\(\int \frac {(-8 x^2+2 x^3-2 x^4) \log (5) \log (\frac {1}{4} (-4 x+x^2-x^3))+(16-8 x+12 x^2+(-4 x^2+2 x^3-3 x^4) \log (5)) \log (-4+x^2 \log (5))+(8 x^2-2 x^3+2 x^4) \log (5) \log (\frac {1}{4} (-4 x+x^2-x^3)) \log (\log (\frac {1}{4} (-4 x+x^2-x^3)))}{(-16 x+4 x^2-4 x^3+(4 x^3-x^4+x^5) \log (5)) \log (\frac {1}{4} (-4 x+x^2-x^3)) \log ^2(-4+x^2 \log (5))} \, dx\) [700]

Optimal result

Integrand size = 194, antiderivative size = 34 \[ \int \frac {\left (-8 x^2+2 x^3-2 x^4\right ) \log (5) \log \left (\frac {1}{4} \left (-4 x+x^2-x^3\right )\right )+\left (16-8 x+12 x^2+\left (-4 x^2+2 x^3-3 x^4\right ) \log (5)\right ) \log \left (-4+x^2 \log (5)\right )+\left (8 x^2-2 x^3+2 x^4\right ) \log (5) \log \left (\frac {1}{4} \left (-4 x+x^2-x^3\right )\right ) \log \left (\log \left (\frac {1}{4} \left (-4 x+x^2-x^3\right )\right )\right )}{\left (-16 x+4 x^2-4 x^3+\left (4 x^3-x^4+x^5\right ) \log (5)\right ) \log \left (\frac {1}{4} \left (-4 x+x^2-x^3\right )\right ) \log ^2\left (-4+x^2 \log (5)\right )} \, dx=\frac {1-\log \left (\log \left (-x+\frac {1}{4} x \left (x-x^2\right )\right )\right )}{\log \left (-4+x^2 \log (5)\right )} \] Output:

(1-ln(ln(1/4*x*(-x^2+x)-x)))/ln(x^2*ln(5)-4)
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.18 \[ \int \frac {\left (-8 x^2+2 x^3-2 x^4\right ) \log (5) \log \left (\frac {1}{4} \left (-4 x+x^2-x^3\right )\right )+\left (16-8 x+12 x^2+\left (-4 x^2+2 x^3-3 x^4\right ) \log (5)\right ) \log \left (-4+x^2 \log (5)\right )+\left (8 x^2-2 x^3+2 x^4\right ) \log (5) \log \left (\frac {1}{4} \left (-4 x+x^2-x^3\right )\right ) \log \left (\log \left (\frac {1}{4} \left (-4 x+x^2-x^3\right )\right )\right )}{\left (-16 x+4 x^2-4 x^3+\left (4 x^3-x^4+x^5\right ) \log (5)\right ) \log \left (\frac {1}{4} \left (-4 x+x^2-x^3\right )\right ) \log ^2\left (-4+x^2 \log (5)\right )} \, dx=\frac {1}{\log \left (-4+x^2 \log (5)\right )}-\frac {\log \left (\log \left (-\frac {1}{4} x \left (4-x+x^2\right )\right )\right )}{\log \left (-4+x^2 \log (5)\right )} \] Input:

Integrate[((-8*x^2 + 2*x^3 - 2*x^4)*Log[5]*Log[(-4*x + x^2 - x^3)/4] + (16 
 - 8*x + 12*x^2 + (-4*x^2 + 2*x^3 - 3*x^4)*Log[5])*Log[-4 + x^2*Log[5]] + 
(8*x^2 - 2*x^3 + 2*x^4)*Log[5]*Log[(-4*x + x^2 - x^3)/4]*Log[Log[(-4*x + x 
^2 - x^3)/4]])/((-16*x + 4*x^2 - 4*x^3 + (4*x^3 - x^4 + x^5)*Log[5])*Log[( 
-4*x + x^2 - x^3)/4]*Log[-4 + x^2*Log[5]]^2),x]
 

Output:

Log[-4 + x^2*Log[5]]^(-1) - Log[Log[-1/4*(x*(4 - x + x^2))]]/Log[-4 + x^2* 
Log[5]]
 

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.

\(\displaystyle \int \frac {\left (2 x^4-2 x^3+8 x^2\right ) \log (5) \log \left (\log \left (\frac {1}{4} \left (-x^3+x^2-4 x\right )\right )\right ) \log \left (\frac {1}{4} \left (-x^3+x^2-4 x\right )\right )+\left (-2 x^4+2 x^3-8 x^2\right ) \log (5) \log \left (\frac {1}{4} \left (-x^3+x^2-4 x\right )\right )+\left (12 x^2+\left (-3 x^4+2 x^3-4 x^2\right ) \log (5)-8 x+16\right ) \log \left (x^2 \log (5)-4\right )}{\left (-4 x^3+4 x^2+\left (x^5-x^4+4 x^3\right ) \log (5)-16 x\right ) \log \left (\frac {1}{4} \left (-x^3+x^2-4 x\right )\right ) \log ^2\left (x^2 \log (5)-4\right )} \, dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {\left (2 x^4-2 x^3+8 x^2\right ) \log (5) \log \left (\log \left (\frac {1}{4} \left (-x^3+x^2-4 x\right )\right )\right ) \log \left (\frac {1}{4} \left (-x^3+x^2-4 x\right )\right )+\left (-2 x^4+2 x^3-8 x^2\right ) \log (5) \log \left (\frac {1}{4} \left (-x^3+x^2-4 x\right )\right )+\left (12 x^2+\left (-3 x^4+2 x^3-4 x^2\right ) \log (5)-8 x+16\right ) \log \left (x^2 \log (5)-4\right )}{x \left (x^4 \log (5)-x^3 \log (5)-4 x^2 (1-\log (5))+4 x-16\right ) \log \left (\frac {1}{4} \left (-x^3+x^2-4 x\right )\right ) \log ^2\left (x^2 \log (5)-4\right )}dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \left (\frac {(x (-\log (5))-4-\log (125)) \left (\left (2 x^4-2 x^3+8 x^2\right ) \log (5) \log \left (\log \left (\frac {1}{4} \left (-x^3+x^2-4 x\right )\right )\right ) \log \left (\frac {1}{4} \left (-x^3+x^2-4 x\right )\right )+\left (-2 x^4+2 x^3-8 x^2\right ) \log (5) \log \left (\frac {1}{4} \left (-x^3+x^2-4 x\right )\right )+\left (12 x^2+\left (-3 x^4+2 x^3-4 x^2\right ) \log (5)-8 x+16\right ) \log \left (x^2 \log (5)-4\right )\right )}{4 x \left (x^2-x+4\right ) \left (4+4 \log ^2(5)+7 \log (5)\right ) \log \left (\frac {1}{4} \left (-x^3+x^2-4 x\right )\right ) \log ^2\left (x^2 \log (5)-4\right )}+\frac {\log (5) (x \log (5)+4+\log (625)) \left (\left (2 x^4-2 x^3+8 x^2\right ) \log (5) \log \left (\log \left (\frac {1}{4} \left (-x^3+x^2-4 x\right )\right )\right ) \log \left (\frac {1}{4} \left (-x^3+x^2-4 x\right )\right )+\left (-2 x^4+2 x^3-8 x^2\right ) \log (5) \log \left (\frac {1}{4} \left (-x^3+x^2-4 x\right )\right )+\left (12 x^2+\left (-3 x^4+2 x^3-4 x^2\right ) \log (5)-8 x+16\right ) \log \left (x^2 \log (5)-4\right )\right )}{4 x \left (4+4 \log ^2(5)+7 \log (5)\right ) \left (x^2 \log (5)-4\right ) \log \left (\frac {1}{4} \left (-x^3+x^2-4 x\right )\right ) \log ^2\left (x^2 \log (5)-4\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {(4+\log (5) \log (625)+\log (78125)) \left (-2 \left (x^2-x+4\right ) x^2 \log (5) \log \left (-\frac {1}{4} x \left (x^2-x+4\right )\right ) \left (\log \left (\log \left (-\frac {1}{4} x \left (x^2-x+4\right )\right )\right )-1\right )-\left (-3 x^4 \log (5)+x^3 \log (25)-4 x^2 (\log (5)-3)-8 x+16\right ) \log \left (x^2 \log (5)-4\right )\right )}{x \left (x^2-x+4\right ) \left (4+4 \log ^2(5)+7 \log (5)\right ) \left (4-x^2 \log (5)\right ) \log \left (-\frac {1}{4} x \left (x^2-x+4\right )\right ) \log ^2\left (x^2 \log (5)-4\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(4+\log (5) \log (625)+\log (78125)) \int -\frac {\left (-3 \log (5) x^4+\log (25) x^3+4 (3-\log (5)) x^2-8 x+16\right ) \log \left (x^2 \log (5)-4\right )-2 x^2 \left (x^2-x+4\right ) \log (5) \log \left (-\frac {1}{4} x \left (x^2-x+4\right )\right ) \left (1-\log \left (\log \left (-\frac {1}{4} x \left (x^2-x+4\right )\right )\right )\right )}{x \left (x^2-x+4\right ) \left (4-x^2 \log (5)\right ) \log \left (-\frac {1}{4} x \left (x^2-x+4\right )\right ) \log ^2\left (x^2 \log (5)-4\right )}dx}{4+4 \log ^2(5)+7 \log (5)}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {(4+\log (5) \log (625)+\log (78125)) \int \frac {\left (-3 \log (5) x^4+\log (25) x^3+4 (3-\log (5)) x^2-8 x+16\right ) \log \left (x^2 \log (5)-4\right )-2 x^2 \left (x^2-x+4\right ) \log (5) \log \left (-\frac {1}{4} x \left (x^2-x+4\right )\right ) \left (1-\log \left (\log \left (-\frac {1}{4} x \left (x^2-x+4\right )\right )\right )\right )}{x \left (x^2-x+4\right ) \left (4-x^2 \log (5)\right ) \log \left (-\frac {1}{4} x \left (x^2-x+4\right )\right ) \log ^2\left (x^2 \log (5)-4\right )}dx}{4+4 \log ^2(5)+7 \log (5)}\)

\(\Big \downarrow \) 7276

\(\displaystyle -\frac {(4+\log (5) \log (625)+\log (78125)) \int \left (\frac {-\log (25) \log \left (-\frac {1}{4} x \left (x^2-x+4\right )\right ) x^4-3 \log (5) \log \left (x^2 \log (5)-4\right ) x^4+\log (25) \log \left (-\frac {1}{4} x \left (x^2-x+4\right )\right ) x^3+\log (25) \log \left (x^2 \log (5)-4\right ) x^3-4 \log (25) \log \left (-\frac {1}{4} x \left (x^2-x+4\right )\right ) x^2+12 \left (1-\frac {\log (5)}{3}\right ) \log \left (x^2 \log (5)-4\right ) x^2-8 \log \left (x^2 \log (5)-4\right ) x+16 \log \left (x^2 \log (5)-4\right )}{x \left (x^2-x+4\right ) \left (4-x^2 \log (5)\right ) \log \left (-\frac {1}{4} x \left (x^2-x+4\right )\right ) \log ^2\left (x^2 \log (5)-4\right )}-\frac {2 x \log (5) \log \left (\log \left (-\frac {1}{4} x \left (x^2-x+4\right )\right )\right )}{\left (x^2 \log (5)-4\right ) \log ^2\left (x^2 \log (5)-4\right )}\right )dx}{4+4 \log ^2(5)+7 \log (5)}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {(4+\log (5) \log (625)+\log (78125)) \left (\sqrt {\log (5)} \int \frac {\log \left (\log \left (-\frac {1}{4} x \left (x^2-x+4\right )\right )\right )}{\left (2-x \sqrt {\log (5)}\right ) \log ^2\left (x^2 \log (5)-4\right )}dx-\sqrt {\log (5)} \int \frac {\log \left (\log \left (-\frac {1}{4} x \left (x^2-x+4\right )\right )\right )}{\left (\sqrt {\log (5)} x+2\right ) \log ^2\left (x^2 \log (5)-4\right )}dx-\frac {2 i \int \frac {1}{\left (-2 x+i \sqrt {15}+1\right ) \log \left (-\frac {1}{4} x \left (x^2-x+4\right )\right ) \log \left (x^2 \log (5)-4\right )}dx}{\sqrt {15}}+\int \frac {1}{x \log \left (-\frac {1}{4} x \left (x^2-x+4\right )\right ) \log \left (x^2 \log (5)-4\right )}dx+\frac {\left (15-i \sqrt {15}\right ) \log (25) \int \frac {1}{\left (2 x-i \sqrt {15}-1\right ) \log \left (-\frac {1}{4} x \left (x^2-x+4\right )\right ) \log \left (x^2 \log (5)-4\right )}dx}{15 \log (5)}+\frac {\left (15+i \sqrt {15}\right ) \log (25) \int \frac {1}{\left (2 x+i \sqrt {15}-1\right ) \log \left (-\frac {1}{4} x \left (x^2-x+4\right )\right ) \log \left (x^2 \log (5)-4\right )}dx}{15 \log (5)}-\frac {2 i \int \frac {1}{\left (2 x+i \sqrt {15}-1\right ) \log \left (-\frac {1}{4} x \left (x^2-x+4\right )\right ) \log \left (x^2 \log (5)-4\right )}dx}{\sqrt {15}}-\frac {\log (25)}{2 \log (5) \log \left (x^2 \log (5)-4\right )}\right )}{4+4 \log ^2(5)+7 \log (5)}\)

Input:

Int[((-8*x^2 + 2*x^3 - 2*x^4)*Log[5]*Log[(-4*x + x^2 - x^3)/4] + (16 - 8*x 
 + 12*x^2 + (-4*x^2 + 2*x^3 - 3*x^4)*Log[5])*Log[-4 + x^2*Log[5]] + (8*x^2 
 - 2*x^3 + 2*x^4)*Log[5]*Log[(-4*x + x^2 - x^3)/4]*Log[Log[(-4*x + x^2 - x 
^3)/4]])/((-16*x + 4*x^2 - 4*x^3 + (4*x^3 - x^4 + x^5)*Log[5])*Log[(-4*x + 
 x^2 - x^3)/4]*Log[-4 + x^2*Log[5]]^2),x]
 

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

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

Time = 0.39 (sec) , antiderivative size = 147, normalized size of antiderivative = 4.32

\[-\frac {\ln \left (-2 \ln \left (2\right )+i \pi +\ln \left (x \right )+\ln \left (x^{2}-x +4\right )-\frac {i \pi \,\operatorname {csgn}\left (i x \left (x^{2}-x +4\right )\right ) \left (-\operatorname {csgn}\left (i x \left (x^{2}-x +4\right )\right )+\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i x \left (x^{2}-x +4\right )\right )+\operatorname {csgn}\left (i \left (x^{2}-x +4\right )\right )\right )}{2}+i \pi {\operatorname {csgn}\left (i x \left (x^{2}-x +4\right )\right )}^{2} \left (\operatorname {csgn}\left (i x \left (x^{2}-x +4\right )\right )-1\right )\right )}{\ln \left (x^{2} \ln \left (5\right )-4\right )}+\frac {1}{\ln \left (x^{2} \ln \left (5\right )-4\right )}\]

Input:

int(((2*x^4-2*x^3+8*x^2)*ln(5)*ln(-1/4*x^3+1/4*x^2-x)*ln(ln(-1/4*x^3+1/4*x 
^2-x))+((-3*x^4+2*x^3-4*x^2)*ln(5)+12*x^2-8*x+16)*ln(x^2*ln(5)-4)+(-2*x^4+ 
2*x^3-8*x^2)*ln(5)*ln(-1/4*x^3+1/4*x^2-x))/((x^5-x^4+4*x^3)*ln(5)-4*x^3+4* 
x^2-16*x)/ln(-1/4*x^3+1/4*x^2-x)/ln(x^2*ln(5)-4)^2,x)
 

Output:

-1/ln(x^2*ln(5)-4)*ln(-2*ln(2)+I*Pi+ln(x)+ln(x^2-x+4)-1/2*I*Pi*csgn(I*x*(x 
^2-x+4))*(-csgn(I*x*(x^2-x+4))+csgn(I*x))*(-csgn(I*x*(x^2-x+4))+csgn(I*(x^ 
2-x+4)))+I*Pi*csgn(I*x*(x^2-x+4))^2*(csgn(I*x*(x^2-x+4))-1))+1/ln(x^2*ln(5 
)-4)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91 \[ \int \frac {\left (-8 x^2+2 x^3-2 x^4\right ) \log (5) \log \left (\frac {1}{4} \left (-4 x+x^2-x^3\right )\right )+\left (16-8 x+12 x^2+\left (-4 x^2+2 x^3-3 x^4\right ) \log (5)\right ) \log \left (-4+x^2 \log (5)\right )+\left (8 x^2-2 x^3+2 x^4\right ) \log (5) \log \left (\frac {1}{4} \left (-4 x+x^2-x^3\right )\right ) \log \left (\log \left (\frac {1}{4} \left (-4 x+x^2-x^3\right )\right )\right )}{\left (-16 x+4 x^2-4 x^3+\left (4 x^3-x^4+x^5\right ) \log (5)\right ) \log \left (\frac {1}{4} \left (-4 x+x^2-x^3\right )\right ) \log ^2\left (-4+x^2 \log (5)\right )} \, dx=-\frac {\log \left (\log \left (-\frac {1}{4} \, x^{3} + \frac {1}{4} \, x^{2} - x\right )\right ) - 1}{\log \left (x^{2} \log \left (5\right ) - 4\right )} \] Input:

integrate(((2*x^4-2*x^3+8*x^2)*log(5)*log(-1/4*x^3+1/4*x^2-x)*log(log(-1/4 
*x^3+1/4*x^2-x))+((-3*x^4+2*x^3-4*x^2)*log(5)+12*x^2-8*x+16)*log(x^2*log(5 
)-4)+(-2*x^4+2*x^3-8*x^2)*log(5)*log(-1/4*x^3+1/4*x^2-x))/((x^5-x^4+4*x^3) 
*log(5)-4*x^3+4*x^2-16*x)/log(-1/4*x^3+1/4*x^2-x)/log(x^2*log(5)-4)^2,x, a 
lgorithm="fricas")
 

Output:

-(log(log(-1/4*x^3 + 1/4*x^2 - x)) - 1)/log(x^2*log(5) - 4)
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {\left (-8 x^2+2 x^3-2 x^4\right ) \log (5) \log \left (\frac {1}{4} \left (-4 x+x^2-x^3\right )\right )+\left (16-8 x+12 x^2+\left (-4 x^2+2 x^3-3 x^4\right ) \log (5)\right ) \log \left (-4+x^2 \log (5)\right )+\left (8 x^2-2 x^3+2 x^4\right ) \log (5) \log \left (\frac {1}{4} \left (-4 x+x^2-x^3\right )\right ) \log \left (\log \left (\frac {1}{4} \left (-4 x+x^2-x^3\right )\right )\right )}{\left (-16 x+4 x^2-4 x^3+\left (4 x^3-x^4+x^5\right ) \log (5)\right ) \log \left (\frac {1}{4} \left (-4 x+x^2-x^3\right )\right ) \log ^2\left (-4+x^2 \log (5)\right )} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(((2*x**4-2*x**3+8*x**2)*ln(5)*ln(-1/4*x**3+1/4*x**2-x)*ln(ln(-1/ 
4*x**3+1/4*x**2-x))+((-3*x**4+2*x**3-4*x**2)*ln(5)+12*x**2-8*x+16)*ln(x**2 
*ln(5)-4)+(-2*x**4+2*x**3-8*x**2)*ln(5)*ln(-1/4*x**3+1/4*x**2-x))/((x**5-x 
**4+4*x**3)*ln(5)-4*x**3+4*x**2-16*x)/ln(-1/4*x**3+1/4*x**2-x)/ln(x**2*ln( 
5)-4)**2,x)
 

Output:

Exception raised: TypeError >> '>' not supported between instances of 'Pol 
y' and 'int'
 

Maxima [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \frac {\left (-8 x^2+2 x^3-2 x^4\right ) \log (5) \log \left (\frac {1}{4} \left (-4 x+x^2-x^3\right )\right )+\left (16-8 x+12 x^2+\left (-4 x^2+2 x^3-3 x^4\right ) \log (5)\right ) \log \left (-4+x^2 \log (5)\right )+\left (8 x^2-2 x^3+2 x^4\right ) \log (5) \log \left (\frac {1}{4} \left (-4 x+x^2-x^3\right )\right ) \log \left (\log \left (\frac {1}{4} \left (-4 x+x^2-x^3\right )\right )\right )}{\left (-16 x+4 x^2-4 x^3+\left (4 x^3-x^4+x^5\right ) \log (5)\right ) \log \left (\frac {1}{4} \left (-4 x+x^2-x^3\right )\right ) \log ^2\left (-4+x^2 \log (5)\right )} \, dx=-\frac {\log \left (-2 \, \log \left (2\right ) + \log \left (-x^{2} + x - 4\right ) + \log \left (x\right )\right ) - 1}{\log \left (x^{2} \log \left (5\right ) - 4\right )} \] Input:

integrate(((2*x^4-2*x^3+8*x^2)*log(5)*log(-1/4*x^3+1/4*x^2-x)*log(log(-1/4 
*x^3+1/4*x^2-x))+((-3*x^4+2*x^3-4*x^2)*log(5)+12*x^2-8*x+16)*log(x^2*log(5 
)-4)+(-2*x^4+2*x^3-8*x^2)*log(5)*log(-1/4*x^3+1/4*x^2-x))/((x^5-x^4+4*x^3) 
*log(5)-4*x^3+4*x^2-16*x)/log(-1/4*x^3+1/4*x^2-x)/log(x^2*log(5)-4)^2,x, a 
lgorithm="maxima")
 

Output:

-(log(-2*log(2) + log(-x^2 + x - 4) + log(x)) - 1)/log(x^2*log(5) - 4)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (-8 x^2+2 x^3-2 x^4\right ) \log (5) \log \left (\frac {1}{4} \left (-4 x+x^2-x^3\right )\right )+\left (16-8 x+12 x^2+\left (-4 x^2+2 x^3-3 x^4\right ) \log (5)\right ) \log \left (-4+x^2 \log (5)\right )+\left (8 x^2-2 x^3+2 x^4\right ) \log (5) \log \left (\frac {1}{4} \left (-4 x+x^2-x^3\right )\right ) \log \left (\log \left (\frac {1}{4} \left (-4 x+x^2-x^3\right )\right )\right )}{\left (-16 x+4 x^2-4 x^3+\left (4 x^3-x^4+x^5\right ) \log (5)\right ) \log \left (\frac {1}{4} \left (-4 x+x^2-x^3\right )\right ) \log ^2\left (-4+x^2 \log (5)\right )} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(((2*x^4-2*x^3+8*x^2)*log(5)*log(-1/4*x^3+1/4*x^2-x)*log(log(-1/4 
*x^3+1/4*x^2-x))+((-3*x^4+2*x^3-4*x^2)*log(5)+12*x^2-8*x+16)*log(x^2*log(5 
)-4)+(-2*x^4+2*x^3-8*x^2)*log(5)*log(-1/4*x^3+1/4*x^2-x))/((x^5-x^4+4*x^3) 
*log(5)-4*x^3+4*x^2-16*x)/log(-1/4*x^3+1/4*x^2-x)/log(x^2*log(5)-4)^2,x, a 
lgorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error index.cc index_gcd Error: Bad 
 Argument ValueError index.cc index_gcd Error: Bad Argument Value
 

Mupad [B] (verification not implemented)

Time = 5.39 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91 \[ \int \frac {\left (-8 x^2+2 x^3-2 x^4\right ) \log (5) \log \left (\frac {1}{4} \left (-4 x+x^2-x^3\right )\right )+\left (16-8 x+12 x^2+\left (-4 x^2+2 x^3-3 x^4\right ) \log (5)\right ) \log \left (-4+x^2 \log (5)\right )+\left (8 x^2-2 x^3+2 x^4\right ) \log (5) \log \left (\frac {1}{4} \left (-4 x+x^2-x^3\right )\right ) \log \left (\log \left (\frac {1}{4} \left (-4 x+x^2-x^3\right )\right )\right )}{\left (-16 x+4 x^2-4 x^3+\left (4 x^3-x^4+x^5\right ) \log (5)\right ) \log \left (\frac {1}{4} \left (-4 x+x^2-x^3\right )\right ) \log ^2\left (-4+x^2 \log (5)\right )} \, dx=-\frac {\ln \left (\ln \left (-\frac {x^3}{4}+\frac {x^2}{4}-x\right )\right )-1}{\ln \left (x^2\,\ln \left (5\right )-4\right )} \] Input:

int((log(x^2*log(5) - 4)*(8*x + log(5)*(4*x^2 - 2*x^3 + 3*x^4) - 12*x^2 - 
16) + log(x^2/4 - x - x^3/4)*log(5)*(8*x^2 - 2*x^3 + 2*x^4) - log(x^2/4 - 
x - x^3/4)*log(5)*log(log(x^2/4 - x - x^3/4))*(8*x^2 - 2*x^3 + 2*x^4))/(lo 
g(x^2/4 - x - x^3/4)*log(x^2*log(5) - 4)^2*(16*x - log(5)*(4*x^3 - x^4 + x 
^5) - 4*x^2 + 4*x^3)),x)
 

Output:

-(log(log(x^2/4 - x - x^3/4)) - 1)/log(x^2*log(5) - 4)
 

Reduce [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \frac {\left (-8 x^2+2 x^3-2 x^4\right ) \log (5) \log \left (\frac {1}{4} \left (-4 x+x^2-x^3\right )\right )+\left (16-8 x+12 x^2+\left (-4 x^2+2 x^3-3 x^4\right ) \log (5)\right ) \log \left (-4+x^2 \log (5)\right )+\left (8 x^2-2 x^3+2 x^4\right ) \log (5) \log \left (\frac {1}{4} \left (-4 x+x^2-x^3\right )\right ) \log \left (\log \left (\frac {1}{4} \left (-4 x+x^2-x^3\right )\right )\right )}{\left (-16 x+4 x^2-4 x^3+\left (4 x^3-x^4+x^5\right ) \log (5)\right ) \log \left (\frac {1}{4} \left (-4 x+x^2-x^3\right )\right ) \log ^2\left (-4+x^2 \log (5)\right )} \, dx=\frac {-\mathrm {log}\left (\mathrm {log}\left (-\frac {1}{4} x^{3}+\frac {1}{4} x^{2}-x \right )\right )+1}{\mathrm {log}\left (\mathrm {log}\left (5\right ) x^{2}-4\right )} \] Input:

int(((2*x^4-2*x^3+8*x^2)*log(5)*log(-1/4*x^3+1/4*x^2-x)*log(log(-1/4*x^3+1 
/4*x^2-x))+((-3*x^4+2*x^3-4*x^2)*log(5)+12*x^2-8*x+16)*log(x^2*log(5)-4)+( 
-2*x^4+2*x^3-8*x^2)*log(5)*log(-1/4*x^3+1/4*x^2-x))/((x^5-x^4+4*x^3)*log(5 
)-4*x^3+4*x^2-16*x)/log(-1/4*x^3+1/4*x^2-x)/log(x^2*log(5)-4)^2,x)
 

Output:

( - log(log(( - x**3 + x**2 - 4*x)/4)) + 1)/log(log(5)*x**2 - 4)