Integrand size = 88, antiderivative size = 25 \[ \int \frac {-80 x^2+80 x^3-23 x^4+2 x^5+\left (50-20 x+2 x^2\right ) \log (x)+\left (-25+10 x-x^2\right ) \log ^2(x)}{-80 x^3+56 x^4-13 x^5+x^6+\left (25 x-10 x^2+x^3\right ) \log ^2(x)} \, dx=\log \left (-\frac {x}{5-x}+(-3+x) x+\frac {\log ^2(x)}{x}\right ) \] Output:
ln(ln(x)^2/x-x/(5-x)+x*(-3+x))
Time = 0.18 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.60 \[ \int \frac {-80 x^2+80 x^3-23 x^4+2 x^5+\left (50-20 x+2 x^2\right ) \log (x)+\left (-25+10 x-x^2\right ) \log ^2(x)}{-80 x^3+56 x^4-13 x^5+x^6+\left (25 x-10 x^2+x^3\right ) \log ^2(x)} \, dx=-\log (5-x)-\log (x)+\log \left (16 x^2-8 x^3+x^4-5 \log ^2(x)+x \log ^2(x)\right ) \] Input:
Integrate[(-80*x^2 + 80*x^3 - 23*x^4 + 2*x^5 + (50 - 20*x + 2*x^2)*Log[x] + (-25 + 10*x - x^2)*Log[x]^2)/(-80*x^3 + 56*x^4 - 13*x^5 + x^6 + (25*x - 10*x^2 + x^3)*Log[x]^2),x]
Output:
-Log[5 - x] - Log[x] + Log[16*x^2 - 8*x^3 + x^4 - 5*Log[x]^2 + x*Log[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.
| \(\displaystyle \int \frac {2 x^5-23 x^4+80 x^3-80 x^2+\left (-x^2+10 x-25\right ) \log ^2(x)+\left (2 x^2-20 x+50\right ) \log (x)}{x^6-13 x^5+56 x^4-80 x^3+\left (x^3-10 x^2+25 x\right ) \log ^2(x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-2 x^5+23 x^4-80 x^3+80 x^2-\left (-x^2+10 x-25\right ) \log ^2(x)-\left (2 x^2-20 x+50\right ) \log (x)}{(5-x) x \left (x^4-8 x^3+16 x^2+x \log ^2(x)-5 \log ^2(x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {3 x^5-36 x^4+136 x^3-160 x^2+2 x^2 \log (x)-20 x \log (x)+50 \log (x)}{(x-5) x \left (x^4-8 x^3+16 x^2+x \log ^2(x)-5 \log ^2(x)\right )}-\frac {1}{x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -5 \int \frac {1}{x^4-8 x^3+16 x^2+\log ^2(x) x-5 \log ^2(x)}dx-25 \int \frac {1}{(x-5) \left (x^4-8 x^3+16 x^2+\log ^2(x) x-5 \log ^2(x)\right )}dx+31 \int \frac {x}{x^4-8 x^3+16 x^2+\log ^2(x) x-5 \log ^2(x)}dx-21 \int \frac {x^2}{x^4-8 x^3+16 x^2+\log ^2(x) x-5 \log ^2(x)}dx+3 \int \frac {x^3}{x^4-8 x^3+16 x^2+\log ^2(x) x-5 \log ^2(x)}dx+2 \int \frac {\log (x)}{x^4-8 x^3+16 x^2+\log ^2(x) x-5 \log ^2(x)}dx-10 \int \frac {\log (x)}{x \left (x^4-8 x^3+16 x^2+\log ^2(x) x-5 \log ^2(x)\right )}dx-\log (x)\) |
Input:
Int[(-80*x^2 + 80*x^3 - 23*x^4 + 2*x^5 + (50 - 20*x + 2*x^2)*Log[x] + (-25 + 10*x - x^2)*Log[x]^2)/(-80*x^3 + 56*x^4 - 13*x^5 + x^6 + (25*x - 10*x^2 + x^3)*Log[x]^2),x]
Output:
$Aborted
Time = 0.49 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16
| method | result | size |
| risch | \(-\ln \left (x \right )+\ln \left (\ln \left (x \right )^{2}+\frac {x^{2} \left (x^{2}-8 x +16\right )}{-5+x}\right )\) | \(29\) |
| default | \(-\ln \left (x \right )+\ln \left (x^{4}+x \ln \left (x \right )^{2}-8 x^{3}-5 \ln \left (x \right )^{2}+16 x^{2}\right )-\ln \left (-5+x \right )\) | \(39\) |
| norman | \(-\ln \left (x \right )+\ln \left (x^{4}+x \ln \left (x \right )^{2}-8 x^{3}-5 \ln \left (x \right )^{2}+16 x^{2}\right )-\ln \left (-5+x \right )\) | \(39\) |
| parallelrisch | \(-\ln \left (x \right )+\ln \left (x^{4}+x \ln \left (x \right )^{2}-8 x^{3}-5 \ln \left (x \right )^{2}+16 x^{2}\right )-\ln \left (-5+x \right )\) | \(39\) |
Input:
int(((-x^2+10*x-25)*ln(x)^2+(2*x^2-20*x+50)*ln(x)+2*x^5-23*x^4+80*x^3-80*x ^2)/((x^3-10*x^2+25*x)*ln(x)^2+x^6-13*x^5+56*x^4-80*x^3),x,method=_RETURNV ERBOSE)
Output:
-ln(x)+ln(ln(x)^2+x^2*(x^2-8*x+16)/(-5+x))
Time = 0.10 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {-80 x^2+80 x^3-23 x^4+2 x^5+\left (50-20 x+2 x^2\right ) \log (x)+\left (-25+10 x-x^2\right ) \log ^2(x)}{-80 x^3+56 x^4-13 x^5+x^6+\left (25 x-10 x^2+x^3\right ) \log ^2(x)} \, dx=-\log \left (x\right ) + \log \left (\frac {x^{4} - 8 \, x^{3} + {\left (x - 5\right )} \log \left (x\right )^{2} + 16 \, x^{2}}{x - 5}\right ) \] Input:
integrate(((-x^2+10*x-25)*log(x)^2+(2*x^2-20*x+50)*log(x)+2*x^5-23*x^4+80* x^3-80*x^2)/((x^3-10*x^2+25*x)*log(x)^2+x^6-13*x^5+56*x^4-80*x^3),x, algor ithm="fricas")
Output:
-log(x) + log((x^4 - 8*x^3 + (x - 5)*log(x)^2 + 16*x^2)/(x - 5))
Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {-80 x^2+80 x^3-23 x^4+2 x^5+\left (50-20 x+2 x^2\right ) \log (x)+\left (-25+10 x-x^2\right ) \log ^2(x)}{-80 x^3+56 x^4-13 x^5+x^6+\left (25 x-10 x^2+x^3\right ) \log ^2(x)} \, dx=- \log {\left (x \right )} + \log {\left (\log {\left (x \right )}^{2} + \frac {x^{4} - 8 x^{3} + 16 x^{2}}{x - 5} \right )} \] Input:
integrate(((-x**2+10*x-25)*ln(x)**2+(2*x**2-20*x+50)*ln(x)+2*x**5-23*x**4+ 80*x**3-80*x**2)/((x**3-10*x**2+25*x)*ln(x)**2+x**6-13*x**5+56*x**4-80*x** 3),x)
Output:
-log(x) + log(log(x)**2 + (x**4 - 8*x**3 + 16*x**2)/(x - 5))
Time = 0.07 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {-80 x^2+80 x^3-23 x^4+2 x^5+\left (50-20 x+2 x^2\right ) \log (x)+\left (-25+10 x-x^2\right ) \log ^2(x)}{-80 x^3+56 x^4-13 x^5+x^6+\left (25 x-10 x^2+x^3\right ) \log ^2(x)} \, dx=-\log \left (x\right ) + \log \left (\frac {x^{4} - 8 \, x^{3} + {\left (x - 5\right )} \log \left (x\right )^{2} + 16 \, x^{2}}{x - 5}\right ) \] Input:
integrate(((-x^2+10*x-25)*log(x)^2+(2*x^2-20*x+50)*log(x)+2*x^5-23*x^4+80* x^3-80*x^2)/((x^3-10*x^2+25*x)*log(x)^2+x^6-13*x^5+56*x^4-80*x^3),x, algor ithm="maxima")
Output:
-log(x) + log((x^4 - 8*x^3 + (x - 5)*log(x)^2 + 16*x^2)/(x - 5))
Time = 0.15 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.52 \[ \int \frac {-80 x^2+80 x^3-23 x^4+2 x^5+\left (50-20 x+2 x^2\right ) \log (x)+\left (-25+10 x-x^2\right ) \log ^2(x)}{-80 x^3+56 x^4-13 x^5+x^6+\left (25 x-10 x^2+x^3\right ) \log ^2(x)} \, dx=\log \left (x^{4} - 8 \, x^{3} + x \log \left (x\right )^{2} + 16 \, x^{2} - 5 \, \log \left (x\right )^{2}\right ) - \log \left (x - 5\right ) - \log \left (x\right ) \] Input:
integrate(((-x^2+10*x-25)*log(x)^2+(2*x^2-20*x+50)*log(x)+2*x^5-23*x^4+80* x^3-80*x^2)/((x^3-10*x^2+25*x)*log(x)^2+x^6-13*x^5+56*x^4-80*x^3),x, algor ithm="giac")
Output:
log(x^4 - 8*x^3 + x*log(x)^2 + 16*x^2 - 5*log(x)^2) - log(x - 5) - log(x)
Timed out. \[ \int \frac {-80 x^2+80 x^3-23 x^4+2 x^5+\left (50-20 x+2 x^2\right ) \log (x)+\left (-25+10 x-x^2\right ) \log ^2(x)}{-80 x^3+56 x^4-13 x^5+x^6+\left (25 x-10 x^2+x^3\right ) \log ^2(x)} \, dx=\int \frac {\ln \left (x\right )\,\left (2\,x^2-20\,x+50\right )-{\ln \left (x\right )}^2\,\left (x^2-10\,x+25\right )-80\,x^2+80\,x^3-23\,x^4+2\,x^5}{56\,x^4-80\,x^3-13\,x^5+x^6+{\ln \left (x\right )}^2\,\left (x^3-10\,x^2+25\,x\right )} \,d x \] Input:
int((log(x)*(2*x^2 - 20*x + 50) - log(x)^2*(x^2 - 10*x + 25) - 80*x^2 + 80 *x^3 - 23*x^4 + 2*x^5)/(56*x^4 - 80*x^3 - 13*x^5 + x^6 + log(x)^2*(25*x - 10*x^2 + x^3)),x)
Output:
int((log(x)*(2*x^2 - 20*x + 50) - log(x)^2*(x^2 - 10*x + 25) - 80*x^2 + 80 *x^3 - 23*x^4 + 2*x^5)/(56*x^4 - 80*x^3 - 13*x^5 + x^6 + log(x)^2*(25*x - 10*x^2 + x^3)), x)
\[ \int \frac {-80 x^2+80 x^3-23 x^4+2 x^5+\left (50-20 x+2 x^2\right ) \log (x)+\left (-25+10 x-x^2\right ) \log ^2(x)}{-80 x^3+56 x^4-13 x^5+x^6+\left (25 x-10 x^2+x^3\right ) \log ^2(x)} \, dx=-25 \left (\int \frac {\mathrm {log}\left (x \right )^{2}}{\mathrm {log}\left (x \right )^{2} x^{3}-10 \mathrm {log}\left (x \right )^{2} x^{2}+25 \mathrm {log}\left (x \right )^{2} x +x^{6}-13 x^{5}+56 x^{4}-80 x^{3}}d x \right )+10 \left (\int \frac {\mathrm {log}\left (x \right )^{2}}{\mathrm {log}\left (x \right )^{2} x^{2}-10 \mathrm {log}\left (x \right )^{2} x +25 \mathrm {log}\left (x \right )^{2}+x^{5}-13 x^{4}+56 x^{3}-80 x^{2}}d x \right )+2 \left (\int \frac {x^{4}}{\mathrm {log}\left (x \right )^{2} x^{2}-10 \mathrm {log}\left (x \right )^{2} x +25 \mathrm {log}\left (x \right )^{2}+x^{5}-13 x^{4}+56 x^{3}-80 x^{2}}d x \right )-23 \left (\int \frac {x^{3}}{\mathrm {log}\left (x \right )^{2} x^{2}-10 \mathrm {log}\left (x \right )^{2} x +25 \mathrm {log}\left (x \right )^{2}+x^{5}-13 x^{4}+56 x^{3}-80 x^{2}}d x \right )+80 \left (\int \frac {x^{2}}{\mathrm {log}\left (x \right )^{2} x^{2}-10 \mathrm {log}\left (x \right )^{2} x +25 \mathrm {log}\left (x \right )^{2}+x^{5}-13 x^{4}+56 x^{3}-80 x^{2}}d x \right )+50 \left (\int \frac {\mathrm {log}\left (x \right )}{\mathrm {log}\left (x \right )^{2} x^{3}-10 \mathrm {log}\left (x \right )^{2} x^{2}+25 \mathrm {log}\left (x \right )^{2} x +x^{6}-13 x^{5}+56 x^{4}-80 x^{3}}d x \right )-20 \left (\int \frac {\mathrm {log}\left (x \right )}{\mathrm {log}\left (x \right )^{2} x^{2}-10 \mathrm {log}\left (x \right )^{2} x +25 \mathrm {log}\left (x \right )^{2}+x^{5}-13 x^{4}+56 x^{3}-80 x^{2}}d x \right )-\left (\int \frac {\mathrm {log}\left (x \right )^{2} x}{\mathrm {log}\left (x \right )^{2} x^{2}-10 \mathrm {log}\left (x \right )^{2} x +25 \mathrm {log}\left (x \right )^{2}+x^{5}-13 x^{4}+56 x^{3}-80 x^{2}}d x \right )+2 \left (\int \frac {\mathrm {log}\left (x \right ) x}{\mathrm {log}\left (x \right )^{2} x^{2}-10 \mathrm {log}\left (x \right )^{2} x +25 \mathrm {log}\left (x \right )^{2}+x^{5}-13 x^{4}+56 x^{3}-80 x^{2}}d x \right )-80 \left (\int \frac {x}{\mathrm {log}\left (x \right )^{2} x^{2}-10 \mathrm {log}\left (x \right )^{2} x +25 \mathrm {log}\left (x \right )^{2}+x^{5}-13 x^{4}+56 x^{3}-80 x^{2}}d x \right ) \] Input:
int(((-x^2+10*x-25)*log(x)^2+(2*x^2-20*x+50)*log(x)+2*x^5-23*x^4+80*x^3-80 *x^2)/((x^3-10*x^2+25*x)*log(x)^2+x^6-13*x^5+56*x^4-80*x^3),x)
Output:
- 25*int(log(x)**2/(log(x)**2*x**3 - 10*log(x)**2*x**2 + 25*log(x)**2*x + x**6 - 13*x**5 + 56*x**4 - 80*x**3),x) + 10*int(log(x)**2/(log(x)**2*x**2 - 10*log(x)**2*x + 25*log(x)**2 + x**5 - 13*x**4 + 56*x**3 - 80*x**2),x) + 2*int(x**4/(log(x)**2*x**2 - 10*log(x)**2*x + 25*log(x)**2 + x**5 - 13*x **4 + 56*x**3 - 80*x**2),x) - 23*int(x**3/(log(x)**2*x**2 - 10*log(x)**2*x + 25*log(x)**2 + x**5 - 13*x**4 + 56*x**3 - 80*x**2),x) + 80*int(x**2/(lo g(x)**2*x**2 - 10*log(x)**2*x + 25*log(x)**2 + x**5 - 13*x**4 + 56*x**3 - 80*x**2),x) + 50*int(log(x)/(log(x)**2*x**3 - 10*log(x)**2*x**2 + 25*log(x )**2*x + x**6 - 13*x**5 + 56*x**4 - 80*x**3),x) - 20*int(log(x)/(log(x)**2 *x**2 - 10*log(x)**2*x + 25*log(x)**2 + x**5 - 13*x**4 + 56*x**3 - 80*x**2 ),x) - int((log(x)**2*x)/(log(x)**2*x**2 - 10*log(x)**2*x + 25*log(x)**2 + x**5 - 13*x**4 + 56*x**3 - 80*x**2),x) + 2*int((log(x)*x)/(log(x)**2*x**2 - 10*log(x)**2*x + 25*log(x)**2 + x**5 - 13*x**4 + 56*x**3 - 80*x**2),x) - 80*int(x/(log(x)**2*x**2 - 10*log(x)**2*x + 25*log(x)**2 + x**5 - 13*x** 4 + 56*x**3 - 80*x**2),x)