\(\int \frac {-16-27 x-2 x^2+(3 x+x^2) \log (e^{4 x} x^4)+(5 x-2 x^2+(-4 x-x^2) \log (e^{4 x} x^4)+(-5+2 x+(4+x) \log (e^{4 x} x^4)) \log (-5+2 x+(4+x) \log (e^{4 x} x^4))) \log (x-\log (-5+2 x+(4+x) \log (e^{4 x} x^4)))}{(5 x-2 x^2+(-4 x-x^2) \log (e^{4 x} x^4)+(-5+2 x+(4+x) \log (e^{4 x} x^4)) \log (-5+2 x+(4+x) \log (e^{4 x} x^4))) \log ^2(x-\log (-5+2 x+(4+x) \log (e^{4 x} x^4)))} \, dx\) [2484]

Optimal result

Integrand size = 222, antiderivative size = 31 \[ \int \frac {-16-27 x-2 x^2+\left (3 x+x^2\right ) \log \left (e^{4 x} x^4\right )+\left (5 x-2 x^2+\left (-4 x-x^2\right ) \log \left (e^{4 x} x^4\right )+\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )}{\left (5 x-2 x^2+\left (-4 x-x^2\right ) \log \left (e^{4 x} x^4\right )+\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx=-5+\frac {x}{\log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \] Output:

x/ln(-ln((4+x)*ln(x^4*exp(x)^4)+2*x-5)+x)-5
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int \frac {-16-27 x-2 x^2+\left (3 x+x^2\right ) \log \left (e^{4 x} x^4\right )+\left (5 x-2 x^2+\left (-4 x-x^2\right ) \log \left (e^{4 x} x^4\right )+\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )}{\left (5 x-2 x^2+\left (-4 x-x^2\right ) \log \left (e^{4 x} x^4\right )+\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx=\frac {x}{\log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \] Input:

Integrate[(-16 - 27*x - 2*x^2 + (3*x + x^2)*Log[E^(4*x)*x^4] + (5*x - 2*x^ 
2 + (-4*x - x^2)*Log[E^(4*x)*x^4] + (-5 + 2*x + (4 + x)*Log[E^(4*x)*x^4])* 
Log[-5 + 2*x + (4 + x)*Log[E^(4*x)*x^4]])*Log[x - Log[-5 + 2*x + (4 + x)*L 
og[E^(4*x)*x^4]]])/((5*x - 2*x^2 + (-4*x - x^2)*Log[E^(4*x)*x^4] + (-5 + 2 
*x + (4 + x)*Log[E^(4*x)*x^4])*Log[-5 + 2*x + (4 + x)*Log[E^(4*x)*x^4]])*L 
og[x - Log[-5 + 2*x + (4 + x)*Log[E^(4*x)*x^4]]]^2),x]
 

Output:

x/Log[x - Log[-5 + 2*x + (4 + x)*Log[E^(4*x)*x^4]]]
 

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.

\(\displaystyle \int \frac {-2 x^2+\left (x^2+3 x\right ) \log \left (e^{4 x} x^4\right )+\left (\left ((x+4) \log \left (e^{4 x} x^4\right )+2 x-5\right ) \log \left ((x+4) \log \left (e^{4 x} x^4\right )+2 x-5\right )-2 x^2+\left (-x^2-4 x\right ) \log \left (e^{4 x} x^4\right )+5 x\right ) \log \left (x-\log \left ((x+4) \log \left (e^{4 x} x^4\right )+2 x-5\right )\right )-27 x-16}{\left (\left ((x+4) \log \left (e^{4 x} x^4\right )+2 x-5\right ) \log \left ((x+4) \log \left (e^{4 x} x^4\right )+2 x-5\right )-2 x^2+\left (-x^2-4 x\right ) \log \left (e^{4 x} x^4\right )+5 x\right ) \log ^2\left (x-\log \left ((x+4) \log \left (e^{4 x} x^4\right )+2 x-5\right )\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {-2 x^2+\left (x^2+3 x\right ) \log \left (e^{4 x} x^4\right )+\left (\left ((x+4) \log \left (e^{4 x} x^4\right )+2 x-5\right ) \log \left ((x+4) \log \left (e^{4 x} x^4\right )+2 x-5\right )-2 x^2+\left (-x^2-4 x\right ) \log \left (e^{4 x} x^4\right )+5 x\right ) \log \left (x-\log \left ((x+4) \log \left (e^{4 x} x^4\right )+2 x-5\right )\right )-27 x-16}{\left (x \left (-\log \left (e^{4 x} x^4\right )\right )-4 \log \left (e^{4 x} x^4\right )-2 x+5\right ) \left (x-\log \left ((x+4) \log \left (e^{4 x} x^4\right )+2 x-5\right )\right ) \log ^2\left (x-\log \left ((x+4) \log \left (e^{4 x} x^4\right )+2 x-5\right )\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {1}{\log \left (x-\log \left ((x+4) \log \left (e^{4 x} x^4\right )+2 x-5\right )\right )}+\frac {-3 x \log \left (e^{4 x} x^4\right )+2 x^2+x^2 \left (-\log \left (e^{4 x} x^4\right )\right )+27 x+16}{\left (x \log \left (e^{4 x} x^4\right )+4 \log \left (e^{4 x} x^4\right )+2 x-5\right ) \left (x-\log \left ((x+4) \log \left (e^{4 x} x^4\right )+2 x-5\right )\right ) \log ^2\left (x-\log \left ((x+4) \log \left (e^{4 x} x^4\right )+2 x-5\right )\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 16 \int \frac {1}{\left (\log \left (e^{4 x} x^4\right ) x+2 x+4 \log \left (e^{4 x} x^4\right )-5\right ) \left (x-\log \left (2 x+(x+4) \log \left (e^{4 x} x^4\right )-5\right )\right ) \log ^2\left (x-\log \left (2 x+(x+4) \log \left (e^{4 x} x^4\right )-5\right )\right )}dx+27 \int \frac {x}{\left (\log \left (e^{4 x} x^4\right ) x+2 x+4 \log \left (e^{4 x} x^4\right )-5\right ) \left (x-\log \left (2 x+(x+4) \log \left (e^{4 x} x^4\right )-5\right )\right ) \log ^2\left (x-\log \left (2 x+(x+4) \log \left (e^{4 x} x^4\right )-5\right )\right )}dx-3 \int \frac {x \log \left (e^{4 x} x^4\right )}{\left (\log \left (e^{4 x} x^4\right ) x+2 x+4 \log \left (e^{4 x} x^4\right )-5\right ) \left (x-\log \left (2 x+(x+4) \log \left (e^{4 x} x^4\right )-5\right )\right ) \log ^2\left (x-\log \left (2 x+(x+4) \log \left (e^{4 x} x^4\right )-5\right )\right )}dx+\int \frac {1}{\log \left (x-\log \left (2 x+(x+4) \log \left (e^{4 x} x^4\right )-5\right )\right )}dx+2 \int \frac {x^2}{\left (\log \left (e^{4 x} x^4\right ) x+2 x+4 \log \left (e^{4 x} x^4\right )-5\right ) \left (x-\log \left (2 x+(x+4) \log \left (e^{4 x} x^4\right )-5\right )\right ) \log ^2\left (x-\log \left (2 x+(x+4) \log \left (e^{4 x} x^4\right )-5\right )\right )}dx-\int \frac {x^2 \log \left (e^{4 x} x^4\right )}{\left (\log \left (e^{4 x} x^4\right ) x+2 x+4 \log \left (e^{4 x} x^4\right )-5\right ) \left (x-\log \left (2 x+(x+4) \log \left (e^{4 x} x^4\right )-5\right )\right ) \log ^2\left (x-\log \left (2 x+(x+4) \log \left (e^{4 x} x^4\right )-5\right )\right )}dx\)

Input:

Int[(-16 - 27*x - 2*x^2 + (3*x + x^2)*Log[E^(4*x)*x^4] + (5*x - 2*x^2 + (- 
4*x - x^2)*Log[E^(4*x)*x^4] + (-5 + 2*x + (4 + x)*Log[E^(4*x)*x^4])*Log[-5 
 + 2*x + (4 + x)*Log[E^(4*x)*x^4]])*Log[x - Log[-5 + 2*x + (4 + x)*Log[E^( 
4*x)*x^4]]])/((5*x - 2*x^2 + (-4*x - x^2)*Log[E^(4*x)*x^4] + (-5 + 2*x + ( 
4 + x)*Log[E^(4*x)*x^4])*Log[-5 + 2*x + (4 + x)*Log[E^(4*x)*x^4]])*Log[x - 
 Log[-5 + 2*x + (4 + x)*Log[E^(4*x)*x^4]]]^2),x]
 

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

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

Time = 15.86 (sec) , antiderivative size = 329, normalized size of antiderivative = 10.61

\[\frac {x}{\ln \left (-\ln \left (\left (4+x \right ) \left (4 \ln \left (x \right )+4 \ln \left ({\mathrm e}^{x}\right )-\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right ) {\left (-\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )+\operatorname {csgn}\left (i {\mathrm e}^{x}\right )\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{3 x}\right ) \left (-\operatorname {csgn}\left (i {\mathrm e}^{3 x}\right )+\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )\right ) \left (-\operatorname {csgn}\left (i {\mathrm e}^{3 x}\right )+\operatorname {csgn}\left (i {\mathrm e}^{x}\right )\right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{4 x}\right ) \left (-\operatorname {csgn}\left (i {\mathrm e}^{4 x}\right )+\operatorname {csgn}\left (i {\mathrm e}^{3 x}\right )\right ) \left (-\operatorname {csgn}\left (i {\mathrm e}^{4 x}\right )+\operatorname {csgn}\left (i {\mathrm e}^{x}\right )\right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) {\left (-\operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x \right )\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i x^{3}\right ) \left (-\operatorname {csgn}\left (i x^{3}\right )+\operatorname {csgn}\left (i x^{2}\right )\right ) \left (-\operatorname {csgn}\left (i x^{3}\right )+\operatorname {csgn}\left (i x \right )\right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (i x^{4}\right ) \left (-\operatorname {csgn}\left (i x^{4}\right )+\operatorname {csgn}\left (i x^{3}\right )\right ) \left (-\operatorname {csgn}\left (i x^{4}\right )+\operatorname {csgn}\left (i x \right )\right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (i x^{4} {\mathrm e}^{4 x}\right ) \left (-\operatorname {csgn}\left (i x^{4} {\mathrm e}^{4 x}\right )+\operatorname {csgn}\left (i x^{4}\right )\right ) \left (-\operatorname {csgn}\left (i x^{4} {\mathrm e}^{4 x}\right )+\operatorname {csgn}\left (i {\mathrm e}^{4 x}\right )\right )}{2}\right )+2 x -5\right )+x \right )}\]

Input:

int(((((4+x)*ln(x^4*exp(x)^4)+2*x-5)*ln((4+x)*ln(x^4*exp(x)^4)+2*x-5)+(-x^ 
2-4*x)*ln(x^4*exp(x)^4)-2*x^2+5*x)*ln(-ln((4+x)*ln(x^4*exp(x)^4)+2*x-5)+x) 
+(x^2+3*x)*ln(x^4*exp(x)^4)-2*x^2-27*x-16)/(((4+x)*ln(x^4*exp(x)^4)+2*x-5) 
*ln((4+x)*ln(x^4*exp(x)^4)+2*x-5)+(-x^2-4*x)*ln(x^4*exp(x)^4)-2*x^2+5*x)/l 
n(-ln((4+x)*ln(x^4*exp(x)^4)+2*x-5)+x)^2,x)
 

Output:

x/ln(-ln((4+x)*(4*ln(x)+4*ln(exp(x))-1/2*I*Pi*csgn(I*exp(2*x))*(-csgn(I*ex 
p(2*x))+csgn(I*exp(x)))^2-1/2*I*Pi*csgn(I*exp(3*x))*(-csgn(I*exp(3*x))+csg 
n(I*exp(2*x)))*(-csgn(I*exp(3*x))+csgn(I*exp(x)))-1/2*I*Pi*csgn(I*exp(4*x) 
)*(-csgn(I*exp(4*x))+csgn(I*exp(3*x)))*(-csgn(I*exp(4*x))+csgn(I*exp(x)))- 
1/2*I*Pi*csgn(I*x^2)*(-csgn(I*x^2)+csgn(I*x))^2-1/2*I*Pi*csgn(I*x^3)*(-csg 
n(I*x^3)+csgn(I*x^2))*(-csgn(I*x^3)+csgn(I*x))-1/2*I*Pi*csgn(I*x^4)*(-csgn 
(I*x^4)+csgn(I*x^3))*(-csgn(I*x^4)+csgn(I*x))-1/2*I*Pi*csgn(I*x^4*exp(4*x) 
)*(-csgn(I*x^4*exp(4*x))+csgn(I*x^4))*(-csgn(I*x^4*exp(4*x))+csgn(I*exp(4* 
x))))+2*x-5)+x)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \[ \int \frac {-16-27 x-2 x^2+\left (3 x+x^2\right ) \log \left (e^{4 x} x^4\right )+\left (5 x-2 x^2+\left (-4 x-x^2\right ) \log \left (e^{4 x} x^4\right )+\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )}{\left (5 x-2 x^2+\left (-4 x-x^2\right ) \log \left (e^{4 x} x^4\right )+\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx=\frac {x}{\log \left (x - \log \left ({\left (x + 4\right )} \log \left (x^{4} e^{\left (4 \, x\right )}\right ) + 2 \, x - 5\right )\right )} \] Input:

integrate(((((4+x)*log(x^4*exp(x)^4)+2*x-5)*log((4+x)*log(x^4*exp(x)^4)+2* 
x-5)+(-x^2-4*x)*log(x^4*exp(x)^4)-2*x^2+5*x)*log(-log((4+x)*log(x^4*exp(x) 
^4)+2*x-5)+x)+(x^2+3*x)*log(x^4*exp(x)^4)-2*x^2-27*x-16)/(((4+x)*log(x^4*e 
xp(x)^4)+2*x-5)*log((4+x)*log(x^4*exp(x)^4)+2*x-5)+(-x^2-4*x)*log(x^4*exp( 
x)^4)-2*x^2+5*x)/log(-log((4+x)*log(x^4*exp(x)^4)+2*x-5)+x)^2,x, algorithm 
="fricas")
 

Output:

x/log(x - log((x + 4)*log(x^4*e^(4*x)) + 2*x - 5))
 

Sympy [F(-1)]

Timed out. \[ \int \frac {-16-27 x-2 x^2+\left (3 x+x^2\right ) \log \left (e^{4 x} x^4\right )+\left (5 x-2 x^2+\left (-4 x-x^2\right ) \log \left (e^{4 x} x^4\right )+\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )}{\left (5 x-2 x^2+\left (-4 x-x^2\right ) \log \left (e^{4 x} x^4\right )+\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx=\text {Timed out} \] Input:

integrate(((((4+x)*ln(x**4*exp(x)**4)+2*x-5)*ln((4+x)*ln(x**4*exp(x)**4)+2 
*x-5)+(-x**2-4*x)*ln(x**4*exp(x)**4)-2*x**2+5*x)*ln(-ln((4+x)*ln(x**4*exp( 
x)**4)+2*x-5)+x)+(x**2+3*x)*ln(x**4*exp(x)**4)-2*x**2-27*x-16)/(((4+x)*ln( 
x**4*exp(x)**4)+2*x-5)*ln((4+x)*ln(x**4*exp(x)**4)+2*x-5)+(-x**2-4*x)*ln(x 
**4*exp(x)**4)-2*x**2+5*x)/ln(-ln((4+x)*ln(x**4*exp(x)**4)+2*x-5)+x)**2,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {-16-27 x-2 x^2+\left (3 x+x^2\right ) \log \left (e^{4 x} x^4\right )+\left (5 x-2 x^2+\left (-4 x-x^2\right ) \log \left (e^{4 x} x^4\right )+\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )}{\left (5 x-2 x^2+\left (-4 x-x^2\right ) \log \left (e^{4 x} x^4\right )+\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx=\frac {x}{\log \left (x - \log \left (4 \, x^{2} + 4 \, {\left (x + 4\right )} \log \left (x\right ) + 18 \, x - 5\right )\right )} \] Input:

integrate(((((4+x)*log(x^4*exp(x)^4)+2*x-5)*log((4+x)*log(x^4*exp(x)^4)+2* 
x-5)+(-x^2-4*x)*log(x^4*exp(x)^4)-2*x^2+5*x)*log(-log((4+x)*log(x^4*exp(x) 
^4)+2*x-5)+x)+(x^2+3*x)*log(x^4*exp(x)^4)-2*x^2-27*x-16)/(((4+x)*log(x^4*e 
xp(x)^4)+2*x-5)*log((4+x)*log(x^4*exp(x)^4)+2*x-5)+(-x^2-4*x)*log(x^4*exp( 
x)^4)-2*x^2+5*x)/log(-log((4+x)*log(x^4*exp(x)^4)+2*x-5)+x)^2,x, algorithm 
="maxima")
 

Output:

x/log(x - log(4*x^2 + 4*(x + 4)*log(x) + 18*x - 5))
 

Giac [A] (verification not implemented)

Time = 1.79 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {-16-27 x-2 x^2+\left (3 x+x^2\right ) \log \left (e^{4 x} x^4\right )+\left (5 x-2 x^2+\left (-4 x-x^2\right ) \log \left (e^{4 x} x^4\right )+\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )}{\left (5 x-2 x^2+\left (-4 x-x^2\right ) \log \left (e^{4 x} x^4\right )+\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx=\frac {x}{\log \left (x - \log \left (4 \, x^{2} + x \log \left (x^{4}\right ) + 18 \, x + 4 \, \log \left (x^{4}\right ) - 5\right )\right )} \] Input:

integrate(((((4+x)*log(x^4*exp(x)^4)+2*x-5)*log((4+x)*log(x^4*exp(x)^4)+2* 
x-5)+(-x^2-4*x)*log(x^4*exp(x)^4)-2*x^2+5*x)*log(-log((4+x)*log(x^4*exp(x) 
^4)+2*x-5)+x)+(x^2+3*x)*log(x^4*exp(x)^4)-2*x^2-27*x-16)/(((4+x)*log(x^4*e 
xp(x)^4)+2*x-5)*log((4+x)*log(x^4*exp(x)^4)+2*x-5)+(-x^2-4*x)*log(x^4*exp( 
x)^4)-2*x^2+5*x)/log(-log((4+x)*log(x^4*exp(x)^4)+2*x-5)+x)^2,x, algorithm 
="giac")
 

Output:

x/log(x - log(4*x^2 + x*log(x^4) + 18*x + 4*log(x^4) - 5))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-16-27 x-2 x^2+\left (3 x+x^2\right ) \log \left (e^{4 x} x^4\right )+\left (5 x-2 x^2+\left (-4 x-x^2\right ) \log \left (e^{4 x} x^4\right )+\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )}{\left (5 x-2 x^2+\left (-4 x-x^2\right ) \log \left (e^{4 x} x^4\right )+\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx=\int -\frac {27\,x-\ln \left (x^4\,{\mathrm {e}}^{4\,x}\right )\,\left (x^2+3\,x\right )+2\,x^2-\ln \left (x-\ln \left (2\,x+\ln \left (x^4\,{\mathrm {e}}^{4\,x}\right )\,\left (x+4\right )-5\right )\right )\,\left (5\,x-\ln \left (x^4\,{\mathrm {e}}^{4\,x}\right )\,\left (x^2+4\,x\right )+\ln \left (2\,x+\ln \left (x^4\,{\mathrm {e}}^{4\,x}\right )\,\left (x+4\right )-5\right )\,\left (2\,x+\ln \left (x^4\,{\mathrm {e}}^{4\,x}\right )\,\left (x+4\right )-5\right )-2\,x^2\right )+16}{{\ln \left (x-\ln \left (2\,x+\ln \left (x^4\,{\mathrm {e}}^{4\,x}\right )\,\left (x+4\right )-5\right )\right )}^2\,\left (5\,x-\ln \left (x^4\,{\mathrm {e}}^{4\,x}\right )\,\left (x^2+4\,x\right )+\ln \left (2\,x+\ln \left (x^4\,{\mathrm {e}}^{4\,x}\right )\,\left (x+4\right )-5\right )\,\left (2\,x+\ln \left (x^4\,{\mathrm {e}}^{4\,x}\right )\,\left (x+4\right )-5\right )-2\,x^2\right )} \,d x \] Input:

int(-(27*x - log(x^4*exp(4*x))*(3*x + x^2) + 2*x^2 - log(x - log(2*x + log 
(x^4*exp(4*x))*(x + 4) - 5))*(5*x - log(x^4*exp(4*x))*(4*x + x^2) + log(2* 
x + log(x^4*exp(4*x))*(x + 4) - 5)*(2*x + log(x^4*exp(4*x))*(x + 4) - 5) - 
 2*x^2) + 16)/(log(x - log(2*x + log(x^4*exp(4*x))*(x + 4) - 5))^2*(5*x - 
log(x^4*exp(4*x))*(4*x + x^2) + log(2*x + log(x^4*exp(4*x))*(x + 4) - 5)*( 
2*x + log(x^4*exp(4*x))*(x + 4) - 5) - 2*x^2)),x)
 

Output:

int(-(27*x - log(x^4*exp(4*x))*(3*x + x^2) + 2*x^2 - log(x - log(2*x + log 
(x^4*exp(4*x))*(x + 4) - 5))*(5*x - log(x^4*exp(4*x))*(4*x + x^2) + log(2* 
x + log(x^4*exp(4*x))*(x + 4) - 5)*(2*x + log(x^4*exp(4*x))*(x + 4) - 5) - 
 2*x^2) + 16)/(log(x - log(2*x + log(x^4*exp(4*x))*(x + 4) - 5))^2*(5*x - 
log(x^4*exp(4*x))*(4*x + x^2) + log(2*x + log(x^4*exp(4*x))*(x + 4) - 5)*( 
2*x + log(x^4*exp(4*x))*(x + 4) - 5) - 2*x^2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.26 \[ \int \frac {-16-27 x-2 x^2+\left (3 x+x^2\right ) \log \left (e^{4 x} x^4\right )+\left (5 x-2 x^2+\left (-4 x-x^2\right ) \log \left (e^{4 x} x^4\right )+\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log \left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )}{\left (5 x-2 x^2+\left (-4 x-x^2\right ) \log \left (e^{4 x} x^4\right )+\left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right ) \log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right ) \log ^2\left (x-\log \left (-5+2 x+(4+x) \log \left (e^{4 x} x^4\right )\right )\right )} \, dx=\frac {x}{\mathrm {log}\left (-\mathrm {log}\left (\mathrm {log}\left (e^{4 x} x^{4}\right ) x +4 \,\mathrm {log}\left (e^{4 x} x^{4}\right )+2 x -5\right )+x \right )} \] Input:

int(((((4+x)*log(x^4*exp(x)^4)+2*x-5)*log((4+x)*log(x^4*exp(x)^4)+2*x-5)+( 
-x^2-4*x)*log(x^4*exp(x)^4)-2*x^2+5*x)*log(-log((4+x)*log(x^4*exp(x)^4)+2* 
x-5)+x)+(x^2+3*x)*log(x^4*exp(x)^4)-2*x^2-27*x-16)/(((4+x)*log(x^4*exp(x)^ 
4)+2*x-5)*log((4+x)*log(x^4*exp(x)^4)+2*x-5)+(-x^2-4*x)*log(x^4*exp(x)^4)- 
2*x^2+5*x)/log(-log((4+x)*log(x^4*exp(x)^4)+2*x-5)+x)^2,x)
 

Output:

x/log( - log(log(e**(4*x)*x**4)*x + 4*log(e**(4*x)*x**4) + 2*x - 5) + x)