Integrand size = 102, antiderivative size = 23 \[ \int \frac {-4 x+2 x^2+\left (-134 x-2 x^2+2 x \log \left (4 x^2\right )\right ) \log \left (67+x-\log \left (4 x^2\right )\right )+\left (134+2 x-2 \log \left (4 x^2\right )\right ) \log ^2\left (67+x-\log \left (4 x^2\right )\right )}{\left (-67 x-x^2+x \log \left (4 x^2\right )\right ) \log ^2\left (67+x-\log \left (4 x^2\right )\right )} \, dx=2 \left (-\log (x)+\frac {x}{\log \left (67+x-\log \left (4 x^2\right )\right )}\right ) \] Output:
2*x/ln(-ln(4*x^2)+x+67)-2*ln(x)
Time = 0.08 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {-4 x+2 x^2+\left (-134 x-2 x^2+2 x \log \left (4 x^2\right )\right ) \log \left (67+x-\log \left (4 x^2\right )\right )+\left (134+2 x-2 \log \left (4 x^2\right )\right ) \log ^2\left (67+x-\log \left (4 x^2\right )\right )}{\left (-67 x-x^2+x \log \left (4 x^2\right )\right ) \log ^2\left (67+x-\log \left (4 x^2\right )\right )} \, dx=2 \left (-\log (x)+\frac {x}{\log \left (67+x-\log \left (4 x^2\right )\right )}\right ) \] Input:
Integrate[(-4*x + 2*x^2 + (-134*x - 2*x^2 + 2*x*Log[4*x^2])*Log[67 + x - L og[4*x^2]] + (134 + 2*x - 2*Log[4*x^2])*Log[67 + x - Log[4*x^2]]^2)/((-67* x - x^2 + x*Log[4*x^2])*Log[67 + x - Log[4*x^2]]^2),x]
Output:
2*(-Log[x] + x/Log[67 + x - Log[4*x^2]])
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.
\(\displaystyle \int \frac {2 x^2+\left (-2 \log \left (4 x^2\right )+2 x+134\right ) \log ^2\left (-\log \left (4 x^2\right )+x+67\right )+\left (-2 x^2+2 x \log \left (4 x^2\right )-134 x\right ) \log \left (-\log \left (4 x^2\right )+x+67\right )-4 x}{\left (-x^2+x \log \left (4 x^2\right )-67 x\right ) \log ^2\left (-\log \left (4 x^2\right )+x+67\right )} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int 2 \left (\frac {2-x}{\left (-\log \left (4 x^2\right )+x+67\right ) \log ^2\left (-\log \left (4 x^2\right )+x+67\right )}+\frac {1}{\log \left (-\log \left (4 x^2\right )+x+67\right )}-\frac {1}{x}\right )dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int \left (\frac {2-x}{\left (x-\log \left (4 x^2\right )+67\right ) \log ^2\left (x-\log \left (4 x^2\right )+67\right )}-\frac {1}{x}+\frac {1}{\log \left (x-\log \left (4 x^2\right )+67\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (2 \int \frac {1}{\left (x-\log \left (4 x^2\right )+67\right ) \log ^2\left (x-\log \left (4 x^2\right )+67\right )}dx-\int \frac {x}{\left (x-\log \left (4 x^2\right )+67\right ) \log ^2\left (x-\log \left (4 x^2\right )+67\right )}dx+\int \frac {1}{\log \left (x-\log \left (4 x^2\right )+67\right )}dx-\log (x)\right )\) |
Input:
Int[(-4*x + 2*x^2 + (-134*x - 2*x^2 + 2*x*Log[4*x^2])*Log[67 + x - Log[4*x ^2]] + (134 + 2*x - 2*Log[4*x^2])*Log[67 + x - Log[4*x^2]]^2)/((-67*x - x^ 2 + x*Log[4*x^2])*Log[67 + x - Log[4*x^2]]^2),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(54\) vs. \(2(22)=44\).
Time = 0.39 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.39
method | result | size |
parallelrisch | \(\frac {-2 \ln \left (-\ln \left (4 x^{2}\right )+x +67\right ) \ln \left (4 x^{2}\right )+4 x -134 \ln \left (-\ln \left (4 x^{2}\right )+x +67\right )}{2 \ln \left (-\ln \left (4 x^{2}\right )+x +67\right )}\) | \(55\) |
Input:
int(((-2*ln(4*x^2)+2*x+134)*ln(-ln(4*x^2)+x+67)^2+(2*x*ln(4*x^2)-2*x^2-134 *x)*ln(-ln(4*x^2)+x+67)+2*x^2-4*x)/(x*ln(4*x^2)-x^2-67*x)/ln(-ln(4*x^2)+x+ 67)^2,x,method=_RETURNVERBOSE)
Output:
1/2*(-2*ln(-ln(4*x^2)+x+67)*ln(4*x^2)+4*x-134*ln(-ln(4*x^2)+x+67))/ln(-ln( 4*x^2)+x+67)
Time = 0.07 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.70 \[ \int \frac {-4 x+2 x^2+\left (-134 x-2 x^2+2 x \log \left (4 x^2\right )\right ) \log \left (67+x-\log \left (4 x^2\right )\right )+\left (134+2 x-2 \log \left (4 x^2\right )\right ) \log ^2\left (67+x-\log \left (4 x^2\right )\right )}{\left (-67 x-x^2+x \log \left (4 x^2\right )\right ) \log ^2\left (67+x-\log \left (4 x^2\right )\right )} \, dx=-\frac {\log \left (4 \, x^{2}\right ) \log \left (x - \log \left (4 \, x^{2}\right ) + 67\right ) - 2 \, x}{\log \left (x - \log \left (4 \, x^{2}\right ) + 67\right )} \] Input:
integrate(((-2*log(4*x^2)+2*x+134)*log(-log(4*x^2)+x+67)^2+(2*x*log(4*x^2) -2*x^2-134*x)*log(-log(4*x^2)+x+67)+2*x^2-4*x)/(x*log(4*x^2)-x^2-67*x)/log (-log(4*x^2)+x+67)^2,x, algorithm="fricas")
Output:
-(log(4*x^2)*log(x - log(4*x^2) + 67) - 2*x)/log(x - log(4*x^2) + 67)
Time = 0.11 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {-4 x+2 x^2+\left (-134 x-2 x^2+2 x \log \left (4 x^2\right )\right ) \log \left (67+x-\log \left (4 x^2\right )\right )+\left (134+2 x-2 \log \left (4 x^2\right )\right ) \log ^2\left (67+x-\log \left (4 x^2\right )\right )}{\left (-67 x-x^2+x \log \left (4 x^2\right )\right ) \log ^2\left (67+x-\log \left (4 x^2\right )\right )} \, dx=\frac {2 x}{\log {\left (x - \log {\left (4 x^{2} \right )} + 67 \right )}} - 2 \log {\left (x \right )} \] Input:
integrate(((-2*ln(4*x**2)+2*x+134)*ln(-ln(4*x**2)+x+67)**2+(2*x*ln(4*x**2) -2*x**2-134*x)*ln(-ln(4*x**2)+x+67)+2*x**2-4*x)/(x*ln(4*x**2)-x**2-67*x)/l n(-ln(4*x**2)+x+67)**2,x)
Output:
2*x/log(x - log(4*x**2) + 67) - 2*log(x)
Time = 0.14 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {-4 x+2 x^2+\left (-134 x-2 x^2+2 x \log \left (4 x^2\right )\right ) \log \left (67+x-\log \left (4 x^2\right )\right )+\left (134+2 x-2 \log \left (4 x^2\right )\right ) \log ^2\left (67+x-\log \left (4 x^2\right )\right )}{\left (-67 x-x^2+x \log \left (4 x^2\right )\right ) \log ^2\left (67+x-\log \left (4 x^2\right )\right )} \, dx=\frac {2 \, x}{\log \left (x - 2 \, \log \left (2\right ) - 2 \, \log \left (x\right ) + 67\right )} - 2 \, \log \left (x\right ) \] Input:
integrate(((-2*log(4*x^2)+2*x+134)*log(-log(4*x^2)+x+67)^2+(2*x*log(4*x^2) -2*x^2-134*x)*log(-log(4*x^2)+x+67)+2*x^2-4*x)/(x*log(4*x^2)-x^2-67*x)/log (-log(4*x^2)+x+67)^2,x, algorithm="maxima")
Output:
2*x/log(x - 2*log(2) - 2*log(x) + 67) - 2*log(x)
Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {-4 x+2 x^2+\left (-134 x-2 x^2+2 x \log \left (4 x^2\right )\right ) \log \left (67+x-\log \left (4 x^2\right )\right )+\left (134+2 x-2 \log \left (4 x^2\right )\right ) \log ^2\left (67+x-\log \left (4 x^2\right )\right )}{\left (-67 x-x^2+x \log \left (4 x^2\right )\right ) \log ^2\left (67+x-\log \left (4 x^2\right )\right )} \, dx=\frac {2 \, x}{\log \left (x - \log \left (4 \, x^{2}\right ) + 67\right )} - 2 \, \log \left (x\right ) \] Input:
integrate(((-2*log(4*x^2)+2*x+134)*log(-log(4*x^2)+x+67)^2+(2*x*log(4*x^2) -2*x^2-134*x)*log(-log(4*x^2)+x+67)+2*x^2-4*x)/(x*log(4*x^2)-x^2-67*x)/log (-log(4*x^2)+x+67)^2,x, algorithm="giac")
Output:
2*x/log(x - log(4*x^2) + 67) - 2*log(x)
Time = 2.26 (sec) , antiderivative size = 78, normalized size of antiderivative = 3.39 \[ \int \frac {-4 x+2 x^2+\left (-134 x-2 x^2+2 x \log \left (4 x^2\right )\right ) \log \left (67+x-\log \left (4 x^2\right )\right )+\left (134+2 x-2 \log \left (4 x^2\right )\right ) \log ^2\left (67+x-\log \left (4 x^2\right )\right )}{\left (-67 x-x^2+x \log \left (4 x^2\right )\right ) \log ^2\left (67+x-\log \left (4 x^2\right )\right )} \, dx=2\,x-6\,\ln \left (x\right )+\frac {276}{x-2}-\frac {4\,\ln \left (4\,x^2\right )}{x-2}+\frac {2\,x-\frac {2\,x\,\ln \left (x-\ln \left (4\,x^2\right )+67\right )\,\left (x-\ln \left (4\,x^2\right )+67\right )}{x-2}}{\ln \left (x-\ln \left (4\,x^2\right )+67\right )} \] Input:
int((4*x - log(x - log(4*x^2) + 67)^2*(2*x - 2*log(4*x^2) + 134) + log(x - log(4*x^2) + 67)*(134*x - 2*x*log(4*x^2) + 2*x^2) - 2*x^2)/(log(x - log(4 *x^2) + 67)^2*(67*x - x*log(4*x^2) + x^2)),x)
Output:
2*x - 6*log(x) + 276/(x - 2) - (4*log(4*x^2))/(x - 2) + (2*x - (2*x*log(x - log(4*x^2) + 67)*(x - log(4*x^2) + 67))/(x - 2))/log(x - log(4*x^2) + 67 )
Time = 0.18 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.48 \[ \int \frac {-4 x+2 x^2+\left (-134 x-2 x^2+2 x \log \left (4 x^2\right )\right ) \log \left (67+x-\log \left (4 x^2\right )\right )+\left (134+2 x-2 \log \left (4 x^2\right )\right ) \log ^2\left (67+x-\log \left (4 x^2\right )\right )}{\left (-67 x-x^2+x \log \left (4 x^2\right )\right ) \log ^2\left (67+x-\log \left (4 x^2\right )\right )} \, dx=\frac {-2 \,\mathrm {log}\left (-\mathrm {log}\left (4 x^{2}\right )+x +67\right ) \mathrm {log}\left (x \right )+2 x}{\mathrm {log}\left (-\mathrm {log}\left (4 x^{2}\right )+x +67\right )} \] Input:
int(((-2*log(4*x^2)+2*x+134)*log(-log(4*x^2)+x+67)^2+(2*x*log(4*x^2)-2*x^2 -134*x)*log(-log(4*x^2)+x+67)+2*x^2-4*x)/(x*log(4*x^2)-x^2-67*x)/log(-log( 4*x^2)+x+67)^2,x)
Output:
(2*( - log( - log(4*x**2) + x + 67)*log(x) + x))/log( - log(4*x**2) + x + 67)