\(\int \frac {-x^4+(200 x+30 x^2-40 x^3-x^4+2 x^5-4 x^8) \log (x)+(40 x^3+x^4-4 x^5) \log (x) \log (e^{x^4} \log (x))+2 x^5 \log (x) \log ^2(e^{x^4} \log (x))}{(300-60 x^2+3 x^4) \log (x)+(60 x^2-6 x^4) \log (x) \log (e^{x^4} \log (x))+3 x^4 \log (x) \log ^2(e^{x^4} \log (x))} \, dx\) [879]

Optimal result

Integrand size = 142, antiderivative size = 28 \[ \int \frac {-x^4+\left (200 x+30 x^2-40 x^3-x^4+2 x^5-4 x^8\right ) \log (x)+\left (40 x^3+x^4-4 x^5\right ) \log (x) \log \left (e^{x^4} \log (x)\right )+2 x^5 \log (x) \log ^2\left (e^{x^4} \log (x)\right )}{\left (300-60 x^2+3 x^4\right ) \log (x)+\left (60 x^2-6 x^4\right ) \log (x) \log \left (e^{x^4} \log (x)\right )+3 x^4 \log (x) \log ^2\left (e^{x^4} \log (x)\right )} \, dx=\frac {1}{3} \left (x^2+\frac {x}{-1+\frac {10}{x^2}+\log \left (e^{x^4} \log (x)\right )}\right ) \] Output:

1/3*x^2+1/3*x/(10/x^2+ln(exp(x^4)*ln(x))-1)
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.79 \[ \int \frac {-x^4+\left (200 x+30 x^2-40 x^3-x^4+2 x^5-4 x^8\right ) \log (x)+\left (40 x^3+x^4-4 x^5\right ) \log (x) \log \left (e^{x^4} \log (x)\right )+2 x^5 \log (x) \log ^2\left (e^{x^4} \log (x)\right )}{\left (300-60 x^2+3 x^4\right ) \log (x)+\left (60 x^2-6 x^4\right ) \log (x) \log \left (e^{x^4} \log (x)\right )+3 x^4 \log (x) \log ^2\left (e^{x^4} \log (x)\right )} \, dx=\frac {x^2 \left (10+x-x^2+x^2 \log \left (e^{x^4} \log (x)\right )\right )}{3 \left (10-x^2+x^2 \log \left (e^{x^4} \log (x)\right )\right )} \] Input:

Integrate[(-x^4 + (200*x + 30*x^2 - 40*x^3 - x^4 + 2*x^5 - 4*x^8)*Log[x] + 
 (40*x^3 + x^4 - 4*x^5)*Log[x]*Log[E^x^4*Log[x]] + 2*x^5*Log[x]*Log[E^x^4* 
Log[x]]^2)/((300 - 60*x^2 + 3*x^4)*Log[x] + (60*x^2 - 6*x^4)*Log[x]*Log[E^ 
x^4*Log[x]] + 3*x^4*Log[x]*Log[E^x^4*Log[x]]^2),x]
 

Output:

(x^2*(10 + x - x^2 + x^2*Log[E^x^4*Log[x]]))/(3*(10 - x^2 + x^2*Log[E^x^4* 
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.

\(\displaystyle \int \frac {-x^4+2 x^5 \log (x) \log ^2\left (e^{x^4} \log (x)\right )+\left (-4 x^5+x^4+40 x^3\right ) \log (x) \log \left (e^{x^4} \log (x)\right )+\left (-4 x^8+2 x^5-x^4-40 x^3+30 x^2+200 x\right ) \log (x)}{3 x^4 \log (x) \log ^2\left (e^{x^4} \log (x)\right )+\left (3 x^4-60 x^2+300\right ) \log (x)+\left (60 x^2-6 x^4\right ) \log (x) \log \left (e^{x^4} \log (x)\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {-x^4+2 x^5 \log (x) \log ^2\left (e^{x^4} \log (x)\right )+\left (-4 x^5+x^4+40 x^3\right ) \log (x) \log \left (e^{x^4} \log (x)\right )+\left (-4 x^8+2 x^5-x^4-40 x^3+30 x^2+200 x\right ) \log (x)}{3 \log (x) \left (-x^2+x^2 \log \left (e^{x^4} \log (x)\right )+10\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \int -\frac {-2 \log (x) \log ^2\left (e^{x^4} \log (x)\right ) x^5+x^4-\left (-4 x^8+2 x^5-x^4-40 x^3+30 x^2+200 x\right ) \log (x)-\left (-4 x^5+x^4+40 x^3\right ) \log (x) \log \left (e^{x^4} \log (x)\right )}{\log (x) \left (\log \left (e^{x^4} \log (x)\right ) x^2-x^2+10\right )^2}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {1}{3} \int \frac {-2 \log (x) \log ^2\left (e^{x^4} \log (x)\right ) x^5+x^4-\left (-4 x^8+2 x^5-x^4-40 x^3+30 x^2+200 x\right ) \log (x)-\left (-4 x^5+x^4+40 x^3\right ) \log (x) \log \left (e^{x^4} \log (x)\right )}{\log (x) \left (\log \left (e^{x^4} \log (x)\right ) x^2-x^2+10\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{3} \int \left (-\frac {x^2}{\log \left (e^{x^4} \log (x)\right ) x^2-x^2+10}-2 x+\frac {4 \log (x) x^8+x^4-20 \log (x) x^2}{\log (x) \left (\log \left (e^{x^4} \log (x)\right ) x^2-x^2+10\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{3} \left (20 \int \frac {x^2}{\left (\log \left (e^{x^4} \log (x)\right ) x^2-x^2+10\right )^2}dx-\int \frac {x^4}{\log (x) \left (\log \left (e^{x^4} \log (x)\right ) x^2-x^2+10\right )^2}dx+\int \frac {x^2}{\log \left (e^{x^4} \log (x)\right ) x^2-x^2+10}dx-4 \int \frac {x^8}{\left (\log \left (e^{x^4} \log (x)\right ) x^2-x^2+10\right )^2}dx+x^2\right )\)

Input:

Int[(-x^4 + (200*x + 30*x^2 - 40*x^3 - x^4 + 2*x^5 - 4*x^8)*Log[x] + (40*x 
^3 + x^4 - 4*x^5)*Log[x]*Log[E^x^4*Log[x]] + 2*x^5*Log[x]*Log[E^x^4*Log[x] 
]^2)/((300 - 60*x^2 + 3*x^4)*Log[x] + (60*x^2 - 6*x^4)*Log[x]*Log[E^x^4*Lo 
g[x]] + 3*x^4*Log[x]*Log[E^x^4*Log[x]]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 17.87 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.82

method	result	size
parallelrisch	\(-\frac {-x^{4} \ln \left ({\mathrm e}^{x^{4}} \ln \left (x \right )\right )+x^{4}-x^{3}-10 x^{2}}{3 \left (x^{2} \ln \left ({\mathrm e}^{x^{4}} \ln \left (x \right )\right )-x^{2}+10\right )}\)	\(51\)
risch	\(\frac {x^{2}}{3}+\frac {2 i x^{3}}{3 \left (\pi \,x^{2} \operatorname {csgn}\left (i {\mathrm e}^{x^{4}}\right ) \operatorname {csgn}\left (i \ln \left (x \right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{x^{4}} \ln \left (x \right )\right )-\pi \,x^{2} \operatorname {csgn}\left (i {\mathrm e}^{x^{4}}\right ) \operatorname {csgn}\left (i {\mathrm e}^{x^{4}} \ln \left (x \right )\right )^{2}-\pi \,x^{2} \operatorname {csgn}\left (i \ln \left (x \right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{x^{4}} \ln \left (x \right )\right )^{2}+\pi \,x^{2} \operatorname {csgn}\left (i {\mathrm e}^{x^{4}} \ln \left (x \right )\right )^{3}+2 i x^{2} \ln \left (\ln \left (x \right )\right )+2 i x^{2} \ln \left ({\mathrm e}^{x^{4}}\right )-2 i x^{2}+20 i\right )}\)	\(140\)

Input:

int((2*x^5*ln(x)*ln(exp(x^4)*ln(x))^2+(-4*x^5+x^4+40*x^3)*ln(x)*ln(exp(x^4 
)*ln(x))+(-4*x^8+2*x^5-x^4-40*x^3+30*x^2+200*x)*ln(x)-x^4)/(3*x^4*ln(x)*ln 
(exp(x^4)*ln(x))^2+(-6*x^4+60*x^2)*ln(x)*ln(exp(x^4)*ln(x))+(3*x^4-60*x^2+ 
300)*ln(x)),x,method=_RETURNVERBOSE)
 

Output:

-1/3*(-x^4*ln(exp(x^4)*ln(x))+x^4-x^3-10*x^2)/(x^2*ln(exp(x^4)*ln(x))-x^2+ 
10)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75 \[ \int \frac {-x^4+\left (200 x+30 x^2-40 x^3-x^4+2 x^5-4 x^8\right ) \log (x)+\left (40 x^3+x^4-4 x^5\right ) \log (x) \log \left (e^{x^4} \log (x)\right )+2 x^5 \log (x) \log ^2\left (e^{x^4} \log (x)\right )}{\left (300-60 x^2+3 x^4\right ) \log (x)+\left (60 x^2-6 x^4\right ) \log (x) \log \left (e^{x^4} \log (x)\right )+3 x^4 \log (x) \log ^2\left (e^{x^4} \log (x)\right )} \, dx=\frac {x^{4} \log \left (e^{\left (x^{4}\right )} \log \left (x\right )\right ) - x^{4} + x^{3} + 10 \, x^{2}}{3 \, {\left (x^{2} \log \left (e^{\left (x^{4}\right )} \log \left (x\right )\right ) - x^{2} + 10\right )}} \] Input:

integrate((2*x^5*log(x)*log(exp(x^4)*log(x))^2+(-4*x^5+x^4+40*x^3)*log(x)* 
log(exp(x^4)*log(x))+(-4*x^8+2*x^5-x^4-40*x^3+30*x^2+200*x)*log(x)-x^4)/(3 
*x^4*log(x)*log(exp(x^4)*log(x))^2+(-6*x^4+60*x^2)*log(x)*log(exp(x^4)*log 
(x))+(3*x^4-60*x^2+300)*log(x)),x, algorithm="fricas")
 

Output:

1/3*(x^4*log(e^(x^4)*log(x)) - x^4 + x^3 + 10*x^2)/(x^2*log(e^(x^4)*log(x) 
) - x^2 + 10)
 

Sympy [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {-x^4+\left (200 x+30 x^2-40 x^3-x^4+2 x^5-4 x^8\right ) \log (x)+\left (40 x^3+x^4-4 x^5\right ) \log (x) \log \left (e^{x^4} \log (x)\right )+2 x^5 \log (x) \log ^2\left (e^{x^4} \log (x)\right )}{\left (300-60 x^2+3 x^4\right ) \log (x)+\left (60 x^2-6 x^4\right ) \log (x) \log \left (e^{x^4} \log (x)\right )+3 x^4 \log (x) \log ^2\left (e^{x^4} \log (x)\right )} \, dx=\frac {x^{3}}{3 x^{2} \log {\left (e^{x^{4}} \log {\left (x \right )} \right )} - 3 x^{2} + 30} + \frac {x^{2}}{3} \] Input:

integrate((2*x**5*ln(x)*ln(exp(x**4)*ln(x))**2+(-4*x**5+x**4+40*x**3)*ln(x 
)*ln(exp(x**4)*ln(x))+(-4*x**8+2*x**5-x**4-40*x**3+30*x**2+200*x)*ln(x)-x* 
*4)/(3*x**4*ln(x)*ln(exp(x**4)*ln(x))**2+(-6*x**4+60*x**2)*ln(x)*ln(exp(x* 
*4)*ln(x))+(3*x**4-60*x**2+300)*ln(x)),x)
 

Output:

x**3/(3*x**2*log(exp(x**4)*log(x)) - 3*x**2 + 30) + x**2/3
 

Maxima [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.61 \[ \int \frac {-x^4+\left (200 x+30 x^2-40 x^3-x^4+2 x^5-4 x^8\right ) \log (x)+\left (40 x^3+x^4-4 x^5\right ) \log (x) \log \left (e^{x^4} \log (x)\right )+2 x^5 \log (x) \log ^2\left (e^{x^4} \log (x)\right )}{\left (300-60 x^2+3 x^4\right ) \log (x)+\left (60 x^2-6 x^4\right ) \log (x) \log \left (e^{x^4} \log (x)\right )+3 x^4 \log (x) \log ^2\left (e^{x^4} \log (x)\right )} \, dx=\frac {x^{8} + x^{4} \log \left (\log \left (x\right )\right ) - x^{4} + x^{3} + 10 \, x^{2}}{3 \, {\left (x^{6} + x^{2} \log \left (\log \left (x\right )\right ) - x^{2} + 10\right )}} \] Input:

integrate((2*x^5*log(x)*log(exp(x^4)*log(x))^2+(-4*x^5+x^4+40*x^3)*log(x)* 
log(exp(x^4)*log(x))+(-4*x^8+2*x^5-x^4-40*x^3+30*x^2+200*x)*log(x)-x^4)/(3 
*x^4*log(x)*log(exp(x^4)*log(x))^2+(-6*x^4+60*x^2)*log(x)*log(exp(x^4)*log 
(x))+(3*x^4-60*x^2+300)*log(x)),x, algorithm="maxima")
 

Output:

1/3*(x^8 + x^4*log(log(x)) - x^4 + x^3 + 10*x^2)/(x^6 + x^2*log(log(x)) - 
x^2 + 10)
 

Giac [A] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {-x^4+\left (200 x+30 x^2-40 x^3-x^4+2 x^5-4 x^8\right ) \log (x)+\left (40 x^3+x^4-4 x^5\right ) \log (x) \log \left (e^{x^4} \log (x)\right )+2 x^5 \log (x) \log ^2\left (e^{x^4} \log (x)\right )}{\left (300-60 x^2+3 x^4\right ) \log (x)+\left (60 x^2-6 x^4\right ) \log (x) \log \left (e^{x^4} \log (x)\right )+3 x^4 \log (x) \log ^2\left (e^{x^4} \log (x)\right )} \, dx=\frac {1}{3} \, x^{2} + \frac {x^{3}}{3 \, {\left (x^{6} + x^{2} \log \left (\log \left (x\right )\right ) - x^{2} + 10\right )}} \] Input:

integrate((2*x^5*log(x)*log(exp(x^4)*log(x))^2+(-4*x^5+x^4+40*x^3)*log(x)* 
log(exp(x^4)*log(x))+(-4*x^8+2*x^5-x^4-40*x^3+30*x^2+200*x)*log(x)-x^4)/(3 
*x^4*log(x)*log(exp(x^4)*log(x))^2+(-6*x^4+60*x^2)*log(x)*log(exp(x^4)*log 
(x))+(3*x^4-60*x^2+300)*log(x)),x, algorithm="giac")
 

Output:

1/3*x^2 + 1/3*x^3/(x^6 + x^2*log(log(x)) - x^2 + 10)
 

Mupad [B] (verification not implemented)

Time = 2.84 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {-x^4+\left (200 x+30 x^2-40 x^3-x^4+2 x^5-4 x^8\right ) \log (x)+\left (40 x^3+x^4-4 x^5\right ) \log (x) \log \left (e^{x^4} \log (x)\right )+2 x^5 \log (x) \log ^2\left (e^{x^4} \log (x)\right )}{\left (300-60 x^2+3 x^4\right ) \log (x)+\left (60 x^2-6 x^4\right ) \log (x) \log \left (e^{x^4} \log (x)\right )+3 x^4 \log (x) \log ^2\left (e^{x^4} \log (x)\right )} \, dx=\frac {x^3}{3\,\left (x^2\,\ln \left (\ln \left (x\right )\right )-x^2+x^6+10\right )}+\frac {x^2}{3} \] Input:

int((log(x)*(200*x + 30*x^2 - 40*x^3 - x^4 + 2*x^5 - 4*x^8) - x^4 + 2*x^5* 
log(x)*log(exp(x^4)*log(x))^2 + log(x)*log(exp(x^4)*log(x))*(40*x^3 + x^4 
- 4*x^5))/(log(x)*(3*x^4 - 60*x^2 + 300) + 3*x^4*log(x)*log(exp(x^4)*log(x 
))^2 + log(x)*log(exp(x^4)*log(x))*(60*x^2 - 6*x^4)),x)
 

Output:

x^3/(3*(x^2*log(log(x)) - x^2 + x^6 + 10)) + x^2/3
 

Reduce [B] (verification not implemented)

Time = 21.18 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.71 \[ \int \frac {-x^4+\left (200 x+30 x^2-40 x^3-x^4+2 x^5-4 x^8\right ) \log (x)+\left (40 x^3+x^4-4 x^5\right ) \log (x) \log \left (e^{x^4} \log (x)\right )+2 x^5 \log (x) \log ^2\left (e^{x^4} \log (x)\right )}{\left (300-60 x^2+3 x^4\right ) \log (x)+\left (60 x^2-6 x^4\right ) \log (x) \log \left (e^{x^4} \log (x)\right )+3 x^4 \log (x) \log ^2\left (e^{x^4} \log (x)\right )} \, dx=\frac {x^{2} \left (\mathrm {log}\left (e^{x^{4}} \mathrm {log}\left (x \right )\right ) x^{2}-x^{2}+x +10\right )}{3 \,\mathrm {log}\left (e^{x^{4}} \mathrm {log}\left (x \right )\right ) x^{2}-3 x^{2}+30} \] Input:

int((2*x^5*log(x)*log(exp(x^4)*log(x))^2+(-4*x^5+x^4+40*x^3)*log(x)*log(ex 
p(x^4)*log(x))+(-4*x^8+2*x^5-x^4-40*x^3+30*x^2+200*x)*log(x)-x^4)/(3*x^4*l 
og(x)*log(exp(x^4)*log(x))^2+(-6*x^4+60*x^2)*log(x)*log(exp(x^4)*log(x))+( 
3*x^4-60*x^2+300)*log(x)),x)
 

Output:

(x**2*(log(e**(x**4)*log(x))*x**2 - x**2 + x + 10))/(3*(log(e**(x**4)*log( 
x))*x**2 - x**2 + 10))