^2\left (2-e^{4 x}\right )-96 \log ^2\left (2-e^{4 x}\right )+128 e^x \log \left (2-e^{4 x}\right )+4 e^x \log ^5(x) \log \left (2-e^{4 x}\right )+40 e^x \log ^4(x) \log \left (2-e^{4 x}\right )+160 e^x \log ^3(x) \log \left (2-e^{4 x}\right )+320 e^x \log ^2(x) \log \left (2-e^{4 x}\right )+320 e^x \log (x) \log \left (2-e^{4 x}\right )-128 e^{2 x}-2 e^{2 x} \log ^6(x)-24 e^{2 x} \log ^5(x)-120 e^{2 x} \log ^4(x)-320 e^{2 x} \log ^3(x)-480 e^{2 x} \log ^2(x)-384 e^{2 x} \log (x)\right )}{\left (-2+e^{4 x}\right ) \left (\log ^4\left (2-e^{4 x}\right )-2 \log ^4(x)-16 \log ^3(x)-48 \log ^2(x)-64 \log (x)-32\right )^2}+\frac {-4 x \log ^9(x)+4 \log ^9(x)-72 x \log ^8(x)+68 \log ^8(x)-576 x \log ^7(x)+512 \log ^7(x)-2688 x \log ^6(x)+2240 \log ^6(x)+4 x \log ^4\left (2-e^{4 x}\right ) \log ^5(x)-4 \log ^4\left (2-e^{4 x}\right ) \log ^5(x)-32 x \log ^3\left (2-e^{4 x}\right ) \log ^5(x)-8064 x \log ^5(x)+6272 \log ^5(x)+40 x \log ^4\left (2-e^{4 x}\right ) \log ^4(x)-36 \log ^4\left (2-e^{4 x}\right ) \log ^4(x)-320 x \log ^3\left (2-e^{4 x}\right ) \log ^4(x)-16128 x \log ^4(x)+11648 \log ^4(x)+160 x \log ^4\left (2-e^{4 x}\right ) \log ^3(x)-128 \log ^4\left (2-e^{4 x}\right ) \log ^3(x)-1280 x \log ^3\left (2-e^{4 x}\right ) \log ^3(x)-21504 x \log ^3(x)+14336 \log ^3(x)+320 x \log ^4\left (2-e^{4 x}\right ) \log ^2(x)-224 \log ^4\left (2-e^{4 x}\right ) \log ^2(x)-2560 x \log ^3\left (2-e^{4 x}\right ) \log ^2(x)-18432 x \log ^2(x)+11264 \log ^2(x)-x \log ^8\left (2-e^{4 x}\right ) \log (x)+\log ^8\left (2-e^{4 x}\right ) \log (x)+320 x \log ^4\left (2-e^{4 x}\right ) \log (x)-192 \log ^4\left (2-e^{4 x}\right ) \log (x)-2560 x \log ^3\left (2-e^{4 x}\right ) \log (x)-9216 x \log (x)+5120 \log (x)-2 x \log ^8\left (2-e^{4 x}\right )+\log ^8\left (2-e^{4 x}\right )+128 x \log ^4\left (2-e^{4 x}\right )-64 \log ^4\left (2-e^{4 x}\right )-1024 x \log ^3\left (2-e^{4 x}\right )-2048 x+1024}{(\log (x)+2) \left (\log \left (2-e^{4 x}\right )+2 e^x+e^x \log (x)\right ) \left (-\log ^4\left (2-e^{4 x}\right )+2 \log ^4(x)+16 \log ^3(x)+48 \log ^2(x)+64 \log (x)+32\right )^2}+1\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-\left (\left (e^{4 x}-2\right ) \left (e^{x+e^x}+1\right ) \log ^2\left (2-e^{4 x}\right )\right )-e^x \left (-8 e^x+4 e^{5 x}-8 e^{2 x+e^x}+4 e^{6 x+e^x}+e^{4 x} (1-2 x)-4 e^{3 x} x+4 x+e^x \left (e^{4 x}-2\right ) \left (e^{x+e^x}+1\right ) \log ^2(x)+\left (e^{4 x}-2\right ) \left (-x+4 e^x+4 e^{2 x+e^x}+1\right ) \log (x)-2\right )-\left (e^{4 x}-2\right ) \left (4 e^x+4 e^{2 x+e^x}+2 e^x \left (e^{x+e^x}+1\right ) \log (x)+1\right ) \log \left (2-e^{4 x}\right )}{\left (2-e^{4 x}\right ) \left (\log \left (2-e^{4 x}\right )+e^x (\log (x)+2)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\log \left (2-e^{4 x}\right ) \left (-2 x \log ^5(x)-20 x \log ^4(x)-2 \log ^4(x)-80 x \log ^3(x)-16 \log ^3(x)-160 x \log ^2(x)-48 \log ^2(x)+x \log ^4\left (2-e^{4 x}\right ) \log (x)-4 x \log ^3\left (2-e^{4 x}\right ) \log (x)-160 x \log (x)-64 \log (x)+2 x \log ^4\left (2-e^{4 x}\right )+\log ^4\left (2-e^{4 x}\right )-8 x \log ^3\left (2-e^{4 x}\right )-64 x-32\right )}{(\log (x)+2) \left (\log \left (2-e^{4 x}\right )+2 e^x+e^x \log (x)\right )^2 \left (\log ^4\left (2-e^{4 x}\right )-2 \log ^4(x)-16 \log ^3(x)-48 \log ^2(x)-64 \log (x)-32\right )}+e^{x+e^x}+\frac {8 x \left (-\log ^6\left (2-e^{4 x}\right )+4 e^x \log ^5\left (2-e^{4 x}\right )+2 e^x \log (x) \log ^5\left (2-e^{4 x}\right )-12 e^{2 x} \log ^4\left (2-e^{4 x}\right )-3 e^{2 x} \log ^2(x) \log ^4\left (2-e^{4 x}\right )-12 e^{2 x} \log (x) \log ^4\left (2-e^{4 x}\right )+32 e^{3 x} \log ^3\left (2-e^{4 x}\right )+4 e^{3 x} \log ^3(x) \log ^3\left (2-e^{4 x}\right )+24 e^{3 x} \log ^2(x) \log ^3\left (2-e^{4 x}\right )+48 e^{3 x} \log (x) \log ^3\left (2-e^{4 x}\right )-6 \log ^4(x) \log ^2\left (2-e^{4 x}\right )-48 \log ^3(x) \log ^2\left (2-e^{4 x}\right )-144 \log ^2(x) \log ^2\left (2-e^{4 x}\right )-192 \log (x) \log ^2\left (2-e^{4 x}\right )-96 \log ^2\left (2-e^{4 x}\right )+128 e^x \log \left (2-e^{4 x}\right )+4 e^x \log ^5(x) \log \left (2-e^{4 x}\right )+40 e^x \log ^4(x) \log \left (2-e^{4 x}\right )+160 e^x \log ^3(x) \log \left (2-e^{4 x}\right )+320 e^x \log ^2(x) \log \left (2-e^{4 x}\right )+320 e^x \log (x) \log \left (2-e^{4 x}\right )-128 e^{2 x}-2 e^{2 x} \log ^6(x)-24 e^{2 x} \log ^5(x)-120 e^{2 x} \log ^4(x)-320 e^{2 x} \log ^3(x)-480 e^{2 x} \log ^2(x)-384 e^{2 x} \log (x)\right )}{\left (-2+e^{4 x}\right ) \left (\log ^4\left (2-e^{4 x}\right )-2 \log ^4(x)-16 \log ^3(x)-48 \log ^2(x)-64 \log (x)-32\right )^2}+\frac {-4 x \log ^9(x)+4 \log ^9(x)-72 x \log ^8(x)+68 \log ^8(x)-576 x \log ^7(x)+512 \log ^7(x)-2688 x \log ^6(x)+2240 \log ^6(x)+4 x \log ^4\left (2-e^{4 x}\right ) \log ^5(x)-4 \log ^4\left (2-e^{4 x}\right ) \log ^5(x)-32 x \log ^3\left (2-e^{4 x}\right ) \log ^5(x)-8064 x \log ^5(x)+6272 \log ^5(x)+40 x \log ^4\left (2-e^{4 x}\right ) \log ^4(x)-36 \log ^4\left (2-e^{4 x}\right ) \log ^4(x)-320 x \log ^3\left (2-e^{4 x}\right ) \log ^4(x)-16128 x \log ^4(x)+11648 \log ^4(x)+160 x \log ^4\left (2-e^{4 x}\right ) \log ^3(x)-128 \log ^4\left (2-e^{4 x}\right ) \log ^3(x)-1280 x \log ^3\left (2-e^{4 x}\right ) \log ^3(x)-21504 x \log ^3(x)+14336 \log ^3(x)+320 x \log ^4\left (2-e^{4 x}\right ) \log ^2(x)-224 \log ^4\left (2-e^{4 x}\right ) \log ^2(x)-2560 x \log ^3\left (2-e^{4 x}\right ) \log ^2(x)-18432 x \log ^2(x)+11264 \log ^2(x)-x \log ^8\left (2-e^{4 x}\right ) \log (x)+\log ^8\left (2-e^{4 x}\right ) \log (x)+320 x \log ^4\left (2-e^{4 x}\right ) \log (x)-192 \log ^4\left (2-e^{4 x}\right ) \log (x)-2560 x \log ^3\left (2-e^{4 x}\right ) \log (x)-9216 x \log (x)+5120 \log (x)-2 x \log ^8\left (2-e^{4 x}\right )+\log ^8\left (2-e^{4 x}\right )+128 x \log ^4\left (2-e^{4 x}\right )-64 \log ^4\left (2-e^{4 x}\right )-1024 x \log ^3\left (2-e^{4 x}\right )-2048 x+1024}{(\log (x)+2) \left (\log \left (2-e^{4 x}\right )+2 e^x+e^x \log (x)\right ) \left (-\log ^4\left (2-e^{4 x}\right )+2 \log ^4(x)+16 \log ^3(x)+48 \log ^2(x)+64 \log (x)+32\right )^2}+1\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-\left (\left (e^{4 x}-2\right ) \left (e^{x+e^x}+1\right ) \log ^2\left (2-e^{4 x}\right )\right )-e^x \left (-8 e^x+4 e^{5 x}-8 e^{2 x+e^x}+4 e^{6 x+e^x}+e^{4 x} (1-2 x)-4 e^{3 x} x+4 x+e^x \left (e^{4 x}-2\right ) \left (e^{x+e^x}+1\right ) \log ^2(x)+\left (e^{4 x}-2\right ) \left (-x+4 e^x+4 e^{2 x+e^x}+1\right ) \log (x)-2\right )-\left (e^{4 x}-2\right ) \left (4 e^x+4 e^{2 x+e^x}+2 e^x \left (e^{x+e^x}+1\right ) \log (x)+1\right ) \log \left (2-e^{4 x}\right )}{\left (2-e^{4 x}\right ) \left (\log \left (2-e^{4 x}\right )+e^x (\log (x)+2)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\log \left (2-e^{4 x}\right ) \left (-2 x \log ^5(x)-20 x \log ^4(x)-2 \log ^4(x)-80 x \log ^3(x)-16 \log ^3(x)-160 x \log ^2(x)-48 \log ^2(x)+x \log ^4\left (2-e^{4 x}\right ) \log (x)-4 x \log ^3\left (2-e^{4 x}\right ) \log (x)-160 x \log (x)-64 \log (x)+2 x \log ^4\left (2-e^{4 x}\right )+\log ^4\left (2-e^{4 x}\right )-8 x \log ^3\left (2-e^{4 x}\right )-64 x-32\right )}{(\log (x)+2) \left (\log \left (2-e^{4 x}\right )+2 e^x+e^x \log (x)\right )^2 \left (\log ^4\left (2-e^{4 x}\right )-2 \log ^4(x)-16 \log ^3(x)-48 \log ^2(x)-64 \log (x)-32\right )}+e^{x+e^x}+\frac {8 x \left (-\log ^6\left (2-e^{4 x}\right )+4 e^x \log ^5\left (2-e^{4 x}\right )+2 e^x \log (x) \log ^5\left (2-e^{4 x}\right )-12 e^{2 x} \log ^4\left (2-e^{4 x}\right )-3 e^{2 x} \log ^2(x) \log ^4\left (2-e^{4 x}\right )-12 e^{2 x} \log (x) \log ^4\left (2-e^{4 x}\right )+32 e^{3 x} \log ^3\left (2-e^{4 x}\right )+4 e^{3 x} \log ^3(x) \log ^3\left (2-e^{4 x}\right )+24 e^{3 x} \log ^2(x) \log ^3\left (2-e^{4 x}\right )+48 e^{3 x} \log (x) \log ^3\left (2-e^{4 x}\right )-6 \log ^4(x) \log ^2\left (2-e^{4 x}\right )-48 \log ^3(x) \log ^2\left (2-e^{4 x}\right )-144 \log ^2(x) \log ^2\left (2-e^{4 x}\right )-192 \log (x) \log ^2\left (2-e^{4 x}\right )-96 \log ^2\left (2-e^{4 x}\right )+128 e^x \log \left (2-e^{4 x}\right )+4 e^x \log ^5(x) \log \left (2-e^{4 x}\right )+40 e^x \log ^4(x) \log \left (2-e^{4 x}\right )+160 e^x \log ^3(x) \log \left (2-e^{4 x}\right )+320 e^x \log ^2(x) \log \left (2-e^{4 x}\right )+320 e^x \log (x) \log \left (2-e^{4 x}\right )-128 e^{2 x}-2 e^{2 x} \log ^6(x)-24 e^{2 x} \log ^5(x)-120 e^{2 x} \log ^4(x)-320 e^{2 x} \log ^3(x)-480 e^{2 x} \log ^2(x)-384 e^{2 x} \log (x)\right )}{\left (-2+e^{4 x}\right ) \left (\log ^4\left (2-e^{4 x}\right )-2 \log ^4(x)-16 \log ^3(x)-48 \log ^2(x)-64 \log (x)-32\right )^2}+\frac {-4 x \log ^9(x)+4 \log ^9(x)-72 x \log ^8(x)+68 \log ^8(x)-576 x \log ^7(x)+512 \log ^7(x)-2688 x \log ^6(x)+2240 \log ^6(x)+4 x \log ^4\left (2-e^{4 x}\right ) \log ^5(x)-4 \log ^4\left (2-e^{4 x}\right ) \log ^5(x)-32 x \log ^3\left (2-e^{4 x}\right ) \log ^5(x)-8064 x \log ^5(x)+6272 \log ^5(x)+40 x \log ^4\left (2-e^{4 x}\right ) \log ^4(x)-36 \log ^4\left (2-e^{4 x}\right ) \log ^4(x)-320 x \log ^3\left (2-e^{4 x}\right ) \log ^4(x)-16128 x \log ^4(x)+11648 \log ^4(x)+160 x \log ^4\left (2-e^{4 x}\right ) \log ^3(x)-128 \log ^4\left (2-e^{4 x}\right ) \log ^3(x)-1280 x \log ^3\left (2-e^{4 x}\right ) \log ^3(x)-21504 x \log ^3(x)+14336 \log ^3(x)+320 x \log ^4\left (2-e^{4 x}\right ) \log ^2(x)-224 \log ^4\left (2-e^{4 x}\right ) \log ^2(x)-2560 x \log ^3\left (2-e^{4 x}\right ) \log ^2(x)-18432 x \log ^2(x)+11264 \log ^2(x)-x \log ^8\left (2-e^{4 x}\right ) \log (x)+\log ^8\left (2-e^{4 x}\right ) \log (x)+320 x \log ^4\left (2-e^{4 x}\right ) \log (x)-192 \log ^4\left (2-e^{4 x}\right ) \log (x)-2560 x \log ^3\left (2-e^{4 x}\right ) \log (x)-9216 x \log (x)+5120 \log (x)-2 x \log ^8\left (2-e^{4 x}\right )+\log ^8\left (2-e^{4 x}\right )+128 x \log ^4\left (2-e^{4 x}\right )-64 \log ^4\left (2-e^{4 x}\right )-1024 x \log ^3\left (2-e^{4 x}\right )-2048 x+1024}{(\log (x)+2) \left (\log \left (2-e^{4 x}\right )+2 e^x+e^x \log (x)\right ) \left (-\log ^4\left (2-e^{4 x}\right )+2 \log ^4(x)+16 \log ^3(x)+48 \log ^2(x)+64 \log (x)+32\right )^2}+1\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-\left (\left (e^{4 x}-2\right ) \left (e^{x+e^x}+1\right ) \log ^2\left (2-e^{4 x}\right )\right )-e^x \left (-8 e^x+4 e^{5 x}-8 e^{2 x+e^x}+4 e^{6 x+e^x}+e^{4 x} (1-2 x)-4 e^{3 x} x+4 x+e^x \left (e^{4 x}-2\right ) \left (e^{x+e^x}+1\right ) \log ^2(x)+\left (e^{4 x}-2\right ) \left (-x+4 e^x+4 e^{2 x+e^x}+1\right ) \log (x)-2\right )-\left (e^{4 x}-2\right ) \left (4 e^x+4 e^{2 x+e^x}+2 e^x \left (e^{x+e^x}+1\right ) \log (x)+1\right ) \log \left (2-e^{4 x}\right )}{\left (2-e^{4 x}\right ) \left (\log \left (2-e^{4 x}\right )+e^x (\log (x)+2)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\log \left (2-e^{4 x}\right ) \left (-2 x \log ^5(x)-20 x \log ^4(x)-2 \log ^4(x)-80 x \log ^3(x)-16 \log ^3(x)-160 x \log ^2(x)-48 \log ^2(x)+x \log ^4\left (2-e^{4 x}\right ) \log (x)-4 x \log ^3\left (2-e^{4 x}\right ) \log (x)-160 x \log (x)-64 \log (x)+2 x \log ^4\left (2-e^{4 x}\right )+\log ^4\left (2-e^{4 x}\right )-8 x \log ^3\left (2-e^{4 x}\right )-64 x-32\right )}{(\log (x)+2) \left (\log \left (2-e^{4 x}\right )+2 e^x+e^x \log (x)\right )^2 \left (\log ^4\left (2-e^{4 x}\right )-2 \log ^4(x)-16 \log ^3(x)-48 \log ^2(x)-64 \log (x)-32\right )}+e^{x+e^x}+\frac {8 x \left (-\log ^6\left (2-e^{4 x}\right )+4 e^x \log ^5\left (2-e^{4 x}\right )+2 e^x \log (x) \log ^5\left (2-e^{4 x}\right )-12 e^{2 x} \log ^4\left (2-e^{4 x}\right )-3 e^{2 x} \log ^2(x) \log ^4\left (2-e^{4 x}\right )-12 e^{2 x} \log (x) \log ^4\left (2-e^{4 x}\right )+32 e^{3 x} \log ^3\left (2-e^{4 x}\right )+4 e^{3 x} \log ^3(x) \log ^3\left (2-e^{4 x}\right )+24 e^{3 x} \log ^2(x) \log ^3\left (2-e^{4 x}\right )+48 e^{3 x} \log (x) \log ^3\left (2-e^{4 x}\right )-6 \log ^4(x) \log ^2\left (2-e^{4 x}\right )-48 \log ^3(x) \log ^2\left (2-e^{4 x}\right )-144 \log ^2(x) \log ^2\left (2-e^{4 x}\right )-192 \log (x) \log ^2\left (2-e^{4 x}\right )-96 \log ^2\left (2-e^{4 x}\right )+128 e^x \log \left (2-e^{4 x}\right )+4 e^x \log ^5(x) \log \left (2-e^{4 x}\right )+40 e^x \log ^4(x) \log \left (2-e^{4 x}\right )+160 e^x \log ^3(x) \log \left (2-e^{4 x}\right )+320 e^x \log ^2(x) \log \left (2-e^{4 x}\right )+320 e^x \log (x) \log \left (2-e^{4 x}\right )-128 e^{2 x}-2 e^{2 x} \log ^6(x)-24 e^{2 x} \log ^5(x)-120 e^{2 x} \log ^4(x)-320 e^{2 x} \log ^3(x)-480 e^{2 x} \log ^2(x)-384 e^{2 x} \log (x)\right )}{\left (-2+e^{4 x}\right ) \left (\log ^4\left (2-e^{4 x}\right )-2 \log ^4(x)-16 \log ^3(x)-48 \log ^2(x)-64 \log (x)-32\right )^2}+\frac {-4 x \log ^9(x)+4 \log ^9(x)-72 x \log ^8(x)+68 \log ^8(x)-576 x \log ^7(x)+512 \log ^7(x)-2688 x \log ^6(x)+2240 \log ^6(x)+4 x \log ^4\left (2-e^{4 x}\right ) \log ^5(x)-4 \log ^4\left (2-e^{4 x}\right ) \log ^5(x)-32 x \log ^3\left (2-e^{4 x}\right ) \log ^5(x)-8064 x \log ^5(x)+6272 \log ^5(x)+40 x \log ^4\left (2-e^{4 x}\right ) \log ^4(x)-36 \log ^4\left (2-e^{4 x}\right ) \log ^4(x)-320 x \log ^3\left (2-e^{4 x}\right ) \log ^4(x)-16128 x \log ^4(x)+11648 \log ^4(x)+160 x \log ^4\left (2-e^{4 x}\right ) \log ^3(x)-128 \log ^4\left (2-e^{4 x}\right ) \log ^3(x)-1280 x \log ^3\left (2-e^{4 x}\right ) \log ^3(x)-21504 x \log ^3(x)+14336 \log ^3(x)+320 x \log ^4\left (2-e^{4 x}\right ) \log ^2(x)-224 \log ^4\left (2-e^{4 x}\right ) \log ^2(x)-2560 x \log ^3\left (2-e^{4 x}\right ) \log ^2(x)-18432 x \log ^2(x)+11264 \log ^2(x)-x \log ^8\left (2-e^{4 x}\right ) \log (x)+\log ^8\left (2-e^{4 x}\right ) \log (x)+320 x \log ^4\left (2-e^{4 x}\right ) \log (x)-192 \log ^4\left (2-e^{4 x}\right ) \log (x)-2560 x \log ^3\left (2-e^{4 x}\right ) \log (x)-9216 x \log (x)+5120 \log (x)-2 x \log ^8\left (2-e^{4 x}\right )+\log ^8\left (2-e^{4 x}\right )+128 x \log ^4\left (2-e^{4 x}\right )-64 \log ^4\left (2-e^{4 x}\right )-1024 x \log ^3\left (2-e^{4 x}\right )-2048 x+1024}{(\log (x)+2) \left (\log \left (2-e^{4 x}\right )+2 e^x+e^x \log (x)\right ) \left (-\log ^4\left (2-e^{4 x}\right )+2 \log ^4(x)+16 \log ^3(x)+48 \log ^2(x)+64 \log (x)+32\right )^2}+1\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-\left (\left (e^{4 x}-2\right ) \left (e^{x+e^x}+1\right ) \log ^2\left (2-e^{4 x}\right )\right )-e^x \left (-8 e^x+4 e^{5 x}-8 e^{2 x+e^x}+4 e^{6 x+e^x}+e^{4 x} (1-2 x)-4 e^{3 x} x+4 x+e^x \left (e^{4 x}-2\right ) \left (e^{x+e^x}+1\right ) \log ^2(x)+\left (e^{4 x}-2\right ) \left (-x+4 e^x+4 e^{2 x+e^x}+1\right ) \log (x)-2\right )-\left (e^{4 x}-2\right ) \left (4 e^x+4 e^{2 x+e^x}+2 e^x \left (e^{x+e^x}+1\right ) \log (x)+1\right ) \log \left (2-e^{4 x}\right )}{\left (2-e^{4 x}\right ) \left (\log \left (2-e^{4 x}\right )+e^x (\log (x)+2)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\log \left (2-e^{4 x}\right ) \left (-2 x \log ^5(x)-20 x \log ^4(x)-2 \log ^4(x)-80 x \log ^3(x)-16 \log ^3(x)-160 x \log ^2(x)-48 \log ^2(x)+x \log ^4\left (2-e^{4 x}\right ) \log (x)-4 x \log ^3\left (2-e^{4 x}\right ) \log (x)-160 x \log (x)-64 \log (x)+2 x \log ^4\left (2-e^{4 x}\right )+\log ^4\left (2-e^{4 x}\right )-8 x \log ^3\left (2-e^{4 x}\right )-64 x-32\right )}{(\log (x)+2) \left (\log \left (2-e^{4 x}\right )+2 e^x+e^x \log (x)\right )^2 \left (\log ^4\left (2-e^{4 x}\right )-2 \log ^4(x)-16 \log ^3(x)-48 \log ^2(x)-64 \log (x)-32\right )}+e^{x+e^x}+\frac {8 x \left (-\log ^6\left (2-e^{4 x}\right )+4 e^x \log ^5\left (2-e^{4 x}\right )+2 e^x \log (x) \log ^5\left (2-e^{4 x}\right )-12 e^{2 x} \log ^4\left (2-e^{4 x}\right )-3 e^{2 x} \log ^2(x) \log ^4\left (2-e^{4 x}\right )-12 e^{2 x} \log (x) \log ^4\left (2-e^{4 x}\right )+32 e^{3 x} \log ^3\left (2-e^{4 x}\right )+4 e^{3 x} \log ^3(x) \log ^3\left (2-e^{4 x}\right )+24 e^{3 x} \log ^2(x) \log ^3\left (2-e^{4 x}\right )+48 e^{3 x} \log (x) \log ^3\left (2-e^{4 x}\right )-6 \log ^4(x) \log ^2\left (2-e^{4 x}\right )-48 \log ^3(x) \log ^2\left (2-e^{4 x}\right )-144 \log ^2(x) \log ^2\left (2-e^{4 x}\right )-192 \log (x) \log ^2\left (2-e^{4 x}\right )-96 \log ^2\left (2-e^{4 x}\right )+128 e^x \log \left (2-e^{4 x}\right )+4 e^x \log ^5(x) \log \left (2-e^{4 x}\right )+40 e^x \log ^4(x) \log \left (2-e^{4 x}\right )+160 e^x \log ^3(x) \log \left (2-e^{4 x}\right )+320 e^x \log ^2(x) \log \left (2-e^{4 x}\right )+320 e^x \log (x) \log \left (2-e^{4 x}\right )-128 e^{2 x}-2 e^{2 x} \log ^6(x)-24 e^{2 x} \log ^5(x)-120 e^{2 x} \log ^4(x)-320 e^{2 x} \log ^3(x)-480 e^{2 x} \log ^2(x)-384 e^{2 x} \log (x)\right )}{\left (-2+e^{4 x}\right ) \left (\log ^4\left (2-e^{4 x}\right )-2 \log ^4(x)-16 \log ^3(x)-48 \log ^2(x)-64 \log (x)-32\right )^2}+\frac {-4 x \log ^9(x)+4 \log ^9(x)-72 x \log ^8(x)+68 \log ^8(x)-576 x \log ^7(x)+512 \log ^7(x)-2688 x \log ^6(x)+2240 \log ^6(x)+4 x \log ^4\left (2-e^{4 x}\right ) \log ^5(x)-4 \log ^4\left (2-e^{4 x}\right ) \log ^5(x)-32 x \log ^3\left (2-e^{4 x}\right ) \log ^5(x)-8064 x \log ^5(x)+6272 \log ^5(x)+40 x \log ^4\left (2-e^{4 x}\right ) \log ^4(x)-36 \log ^4\left (2-e^{4 x}\right ) \log ^4(x)-320 x \log ^3\left (2-e^{4 x}\right ) \log ^4(x)-16128 x \log ^4(x)+11648 \log ^4(x)+160 x \log ^4\left (2-e^{4 x}\right ) \log ^3(x)-128 \log ^4\left (2-e^{4 x}\right ) \log ^3(x)-1280 x \log ^3\left (2-e^{4 x}\right ) \log ^3(x)-21504 x \log ^3(x)+14336 \log ^3(x)+320 x \log ^4\left (2-e^{4 x}\right ) \log ^2(x)-224 \log ^4\left (2-e^{4 x}\right ) \log ^2(x)-2560 x \log ^3\left (2-e^{4 x}\right ) \log ^2(x)-18432 x \log ^2(x)+11264 \log ^2(x)-x \log ^8\left (2-e^{4 x}\right ) \log (x)+\log ^8\left (2-e^{4 x}\right ) \log (x)+320 x \log ^4\left (2-e^{4 x}\right ) \log (x)-192 \log ^4\left (2-e^{4 x}\right ) \log (x)-2560 x \log ^3\left (2-e^{4 x}\right ) \log (x)-9216 x \log (x)+5120 \log (x)-2 x \log ^8\left (2-e^{4 x}\right )+\log ^8\left (2-e^{4 x}\right )+128 x \log ^4\left (2-e^{4 x}\right )-64 \log ^4\left (2-e^{4 x}\right )-1024 x \log ^3\left (2-e^{4 x}\right )-2048 x+1024}{(\log (x)+2) \left (\log \left (2-e^{4 x}\right )+2 e^x+e^x \log (x)\right ) \left (-\log ^4\left (2-e^{4 x}\right )+2 \log ^4(x)+16 \log ^3(x)+48 \log ^2(x)+64 \log (x)+32\right )^2}+1\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-\left (\left (e^{4 x}-2\right ) \left (e^{x+e^x}+1\right ) \log ^2\left (2-e^{4 x}\right )\right )-e^x \left (-8 e^x+4 e^{5 x}-8 e^{2 x+e^x}+4 e^{6 x+e^x}+e^{4 x} (1-2 x)-4 e^{3 x} x+4 x+e^x \left (e^{4 x}-2\right ) \left (e^{x+e^x}+1\right ) \log ^2(x)+\left (e^{4 x}-2\right ) \left (-x+4 e^x+4 e^{2 x+e^x}+1\right ) \log (x)-2\right )-\left (e^{4 x}-2\right ) \left (4 e^x+4 e^{2 x+e^x}+2 e^x \left (e^{x+e^x}+1\right ) \log (x)+1\right ) \log \left (2-e^{4 x}\right )}{\left (2-e^{4 x}\right ) \left (\log \left (2-e^{4 x}\right )+e^x (\log (x)+2)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\log \left (2-e^{4 x}\right ) \left (-2 x \log ^5(x)-20 x \log ^4(x)-2 \log ^4(x)-80 x \log ^3(x)-16 \log ^3(x)-160 x \log ^2(x)-48 \log ^2(x)+x \log ^4\left (2-e^{4 x}\right ) \log (x)-4 x \log ^3\left (2-e^{4 x}\right ) \log (x)-160 x \log (x)-64 \log (x)+2 x \log ^4\left (2-e^{4 x}\right )+\log ^4\left (2-e^{4 x}\right )-8 x \log ^3\left (2-e^{4 x}\right )-64 x-32\right )}{(\log (x)+2) \left (\log \left (2-e^{4 x}\right )+2 e^x+e^x \log (x)\right )^2 \left (\log ^4\left (2-e^{4 x}\right )-2 \log ^4(x)-16 \log ^3(x)-48 \log ^2(x)-64 \log (x)-32\right )}+e^{x+e^x}+\frac {8 x \left (-\log ^6\left (2-e^{4 x}\right )+4 e^x \log ^5\left (2-e^{4 x}\right )+2 e^x \log (x) \log ^5\left (2-e^{4 x}\right )-12 e^{2 x} \log ^4\left (2-e^{4 x}\right )-3 e^{2 x} \log ^2(x) \log ^4\left (2-e^{4 x}\right )-12 e^{2 x} \log (x) \log ^4\left (2-e^{4 x}\right )+32 e^{3 x} \log ^3\left (2-e^{4 x}\right )+4 e^{3 x} \log ^3(x) \log ^3\left (2-e^{4 x}\right )+24 e^{3 x} \log ^2(x) \log ^3\left (2-e^{4 x}\right )+48 e^{3 x} \log (x) \log ^3\left (2-e^{4 x}\right )-6 \log ^4(x) \log ^2\left (2-e^{4 x}\right )-48 \log ^3(x) \log ^2\left (2-e^{4 x}\right )-144 \log ^2(x) \log ^2\left (2-e^{4 x}\right )-192 \log (x) \log ^2\left (2-e^{4 x}\right )-96 \log ^2\left (2-e^{4 x}\right )+128 e^x \log \left (2-e^{4 x}\right )+4 e^x \log ^5(x) \log \left (2-e^{4 x}\right )+40 e^x \log ^4(x) \log \left (2-e^{4 x}\right )+160 e^x \log ^3(x) \log \left (2-e^{4 x}\right )+320 e^x \log ^2(x) \log \left (2-e^{4 x}\right )+320 e^x \log (x) \log \left (2-e^{4 x}\right )-128 e^{2 x}-2 e^{2 x} \log ^6(x)-24 e^{2 x} \log ^5(x)-120 e^{2 x} \log ^4(x)-320 e^{2 x} \log ^3(x)-480 e^{2 x} \log ^2(x)-384 e^{2 x} \log (x)\right )}{\left (-2+e^{4 x}\right ) \left (\log ^4\left (2-e^{4 x}\right )-2 \log ^4(x)-16 \log ^3(x)-48 \log ^2(x)-64 \log (x)-32\right )^2}+\frac {-4 x \log ^9(x)+4 \log ^9(x)-72 x \log ^8(x)+68 \log ^8(x)-576 x \log ^7(x)+512 \log ^7(x)-2688 x \log ^6(x)+2240 \log ^6(x)+4 x \log ^4\left (2-e^{4 x}\right ) \log ^5(x)-4 \log ^4\left (2-e^{4 x}\right ) \log ^5(x)-32 x \log ^3\left (2-e^{4 x}\right ) \log ^5(x)-8064 x \log ^5(x)+6272 \log ^5(x)+40 x \log ^4\left (2-e^{4 x}\right ) \log ^4(x)-36 \log ^4\left (2-e^{4 x}\right ) \log ^4(x)-320 x \log ^3\left (2-e^{4 x}\right ) \log ^4(x)-16128 x \log ^4(x)+11648 \log ^4(x)+160 x \log ^4\left (2-e^{4 x}\right ) \log ^3(x)-128 \log ^4\left (2-e^{4 x}\right ) \log ^3(x)-1280 x \log ^3\left (2-e^{4 x}\right ) \log ^3(x)-21504 x \log ^3(x)+14336 \log ^3(x)+320 x \log ^4\left (2-e^{4 x}\right ) \log ^2(x)-224 \log ^4\left (2-e^{4 x}\right ) \log ^2(x)-2560 x \log ^3\left (2-e^{4 x}\right ) \log ^2(x)-18432 x \log ^2(x)+11264 \log ^2(x)-x \log ^8\left (2-e^{4 x}\right ) \log (x)+\log ^8\left (2-e^{4 x}\right ) \log (x)+320 x \log ^4\left (2-e^{4 x}\right ) \log (x)-192 \log ^4\left (2-e^{4 x}\right ) \log (x)-2560 x \log ^3\left (2-e^{4 x}\right ) \log (x)-9216 x \log (x)+5120 \log (x)-2 x \log ^8\left (2-e^{4 x}\right )+\log ^8\left (2-e^{4 x}\right )+128 x \log ^4\left (2-e^{4 x}\right )-64 \log ^4\left (2-e^{4 x}\right )-1024 x \log ^3\left (2-e^{4 x}\right )-2048 x+1024}{(\log (x)+2) \left (\log \left (2-e^{4 x}\right )+2 e^x+e^x \log (x)\right ) \left (-\log ^4\left (2-e^{4 x}\right )+2 \log ^4(x)+16 \log ^3(x)+48 \log ^2(x)+64 \log (x)+32\right )^2}+1\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-\left (\left (e^{4 x}-2\right ) \left (e^{x+e^x}+1\right ) \log ^2\left (2-e^{4 x}\right )\right )-e^x \left (-8 e^x+4 e^{5 x}-8 e^{2 x+e^x}+4 e^{6 x+e^x}+e^{4 x} (1-2 x)-4 e^{3 x} x+4 x+e^x \left (e^{4 x}-2\right ) \left (e^{x+e^x}+1\right ) \log ^2(x)+\left (e^{4 x}-2\right ) \left (-x+4 e^x+4 e^{2 x+e^x}+1\right ) \log (x)-2\right )-\left (e^{4 x}-2\right ) \left (4 e^x+4 e^{2 x+e^x}+2 e^x \left (e^{x+e^x}+1\right ) \log (x)+1\right ) \log \left (2-e^{4 x}\right )}{\left (2-e^{4 x}\right ) \left (\log \left (2-e^{4 x}\right )+e^x (\log (x)+2)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\log \left (2-e^{4 x}\right ) \left (-2 x \log ^5(x)-20 x \log ^4(x)-2 \log ^4(x)-80 x \log ^3(x)-16 \log ^3(x)-160 x \log ^2(x)-48 \log ^2(x)+x \log ^4\left (2-e^{4 x}\right ) \log (x)-4 x \log ^3\left (2-e^{4 x}\right ) \log (x)-160 x \log (x)-64 \log (x)+2 x \log ^4\left (2-e^{4 x}\right )+\log ^4\left (2-e^{4 x}\right )-8 x \log ^3\left (2-e^{4 x}\right )-64 x-32\right )}{(\log (x)+2) \left (\log \left (2-e^{4 x}\right )+2 e^x+e^x \log (x)\right )^2 \left (\log ^4\left (2-e^{4 x}\right )-2 \log ^4(x)-16 \log ^3(x)-48 \log ^2(x)-64 \log (x)-32\right )}+e^{x+e^x}+\frac {8 x \left (-\log ^6\left (2-e^{4 x}\right )+4 e^x \log ^5\left (2-e^{4 x}\right )+2 e^x \log (x) \log ^5\left (2-e^{4 x}\right )-12 e^{2 x} \log ^4\left (2-e^{4 x}\right )-3 e^{2 x} \log ^2(x) \log ^4\left (2-e^{4 x}\right )-12 e^{2 x} \log (x) \log ^4\left (2-e^{4 x}\right )+32 e^{3 x} \log ^3\left (2-e^{4 x}\right )+4 e^{3 x} \log ^3(x) \log ^3\left (2-e^{4 x}\right )+24 e^{3 x} \log ^2(x) \log ^3\left (2-e^{4 x}\right )+48 e^{3 x} \log (x) \log ^3\left (2-e^{4 x}\right )-6 \log ^4(x) \log ^2\left (2-e^{4 x}\right )-48 \log ^3(x) \log ^2\left (2-e^{4 x}\right )-144 \log ^2(x) \log ^2\left (2-e^{4 x}\right )-192 \log (x) \log ^2\left (2-e^{4 x}\right )-96 \log ^2\left (2-e^{4 x}\right )+128 e^x \log \left (2-e^{4 x}\right )+4 e^x \log ^5(x) \log \left (2-e^{4 x}\right )+40 e^x \log ^4(x) \log \left (2-e^{4 x}\right )+160 e^x \log ^3(x) \log \left (2-e^{4 x}\right )+320 e^x \log ^2(x) \log \left (2-e^{4 x}\right )+320 e^x \log (x) \log \left (2-e^{4 x}\right )-128 e^{2 x}-2 e^{2 x} \log ^6(x)-24 e^{2 x} \log ^5(x)-120 e^{2 x} \log ^4(x)-320 e^{2 x} \log ^3(x)-480 e^{2 x} \log ^2(x)-384 e^{2 x} \log (x)\right )}{\left (-2+e^{4 x}\right ) \left (\log ^4\left (2-e^{4 x}\right )-2 \log ^4(x)-16 \log ^3(x)-48 \log ^2(x)-64 \log (x)-32\right )^2}+\frac {-4 x \log ^9(x)+4 \log ^9(x)-72 x \log ^8(x)+68 \log ^8(x)-576 x \log ^7(x)+512 \log ^7(x)-2688 x \log ^6(x)+2240 \log ^6(x)+4 x \log ^4\left (2-e^{4 x}\right ) \log ^5(x)-4 \log ^4\left (2-e^{4 x}\right ) \log ^5(x)-32 x \log ^3\left (2-e^{4 x}\right ) \log ^5(x)-8064 x \log ^5(x)+6272 \log ^5(x)+40 x \log ^4\left (2-e^{4 x}\right ) \log ^4(x)-36 \log ^4\left (2-e^{4 x}\right ) \log ^4(x)-320 x \log ^3\left (2-e^{4 x}\right ) \log ^4(x)-16128 x \log ^4(x)+11648 \log ^4(x)+160 x \log ^4\left (2-e^{4 x}\right ) \log ^3(x)-128 \log ^4\left (2-e^{4 x}\right ) \log ^3(x)-1280 x \log ^3\left (2-e^{4 x}\right ) \log ^3(x)-21504 x \log ^3(x)+14336 \log ^3(x)+320 x \log ^4\left (2-e^{4 x}\right ) \log ^2(x)-224 \log ^4\left (2-e^{4 x}\right ) \log ^2(x)-2560 x \log ^3\left (2-e^{4 x}\right ) \log ^2(x)-18432 x \log ^2(x)+11264 \log ^2(x)-x \log ^8\left (2-e^{4 x}\right ) \log (x)+\log ^8\left (2-e^{4 x}\right ) \log (x)+320 x \log ^4\left (2-e^{4 x}\right ) \log (x)-192 \log ^4\left (2-e^{4 x}\right ) \log (x)-2560 x \log ^3\left (2-e^{4 x}\right ) \log (x)-9216 x \log (x)+5120 \log (x)-2 x \log ^8\left (2-e^{4 x}\right )+\log ^8\left (2-e^{4 x}\right )+128 x \log ^4\left (2-e^{4 x}\right )-64 \log ^4\left (2-e^{4 x}\right )-1024 x \log ^3\left (2-e^{4 x}\right )-2048 x+1024}{(\log (x)+2) \left (\log \left (2-e^{4 x}\right )+2 e^x+e^x \log (x)\right ) \left (-\log ^4\left (2-e^{4 x}\right )+2 \log ^4(x)+16 \log ^3(x)+48 \log ^2(x)+64 \log (x)+32\right )^2}+1\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-\left (\left (e^{4 x}-2\right ) \left (e^{x+e^x}+1\right ) \log ^2\left (2-e^{4 x}\right )\right )-e^x \left (-8 e^x+4 e^{5 x}-8 e^{2 x+e^x}+4 e^{6 x+e^x}+e^{4 x} (1-2 x)-4 e^{3 x} x+4 x+e^x \left (e^{4 x}-2\right ) \left (e^{x+e^x}+1\right ) \log ^2(x)+\left (e^{4 x}-2\right ) \left (-x+4 e^x+4 e^{2 x+e^x}+1\right ) \log (x)-2\right )-\left (e^{4 x}-2\right ) \left (4 e^x+4 e^{2 x+e^x}+2 e^x \left (e^{x+e^x}+1\right ) \log (x)+1\right ) \log \left (2-e^{4 x}\right )}{\left (2-e^{4 x}\right ) \left (\log \left (2-e^{4 x}\right )+e^x (\log (x)+2)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\log \left (2-e^{4 x}\right ) \left (-2 x \log ^5(x)-20 x \log ^4(x)-2 \log ^4(x)-80 x \log ^3(x)-16 \log ^3(x)-160 x \log ^2(x)-48 \log ^2(x)+x \log ^4\left (2-e^{4 x}\right ) \log (x)-4 x \log ^3\left (2-e^{4 x}\right ) \log (x)-160 x \log (x)-64 \log (x)+2 x \log ^4\left (2-e^{4 x}\right )+\log ^4\left (2-e^{4 x}\right )-8 x \log ^3\left (2-e^{4 x}\right )-64 x-32\right )}{(\log (x)+2) \left (\log \left (2-e^{4 x}\right )+2 e^x+e^x \log (x)\right )^2 \left (\log ^4\left (2-e^{4 x}\right )-2 \log ^4(x)-16 \log ^3(x)-48 \log ^2(x)-64 \log (x)-32\right )}+e^{x+e^x}+\frac {8 x \left (-\log ^6\left (2-e^{4 x}\right )+4 e^x \log ^5\left (2-e^{4 x}\right )+2 e^x \log (x) \log ^5\left (2-e^{4 x}\right )-12 e^{2 x} \log ^4\left (2-e^{4 x}\right )-3 e^{2 x} \log ^2(x) \log ^4\left (2-e^{4 x}\right )-12 e^{2 x} \log (x) \log ^4\left (2-e^{4 x}\right )+32 e^{3 x} \log ^3\left (2-e^{4 x}\right )+4 e^{3 x} \log ^3(x) \log ^3\left (2-e^{4 x}\right )+24 e^{3 x} \log ^2(x) \log ^3\left (2-e^{4 x}\right )+48 e^{3 x} \log (x) \log ^3\left (2-e^{4 x}\right )-6 \log ^4(x) \log ^2\left (2-e^{4 x}\right )-48 \log ^3(x) \log ^2\left (2-e^{4 x}\right )-144 \log ^2(x) \log ^2\left (2-e^{4 x}\right )-192 \log (x) \log ^2\left (2-e^{4 x}\right )-96 \log ^2\left (2-e^{4 x}\right )+128 e^x \log \left (2-e^{4 x}\right )+4 e^x \log ^5(x) \log \left (2-e^{4 x}\right )+40 e^x \log ^4(x) \log \left (2-e^{4 x}\right )+160 e^x \log ^3(x) \log \left (2-e^{4 x}\right )+320 e^x \log ^2(x) \log \left (2-e^{4 x}\right )+320 e^x \log (x) \log \left (2-e^{4 x}\right )-128 e^{2 x}-2 e^{2 x} \log ^6(x)-24 e^{2 x} \log ^5(x)-120 e^{2 x} \log ^4(x)-320 e^{2 x} \log ^3(x)-480 e^{2 x} \log ^2(x)-384 e^{2 x} \log (x)\right )}{\left (-2+e^{4 x}\right ) \left (\log ^4\left (2-e^{4 x}\right )-2 \log ^4(x)-16 \log ^3(x)-48 \log ^2(x)-64 \log (x)-32\right )^2}+\frac {-4 x \log ^9(x)+4 \log ^9(x)-72 x \log ^8(x)+68 \log ^8(x)-576 x \log ^7(x)+512 \log ^7(x)-2688 x \log ^6(x)+2240 \log ^6(x)+4 x \log ^4\left (2-e^{4 x}\right ) \log ^5(x)-4 \log ^4\left (2-e^{4 x}\right ) \log ^5(x)-32 x \log ^3\left (2-e^{4 x}\right ) \log ^5(x)-8064 x \log ^5(x)+6272 \log ^5(x)+40 x \log ^4\left (2-e^{4 x}\right ) \log ^4(x)-36 \log ^4\left (2-e^{4 x}\right ) \log ^4(x)-320 x \log ^3\left (2-e^{4 x}\right ) \log ^4(x)-16128 x \log ^4(x)+11648 \log ^4(x)+160 x \log ^4\left (2-e^{4 x}\right ) \log ^3(x)-128 \log ^4\left (2-e^{4 x}\right ) \log ^3(x)-1280 x \log ^3\left (2-e^{4 x}\right ) \log ^3(x)-21504 x \log ^3(x)+14336 \log ^3(x)+320 x \log ^4\left (2-e^{4 x}\right ) \log ^2(x)-224 \log ^4\left (2-e^{4 x}\right ) \log ^2(x)-2560 x \log ^3\left (2-e^{4 x}\right ) \log ^2(x)-18432 x \log ^2(x)+11264 \log ^2(x)-x \log ^8\left (2-e^{4 x}\right ) \log (x)+\log ^8\left (2-e^{4 x}\right ) \log (x)+320 x \log ^4\left (2-e^{4 x}\right ) \log (x)-192 \log ^4\left (2-e^{4 x}\right ) \log (x)-2560 x \log ^3\left (2-e^{4 x}\right ) \log (x)-9216 x \log (x)+5120 \log (x)-2 x \log ^8\left (2-e^{4 x}\right )+\log ^8\left (2-e^{4 x}\right )+128 x \log ^4\left (2-e^{4 x}\right )-64 \log ^4\left (2-e^{4 x}\right )-1024 x \log ^3\left (2-e^{4 x}\right )-2048 x+1024}{(\log (x)+2) \left (\log \left (2-e^{4 x}\right )+2 e^x+e^x \log (x)\right ) \left (-\log ^4\left (2-e^{4 x}\right )+2 \log ^4(x)+16 \log ^3(x)+48 \log ^2(x)+64 \log (x)+32\right )^2}+1\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-\left (\left (e^{4 x}-2\right ) \left (e^{x+e^x}+1\right ) \log ^2\left (2-e^{4 x}\right )\right )-e^x \left (-8 e^x+4 e^{5 x}-8 e^{2 x+e^x}+4 e^{6 x+e^x}+e^{4 x} (1-2 x)-4 e^{3 x} x+4 x+e^x \left (e^{4 x}-2\right ) \left (e^{x+e^x}+1\right ) \log ^2(x)+\left (e^{4 x}-2\right ) \left (-x+4 e^x+4 e^{2 x+e^x}+1\right ) \log (x)-2\right )-\left (e^{4 x}-2\right ) \left (4 e^x+4 e^{2 x+e^x}+2 e^x \left (e^{x+e^x}+1\right ) \log (x)+1\right ) \log \left (2-e^{4 x}\right )}{\left (2-e^{4 x}\right ) \left (\log \left (2-e^{4 x}\right )+e^x (\log (x)+2)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\log \left (2-e^{4 x}\right ) \left (-2 x \log ^5(x)-20 x \log ^4(x)-2 \log ^4(x)-80 x \log ^3(x)-16 \log ^3(x)-160 x \log ^2(x)-48 \log ^2(x)+x \log ^4\left (2-e^{4 x}\right ) \log (x)-4 x \log ^3\left (2-e^{4 x}\right ) \log (x)-160 x \log (x)-64 \log (x)+2 x \log ^4\left (2-e^{4 x}\right )+\log ^4\left (2-e^{4 x}\right )-8 x \log ^3\left (2-e^{4 x}\right )-64 x-32\right )}{(\log (x)+2) \left (\log \left (2-e^{4 x}\right )+2 e^x+e^x \log (x)\right )^2 \left (\log ^4\left (2-e^{4 x}\right )-2 \log ^4(x)-16 \log ^3(x)-48 \log ^2(x)-64 \log (x)-32\right )}+e^{x+e^x}+\frac {8 x \left (-\log ^6\left (2-e^{4 x}\right )+4 e^x \log ^5\left (2-e^{4 x}\right )+2 e^x \log (x) \log ^5\left (2-e^{4 x}\right )-12 e^{2 x} \log ^4\left (2-e^{4 x}\right )-3 e^{2 x} \log ^2(x) \log ^4\left (2-e^{4 x}\right )-12 e^{2 x} \log (x) \log ^4\left (2-e^{4 x}\right )+32 e^{3 x} \log ^3\left (2-e^{4 x}\right )+4 e^{3 x} \log ^3(x) \log ^3\left (2-e^{4 x}\right )+24 e^{3 x} \log ^2(x) \log ^3\left (2-e^{4 x}\right )+48 e^{3 x} \log (x) \log ^3\left (2-e^{4 x}\right )-6 \log ^4(x) \log ^2\left (2-e^{4 x}\right )-48 \log ^3(x) \log ^2\left (2-e^{4 x}\right )-144 \log ^2(x) \log ^2\left (2-e^{4 x}\right )-192 \log (x) \log ^2\left (2-e^{4 x}\right )-96 \log ^2\left (2-e^{4 x}\right )+128 e^x \log \left (2-e^{4 x}\right )+4 e^x \log ^5(x) \log \left (2-e^{4 x}\right )+40 e^x \log ^4(x) \log \left (2-e^{4 x}\right )+160 e^x \log ^3(x) \log \left (2-e^{4 x}\right )+320 e^x \log ^2(x) \log \left (2-e^{4 x}\right )+320 e^x \log (x) \log \left (2-e^{4 x}\right )-128 e^{2 x}-2 e^{2 x} \log ^6(x)-24 e^{2 x} \log ^5(x)-120 e^{2 x} \log ^4(x)-320 e^{2 x} \log ^3(x)-480 e^{2 x} \log ^2(x)-384 e^{2 x} \log (x)\right )}{\left (-2+e^{4 x}\right ) \left (\log ^4\left (2-e^{4 x}\right )-2 \log ^4(x)-16 \log ^3(x)-48 \log ^2(x)-64 \log (x)-32\right )^2}+\frac {-4 x \log ^9(x)+4 \log ^9(x)-72 x \log ^8(x)+68 \log ^8(x)-576 x \log ^7(x)+512 \log ^7(x)-2688 x \log ^6(x)+2240 \log ^6(x)+4 x \log ^4\left (2-e^{4 x}\right ) \log ^5(x)-4 \log ^4\left (2-e^{4 x}\right ) \log ^5(x)-32 x \log ^3\left (2-e^{4 x}\right ) \log ^5(x)-8064 x \log ^5(x)+6272 \log ^5(x)+40 x \log ^4\left (2-e^{4 x}\right ) \log ^4(x)-36 \log ^4\left (2-e^{4 x}\right ) \log ^4(x)-320 x \log ^3\left (2-e^{4 x}\right ) \log ^4(x)-16128 x \log ^4(x)+11648 \log ^4(x)+160 x \log ^4\left (2-e^{4 x}\right ) \log ^3(x)-128 \log ^4\left (2-e^{4 x}\right ) \log ^3(x)-1280 x \log ^3\left (2-e^{4 x}\right ) \log ^3(x)-21504 x \log ^3(x)+14336 \log ^3(x)+320 x \log ^4\left (2-e^{4 x}\right ) \log ^2(x)-224 \log ^4\left (2-e^{4 x}\right ) \log ^2(x)-2560 x \log ^3\left (2-e^{4 x}\right ) \log ^2(x)-18432 x \log ^2(x)+11264 \log ^2(x)-x \log ^8\left (2-e^{4 x}\right ) \log (x)+\log ^8\left (2-e^{4 x}\right ) \log (x)+320 x \log ^4\left (2-e^{4 x}\right ) \log (x)-192 \log ^4\left (2-e^{4 x}\right ) \log (x)-2560 x \log ^3\left (2-e^{4 x}\right ) \log (x)-9216 x \log (x)+5120 \log (x)-2 x \log ^8\left (2-e^{4 x}\right )+\log ^8\left (2-e^{4 x}\right )+128 x \log ^4\left (2-e^{4 x}\right )-64 \log ^4\left (2-e^{4 x}\right )-1024 x \log ^3\left (2-e^{4 x}\right )-2048 x+1024}{(\log (x)+2) \left (\log \left (2-e^{4 x}\right )+2 e^x+e^x \log (x)\right ) \left (-\log ^4\left (2-e^{4 x}\right )+2 \log ^4(x)+16 \log ^3(x)+48 \log ^2(x)+64 \log (x)+32\right )^2}+1\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-\left (\left (e^{4 x}-2\right ) \left (e^{x+e^x}+1\right ) \log ^2\left (2-e^{4 x}\right )\right )-e^x \left (-8 e^x+4 e^{5 x}-8 e^{2 x+e^x}+4 e^{6 x+e^x}+e^{4 x} (1-2 x)-4 e^{3 x} x+4 x+e^x \left (e^{4 x}-2\right ) \left (e^{x+e^x}+1\right ) \log ^2(x)+\left (e^{4 x}-2\right ) \left (-x+4 e^x+4 e^{2 x+e^x}+1\right ) \log (x)-2\right )-\left (e^{4 x}-2\right ) \left (4 e^x+4 e^{2 x+e^x}+2 e^x \left (e^{x+e^x}+1\right ) \log (x)+1\right ) \log \left (2-e^{4 x}\right )}{\left (2-e^{4 x}\right ) \left (\log \left (2-e^{4 x}\right )+e^x (\log (x)+2)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\log \left (2-e^{4 x}\right ) \left (-2 x \log ^5(x)-20 x \log ^4(x)-2 \log ^4(x)-80 x \log ^3(x)-16 \log ^3(x)-160 x \log ^2(x)-48 \log ^2(x)+x \log ^4\left (2-e^{4 x}\right ) \log (x)-4 x \log ^3\left (2-e^{4 x}\right ) \log (x)-160 x \log (x)-64 \log (x)+2 x \log ^4\left (2-e^{4 x}\right )+\log ^4\left (2-e^{4 x}\right )-8 x \log ^3\left (2-e^{4 x}\right )-64 x-32\right )}{(\log (x)+2) \left (\log \left (2-e^{4 x}\right )+2 e^x+e^x \log (x)\right )^2 \left (\log ^4\left (2-e^{4 x}\right )-2 \log ^4(x)-16 \log ^3(x)-48 \log ^2(x)-64 \log (x)-32\right )}+e^{x+e^x}+\frac {8 x \left (-\log ^6\left (2-e^{4 x}\right )+4 e^x \log ^5\left (2-e^{4 x}\right )+2 e^x \log (x) \log ^5\left (2-e^{4 x}\right )-12 e^{2 x} \log ^4\left (2-e^{4 x}\right )-3 e^{2 x} \log ^2(x) \log ^4\left (2-e^{4 x}\right )-12 e^{2 x} \log (x) \log ^4\left (2-e^{4 x}\right )+32 e^{3 x} \log ^3\left (2-e^{4 x}\right )+4 e^{3 x} \log ^3(x) \log ^3\left (2-e^{4 x}\right )+24 e^{3 x} \log ^2(x) \log ^3\left (2-e^{4 x}\right )+48 e^{3 x} \log (x) \log ^3\left (2-e^{4 x}\right )-6 \log ^4(x) \log ^2\left (2-e^{4 x}\right )-48 \log ^3(x) \log ^2\left (2-e^{4 x}\right )-144 \log ^2(x) \log ^2\left (2-e^{4 x}\right )-192 \log (x) \log ^2\left (2-e^{4 x}\right )-96 \log ^2\left (2-e^{4 x}\right )+128 e^x \log \left (2-e^{4 x}\right )+4 e^x \log ^5(x) \log \left (2-e^{4 x}\right )+40 e^x \log ^4(x) \log \left (2-e^{4 x}\right )+160 e^x \log ^3(x) \log \left (2-e^{4 x}\right )+320 e^x \log ^2(x) \log \left (2-e^{4 x}\right )+320 e^x \log (x) \log \left (2-e^{4 x}\right )-128 e^{2 x}-2 e^{2 x} \log ^6(x)-24 e^{2 x} \log ^5(x)-120 e^{2 x} \log ^4(x)-320 e^{2 x} \log ^3(x)-480 e^{2 x} \log ^2(x)-384 e^{2 x} \log (x)\right )}{\left (-2+e^{4 x}\right ) \left (\log ^4\left (2-e^{4 x}\right )-2 \log ^4(x)-16 \log ^3(x)-48 \log ^2(x)-64 \log (x)-32\right )^2}+\frac {-4 x \log ^9(x)+4 \log ^9(x)-72 x \log ^8(x)+68 \log ^8(x)-576 x \log ^7(x)+512 \log ^7(x)-2688 x \log ^6(x)+2240 \log ^6(x)+4 x \log ^4\left (2-e^{4 x}\right ) \log ^5(x)-4 \log ^4\left (2-e^{4 x}\right ) \log ^5(x)-32 x \log ^3\left (2-e^{4 x}\right ) \log ^5(x)-8064 x \log ^5(x)+6272 \log ^5(x)+40 x \log ^4\left (2-e^{4 x}\right ) \log ^4(x)-36 \log ^4\left (2-e^{4 x}\right ) \log ^4(x)-320 x \log ^3\left (2-e^{4 x}\right ) \log ^4(x)-16128 x \log ^4(x)+11648 \log ^4(x)+160 x \log ^4\left (2-e^{4 x}\right ) \log ^3(x)-128 \log ^4\left (2-e^{4 x}\right ) \log ^3(x)-1280 x \log ^3\left (2-e^{4 x}\right ) \log ^3(x)-21504 x \log ^3(x)+14336 \log ^3(x)+320 x \log ^4\left (2-e^{4 x}\right ) \log ^2(x)-224 \log ^4\left (2-e^{4 x}\right ) \log ^2(x)-2560 x \log ^3\left (2-e^{4 x}\right ) \log ^2(x)-18432 x \log ^2(x)+11264 \log ^2(x)-x \log ^8\left (2-e^{4 x}\right ) \log (x)+\log ^8\left (2-e^{4 x}\right ) \log (x)+320 x \log ^4\left (2-e^{4 x}\right ) \log (x)-192 \log ^4\left (2-e^{4 x}\right ) \log (x)-2560 x \log ^3\left (2-e^{4 x}\right ) \log (x)-9216 x \log (x)+5120 \log (x)-2 x \log ^8\left (2-e^{4 x}\right )+\log ^8\left (2-e^{4 x}\right )+128 x \log ^4\left (2-e^{4 x}\right )-64 \log ^4\left (2-e^{4 x}\right )-1024 x \log ^3\left (2-e^{4 x}\right )-2048 x+1024}{(\log (x)+2) \left (\log \left (2-e^{4 x}\right )+2 e^x+e^x \log (x)\right ) \left (-\log ^4\left (2-e^{4 x}\right )+2 \log ^4(x)+16 \log ^3(x)+48 \log ^2(x)+64 \log (x)+32\right )^2}+1\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-\left (\left (e^{4 x}-2\right ) \left (e^{x+e^x}+1\right ) \log ^2\left (2-e^{4 x}\right )\right )-e^x \left (-8 e^x+4 e^{5 x}-8 e^{2 x+e^x}+4 e^{6 x+e^x}+e^{4 x} (1-2 x)-4 e^{3 x} x+4 x+e^x \left (e^{4 x}-2\right ) \left (e^{x+e^x}+1\right ) \log ^2(x)+\left (e^{4 x}-2\right ) \left (-x+4 e^x+4 e^{2 x+e^x}+1\right ) \log (x)-2\right )-\left (e^{4 x}-2\right ) \left (4 e^x+4 e^{2 x+e^x}+2 e^x \left (e^{x+e^x}+1\right ) \log (x)+1\right ) \log \left (2-e^{4 x}\right )}{\left (2-e^{4 x}\right ) \left (\log \left (2-e^{4 x}\right )+e^x (\log (x)+2)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\log \left (2-e^{4 x}\right ) \left (-2 x \log ^5(x)-20 x \log ^4(x)-2 \log ^4(x)-80 x \log ^3(x)-16 \log ^3(x)-160 x \log ^2(x)-48 \log ^2(x)+x \log ^4\left (2-e^{4 x}\right ) \log (x)-4 x \log ^3\left (2-e^{4 x}\right ) \log (x)-160 x \log (x)-64 \log (x)+2 x \log ^4\left (2-e^{4 x}\right )+\log ^4\left (2-e^{4 x}\right )-8 x \log ^3\left (2-e^{4 x}\right )-64 x-32\right )}{(\log (x)+2) \left (\log \left (2-e^{4 x}\right )+2 e^x+e^x \log (x)\right )^2 \left (\log ^4\left (2-e^{4 x}\right )-2 \log ^4(x)-16 \log ^3(x)-48 \log ^2(x)-64 \log (x)-32\right )}+e^{x+e^x}+\frac {8 x \left (-\log ^6\left (2-e^{4 x}\right )+4 e^x \log ^5\left (2-e^{4 x}\right )+2 e^x \log (x) \log ^5\left (2-e^{4 x}\right )-12 e^{2 x} \log ^4\left (2-e^{4 x}\right )-3 e^{2 x} \log ^2(x) \log ^4\left (2-e^{4 x}\right )-12 e^{2 x} \log (x) \log ^4\left (2-e^{4 x}\right )+32 e^{3 x} \log ^3\left (2-e^{4 x}\right )+4 e^{3 x} \log ^3(x) \log ^3\left (2-e^{4 x}\right )+24 e^{3 x} \log ^2(x) \log ^3\left (2-e^{4 x}\right )+48 e^{3 x} \log (x) \log ^3\left (2-e^{4 x}\right )-6 \log ^4(x) \log ^2\left (2-e^{4 x}\right )-48 \log ^3(x) \log ^2\left (2-e^{4 x}\right )-144 \log ^2(x) \log ^2\left (2-e^{4 x}\right )-192 \log (x) \log ^2\left (2-e^{4 x}\right )-96 \log ^2\left (2-e^{4 x}\right )+128 e^x \log \left (2-e^{4 x}\right )+4 e^x \log ^5(x) \log \left (2-e^{4 x}\right )+40 e^x \log ^4(x) \log \left (2-e^{4 x}\right )+160 e^x \log ^3(x) \log \left (2-e^{4 x}\right )+320 e^x \log ^2(x) \log \left (2-e^{4 x}\right )+320 e^x \log (x) \log \left (2-e^{4 x}\right )-128 e^{2 x}-2 e^{2 x} \log ^6(x)-24 e^{2 x} \log ^5(x)-120 e^{2 x} \log ^4(x)-320 e^{2 x} \log ^3(x)-480 e^{2 x} \log ^2(x)-384 e^{2 x} \log (x)\right )}{\left (-2+e^{4 x}\right ) \left (\log ^4\left (2-e^{4 x}\right )-2 \log ^4(x)-16 \log ^3(x)-48 \log ^2(x)-64 \log (x)-32\right )^2}+\frac {-4 x \log ^9(x)+4 \log ^9(x)-72 x \log ^8(x)+68 \log ^8(x)-576 x \log ^7(x)+512 \log ^7(x)-2688 x \log ^6(x)+2240 \log ^6(x)+4 x \log ^4\left (2-e^{4 x}\right ) \log ^5(x)-4 \log ^4\left (2-e^{4 x}\right ) \log ^5(x)-32 x \log ^3\left (2-e^{4 x}\right ) \log ^5(x)-8064 x \log ^5(x)+6272 \log ^5(x)+40 x \log ^4\left (2-e^{4 x}\right ) \log ^4(x)-36 \log ^4\left (2-e^{4 x}\right ) \log ^4(x)-320 x \log ^3\left (2-e^{4 x}\right ) \log ^4(x)-16128 x \log ^4(x)+11648 \log ^4(x)+160 x \log ^4\left (2-e^{4 x}\right ) \log ^3(x)-128 \log ^4\left (2-e^{4 x}\right ) \log ^3(x)-1280 x \log ^3\left (2-e^{4 x}\right ) \log ^3(x)-21504 x \log ^3(x)+14336 \log ^3(x)+320 x \log ^4\left (2-e^{4 x}\right ) \log ^2(x)-224 \log ^4\left (2-e^{4 x}\right ) \log ^2(x)-2560 x \log ^3\left (2-e^{4 x}\right ) \log ^2(x)-18432 x \log ^2(x)+11264 \log ^2(x)-x \log ^8\left (2-e^{4 x}\right ) \log (x)+\log ^8\left (2-e^{4 x}\right ) \log (x)+320 x \log ^4\left (2-e^{4 x}\right ) \log (x)-192 \log ^4\left (2-e^{4 x}\right ) \log (x)-2560 x \log ^3\left (2-e^{4 x}\right ) \log (x)-9216 x \log (x)+5120 \log (x)-2 x \log ^8\left (2-e^{4 x}\right )+\log ^8\left (2-e^{4 x}\right )+128 x \log ^4\left (2-e^{4 x}\right )-64 \log ^4\left (2-e^{4 x}\right )-1024 x \log ^3\left (2-e^{4 x}\right )-2048 x+1024}{(\log (x)+2) \left (\log \left (2-e^{4 x}\right )+2 e^x+e^x \log (x)\right ) \left (-\log ^4\left (2-e^{4 x}\right )+2 \log ^4(x)+16 \log ^3(x)+48 \log ^2(x)+64 \log (x)+32\right )^2}+1\right )dx\)

Input:

Int[(-8*E^(2*x) + E^(4*x)*(4*E^(2*x) + E^x*(1 - 2*x) - 4*x) + E^x*(-2 + 4* 
x) + (-2 + E^(4*x) + E^E^x*(-2*E^x + E^(5*x)))*Log[2 - E^(4*x)]^2 + (-8*E^ 
(2*x) + E^(4*x)*(4*E^(2*x) + E^x*(1 - x)) + E^x*(-2 + 2*x))*Log[x] + (-2*E 
^(2*x) + E^(6*x))*Log[x]^2 + E^E^x*(-8*E^(3*x) + 4*E^(7*x) + (-8*E^(3*x) + 
 4*E^(7*x))*Log[x] + (-2*E^(3*x) + E^(7*x))*Log[x]^2) + Log[2 - E^(4*x)]*( 
-2 - 8*E^x + E^(4*x)*(1 + 4*E^x) + (-4*E^x + 2*E^(5*x))*Log[x] + E^E^x*(-8 
*E^(2*x) + 4*E^(6*x) + (-4*E^(2*x) + 2*E^(6*x))*Log[x])))/(-8*E^(2*x) + 4* 
E^(6*x) + (-2 + E^(4*x))*Log[2 - E^(4*x)]^2 + (-8*E^(2*x) + 4*E^(6*x))*Log 
[x] + (-2*E^(2*x) + E^(6*x))*Log[x]^2 + Log[2 - E^(4*x)]*(-8*E^x + 4*E^(5* 
x) + (-4*E^x + 2*E^(5*x))*Log[x])),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97

\[x +{\mathrm e}^{{\mathrm e}^{x}}+\frac {x}{{\mathrm e}^{x} \ln \left (x \right )+2 \,{\mathrm e}^{x}+\ln \left (-{\mathrm e}^{4 x}+2\right )}\]

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (26) = 52\).

Time = 0.11 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.13 \[ \int \frac {-8 e^{2 x}+e^{4 x} \left (4 e^{2 x}+e^x (1-2 x)-4 x\right )+e^x (-2+4 x)+\left (-2+e^{4 x}+e^{e^x} \left (-2 e^x+e^{5 x}\right )\right ) \log ^2\left (2-e^{4 x}\right )+\left (-8 e^{2 x}+e^{4 x} \left (4 e^{2 x}+e^x (1-x)\right )+e^x (-2+2 x)\right ) \log (x)+\left (-2 e^{2 x}+e^{6 x}\right ) \log ^2(x)+e^{e^x} \left (-8 e^{3 x}+4 e^{7 x}+\left (-8 e^{3 x}+4 e^{7 x}\right ) \log (x)+\left (-2 e^{3 x}+e^{7 x}\right ) \log ^2(x)\right )+\log \left (2-e^{4 x}\right ) \left (-2-8 e^x+e^{4 x} \left (1+4 e^x\right )+\left (-4 e^x+2 e^{5 x}\right ) \log (x)+e^{e^x} \left (-8 e^{2 x}+4 e^{6 x}+\left (-4 e^{2 x}+2 e^{6 x}\right ) \log (x)\right )\right )}{-8 e^{2 x}+4 e^{6 x}+\left (-2+e^{4 x}\right ) \log ^2\left (2-e^{4 x}\right )+\left (-8 e^{2 x}+4 e^{6 x}\right ) \log (x)+\left (-2 e^{2 x}+e^{6 x}\right ) \log ^2(x)+\log \left (2-e^{4 x}\right ) \left (-8 e^x+4 e^{5 x}+\left (-4 e^x+2 e^{5 x}\right ) \log (x)\right )} \, dx=\frac {x e^{x} \log \left (x\right ) + 2 \, x e^{x} + {\left (e^{x} \log \left (x\right ) + 2 \, e^{x}\right )} e^{\left (e^{x}\right )} + {\left (x + e^{\left (e^{x}\right )}\right )} \log \left (-e^{\left (4 \, x\right )} + 2\right ) + x}{e^{x} \log \left (x\right ) + 2 \, e^{x} + \log \left (-e^{\left (4 \, x\right )} + 2\right )} \] Input:

integrate((((exp(x)*exp(4*x)-2*exp(x))*exp(exp(x))+exp(4*x)-2)*log(-exp(4* 
x)+2)^2+(((2*exp(x)^2*exp(4*x)-4*exp(x)^2)*log(x)+4*exp(x)^2*exp(4*x)-8*ex 
p(x)^2)*exp(exp(x))+(2*exp(x)*exp(4*x)-4*exp(x))*log(x)+(4*exp(x)+1)*exp(4 
*x)-8*exp(x)-2)*log(-exp(4*x)+2)+((exp(x)^3*exp(4*x)-2*exp(x)^3)*log(x)^2+ 
(4*exp(x)^3*exp(4*x)-8*exp(x)^3)*log(x)+4*exp(x)^3*exp(4*x)-8*exp(x)^3)*ex 
p(exp(x))+(exp(x)^2*exp(4*x)-2*exp(x)^2)*log(x)^2+((4*exp(x)^2+(1-x)*exp(x 
))*exp(4*x)-8*exp(x)^2+(2*x-2)*exp(x))*log(x)+(4*exp(x)^2+(1-2*x)*exp(x)-4 
*x)*exp(4*x)-8*exp(x)^2+(4*x-2)*exp(x))/((exp(4*x)-2)*log(-exp(4*x)+2)^2+( 
(2*exp(x)*exp(4*x)-4*exp(x))*log(x)+4*exp(x)*exp(4*x)-8*exp(x))*log(-exp(4 
*x)+2)+(exp(x)^2*exp(4*x)-2*exp(x)^2)*log(x)^2+(4*exp(x)^2*exp(4*x)-8*exp( 
x)^2)*log(x)+4*exp(x)^2*exp(4*x)-8*exp(x)^2),x, algorithm="fricas")
 

Output:

(x*e^x*log(x) + 2*x*e^x + (e^x*log(x) + 2*e^x)*e^(e^x) + (x + e^(e^x))*log 
(-e^(4*x) + 2) + x)/(e^x*log(x) + 2*e^x + log(-e^(4*x) + 2))
 

Sympy [A] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90 \[ \int \frac {-8 e^{2 x}+e^{4 x} \left (4 e^{2 x}+e^x (1-2 x)-4 x\right )+e^x (-2+4 x)+\left (-2+e^{4 x}+e^{e^x} \left (-2 e^x+e^{5 x}\right )\right ) \log ^2\left (2-e^{4 x}\right )+\left (-8 e^{2 x}+e^{4 x} \left (4 e^{2 x}+e^x (1-x)\right )+e^x (-2+2 x)\right ) \log (x)+\left (-2 e^{2 x}+e^{6 x}\right ) \log ^2(x)+e^{e^x} \left (-8 e^{3 x}+4 e^{7 x}+\left (-8 e^{3 x}+4 e^{7 x}\right ) \log (x)+\left (-2 e^{3 x}+e^{7 x}\right ) \log ^2(x)\right )+\log \left (2-e^{4 x}\right ) \left (-2-8 e^x+e^{4 x} \left (1+4 e^x\right )+\left (-4 e^x+2 e^{5 x}\right ) \log (x)+e^{e^x} \left (-8 e^{2 x}+4 e^{6 x}+\left (-4 e^{2 x}+2 e^{6 x}\right ) \log (x)\right )\right )}{-8 e^{2 x}+4 e^{6 x}+\left (-2+e^{4 x}\right ) \log ^2\left (2-e^{4 x}\right )+\left (-8 e^{2 x}+4 e^{6 x}\right ) \log (x)+\left (-2 e^{2 x}+e^{6 x}\right ) \log ^2(x)+\log \left (2-e^{4 x}\right ) \left (-8 e^x+4 e^{5 x}+\left (-4 e^x+2 e^{5 x}\right ) \log (x)\right )} \, dx=x + \frac {x}{e^{x} \log {\left (x \right )} + 2 e^{x} + \log {\left (2 - e^{4 x} \right )}} + e^{e^{x}} \] Input:

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (26) = 52\).

Time = 0.35 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.93 \[ \int \frac {-8 e^{2 x}+e^{4 x} \left (4 e^{2 x}+e^x (1-2 x)-4 x\right )+e^x (-2+4 x)+\left (-2+e^{4 x}+e^{e^x} \left (-2 e^x+e^{5 x}\right )\right ) \log ^2\left (2-e^{4 x}\right )+\left (-8 e^{2 x}+e^{4 x} \left (4 e^{2 x}+e^x (1-x)\right )+e^x (-2+2 x)\right ) \log (x)+\left (-2 e^{2 x}+e^{6 x}\right ) \log ^2(x)+e^{e^x} \left (-8 e^{3 x}+4 e^{7 x}+\left (-8 e^{3 x}+4 e^{7 x}\right ) \log (x)+\left (-2 e^{3 x}+e^{7 x}\right ) \log ^2(x)\right )+\log \left (2-e^{4 x}\right ) \left (-2-8 e^x+e^{4 x} \left (1+4 e^x\right )+\left (-4 e^x+2 e^{5 x}\right ) \log (x)+e^{e^x} \left (-8 e^{2 x}+4 e^{6 x}+\left (-4 e^{2 x}+2 e^{6 x}\right ) \log (x)\right )\right )}{-8 e^{2 x}+4 e^{6 x}+\left (-2+e^{4 x}\right ) \log ^2\left (2-e^{4 x}\right )+\left (-8 e^{2 x}+4 e^{6 x}\right ) \log (x)+\left (-2 e^{2 x}+e^{6 x}\right ) \log ^2(x)+\log \left (2-e^{4 x}\right ) \left (-8 e^x+4 e^{5 x}+\left (-4 e^x+2 e^{5 x}\right ) \log (x)\right )} \, dx=\frac {{\left (\log \left (x\right ) + 2\right )} e^{\left (x + e^{x}\right )} + {\left (x \log \left (x\right ) + 2 \, x\right )} e^{x} + {\left (x + e^{\left (e^{x}\right )}\right )} \log \left (-e^{\left (4 \, x\right )} + 2\right ) + x}{{\left (\log \left (x\right ) + 2\right )} e^{x} + \log \left (-e^{\left (4 \, x\right )} + 2\right )} \] Input:

integrate((((exp(x)*exp(4*x)-2*exp(x))*exp(exp(x))+exp(4*x)-2)*log(-exp(4* 
x)+2)^2+(((2*exp(x)^2*exp(4*x)-4*exp(x)^2)*log(x)+4*exp(x)^2*exp(4*x)-8*ex 
p(x)^2)*exp(exp(x))+(2*exp(x)*exp(4*x)-4*exp(x))*log(x)+(4*exp(x)+1)*exp(4 
*x)-8*exp(x)-2)*log(-exp(4*x)+2)+((exp(x)^3*exp(4*x)-2*exp(x)^3)*log(x)^2+ 
(4*exp(x)^3*exp(4*x)-8*exp(x)^3)*log(x)+4*exp(x)^3*exp(4*x)-8*exp(x)^3)*ex 
p(exp(x))+(exp(x)^2*exp(4*x)-2*exp(x)^2)*log(x)^2+((4*exp(x)^2+(1-x)*exp(x 
))*exp(4*x)-8*exp(x)^2+(2*x-2)*exp(x))*log(x)+(4*exp(x)^2+(1-2*x)*exp(x)-4 
*x)*exp(4*x)-8*exp(x)^2+(4*x-2)*exp(x))/((exp(4*x)-2)*log(-exp(4*x)+2)^2+( 
(2*exp(x)*exp(4*x)-4*exp(x))*log(x)+4*exp(x)*exp(4*x)-8*exp(x))*log(-exp(4 
*x)+2)+(exp(x)^2*exp(4*x)-2*exp(x)^2)*log(x)^2+(4*exp(x)^2*exp(4*x)-8*exp( 
x)^2)*log(x)+4*exp(x)^2*exp(4*x)-8*exp(x)^2),x, algorithm="maxima")
 

Output:

((log(x) + 2)*e^(x + e^x) + (x*log(x) + 2*x)*e^x + (x + e^(e^x))*log(-e^(4 
*x) + 2) + x)/((log(x) + 2)*e^x + log(-e^(4*x) + 2))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 736 vs. \(2 (26) = 52\).

Time = 0.40 (sec) , antiderivative size = 736, normalized size of antiderivative = 24.53 \[ \int \frac {-8 e^{2 x}+e^{4 x} \left (4 e^{2 x}+e^x (1-2 x)-4 x\right )+e^x (-2+4 x)+\left (-2+e^{4 x}+e^{e^x} \left (-2 e^x+e^{5 x}\right )\right ) \log ^2\left (2-e^{4 x}\right )+\left (-8 e^{2 x}+e^{4 x} \left (4 e^{2 x}+e^x (1-x)\right )+e^x (-2+2 x)\right ) \log (x)+\left (-2 e^{2 x}+e^{6 x}\right ) \log ^2(x)+e^{e^x} \left (-8 e^{3 x}+4 e^{7 x}+\left (-8 e^{3 x}+4 e^{7 x}\right ) \log (x)+\left (-2 e^{3 x}+e^{7 x}\right ) \log ^2(x)\right )+\log \left (2-e^{4 x}\right ) \left (-2-8 e^x+e^{4 x} \left (1+4 e^x\right )+\left (-4 e^x+2 e^{5 x}\right ) \log (x)+e^{e^x} \left (-8 e^{2 x}+4 e^{6 x}+\left (-4 e^{2 x}+2 e^{6 x}\right ) \log (x)\right )\right )}{-8 e^{2 x}+4 e^{6 x}+\left (-2+e^{4 x}\right ) \log ^2\left (2-e^{4 x}\right )+\left (-8 e^{2 x}+4 e^{6 x}\right ) \log (x)+\left (-2 e^{2 x}+e^{6 x}\right ) \log ^2(x)+\log \left (2-e^{4 x}\right ) \left (-8 e^x+4 e^{5 x}+\left (-4 e^x+2 e^{5 x}\right ) \log (x)\right )} \, dx=\text {Too large to display} \] Input:

integrate((((exp(x)*exp(4*x)-2*exp(x))*exp(exp(x))+exp(4*x)-2)*log(-exp(4* 
x)+2)^2+(((2*exp(x)^2*exp(4*x)-4*exp(x)^2)*log(x)+4*exp(x)^2*exp(4*x)-8*ex 
p(x)^2)*exp(exp(x))+(2*exp(x)*exp(4*x)-4*exp(x))*log(x)+(4*exp(x)+1)*exp(4 
*x)-8*exp(x)-2)*log(-exp(4*x)+2)+((exp(x)^3*exp(4*x)-2*exp(x)^3)*log(x)^2+ 
(4*exp(x)^3*exp(4*x)-8*exp(x)^3)*log(x)+4*exp(x)^3*exp(4*x)-8*exp(x)^3)*ex 
p(exp(x))+(exp(x)^2*exp(4*x)-2*exp(x)^2)*log(x)^2+((4*exp(x)^2+(1-x)*exp(x 
))*exp(4*x)-8*exp(x)^2+(2*x-2)*exp(x))*log(x)+(4*exp(x)^2+(1-2*x)*exp(x)-4 
*x)*exp(4*x)-8*exp(x)^2+(4*x-2)*exp(x))/((exp(4*x)-2)*log(-exp(4*x)+2)^2+( 
(2*exp(x)*exp(4*x)-4*exp(x))*log(x)+4*exp(x)*exp(4*x)-8*exp(x))*log(-exp(4 
*x)+2)+(exp(x)^2*exp(4*x)-2*exp(x)^2)*log(x)^2+(4*exp(x)^2*exp(4*x)-8*exp( 
x)^2)*log(x)+4*exp(x)^2*exp(4*x)-8*exp(x)^2),x, algorithm="giac")
 

Output:

(x^2*e^(6*x)*log(x)^2 - 2*x^2*e^(2*x)*log(x)^2 + x^2*e^(5*x)*log(x)*log(-e 
^(4*x) + 2) - 2*x^2*e^x*log(x)*log(-e^(4*x) + 2) + 4*x^2*e^(6*x)*log(x) + 
5*x^2*e^(5*x)*log(x) - 8*x^2*e^(2*x)*log(x) - 2*x^2*e^x*log(x) + x*e^(6*x 
+ e^x)*log(x)^2 - 2*x*e^(2*x + e^x)*log(x)^2 + 2*x^2*e^(5*x)*log(-e^(4*x) 
+ 2) + 4*x^2*e^(4*x)*log(-e^(4*x) + 2) - 4*x^2*e^x*log(-e^(4*x) + 2) + x*e 
^(5*x + e^x)*log(x)*log(-e^(4*x) + 2) - 2*x*e^(x + e^x)*log(x)*log(-e^(4*x 
) + 2) + 4*x^2*e^(6*x) + 10*x^2*e^(5*x) + 4*x^2*e^(4*x) - 8*x^2*e^(2*x) - 
4*x^2*e^x + x*e^(6*x)*log(x) - 2*x*e^(2*x)*log(x) + 4*x*e^(6*x + e^x)*log( 
x) + 4*x*e^(5*x + e^x)*log(x) - 8*x*e^(2*x + e^x)*log(x) + x*e^(5*x)*log(- 
e^(4*x) + 2) + 2*x*e^(5*x + e^x)*log(-e^(4*x) + 2) + 4*x*e^(4*x + e^x)*log 
(-e^(4*x) + 2) - 4*x*e^(x + e^x)*log(-e^(4*x) + 2) - 2*x*e^x*log(-e^(4*x) 
+ 2) + 2*x*e^(6*x) + x*e^(5*x) - 4*x*e^(2*x) + 4*x*e^(6*x + e^x) + 8*x*e^( 
5*x + e^x) - 8*x*e^(2*x + e^x) - 2*x*e^x + e^(6*x + e^x)*log(x) - 2*e^(2*x 
 + e^x)*log(x) + e^(5*x + e^x)*log(-e^(4*x) + 2) - 2*e^(x + e^x)*log(-e^(4 
*x) + 2) + 2*e^(6*x + e^x) - 4*e^(2*x + e^x))/(x*e^(6*x)*log(x)^2 - 2*x*e^ 
(2*x)*log(x)^2 + x*e^(5*x)*log(x)*log(-e^(4*x) + 2) - 2*x*e^x*log(x)*log(- 
e^(4*x) + 2) + 4*x*e^(6*x)*log(x) + 4*x*e^(5*x)*log(x) - 8*x*e^(2*x)*log(x 
) + 2*x*e^(5*x)*log(-e^(4*x) + 2) + 4*x*e^(4*x)*log(-e^(4*x) + 2) - 4*x*e^ 
x*log(-e^(4*x) + 2) + 4*x*e^(6*x) + 8*x*e^(5*x) - 8*x*e^(2*x) + e^(6*x)*lo 
g(x) - 2*e^(2*x)*log(x) + e^(5*x)*log(-e^(4*x) + 2) - 2*e^x*log(-e^(4*x...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-8 e^{2 x}+e^{4 x} \left (4 e^{2 x}+e^x (1-2 x)-4 x\right )+e^x (-2+4 x)+\left (-2+e^{4 x}+e^{e^x} \left (-2 e^x+e^{5 x}\right )\right ) \log ^2\left (2-e^{4 x}\right )+\left (-8 e^{2 x}+e^{4 x} \left (4 e^{2 x}+e^x (1-x)\right )+e^x (-2+2 x)\right ) \log (x)+\left (-2 e^{2 x}+e^{6 x}\right ) \log ^2(x)+e^{e^x} \left (-8 e^{3 x}+4 e^{7 x}+\left (-8 e^{3 x}+4 e^{7 x}\right ) \log (x)+\left (-2 e^{3 x}+e^{7 x}\right ) \log ^2(x)\right )+\log \left (2-e^{4 x}\right ) \left (-2-8 e^x+e^{4 x} \left (1+4 e^x\right )+\left (-4 e^x+2 e^{5 x}\right ) \log (x)+e^{e^x} \left (-8 e^{2 x}+4 e^{6 x}+\left (-4 e^{2 x}+2 e^{6 x}\right ) \log (x)\right )\right )}{-8 e^{2 x}+4 e^{6 x}+\left (-2+e^{4 x}\right ) \log ^2\left (2-e^{4 x}\right )+\left (-8 e^{2 x}+4 e^{6 x}\right ) \log (x)+\left (-2 e^{2 x}+e^{6 x}\right ) \log ^2(x)+\log \left (2-e^{4 x}\right ) \left (-8 e^x+4 e^{5 x}+\left (-4 e^x+2 e^{5 x}\right ) \log (x)\right )} \, dx=\int \frac {8\,{\mathrm {e}}^{2\,x}+\ln \left (2-{\mathrm {e}}^{4\,x}\right )\,\left (8\,{\mathrm {e}}^x+{\mathrm {e}}^{{\mathrm {e}}^x}\,\left (8\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+\ln \left (x\right )\,\left (4\,{\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^{6\,x}\right )\right )-{\mathrm {e}}^{4\,x}\,\left (4\,{\mathrm {e}}^x+1\right )-\ln \left (x\right )\,\left (2\,{\mathrm {e}}^{5\,x}-4\,{\mathrm {e}}^x\right )+2\right )+{\ln \left (x\right )}^2\,\left (2\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^{6\,x}\right )-{\ln \left (2-{\mathrm {e}}^{4\,x}\right )}^2\,\left ({\mathrm {e}}^{4\,x}+{\mathrm {e}}^{{\mathrm {e}}^x}\,\left ({\mathrm {e}}^{5\,x}-2\,{\mathrm {e}}^x\right )-2\right )+{\mathrm {e}}^{{\mathrm {e}}^x}\,\left (\left (2\,{\mathrm {e}}^{3\,x}-{\mathrm {e}}^{7\,x}\right )\,{\ln \left (x\right )}^2+\left (8\,{\mathrm {e}}^{3\,x}-4\,{\mathrm {e}}^{7\,x}\right )\,\ln \left (x\right )+8\,{\mathrm {e}}^{3\,x}-4\,{\mathrm {e}}^{7\,x}\right )-\ln \left (x\right )\,\left ({\mathrm {e}}^x\,\left (2\,x-2\right )-8\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}\,\left (4\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^x\,\left (x-1\right )\right )\right )-{\mathrm {e}}^x\,\left (4\,x-2\right )+{\mathrm {e}}^{4\,x}\,\left (4\,x-4\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,\left (2\,x-1\right )\right )}{8\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\ln \left (x\right )}^2\,\left (2\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^{6\,x}\right )-\ln \left (2-{\mathrm {e}}^{4\,x}\right )\,\left (4\,{\mathrm {e}}^{5\,x}-8\,{\mathrm {e}}^x+\ln \left (x\right )\,\left (2\,{\mathrm {e}}^{5\,x}-4\,{\mathrm {e}}^x\right )\right )-{\ln \left (2-{\mathrm {e}}^{4\,x}\right )}^2\,\left ({\mathrm {e}}^{4\,x}-2\right )+\ln \left (x\right )\,\left (8\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}\right )} \,d x \] Input:

int((8*exp(2*x) + log(2 - exp(4*x))*(8*exp(x) + exp(exp(x))*(8*exp(2*x) - 
4*exp(6*x) + log(x)*(4*exp(2*x) - 2*exp(6*x))) - exp(4*x)*(4*exp(x) + 1) - 
 log(x)*(2*exp(5*x) - 4*exp(x)) + 2) + log(x)^2*(2*exp(2*x) - exp(6*x)) - 
log(2 - exp(4*x))^2*(exp(4*x) + exp(exp(x))*(exp(5*x) - 2*exp(x)) - 2) + e 
xp(exp(x))*(8*exp(3*x) - 4*exp(7*x) + log(x)^2*(2*exp(3*x) - exp(7*x)) + l 
og(x)*(8*exp(3*x) - 4*exp(7*x))) - log(x)*(exp(x)*(2*x - 2) - 8*exp(2*x) + 
 exp(4*x)*(4*exp(2*x) - exp(x)*(x - 1))) - exp(x)*(4*x - 2) + exp(4*x)*(4* 
x - 4*exp(2*x) + exp(x)*(2*x - 1)))/(8*exp(2*x) - 4*exp(6*x) + log(x)^2*(2 
*exp(2*x) - exp(6*x)) - log(2 - exp(4*x))*(4*exp(5*x) - 8*exp(x) + log(x)* 
(2*exp(5*x) - 4*exp(x))) - log(2 - exp(4*x))^2*(exp(4*x) - 2) + log(x)*(8* 
exp(2*x) - 4*exp(6*x))),x)
 

Output:

int((8*exp(2*x) + log(2 - exp(4*x))*(8*exp(x) + exp(exp(x))*(8*exp(2*x) - 
4*exp(6*x) + log(x)*(4*exp(2*x) - 2*exp(6*x))) - exp(4*x)*(4*exp(x) + 1) - 
 log(x)*(2*exp(5*x) - 4*exp(x)) + 2) + log(x)^2*(2*exp(2*x) - exp(6*x)) - 
log(2 - exp(4*x))^2*(exp(4*x) + exp(exp(x))*(exp(5*x) - 2*exp(x)) - 2) + e 
xp(exp(x))*(8*exp(3*x) - 4*exp(7*x) + log(x)^2*(2*exp(3*x) - exp(7*x)) + l 
og(x)*(8*exp(3*x) - 4*exp(7*x))) - log(x)*(exp(x)*(2*x - 2) - 8*exp(2*x) + 
 exp(4*x)*(4*exp(2*x) - exp(x)*(x - 1))) - exp(x)*(4*x - 2) + exp(4*x)*(4* 
x - 4*exp(2*x) + exp(x)*(2*x - 1)))/(8*exp(2*x) - 4*exp(6*x) + log(x)^2*(2 
*exp(2*x) - exp(6*x)) - log(2 - exp(4*x))*(4*exp(5*x) - 8*exp(x) + log(x)* 
(2*exp(5*x) - 4*exp(x))) - log(2 - exp(4*x))^2*(exp(4*x) - 2) + log(x)*(8* 
exp(2*x) - 4*exp(6*x))), x)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.90 \[ \int \frac {-8 e^{2 x}+e^{4 x} \left (4 e^{2 x}+e^x (1-2 x)-4 x\right )+e^x (-2+4 x)+\left (-2+e^{4 x}+e^{e^x} \left (-2 e^x+e^{5 x}\right )\right ) \log ^2\left (2-e^{4 x}\right )+\left (-8 e^{2 x}+e^{4 x} \left (4 e^{2 x}+e^x (1-x)\right )+e^x (-2+2 x)\right ) \log (x)+\left (-2 e^{2 x}+e^{6 x}\right ) \log ^2(x)+e^{e^x} \left (-8 e^{3 x}+4 e^{7 x}+\left (-8 e^{3 x}+4 e^{7 x}\right ) \log (x)+\left (-2 e^{3 x}+e^{7 x}\right ) \log ^2(x)\right )+\log \left (2-e^{4 x}\right ) \left (-2-8 e^x+e^{4 x} \left (1+4 e^x\right )+\left (-4 e^x+2 e^{5 x}\right ) \log (x)+e^{e^x} \left (-8 e^{2 x}+4 e^{6 x}+\left (-4 e^{2 x}+2 e^{6 x}\right ) \log (x)\right )\right )}{-8 e^{2 x}+4 e^{6 x}+\left (-2+e^{4 x}\right ) \log ^2\left (2-e^{4 x}\right )+\left (-8 e^{2 x}+4 e^{6 x}\right ) \log (x)+\left (-2 e^{2 x}+e^{6 x}\right ) \log ^2(x)+\log \left (2-e^{4 x}\right ) \left (-8 e^x+4 e^{5 x}+\left (-4 e^x+2 e^{5 x}\right ) \log (x)\right )} \, dx=\frac {e^{e^{x}+x} \mathrm {log}\left (x \right )+2 e^{e^{x}+x}+e^{e^{x}} \mathrm {log}\left (-e^{4 x}+2\right )+e^{x} \mathrm {log}\left (x \right ) x +2 e^{x} x +\mathrm {log}\left (-e^{4 x}+2\right ) x +x}{e^{x} \mathrm {log}\left (x \right )+2 e^{x}+\mathrm {log}\left (-e^{4 x}+2\right )} \] Input:

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

Output:

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