Integrand size = 69, antiderivative size = 22 \[ \int \frac {120 x^2-120 x \log \left (\frac {x}{4}\right )+120 x \log ^2\left (\frac {x}{4}\right )}{4 x^2-4 x^3+x^4+\left (4 x-2 x^2\right ) \log ^2\left (\frac {x}{4}\right )+\log ^4\left (\frac {x}{4}\right )} \, dx=\frac {60 x}{2-x+\frac {\log ^2\left (\frac {x}{4}\right )}{x}} \] Output:
60*x/(ln(1/4*x)^2/x-x+2)
Timed out. \[ \int \frac {120 x^2-120 x \log \left (\frac {x}{4}\right )+120 x \log ^2\left (\frac {x}{4}\right )}{4 x^2-4 x^3+x^4+\left (4 x-2 x^2\right ) \log ^2\left (\frac {x}{4}\right )+\log ^4\left (\frac {x}{4}\right )} \, dx=\text {\$Aborted} \] Input:
Integrate[(120*x^2 - 120*x*Log[x/4] + 120*x*Log[x/4]^2)/(4*x^2 - 4*x^3 + x ^4 + (4*x - 2*x^2)*Log[x/4]^2 + Log[x/4]^4),x]
Output:
$Aborted
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.
\(\displaystyle \int \frac {120 x^2+120 x \log ^2\left (\frac {x}{4}\right )-120 x \log \left (\frac {x}{4}\right )}{x^4-4 x^3+4 x^2+\left (4 x-2 x^2\right ) \log ^2\left (\frac {x}{4}\right )+\log ^4\left (\frac {x}{4}\right )} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {120 x \left (x+\log ^2\left (\frac {x}{4}\right )-\log (x)+\log (4)\right )}{\left ((x-2) x-\log ^2\left (\frac {x}{4}\right )\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 120 \int \frac {x \left (\log ^2\left (\frac {x}{4}\right )+x-\log (x)+\log (4)\right )}{\left (\log ^2\left (\frac {x}{4}\right )+(2-x) x\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle 120 \int \left (\frac {x \left (\log ^2\left (\frac {x}{4}\right )+x+\log (4)\right )}{\left (x^2-2 x-\log ^2\left (\frac {x}{4}\right )\right )^2}-\frac {x \log (x)}{\left (x^2-2 x-\log ^2\left (\frac {x}{4}\right )\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 120 \left (\log (4) \int \frac {x}{\left (x^2-2 x-\log ^2\left (\frac {x}{4}\right )\right )^2}dx-\int \frac {x^2}{\left (x^2-2 x-\log ^2\left (\frac {x}{4}\right )\right )^2}dx-\int \frac {x}{x^2-2 x-\log ^2\left (\frac {x}{4}\right )}dx-\int \frac {x \log (x)}{\left (x^2-2 x-\log ^2\left (\frac {x}{4}\right )\right )^2}dx+\int \frac {x^3}{\left (x^2-2 x-\log ^2\left (\frac {x}{4}\right )\right )^2}dx\right )\) |
Input:
Int[(120*x^2 - 120*x*Log[x/4] + 120*x*Log[x/4]^2)/(4*x^2 - 4*x^3 + x^4 + ( 4*x - 2*x^2)*Log[x/4]^2 + Log[x/4]^4),x]
Output:
$Aborted
Time = 0.43 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05
method | result | size |
derivativedivides | \(\frac {60 x^{2}}{\ln \left (\frac {x}{4}\right )^{2}-x^{2}+2 x}\) | \(23\) |
default | \(\frac {60 x^{2}}{\ln \left (\frac {x}{4}\right )^{2}-x^{2}+2 x}\) | \(23\) |
risch | \(-\frac {60 x^{2}}{x^{2}-\ln \left (\frac {x}{4}\right )^{2}-2 x}\) | \(23\) |
parallelrisch | \(-\frac {60 x^{2}}{x^{2}-\ln \left (\frac {x}{4}\right )^{2}-2 x}\) | \(23\) |
norman | \(\frac {-120 x -60 \ln \left (\frac {x}{4}\right )^{2}}{x^{2}-\ln \left (\frac {x}{4}\right )^{2}-2 x}\) | \(31\) |
Input:
int((120*x*ln(1/4*x)^2-120*x*ln(1/4*x)+120*x^2)/(ln(1/4*x)^4+(-2*x^2+4*x)* ln(1/4*x)^2+x^4-4*x^3+4*x^2),x,method=_RETURNVERBOSE)
Output:
60*x^2/(ln(1/4*x)^2-x^2+2*x)
Time = 0.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {120 x^2-120 x \log \left (\frac {x}{4}\right )+120 x \log ^2\left (\frac {x}{4}\right )}{4 x^2-4 x^3+x^4+\left (4 x-2 x^2\right ) \log ^2\left (\frac {x}{4}\right )+\log ^4\left (\frac {x}{4}\right )} \, dx=-\frac {60 \, x^{2}}{x^{2} - \log \left (\frac {1}{4} \, x\right )^{2} - 2 \, x} \] Input:
integrate((120*x*log(1/4*x)^2-120*x*log(1/4*x)+120*x^2)/(log(1/4*x)^4+(-2* x^2+4*x)*log(1/4*x)^2+x^4-4*x^3+4*x^2),x, algorithm="fricas")
Output:
-60*x^2/(x^2 - log(1/4*x)^2 - 2*x)
Time = 0.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {120 x^2-120 x \log \left (\frac {x}{4}\right )+120 x \log ^2\left (\frac {x}{4}\right )}{4 x^2-4 x^3+x^4+\left (4 x-2 x^2\right ) \log ^2\left (\frac {x}{4}\right )+\log ^4\left (\frac {x}{4}\right )} \, dx=\frac {60 x^{2}}{- x^{2} + 2 x + \log {\left (\frac {x}{4} \right )}^{2}} \] Input:
integrate((120*x*ln(1/4*x)**2-120*x*ln(1/4*x)+120*x**2)/(ln(1/4*x)**4+(-2* x**2+4*x)*ln(1/4*x)**2+x**4-4*x**3+4*x**2),x)
Output:
60*x**2/(-x**2 + 2*x + log(x/4)**2)
Time = 0.15 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.45 \[ \int \frac {120 x^2-120 x \log \left (\frac {x}{4}\right )+120 x \log ^2\left (\frac {x}{4}\right )}{4 x^2-4 x^3+x^4+\left (4 x-2 x^2\right ) \log ^2\left (\frac {x}{4}\right )+\log ^4\left (\frac {x}{4}\right )} \, dx=-\frac {60 \, x^{2}}{x^{2} - 4 \, \log \left (2\right )^{2} + 4 \, \log \left (2\right ) \log \left (x\right ) - \log \left (x\right )^{2} - 2 \, x} \] Input:
integrate((120*x*log(1/4*x)^2-120*x*log(1/4*x)+120*x^2)/(log(1/4*x)^4+(-2* x^2+4*x)*log(1/4*x)^2+x^4-4*x^3+4*x^2),x, algorithm="maxima")
Output:
-60*x^2/(x^2 - 4*log(2)^2 + 4*log(2)*log(x) - log(x)^2 - 2*x)
Time = 0.14 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {120 x^2-120 x \log \left (\frac {x}{4}\right )+120 x \log ^2\left (\frac {x}{4}\right )}{4 x^2-4 x^3+x^4+\left (4 x-2 x^2\right ) \log ^2\left (\frac {x}{4}\right )+\log ^4\left (\frac {x}{4}\right )} \, dx=-\frac {60 \, x^{2}}{x^{2} - \log \left (\frac {1}{4} \, x\right )^{2} - 2 \, x} \] Input:
integrate((120*x*log(1/4*x)^2-120*x*log(1/4*x)+120*x^2)/(log(1/4*x)^4+(-2* x^2+4*x)*log(1/4*x)^2+x^4-4*x^3+4*x^2),x, algorithm="giac")
Output:
-60*x^2/(x^2 - log(1/4*x)^2 - 2*x)
Time = 3.14 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {120 x^2-120 x \log \left (\frac {x}{4}\right )+120 x \log ^2\left (\frac {x}{4}\right )}{4 x^2-4 x^3+x^4+\left (4 x-2 x^2\right ) \log ^2\left (\frac {x}{4}\right )+\log ^4\left (\frac {x}{4}\right )} \, dx=\frac {60\,x^2}{-x^2+2\,x+{\ln \left (\frac {x}{4}\right )}^2} \] Input:
int((120*x*log(x/4)^2 - 120*x*log(x/4) + 120*x^2)/(log(x/4)^2*(4*x - 2*x^2 ) + log(x/4)^4 + 4*x^2 - 4*x^3 + x^4),x)
Output:
(60*x^2)/(2*x + log(x/4)^2 - x^2)
\[ \int \frac {120 x^2-120 x \log \left (\frac {x}{4}\right )+120 x \log ^2\left (\frac {x}{4}\right )}{4 x^2-4 x^3+x^4+\left (4 x-2 x^2\right ) \log ^2\left (\frac {x}{4}\right )+\log ^4\left (\frac {x}{4}\right )} \, dx=120 \left (\int \frac {x^{2}}{\mathrm {log}\left (\frac {x}{4}\right )^{4}-2 \mathrm {log}\left (\frac {x}{4}\right )^{2} x^{2}+4 \mathrm {log}\left (\frac {x}{4}\right )^{2} x +x^{4}-4 x^{3}+4 x^{2}}d x \right )+120 \left (\int \frac {\mathrm {log}\left (\frac {x}{4}\right )^{2} x}{\mathrm {log}\left (\frac {x}{4}\right )^{4}-2 \mathrm {log}\left (\frac {x}{4}\right )^{2} x^{2}+4 \mathrm {log}\left (\frac {x}{4}\right )^{2} x +x^{4}-4 x^{3}+4 x^{2}}d x \right )-120 \left (\int \frac {\mathrm {log}\left (\frac {x}{4}\right ) x}{\mathrm {log}\left (\frac {x}{4}\right )^{4}-2 \mathrm {log}\left (\frac {x}{4}\right )^{2} x^{2}+4 \mathrm {log}\left (\frac {x}{4}\right )^{2} x +x^{4}-4 x^{3}+4 x^{2}}d x \right ) \] Input:
int((120*x*log(1/4*x)^2-120*x*log(1/4*x)+120*x^2)/(log(1/4*x)^4+(-2*x^2+4* x)*log(1/4*x)^2+x^4-4*x^3+4*x^2),x)
Output:
120*(int(x**2/(log(x/4)**4 - 2*log(x/4)**2*x**2 + 4*log(x/4)**2*x + x**4 - 4*x**3 + 4*x**2),x) + int((log(x/4)**2*x)/(log(x/4)**4 - 2*log(x/4)**2*x* *2 + 4*log(x/4)**2*x + x**4 - 4*x**3 + 4*x**2),x) - int((log(x/4)*x)/(log( x/4)**4 - 2*log(x/4)**2*x**2 + 4*log(x/4)**2*x + x**4 - 4*x**3 + 4*x**2),x ))