Integrand size = 72, antiderivative size = 24 \[ \int \frac {-8+1260 x-2 x^2+\left (4-630 x+x^2\right ) \log (4 x)+\left (630 x-2 x^2+\left (-630 x+2 x^2\right ) \log (4 x)\right ) \log \left (\frac {x}{-1+\log (4 x)}\right )}{-2 x+2 x \log (4 x)} \, dx=\left (2-\frac {1}{2} (630-x) x\right ) \log \left (\frac {x}{-1+\log (4 x)}\right ) \] Output:
(2-1/2*(-x+630)*x)*ln(x/(ln(4*x)-1))
Time = 0.12 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {-8+1260 x-2 x^2+\left (4-630 x+x^2\right ) \log (4 x)+\left (630 x-2 x^2+\left (-630 x+2 x^2\right ) \log (4 x)\right ) \log \left (\frac {x}{-1+\log (4 x)}\right )}{-2 x+2 x \log (4 x)} \, dx=\frac {1}{2} \left (4 \log (x)-4 \log (1-\log (4 x))+(-630+x) x \log \left (\frac {x}{-1+\log (4 x)}\right )\right ) \] Input:
Integrate[(-8 + 1260*x - 2*x^2 + (4 - 630*x + x^2)*Log[4*x] + (630*x - 2*x ^2 + (-630*x + 2*x^2)*Log[4*x])*Log[x/(-1 + Log[4*x])])/(-2*x + 2*x*Log[4* x]),x]
Output:
(4*Log[x] - 4*Log[1 - Log[4*x]] + (-630 + x)*x*Log[x/(-1 + 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.
\(\displaystyle \int \frac {-2 x^2+\left (x^2-630 x+4\right ) \log (4 x)+\left (-2 x^2+\left (2 x^2-630 x\right ) \log (4 x)+630 x\right ) \log \left (\frac {x}{\log (4 x)-1}\right )+1260 x-8}{2 x \log (4 x)-2 x} \, dx\) |
\(\Big \downarrow \) 3041 |
\(\displaystyle \int \frac {-2 x^2+\left (x^2-630 x+4\right ) \log (4 x)+\left (-2 x^2+\left (2 x^2-630 x\right ) \log (4 x)+630 x\right ) \log \left (\frac {x}{\log (4 x)-1}\right )+1260 x-8}{x (2 \log (4 x)-2)}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {2 x^2-\left (x^2-630 x+4\right ) \log (4 x)-\left (-2 x^2+\left (2 x^2-630 x\right ) \log (4 x)+630 x\right ) \log \left (\frac {x}{\log (4 x)-1}\right )-1260 x+8}{2 x (1-\log (4 x))}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int \frac {2 x^2-1260 x-\left (x^2-630 x+4\right ) \log (4 x)-2 \left (-x^2+315 x-\left (315 x-x^2\right ) \log (4 x)\right ) \log \left (-\frac {x}{1-\log (4 x)}\right )+8}{x (1-\log (4 x))}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{2} \int \left (\frac {\left (x^2-630 x+4\right ) (\log (4 x)-2)}{x (\log (4 x)-1)}+2 (x-315) \log \left (\frac {x}{\log (4 x)-1}\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\int \frac {-x^2+630 x-4}{x (\log (4 x)-1)}dx+\frac {1}{16} e^2 \operatorname {ExpIntegralEi}(-2 (1-\log (4 x)))-\frac {315}{2} e \operatorname {ExpIntegralEi}(\log (4 x)-1)+(315-x)^2 \log \left (-\frac {x}{1-\log (4 x)}\right )-99221 \log (x)+99225 \log (1-\log (4 x))\right )\) |
Input:
Int[(-8 + 1260*x - 2*x^2 + (4 - 630*x + x^2)*Log[4*x] + (630*x - 2*x^2 + ( -630*x + 2*x^2)*Log[4*x])*Log[x/(-1 + Log[4*x])])/(-2*x + 2*x*Log[4*x]),x]
Output:
$Aborted
Time = 1.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.88
method | result | size |
norman | \(2 \ln \left (\frac {x}{\ln \left (4 x \right )-1}\right )-315 x \ln \left (\frac {x}{\ln \left (4 x \right )-1}\right )+\frac {\ln \left (\frac {x}{\ln \left (4 x \right )-1}\right ) x^{2}}{2}\) | \(45\) |
parallelrisch | \(\frac {\ln \left (\frac {x}{\ln \left (4 x \right )-1}\right ) x^{2}}{2}+1-315 x \ln \left (\frac {x}{\ln \left (4 x \right )-1}\right )-2 \ln \left (\ln \left (4 x \right )-1\right )+2 \ln \left (4 x \right )\) | \(48\) |
Input:
int((((2*x^2-630*x)*ln(4*x)-2*x^2+630*x)*ln(x/(ln(4*x)-1))+(x^2-630*x+4)*l n(4*x)-2*x^2+1260*x-8)/(2*x*ln(4*x)-2*x),x,method=_RETURNVERBOSE)
Output:
2*ln(x/(ln(4*x)-1))-315*x*ln(x/(ln(4*x)-1))+1/2*ln(x/(ln(4*x)-1))*x^2
Time = 0.09 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88 \[ \int \frac {-8+1260 x-2 x^2+\left (4-630 x+x^2\right ) \log (4 x)+\left (630 x-2 x^2+\left (-630 x+2 x^2\right ) \log (4 x)\right ) \log \left (\frac {x}{-1+\log (4 x)}\right )}{-2 x+2 x \log (4 x)} \, dx=\frac {1}{2} \, {\left (x^{2} - 630 \, x + 4\right )} \log \left (\frac {x}{\log \left (4 \, x\right ) - 1}\right ) \] Input:
integrate((((2*x^2-630*x)*log(4*x)-2*x^2+630*x)*log(x/(log(4*x)-1))+(x^2-6 30*x+4)*log(4*x)-2*x^2+1260*x-8)/(2*x*log(4*x)-2*x),x, algorithm="fricas")
Output:
1/2*(x^2 - 630*x + 4)*log(x/(log(4*x) - 1))
Time = 0.17 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int \frac {-8+1260 x-2 x^2+\left (4-630 x+x^2\right ) \log (4 x)+\left (630 x-2 x^2+\left (-630 x+2 x^2\right ) \log (4 x)\right ) \log \left (\frac {x}{-1+\log (4 x)}\right )}{-2 x+2 x \log (4 x)} \, dx=\left (\frac {x^{2}}{2} - 315 x\right ) \log {\left (\frac {x}{\log {\left (4 x \right )} - 1} \right )} + 2 \log {\left (x \right )} - 2 \log {\left (\log {\left (4 x \right )} - 1 \right )} \] Input:
integrate((((2*x**2-630*x)*ln(4*x)-2*x**2+630*x)*ln(x/(ln(4*x)-1))+(x**2-6 30*x+4)*ln(4*x)-2*x**2+1260*x-8)/(2*x*ln(4*x)-2*x),x)
Output:
(x**2/2 - 315*x)*log(x/(log(4*x) - 1)) + 2*log(x) - 2*log(log(4*x) - 1)
Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (20) = 40\).
Time = 0.14 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.79 \[ \int \frac {-8+1260 x-2 x^2+\left (4-630 x+x^2\right ) \log (4 x)+\left (630 x-2 x^2+\left (-630 x+2 x^2\right ) \log (4 x)\right ) \log \left (\frac {x}{-1+\log (4 x)}\right )}{-2 x+2 x \log (4 x)} \, dx=\frac {1}{2} \, {\left (x^{2} - 630 \, x + 4\right )} \log \left (x\right ) - \frac {1}{2} \, {\left (x^{2} - 630 \, x - 4\right )} \log \left (2 \, \log \left (2\right ) + \log \left (x\right ) - 1\right ) - 4 \, \log \left (2 \, \log \left (2\right ) + \log \left (x\right ) - 1\right ) \] Input:
integrate((((2*x^2-630*x)*log(4*x)-2*x^2+630*x)*log(x/(log(4*x)-1))+(x^2-6 30*x+4)*log(4*x)-2*x^2+1260*x-8)/(2*x*log(4*x)-2*x),x, algorithm="maxima")
Output:
1/2*(x^2 - 630*x + 4)*log(x) - 1/2*(x^2 - 630*x - 4)*log(2*log(2) + log(x) - 1) - 4*log(2*log(2) + log(x) - 1)
Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (20) = 40\).
Time = 0.14 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.79 \[ \int \frac {-8+1260 x-2 x^2+\left (4-630 x+x^2\right ) \log (4 x)+\left (630 x-2 x^2+\left (-630 x+2 x^2\right ) \log (4 x)\right ) \log \left (\frac {x}{-1+\log (4 x)}\right )}{-2 x+2 x \log (4 x)} \, dx=\frac {1}{2} \, {\left (x^{2} - 630 \, x\right )} \log \left (x\right ) - \frac {1}{2} \, {\left (x^{2} - 630 \, x\right )} \log \left (\log \left (4 \, x\right ) - 1\right ) + 2 \, \log \left (x\right ) - 2 \, \log \left (2 \, \log \left (2\right ) + \log \left (x\right ) - 1\right ) \] Input:
integrate((((2*x^2-630*x)*log(4*x)-2*x^2+630*x)*log(x/(log(4*x)-1))+(x^2-6 30*x+4)*log(4*x)-2*x^2+1260*x-8)/(2*x*log(4*x)-2*x),x, algorithm="giac")
Output:
1/2*(x^2 - 630*x)*log(x) - 1/2*(x^2 - 630*x)*log(log(4*x) - 1) + 2*log(x) - 2*log(2*log(2) + log(x) - 1)
Time = 3.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {-8+1260 x-2 x^2+\left (4-630 x+x^2\right ) \log (4 x)+\left (630 x-2 x^2+\left (-630 x+2 x^2\right ) \log (4 x)\right ) \log \left (\frac {x}{-1+\log (4 x)}\right )}{-2 x+2 x \log (4 x)} \, dx=2\,\ln \left (x\right )-2\,\ln \left (\ln \left (4\,x\right )-1\right )-\ln \left (\frac {x}{\ln \left (4\,x\right )-1}\right )\,\left (315\,x-\frac {x^2}{2}\right ) \] Input:
int((log(x/(log(4*x) - 1))*(log(4*x)*(630*x - 2*x^2) - 630*x + 2*x^2) - 12 60*x - log(4*x)*(x^2 - 630*x + 4) + 2*x^2 + 8)/(2*x - 2*x*log(4*x)),x)
Output:
2*log(x) - 2*log(log(4*x) - 1) - log(x/(log(4*x) - 1))*(315*x - x^2/2)
Time = 0.16 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.83 \[ \int \frac {-8+1260 x-2 x^2+\left (4-630 x+x^2\right ) \log (4 x)+\left (630 x-2 x^2+\left (-630 x+2 x^2\right ) \log (4 x)\right ) \log \left (\frac {x}{-1+\log (4 x)}\right )}{-2 x+2 x \log (4 x)} \, dx=-2 \,\mathrm {log}\left (\mathrm {log}\left (4 x \right )-1\right )+\frac {\mathrm {log}\left (\frac {x}{\mathrm {log}\left (4 x \right )-1}\right ) x^{2}}{2}-315 \,\mathrm {log}\left (\frac {x}{\mathrm {log}\left (4 x \right )-1}\right ) x +2 \,\mathrm {log}\left (x \right ) \] Input:
int((((2*x^2-630*x)*log(4*x)-2*x^2+630*x)*log(x/(log(4*x)-1))+(x^2-630*x+4 )*log(4*x)-2*x^2+1260*x-8)/(2*x*log(4*x)-2*x),x)
Output:
( - 4*log(log(4*x) - 1) + log(x/(log(4*x) - 1))*x**2 - 630*log(x/(log(4*x) - 1))*x + 4*log(x))/2