\(\int \frac {(8 x^2-4 e^x x^3-4 x^5+(-16 x^2-6 x^3+8 x^5+e^x (-2 x-2 x^2+8 x^3)) \log (4 x)+(-4+2 e^x x+2 x^3) \log (2-e^x x-x^3)) \log (\frac {\log (4 x)}{-2 x^2+\log (2-e^x x-x^3)})}{(4 x^3-2 e^x x^4-2 x^6) \log (4 x)+(-2 x+e^x x^2+x^4) \log (4 x) \log (2-e^x x-x^3)} \, dx\) [1717]

Optimal result

Integrand size = 175, antiderivative size = 33 \[ \int \frac {\left (8 x^2-4 e^x x^3-4 x^5+\left (-16 x^2-6 x^3+8 x^5+e^x \left (-2 x-2 x^2+8 x^3\right )\right ) \log (4 x)+\left (-4+2 e^x x+2 x^3\right ) \log \left (2-e^x x-x^3\right )\right ) \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{\left (4 x^3-2 e^x x^4-2 x^6\right ) \log (4 x)+\left (-2 x+e^x x^2+x^4\right ) \log (4 x) \log \left (2-e^x x-x^3\right )} \, dx=\log ^2\left (\frac {\log (4 x)}{-2 x^2+\log \left (2-x^2 \left (\frac {e^x}{x}+x\right )\right )}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.91 \[ \int \frac {\left (8 x^2-4 e^x x^3-4 x^5+\left (-16 x^2-6 x^3+8 x^5+e^x \left (-2 x-2 x^2+8 x^3\right )\right ) \log (4 x)+\left (-4+2 e^x x+2 x^3\right ) \log \left (2-e^x x-x^3\right )\right ) \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{\left (4 x^3-2 e^x x^4-2 x^6\right ) \log (4 x)+\left (-2 x+e^x x^2+x^4\right ) \log (4 x) \log \left (2-e^x x-x^3\right )} \, dx=\log ^2\left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right ) \] Input:

Integrate[((8*x^2 - 4*E^x*x^3 - 4*x^5 + (-16*x^2 - 6*x^3 + 8*x^5 + E^x*(-2 
*x - 2*x^2 + 8*x^3))*Log[4*x] + (-4 + 2*E^x*x + 2*x^3)*Log[2 - E^x*x - x^3 
])*Log[Log[4*x]/(-2*x^2 + Log[2 - E^x*x - x^3])])/((4*x^3 - 2*E^x*x^4 - 2* 
x^6)*Log[4*x] + (-2*x + E^x*x^2 + x^4)*Log[4*x]*Log[2 - E^x*x - x^3]),x]
 

Output:

Log[Log[4*x]/(-2*x^2 + Log[2 - E^x*x - x^3])]^2
 

Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (-4 x^5-4 e^x x^3+\left (2 x^3+2 e^x x-4\right ) \log \left (-x^3-e^x x+2\right )+8 x^2+\left (8 x^5-6 x^3-16 x^2+e^x \left (8 x^3-2 x^2-2 x\right )\right ) \log (4 x)\right ) \log \left (\frac {\log (4 x)}{\log \left (-x^3-e^x x+2\right )-2 x^2}\right )}{\left (-2 x^6-2 e^x x^4+4 x^3\right ) \log (4 x)+\left (x^4+e^x x^2-2 x\right ) \log \left (-x^3-e^x x+2\right ) \log (4 x)} \, dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

Int[((8*x^2 - 4*E^x*x^3 - 4*x^5 + (-16*x^2 - 6*x^3 + 8*x^5 + E^x*(-2*x - 2 
*x^2 + 8*x^3))*Log[4*x] + (-4 + 2*E^x*x + 2*x^3)*Log[2 - E^x*x - x^3])*Log 
[Log[4*x]/(-2*x^2 + Log[2 - E^x*x - x^3])])/((4*x^3 - 2*E^x*x^4 - 2*x^6)*L 
og[4*x] + (-2*x + E^x*x^2 + x^4)*Log[4*x]*Log[2 - E^x*x - x^3]),x]
 

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

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

Time = 0.08 (sec) , antiderivative size = 727, normalized size of antiderivative = 22.03

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

Input:

int(((2*exp(x)*x+2*x^3-4)*ln(-exp(x)*x-x^3+2)+((8*x^3-2*x^2-2*x)*exp(x)+8* 
x^5-6*x^3-16*x^2)*ln(4*x)-4*exp(x)*x^3-4*x^5+8*x^2)*ln(ln(4*x)/(ln(-exp(x) 
*x-x^3+2)-2*x^2))/((exp(x)*x^2+x^4-2*x)*ln(4*x)*ln(-exp(x)*x-x^3+2)+(-2*ex 
p(x)*x^4-2*x^6+4*x^3)*ln(4*x)),x)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int \frac {\left (8 x^2-4 e^x x^3-4 x^5+\left (-16 x^2-6 x^3+8 x^5+e^x \left (-2 x-2 x^2+8 x^3\right )\right ) \log (4 x)+\left (-4+2 e^x x+2 x^3\right ) \log \left (2-e^x x-x^3\right )\right ) \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{\left (4 x^3-2 e^x x^4-2 x^6\right ) \log (4 x)+\left (-2 x+e^x x^2+x^4\right ) \log (4 x) \log \left (2-e^x x-x^3\right )} \, dx=\log \left (-\frac {\log \left (4 \, x\right )}{2 \, x^{2} - \log \left (-x^{3} - x e^{x} + 2\right )}\right )^{2} \] Input:

integrate(((2*exp(x)*x+2*x^3-4)*log(-exp(x)*x-x^3+2)+((8*x^3-2*x^2-2*x)*ex 
p(x)+8*x^5-6*x^3-16*x^2)*log(4*x)-4*exp(x)*x^3-4*x^5+8*x^2)*log(log(4*x)/( 
log(-exp(x)*x-x^3+2)-2*x^2))/((exp(x)*x^2+x^4-2*x)*log(4*x)*log(-exp(x)*x- 
x^3+2)+(-2*exp(x)*x^4-2*x^6+4*x^3)*log(4*x)),x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (8 x^2-4 e^x x^3-4 x^5+\left (-16 x^2-6 x^3+8 x^5+e^x \left (-2 x-2 x^2+8 x^3\right )\right ) \log (4 x)+\left (-4+2 e^x x+2 x^3\right ) \log \left (2-e^x x-x^3\right )\right ) \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{\left (4 x^3-2 e^x x^4-2 x^6\right ) \log (4 x)+\left (-2 x+e^x x^2+x^4\right ) \log (4 x) \log \left (2-e^x x-x^3\right )} \, dx=\text {Timed out} \] Input:

integrate(((2*exp(x)*x+2*x**3-4)*ln(-exp(x)*x-x**3+2)+((8*x**3-2*x**2-2*x) 
*exp(x)+8*x**5-6*x**3-16*x**2)*ln(4*x)-4*exp(x)*x**3-4*x**5+8*x**2)*ln(ln( 
4*x)/(ln(-exp(x)*x-x**3+2)-2*x**2))/((exp(x)*x**2+x**4-2*x)*ln(4*x)*ln(-ex 
p(x)*x-x**3+2)+(-2*exp(x)*x**4-2*x**6+4*x**3)*ln(4*x)),x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 130 vs. \(2 (35) = 70\).

Time = 0.20 (sec) , antiderivative size = 130, normalized size of antiderivative = 3.94 \[ \int \frac {\left (8 x^2-4 e^x x^3-4 x^5+\left (-16 x^2-6 x^3+8 x^5+e^x \left (-2 x-2 x^2+8 x^3\right )\right ) \log (4 x)+\left (-4+2 e^x x+2 x^3\right ) \log \left (2-e^x x-x^3\right )\right ) \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{\left (4 x^3-2 e^x x^4-2 x^6\right ) \log (4 x)+\left (-2 x+e^x x^2+x^4\right ) \log (4 x) \log \left (2-e^x x-x^3\right )} \, dx=-\log \left (-2 \, x^{2} + \log \left (-x^{3} - x e^{x} + 2\right )\right )^{2} - 2 \, {\left (\log \left (-2 \, x^{2} + \log \left (-x^{3} - x e^{x} + 2\right )\right ) - \log \left (2 \, \log \left (2\right ) + \log \left (x\right )\right )\right )} \log \left (-\frac {\log \left (4 \, x\right )}{2 \, x^{2} - \log \left (-x^{3} - x e^{x} + 2\right )}\right ) + 2 \, \log \left (-2 \, x^{2} + \log \left (-x^{3} - x e^{x} + 2\right )\right ) \log \left (2 \, \log \left (2\right ) + \log \left (x\right )\right ) - \log \left (2 \, \log \left (2\right ) + \log \left (x\right )\right )^{2} \] Input:

integrate(((2*exp(x)*x+2*x^3-4)*log(-exp(x)*x-x^3+2)+((8*x^3-2*x^2-2*x)*ex 
p(x)+8*x^5-6*x^3-16*x^2)*log(4*x)-4*exp(x)*x^3-4*x^5+8*x^2)*log(log(4*x)/( 
log(-exp(x)*x-x^3+2)-2*x^2))/((exp(x)*x^2+x^4-2*x)*log(4*x)*log(-exp(x)*x- 
x^3+2)+(-2*exp(x)*x^4-2*x^6+4*x^3)*log(4*x)),x, algorithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int \frac {\left (8 x^2-4 e^x x^3-4 x^5+\left (-16 x^2-6 x^3+8 x^5+e^x \left (-2 x-2 x^2+8 x^3\right )\right ) \log (4 x)+\left (-4+2 e^x x+2 x^3\right ) \log \left (2-e^x x-x^3\right )\right ) \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{\left (4 x^3-2 e^x x^4-2 x^6\right ) \log (4 x)+\left (-2 x+e^x x^2+x^4\right ) \log (4 x) \log \left (2-e^x x-x^3\right )} \, dx=\log \left (-\frac {\log \left (4 \, x\right )}{2 \, x^{2} - \log \left (-x^{3} - x e^{x} + 2\right )}\right )^{2} \] Input:

integrate(((2*exp(x)*x+2*x^3-4)*log(-exp(x)*x-x^3+2)+((8*x^3-2*x^2-2*x)*ex 
p(x)+8*x^5-6*x^3-16*x^2)*log(4*x)-4*exp(x)*x^3-4*x^5+8*x^2)*log(log(4*x)/( 
log(-exp(x)*x-x^3+2)-2*x^2))/((exp(x)*x^2+x^4-2*x)*log(4*x)*log(-exp(x)*x- 
x^3+2)+(-2*exp(x)*x^4-2*x^6+4*x^3)*log(4*x)),x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 3.81 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int \frac {\left (8 x^2-4 e^x x^3-4 x^5+\left (-16 x^2-6 x^3+8 x^5+e^x \left (-2 x-2 x^2+8 x^3\right )\right ) \log (4 x)+\left (-4+2 e^x x+2 x^3\right ) \log \left (2-e^x x-x^3\right )\right ) \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{\left (4 x^3-2 e^x x^4-2 x^6\right ) \log (4 x)+\left (-2 x+e^x x^2+x^4\right ) \log (4 x) \log \left (2-e^x x-x^3\right )} \, dx={\ln \left (\frac {\ln \left (4\,x\right )}{\ln \left (2-x^3-x\,{\mathrm {e}}^x\right )-2\,x^2}\right )}^2 \] Input:

int((log(log(4*x)/(log(2 - x^3 - x*exp(x)) - 2*x^2))*(4*x^3*exp(x) - log(2 
 - x^3 - x*exp(x))*(2*x*exp(x) + 2*x^3 - 4) + log(4*x)*(16*x^2 + 6*x^3 - 8 
*x^5 + exp(x)*(2*x + 2*x^2 - 8*x^3)) - 8*x^2 + 4*x^5))/(log(4*x)*(2*x^4*ex 
p(x) - 4*x^3 + 2*x^6) - log(4*x)*log(2 - x^3 - x*exp(x))*(x^2*exp(x) - 2*x 
 + x^4)),x)
 

Output:

log(log(4*x)/(log(2 - x^3 - x*exp(x)) - 2*x^2))^2
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.91 \[ \int \frac {\left (8 x^2-4 e^x x^3-4 x^5+\left (-16 x^2-6 x^3+8 x^5+e^x \left (-2 x-2 x^2+8 x^3\right )\right ) \log (4 x)+\left (-4+2 e^x x+2 x^3\right ) \log \left (2-e^x x-x^3\right )\right ) \log \left (\frac {\log (4 x)}{-2 x^2+\log \left (2-e^x x-x^3\right )}\right )}{\left (4 x^3-2 e^x x^4-2 x^6\right ) \log (4 x)+\left (-2 x+e^x x^2+x^4\right ) \log (4 x) \log \left (2-e^x x-x^3\right )} \, dx={\mathrm {log}\left (\frac {\mathrm {log}\left (4 x \right )}{\mathrm {log}\left (-e^{x} x -x^{3}+2\right )-2 x^{2}}\right )}^{2} \] Input:

int(((2*exp(x)*x+2*x^3-4)*log(-exp(x)*x-x^3+2)+((8*x^3-2*x^2-2*x)*exp(x)+8 
*x^5-6*x^3-16*x^2)*log(4*x)-4*exp(x)*x^3-4*x^5+8*x^2)*log(log(4*x)/(log(-e 
xp(x)*x-x^3+2)-2*x^2))/((exp(x)*x^2+x^4-2*x)*log(4*x)*log(-exp(x)*x-x^3+2) 
+(-2*exp(x)*x^4-2*x^6+4*x^3)*log(4*x)),x)
 

Output:

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