Integrand size = 235, antiderivative size = 26 \[ \int \frac {\left (\frac {x+4 \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}{\log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}\right )^{\frac {1}{\log \left (x^2\right )}} \left (-x \log \left (x^2\right )+\left (-x+x^2\right ) \log \left (x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+\left (\left (2 x-2 x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+(8-8 x) \log ^2\left (\frac {(-4+4 x) \log (4)}{5 x}\right )\right ) \log \left (\frac {x+4 \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}{\log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}\right )\right )}{\left (-x^2+x^3\right ) \log ^2\left (x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+\left (-4 x+4 x^2\right ) \log ^2\left (x^2\right ) \log ^2\left (\frac {(-4+4 x) \log (4)}{5 x}\right )} \, dx=\left (4+\frac {x}{\log \left (\frac {4 (-1+x) \log (4)}{5 x}\right )}\right )^{\frac {1}{\log \left (x^2\right )}} \] Output:
exp(ln(4+x/ln(8/5*(-1+x)/x*ln(2)))/ln(x^2))
Time = 0.13 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\left (\frac {x+4 \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}{\log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}\right )^{\frac {1}{\log \left (x^2\right )}} \left (-x \log \left (x^2\right )+\left (-x+x^2\right ) \log \left (x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+\left (\left (2 x-2 x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+(8-8 x) \log ^2\left (\frac {(-4+4 x) \log (4)}{5 x}\right )\right ) \log \left (\frac {x+4 \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}{\log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}\right )\right )}{\left (-x^2+x^3\right ) \log ^2\left (x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+\left (-4 x+4 x^2\right ) \log ^2\left (x^2\right ) \log ^2\left (\frac {(-4+4 x) \log (4)}{5 x}\right )} \, dx=\left (4+\frac {x}{\log \left (\frac {4 (-1+x) \log (4)}{5 x}\right )}\right )^{\frac {1}{\log \left (x^2\right )}} \] Input:
Integrate[(((x + 4*Log[((-4 + 4*x)*Log[4])/(5*x)])/Log[((-4 + 4*x)*Log[4]) /(5*x)])^Log[x^2]^(-1)*(-(x*Log[x^2]) + (-x + x^2)*Log[x^2]*Log[((-4 + 4*x )*Log[4])/(5*x)] + ((2*x - 2*x^2)*Log[((-4 + 4*x)*Log[4])/(5*x)] + (8 - 8* x)*Log[((-4 + 4*x)*Log[4])/(5*x)]^2)*Log[(x + 4*Log[((-4 + 4*x)*Log[4])/(5 *x)])/Log[((-4 + 4*x)*Log[4])/(5*x)]]))/((-x^2 + x^3)*Log[x^2]^2*Log[((-4 + 4*x)*Log[4])/(5*x)] + (-4*x + 4*x^2)*Log[x^2]^2*Log[((-4 + 4*x)*Log[4])/ (5*x)]^2),x]
Output:
(4 + x/Log[(4*(-1 + x)*Log[4])/(5*x)])^Log[x^2]^(-1)
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.
\(\displaystyle \int \frac {\left (\frac {x+4 \log \left (\frac {(4 x-4) \log (4)}{5 x}\right )}{\log \left (\frac {(4 x-4) \log (4)}{5 x}\right )}\right )^{\frac {1}{\log \left (x^2\right )}} \left (\left (\left (2 x-2 x^2\right ) \log \left (\frac {(4 x-4) \log (4)}{5 x}\right )+(8-8 x) \log ^2\left (\frac {(4 x-4) \log (4)}{5 x}\right )\right ) \log \left (\frac {x+4 \log \left (\frac {(4 x-4) \log (4)}{5 x}\right )}{\log \left (\frac {(4 x-4) \log (4)}{5 x}\right )}\right )-x \log \left (x^2\right )+\left (x^2-x\right ) \log \left (\frac {(4 x-4) \log (4)}{5 x}\right ) \log \left (x^2\right )\right )}{\left (4 x^2-4 x\right ) \log ^2\left (\frac {(4 x-4) \log (4)}{5 x}\right ) \log ^2\left (x^2\right )+\left (x^3-x^2\right ) \log \left (\frac {(4 x-4) \log (4)}{5 x}\right ) \log ^2\left (x^2\right )} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {\left (\frac {x}{\log \left (\frac {4 (x-1) \log (4)}{5 x}\right )}+4\right )^{\frac {1}{\log \left (x^2\right )}} \left (\frac {\log \left (x^2\right ) \left ((x-1) \log \left (\frac {4 (x-1) \log (4)}{5 x}\right )-1\right )}{(x-1) \log \left (\frac {4 (x-1) \log (4)}{5 x}\right ) \left (x+4 \log \left (\frac {4 (x-1) \log (4)}{5 x}\right )\right )}-\frac {2 \log \left (\frac {x}{\log \left (\frac {4 (x-1) \log (4)}{5 x}\right )}+4\right )}{x}\right )}{\log ^2\left (x^2\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {\left (\frac {x}{\log \left (\frac {4 (x-1) \log (4)}{5 x}\right )}+4\right )^{\frac {1}{\log \left (x^2\right )}} \left (-x \log \left (\frac {4 (x-1) \log (4)}{5 x}\right )+\log \left (\frac {4 (x-1) \log (4)}{5 x}\right )+1\right )}{(1-x) \log \left (x^2\right ) \left (x+4 \log \left (\frac {4 (x-1) \log (4)}{5 x}\right )\right ) \log \left (\frac {4 \log (4)}{5}-\frac {4 \log (4)}{5 x}\right )}-\frac {2 \left (\frac {x}{\log \left (\frac {4 (x-1) \log (4)}{5 x}\right )}+4\right )^{\frac {1}{\log \left (x^2\right )}} \log \left (\frac {x}{\log \left (\frac {4 (x-1) \log (4)}{5 x}\right )}+4\right )}{x \log ^2\left (x^2\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -2 \int \frac {\left (\frac {x}{\log \left (\frac {4 (x-1) \log (4)}{5 x}\right )}+4\right )^{\frac {1}{\log \left (x^2\right )}} \log \left (\frac {x}{\log \left (\frac {4 (x-1) \log (4)}{5 x}\right )}+4\right )}{x \log ^2\left (x^2\right )}dx+\int \frac {\left (\frac {x}{\log \left (\frac {4 (x-1) \log (4)}{5 x}\right )}+4\right )^{\frac {1}{\log \left (x^2\right )}}}{\log \left (x^2\right ) \left (x+4 \log \left (\frac {4 (x-1) \log (4)}{5 x}\right )\right )}dx+\int \frac {\left (\frac {x}{\log \left (\frac {4 (x-1) \log (4)}{5 x}\right )}+4\right )^{\frac {1}{\log \left (x^2\right )}}}{(1-x) \log \left (x^2\right ) \left (x+4 \log \left (\frac {4 (x-1) \log (4)}{5 x}\right )\right ) \log \left (\frac {4 \log (4)}{5}-\frac {4 \log (4)}{5 x}\right )}dx\) |
Input:
Int[(((x + 4*Log[((-4 + 4*x)*Log[4])/(5*x)])/Log[((-4 + 4*x)*Log[4])/(5*x) ])^Log[x^2]^(-1)*(-(x*Log[x^2]) + (-x + x^2)*Log[x^2]*Log[((-4 + 4*x)*Log[ 4])/(5*x)] + ((2*x - 2*x^2)*Log[((-4 + 4*x)*Log[4])/(5*x)] + (8 - 8*x)*Log [((-4 + 4*x)*Log[4])/(5*x)]^2)*Log[(x + 4*Log[((-4 + 4*x)*Log[4])/(5*x)])/ Log[((-4 + 4*x)*Log[4])/(5*x)]]))/((-x^2 + x^3)*Log[x^2]^2*Log[((-4 + 4*x) *Log[4])/(5*x)] + (-4*x + 4*x^2)*Log[x^2]^2*Log[((-4 + 4*x)*Log[4])/(5*x)] ^2),x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.18 (sec) , antiderivative size = 1263, normalized size of antiderivative = 48.58
\[\text {Expression too large to display}\]
Input:
int((((-8*x+8)*ln(2/5*(-4+4*x)*ln(2)/x)^2+(-2*x^2+2*x)*ln(2/5*(-4+4*x)*ln( 2)/x))*ln((4*ln(2/5*(-4+4*x)*ln(2)/x)+x)/ln(2/5*(-4+4*x)*ln(2)/x))+(x^2-x) *ln(x^2)*ln(2/5*(-4+4*x)*ln(2)/x)-x*ln(x^2))*exp(ln((4*ln(2/5*(-4+4*x)*ln( 2)/x)+x)/ln(2/5*(-4+4*x)*ln(2)/x))/ln(x^2))/((4*x^2-4*x)*ln(x^2)^2*ln(2/5* (-4+4*x)*ln(2)/x)^2+(x^3-x^2)*ln(x^2)^2*ln(2/5*(-4+4*x)*ln(2)/x)),x)
Output:
16^(1/2/(-I*Pi*csgn(I*x^2)+I*Pi*csgn(I*x)+2*ln(x)))*(3*ln(2)-ln(5)+ln(ln(2 ))-ln(x)+ln(-1+x)-1/2*I*Pi*csgn(I/x)*csgn(I*(-1+x))*csgn(I*(-1+x)/x)+1/2*I *Pi*csgn(I/x)*csgn(I*(-1+x)/x)^2+1/2*I*Pi*csgn(I*(-1+x))*csgn(I*(-1+x)/x)^ 2-1/2*I*Pi*csgn(I*(-1+x)/x)^3)^(1/(I*Pi*csgn(I*x^2)-I*Pi*csgn(I*x)-2*ln(x) ))*(-1/2*I*Pi*csgn(I/x)*csgn(I*(-1+x))*csgn(I*(-1+x)/x)+1/2*I*Pi*csgn(I/x) *csgn(I*(-1+x)/x)^2+1/2*I*Pi*csgn(I*(-1+x))*csgn(I*(-1+x)/x)^2-1/2*I*Pi*cs gn(I*(-1+x)/x)^3+3*ln(2)-ln(5)+ln(ln(2))+1/4*x-ln(x)+ln(-1+x))^(1/(-I*Pi*c sgn(I*x^2)+I*Pi*csgn(I*x)+2*ln(x)))*exp(-I*Pi*csgn(I*(1/2*I*Pi*csgn(I/x)*c sgn(I*(-1+x))*csgn(I*(-1+x)/x)-1/2*I*Pi*csgn(I/x)*csgn(I*(-1+x)/x)^2-1/2*I *Pi*csgn(I*(-1+x))*csgn(I*(-1+x)/x)^2+1/2*I*Pi*csgn(I*(-1+x)/x)^3-ln(8/5*l n(2))-1/4*x+ln(x)-ln(-1+x))/(-ln(8/5*ln(2))+ln(x)-ln(-1+x)+1/2*I*Pi*csgn(I /x)*csgn(I*(-1+x))*csgn(I*(-1+x)/x)-1/2*I*Pi*csgn(I/x)*csgn(I*(-1+x)/x)^2- 1/2*I*Pi*csgn(I*(-1+x))*csgn(I*(-1+x)/x)^2+1/2*I*Pi*csgn(I*(-1+x)/x)^3))*( csgn(I*(1/2*I*Pi*csgn(I/x)*csgn(I*(-1+x))*csgn(I*(-1+x)/x)-1/2*I*Pi*csgn(I /x)*csgn(I*(-1+x)/x)^2-1/2*I*Pi*csgn(I*(-1+x))*csgn(I*(-1+x)/x)^2+1/2*I*Pi *csgn(I*(-1+x)/x)^3-ln(8/5*ln(2))-1/4*x+ln(x)-ln(-1+x))/(-ln(8/5*ln(2))+ln (x)-ln(-1+x)+1/2*I*Pi*csgn(I/x)*csgn(I*(-1+x))*csgn(I*(-1+x)/x)-1/2*I*Pi*c sgn(I/x)*csgn(I*(-1+x)/x)^2-1/2*I*Pi*csgn(I*(-1+x))*csgn(I*(-1+x)/x)^2+1/2 *I*Pi*csgn(I*(-1+x)/x)^3))+csgn(I*(1/2*I*Pi*csgn(I/x)*csgn(I*(-1+x))*csgn( I*(-1+x)/x)-1/2*I*Pi*csgn(I/x)*csgn(I*(-1+x)/x)^2-1/2*I*Pi*csgn(I*(-1+x...
Time = 0.07 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38 \[ \int \frac {\left (\frac {x+4 \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}{\log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}\right )^{\frac {1}{\log \left (x^2\right )}} \left (-x \log \left (x^2\right )+\left (-x+x^2\right ) \log \left (x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+\left (\left (2 x-2 x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+(8-8 x) \log ^2\left (\frac {(-4+4 x) \log (4)}{5 x}\right )\right ) \log \left (\frac {x+4 \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}{\log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}\right )\right )}{\left (-x^2+x^3\right ) \log ^2\left (x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+\left (-4 x+4 x^2\right ) \log ^2\left (x^2\right ) \log ^2\left (\frac {(-4+4 x) \log (4)}{5 x}\right )} \, dx=\left (\frac {x + 4 \, \log \left (\frac {8 \, {\left (x - 1\right )} \log \left (2\right )}{5 \, x}\right )}{\log \left (\frac {8 \, {\left (x - 1\right )} \log \left (2\right )}{5 \, x}\right )}\right )^{\left (\frac {1}{\log \left (x^{2}\right )}\right )} \] Input:
integrate((((-8*x+8)*log(2/5*(-4+4*x)*log(2)/x)^2+(-2*x^2+2*x)*log(2/5*(-4 +4*x)*log(2)/x))*log((4*log(2/5*(-4+4*x)*log(2)/x)+x)/log(2/5*(-4+4*x)*log (2)/x))+(x^2-x)*log(x^2)*log(2/5*(-4+4*x)*log(2)/x)-x*log(x^2))*exp(log((4 *log(2/5*(-4+4*x)*log(2)/x)+x)/log(2/5*(-4+4*x)*log(2)/x))/log(x^2))/((4*x ^2-4*x)*log(x^2)^2*log(2/5*(-4+4*x)*log(2)/x)^2+(x^3-x^2)*log(x^2)^2*log(2 /5*(-4+4*x)*log(2)/x)),x, algorithm="fricas")
Output:
((x + 4*log(8/5*(x - 1)*log(2)/x))/log(8/5*(x - 1)*log(2)/x))^(1/log(x^2))
Exception generated. \[ \int \frac {\left (\frac {x+4 \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}{\log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}\right )^{\frac {1}{\log \left (x^2\right )}} \left (-x \log \left (x^2\right )+\left (-x+x^2\right ) \log \left (x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+\left (\left (2 x-2 x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+(8-8 x) \log ^2\left (\frac {(-4+4 x) \log (4)}{5 x}\right )\right ) \log \left (\frac {x+4 \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}{\log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}\right )\right )}{\left (-x^2+x^3\right ) \log ^2\left (x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+\left (-4 x+4 x^2\right ) \log ^2\left (x^2\right ) \log ^2\left (\frac {(-4+4 x) \log (4)}{5 x}\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((((-8*x+8)*ln(2/5*(-4+4*x)*ln(2)/x)**2+(-2*x**2+2*x)*ln(2/5*(-4+ 4*x)*ln(2)/x))*ln((4*ln(2/5*(-4+4*x)*ln(2)/x)+x)/ln(2/5*(-4+4*x)*ln(2)/x)) +(x**2-x)*ln(x**2)*ln(2/5*(-4+4*x)*ln(2)/x)-x*ln(x**2))*exp(ln((4*ln(2/5*( -4+4*x)*ln(2)/x)+x)/ln(2/5*(-4+4*x)*ln(2)/x))/ln(x**2))/((4*x**2-4*x)*ln(x **2)**2*ln(2/5*(-4+4*x)*ln(2)/x)**2+(x**3-x**2)*ln(x**2)**2*ln(2/5*(-4+4*x )*ln(2)/x)),x)
Output:
Exception raised: TypeError >> '>' not supported between instances of 'Pol y' and 'int'
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (24) = 48\).
Time = 0.54 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.35 \[ \int \frac {\left (\frac {x+4 \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}{\log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}\right )^{\frac {1}{\log \left (x^2\right )}} \left (-x \log \left (x^2\right )+\left (-x+x^2\right ) \log \left (x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+\left (\left (2 x-2 x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+(8-8 x) \log ^2\left (\frac {(-4+4 x) \log (4)}{5 x}\right )\right ) \log \left (\frac {x+4 \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}{\log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}\right )\right )}{\left (-x^2+x^3\right ) \log ^2\left (x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+\left (-4 x+4 x^2\right ) \log ^2\left (x^2\right ) \log ^2\left (\frac {(-4+4 x) \log (4)}{5 x}\right )} \, dx=e^{\left (\frac {\log \left (x - 4 \, \log \left (5\right ) + 12 \, \log \left (2\right ) + 4 \, \log \left (x - 1\right ) - 4 \, \log \left (x\right ) + 4 \, \log \left (\log \left (2\right )\right )\right )}{2 \, \log \left (x\right )} - \frac {\log \left (-\log \left (5\right ) + 3 \, \log \left (2\right ) + \log \left (x - 1\right ) - \log \left (x\right ) + \log \left (\log \left (2\right )\right )\right )}{2 \, \log \left (x\right )}\right )} \] Input:
integrate((((-8*x+8)*log(2/5*(-4+4*x)*log(2)/x)^2+(-2*x^2+2*x)*log(2/5*(-4 +4*x)*log(2)/x))*log((4*log(2/5*(-4+4*x)*log(2)/x)+x)/log(2/5*(-4+4*x)*log (2)/x))+(x^2-x)*log(x^2)*log(2/5*(-4+4*x)*log(2)/x)-x*log(x^2))*exp(log((4 *log(2/5*(-4+4*x)*log(2)/x)+x)/log(2/5*(-4+4*x)*log(2)/x))/log(x^2))/((4*x ^2-4*x)*log(x^2)^2*log(2/5*(-4+4*x)*log(2)/x)^2+(x^3-x^2)*log(x^2)^2*log(2 /5*(-4+4*x)*log(2)/x)),x, algorithm="maxima")
Output:
e^(1/2*log(x - 4*log(5) + 12*log(2) + 4*log(x - 1) - 4*log(x) + 4*log(log( 2)))/log(x) - 1/2*log(-log(5) + 3*log(2) + log(x - 1) - log(x) + log(log(2 )))/log(x))
\[ \int \frac {\left (\frac {x+4 \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}{\log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}\right )^{\frac {1}{\log \left (x^2\right )}} \left (-x \log \left (x^2\right )+\left (-x+x^2\right ) \log \left (x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+\left (\left (2 x-2 x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+(8-8 x) \log ^2\left (\frac {(-4+4 x) \log (4)}{5 x}\right )\right ) \log \left (\frac {x+4 \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}{\log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}\right )\right )}{\left (-x^2+x^3\right ) \log ^2\left (x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+\left (-4 x+4 x^2\right ) \log ^2\left (x^2\right ) \log ^2\left (\frac {(-4+4 x) \log (4)}{5 x}\right )} \, dx=\int { \frac {{\left ({\left (x^{2} - x\right )} \log \left (x^{2}\right ) \log \left (\frac {8 \, {\left (x - 1\right )} \log \left (2\right )}{5 \, x}\right ) - x \log \left (x^{2}\right ) - 2 \, {\left (4 \, {\left (x - 1\right )} \log \left (\frac {8 \, {\left (x - 1\right )} \log \left (2\right )}{5 \, x}\right )^{2} + {\left (x^{2} - x\right )} \log \left (\frac {8 \, {\left (x - 1\right )} \log \left (2\right )}{5 \, x}\right )\right )} \log \left (\frac {x + 4 \, \log \left (\frac {8 \, {\left (x - 1\right )} \log \left (2\right )}{5 \, x}\right )}{\log \left (\frac {8 \, {\left (x - 1\right )} \log \left (2\right )}{5 \, x}\right )}\right )\right )} \left (\frac {x + 4 \, \log \left (\frac {8 \, {\left (x - 1\right )} \log \left (2\right )}{5 \, x}\right )}{\log \left (\frac {8 \, {\left (x - 1\right )} \log \left (2\right )}{5 \, x}\right )}\right )^{\left (\frac {1}{\log \left (x^{2}\right )}\right )}}{4 \, {\left (x^{2} - x\right )} \log \left (x^{2}\right )^{2} \log \left (\frac {8 \, {\left (x - 1\right )} \log \left (2\right )}{5 \, x}\right )^{2} + {\left (x^{3} - x^{2}\right )} \log \left (x^{2}\right )^{2} \log \left (\frac {8 \, {\left (x - 1\right )} \log \left (2\right )}{5 \, x}\right )} \,d x } \] Input:
integrate((((-8*x+8)*log(2/5*(-4+4*x)*log(2)/x)^2+(-2*x^2+2*x)*log(2/5*(-4 +4*x)*log(2)/x))*log((4*log(2/5*(-4+4*x)*log(2)/x)+x)/log(2/5*(-4+4*x)*log (2)/x))+(x^2-x)*log(x^2)*log(2/5*(-4+4*x)*log(2)/x)-x*log(x^2))*exp(log((4 *log(2/5*(-4+4*x)*log(2)/x)+x)/log(2/5*(-4+4*x)*log(2)/x))/log(x^2))/((4*x ^2-4*x)*log(x^2)^2*log(2/5*(-4+4*x)*log(2)/x)^2+(x^3-x^2)*log(x^2)^2*log(2 /5*(-4+4*x)*log(2)/x)),x, algorithm="giac")
Output:
integrate(((x^2 - x)*log(x^2)*log(8/5*(x - 1)*log(2)/x) - x*log(x^2) - 2*( 4*(x - 1)*log(8/5*(x - 1)*log(2)/x)^2 + (x^2 - x)*log(8/5*(x - 1)*log(2)/x ))*log((x + 4*log(8/5*(x - 1)*log(2)/x))/log(8/5*(x - 1)*log(2)/x)))*((x + 4*log(8/5*(x - 1)*log(2)/x))/log(8/5*(x - 1)*log(2)/x))^(1/log(x^2))/(4*( x^2 - x)*log(x^2)^2*log(8/5*(x - 1)*log(2)/x)^2 + (x^3 - x^2)*log(x^2)^2*l og(8/5*(x - 1)*log(2)/x)), x)
Time = 2.40 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {\left (\frac {x+4 \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}{\log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}\right )^{\frac {1}{\log \left (x^2\right )}} \left (-x \log \left (x^2\right )+\left (-x+x^2\right ) \log \left (x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+\left (\left (2 x-2 x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+(8-8 x) \log ^2\left (\frac {(-4+4 x) \log (4)}{5 x}\right )\right ) \log \left (\frac {x+4 \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}{\log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}\right )\right )}{\left (-x^2+x^3\right ) \log ^2\left (x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+\left (-4 x+4 x^2\right ) \log ^2\left (x^2\right ) \log ^2\left (\frac {(-4+4 x) \log (4)}{5 x}\right )} \, dx={\left (\frac {x}{\ln \left (-\frac {8\,\ln \left (2\right )-8\,x\,\ln \left (2\right )}{5\,x}\right )}+4\right )}^{\frac {1}{\ln \left (x^2\right )}} \] Input:
int((exp(log((x + 4*log((2*log(2)*(4*x - 4))/(5*x)))/log((2*log(2)*(4*x - 4))/(5*x)))/log(x^2))*(x*log(x^2) - log((x + 4*log((2*log(2)*(4*x - 4))/(5 *x)))/log((2*log(2)*(4*x - 4))/(5*x)))*(log((2*log(2)*(4*x - 4))/(5*x))*(2 *x - 2*x^2) - log((2*log(2)*(4*x - 4))/(5*x))^2*(8*x - 8)) + log(x^2)*log( (2*log(2)*(4*x - 4))/(5*x))*(x - x^2)))/(log(x^2)^2*log((2*log(2)*(4*x - 4 ))/(5*x))^2*(4*x - 4*x^2) + log(x^2)^2*log((2*log(2)*(4*x - 4))/(5*x))*(x^ 2 - x^3)),x)
Output:
(x/log(-(8*log(2) - 8*x*log(2))/(5*x)) + 4)^(1/log(x^2))
Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.88 \[ \int \frac {\left (\frac {x+4 \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}{\log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}\right )^{\frac {1}{\log \left (x^2\right )}} \left (-x \log \left (x^2\right )+\left (-x+x^2\right ) \log \left (x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+\left (\left (2 x-2 x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+(8-8 x) \log ^2\left (\frac {(-4+4 x) \log (4)}{5 x}\right )\right ) \log \left (\frac {x+4 \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}{\log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )}\right )\right )}{\left (-x^2+x^3\right ) \log ^2\left (x^2\right ) \log \left (\frac {(-4+4 x) \log (4)}{5 x}\right )+\left (-4 x+4 x^2\right ) \log ^2\left (x^2\right ) \log ^2\left (\frac {(-4+4 x) \log (4)}{5 x}\right )} \, dx=e^{\frac {\mathrm {log}\left (\frac {4 \,\mathrm {log}\left (\frac {8 \,\mathrm {log}\left (2\right ) x -8 \,\mathrm {log}\left (2\right )}{5 x}\right )+x}{\mathrm {log}\left (\frac {8 \,\mathrm {log}\left (2\right ) x -8 \,\mathrm {log}\left (2\right )}{5 x}\right )}\right )}{\mathrm {log}\left (x^{2}\right )}} \] Input:
int((((-8*x+8)*log(2/5*(-4+4*x)*log(2)/x)^2+(-2*x^2+2*x)*log(2/5*(-4+4*x)* log(2)/x))*log((4*log(2/5*(-4+4*x)*log(2)/x)+x)/log(2/5*(-4+4*x)*log(2)/x) )+(x^2-x)*log(x^2)*log(2/5*(-4+4*x)*log(2)/x)-x*log(x^2))*exp(log((4*log(2 /5*(-4+4*x)*log(2)/x)+x)/log(2/5*(-4+4*x)*log(2)/x))/log(x^2))/((4*x^2-4*x )*log(x^2)^2*log(2/5*(-4+4*x)*log(2)/x)^2+(x^3-x^2)*log(x^2)^2*log(2/5*(-4 +4*x)*log(2)/x)),x)
Output:
e**(log((4*log((8*log(2)*x - 8*log(2))/(5*x)) + x)/log((8*log(2)*x - 8*log (2))/(5*x)))/log(x**2))