Integrand size = 91, antiderivative size = 27 \[ \int \frac {1-81 x^2+162 x^3+\left (-1-x^2\right ) \log (x)}{x^2+160 x^3+6238 x^4-12960 x^5+6561 x^6+\left (-2 x-160 x^2+164 x^3+160 x^4-162 x^5\right ) \log (x)+\left (1-2 x^2+x^4\right ) \log ^2(x)} \, dx=\frac {x}{(1-x) \left (x+81 x^2-\log (x)-x \log (x)\right )} \] Output:
x/(81*x^2-ln(x)-x*ln(x)+x)/(1-x)
Time = 0.14 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {1-81 x^2+162 x^3+\left (-1-x^2\right ) \log (x)}{x^2+160 x^3+6238 x^4-12960 x^5+6561 x^6+\left (-2 x-160 x^2+164 x^3+160 x^4-162 x^5\right ) \log (x)+\left (1-2 x^2+x^4\right ) \log ^2(x)} \, dx=\frac {x}{(-1+x) \left (-x-81 x^2+\log (x)+x \log (x)\right )} \] Input:
Integrate[(1 - 81*x^2 + 162*x^3 + (-1 - x^2)*Log[x])/(x^2 + 160*x^3 + 6238 *x^4 - 12960*x^5 + 6561*x^6 + (-2*x - 160*x^2 + 164*x^3 + 160*x^4 - 162*x^ 5)*Log[x] + (1 - 2*x^2 + x^4)*Log[x]^2),x]
Output:
x/((-1 + x)*(-x - 81*x^2 + Log[x] + x*Log[x]))
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.
\(\displaystyle \int \frac {162 x^3-81 x^2+\left (-x^2-1\right ) \log (x)+1}{6561 x^6-12960 x^5+6238 x^4+160 x^3+x^2+\left (x^4-2 x^2+1\right ) \log ^2(x)+\left (-162 x^5+160 x^4+164 x^3-160 x^2-2 x\right ) \log (x)} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {162 x^3-81 x^2-\left (x^2+1\right ) \log (x)+1}{(1-x)^2 (x (81 x+1)-(x+1) \log (x))^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {x^2+1}{(x-1)^2 (x+1) \left (81 x^2+x-x \log (x)-\log (x)\right )}+\frac {81 x^3+161 x^2-x-1}{\left (x^2-1\right ) \left (81 x^2+x-x \log (x)-\log (x)\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 161 \int \frac {1}{\left (81 x^2-\log (x) x+x-\log (x)\right )^2}dx+120 \int \frac {1}{(x-1) \left (81 x^2-\log (x) x+x-\log (x)\right )^2}dx+81 \int \frac {x}{\left (81 x^2-\log (x) x+x-\log (x)\right )^2}dx-40 \int \frac {1}{(x+1) \left (81 x^2-\log (x) x+x-\log (x)\right )^2}dx+\int \frac {1}{(x-1)^2 \left (81 x^2-\log (x) x+x-\log (x)\right )}dx+\frac {1}{2} \int \frac {1}{(x-1) \left (81 x^2-\log (x) x+x-\log (x)\right )}dx+\frac {1}{2} \int \frac {1}{(x+1) \left (81 x^2-\log (x) x+x-\log (x)\right )}dx\) |
Input:
Int[(1 - 81*x^2 + 162*x^3 + (-1 - x^2)*Log[x])/(x^2 + 160*x^3 + 6238*x^4 - 12960*x^5 + 6561*x^6 + (-2*x - 160*x^2 + 164*x^3 + 160*x^4 - 162*x^5)*Log [x] + (1 - 2*x^2 + x^4)*Log[x]^2),x]
Output:
$Aborted
Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93
method | result | size |
default | \(\frac {x}{\left (-1+x \right ) \left (x \ln \left (x \right )-81 x^{2}+\ln \left (x \right )-x \right )}\) | \(25\) |
risch | \(-\frac {x}{\left (-1+x \right ) \left (81 x^{2}-\ln \left (x \right )-x \ln \left (x \right )+x \right )}\) | \(27\) |
norman | \(-\frac {x}{81 x^{3}-x^{2} \ln \left (x \right )-80 x^{2}-x +\ln \left (x \right )}\) | \(29\) |
parallelrisch | \(-\frac {x}{81 x^{3}-x^{2} \ln \left (x \right )-80 x^{2}-x +\ln \left (x \right )}\) | \(29\) |
Input:
int(((-x^2-1)*ln(x)+162*x^3-81*x^2+1)/((x^4-2*x^2+1)*ln(x)^2+(-162*x^5+160 *x^4+164*x^3-160*x^2-2*x)*ln(x)+6561*x^6-12960*x^5+6238*x^4+160*x^3+x^2),x ,method=_RETURNVERBOSE)
Output:
x/(-1+x)/(x*ln(x)-81*x^2+ln(x)-x)
Time = 0.08 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \frac {1-81 x^2+162 x^3+\left (-1-x^2\right ) \log (x)}{x^2+160 x^3+6238 x^4-12960 x^5+6561 x^6+\left (-2 x-160 x^2+164 x^3+160 x^4-162 x^5\right ) \log (x)+\left (1-2 x^2+x^4\right ) \log ^2(x)} \, dx=-\frac {x}{81 \, x^{3} - 80 \, x^{2} - {\left (x^{2} - 1\right )} \log \left (x\right ) - x} \] Input:
integrate(((-x^2-1)*log(x)+162*x^3-81*x^2+1)/((x^4-2*x^2+1)*log(x)^2+(-162 *x^5+160*x^4+164*x^3-160*x^2-2*x)*log(x)+6561*x^6-12960*x^5+6238*x^4+160*x ^3+x^2),x, algorithm="fricas")
Output:
-x/(81*x^3 - 80*x^2 - (x^2 - 1)*log(x) - x)
Time = 0.14 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {1-81 x^2+162 x^3+\left (-1-x^2\right ) \log (x)}{x^2+160 x^3+6238 x^4-12960 x^5+6561 x^6+\left (-2 x-160 x^2+164 x^3+160 x^4-162 x^5\right ) \log (x)+\left (1-2 x^2+x^4\right ) \log ^2(x)} \, dx=\frac {x}{- 81 x^{3} + 80 x^{2} + x + \left (x^{2} - 1\right ) \log {\left (x \right )}} \] Input:
integrate(((-x**2-1)*ln(x)+162*x**3-81*x**2+1)/((x**4-2*x**2+1)*ln(x)**2+( -162*x**5+160*x**4+164*x**3-160*x**2-2*x)*ln(x)+6561*x**6-12960*x**5+6238* x**4+160*x**3+x**2),x)
Output:
x/(-81*x**3 + 80*x**2 + x + (x**2 - 1)*log(x))
Time = 0.07 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \frac {1-81 x^2+162 x^3+\left (-1-x^2\right ) \log (x)}{x^2+160 x^3+6238 x^4-12960 x^5+6561 x^6+\left (-2 x-160 x^2+164 x^3+160 x^4-162 x^5\right ) \log (x)+\left (1-2 x^2+x^4\right ) \log ^2(x)} \, dx=-\frac {x}{81 \, x^{3} - 80 \, x^{2} - {\left (x^{2} - 1\right )} \log \left (x\right ) - x} \] Input:
integrate(((-x^2-1)*log(x)+162*x^3-81*x^2+1)/((x^4-2*x^2+1)*log(x)^2+(-162 *x^5+160*x^4+164*x^3-160*x^2-2*x)*log(x)+6561*x^6-12960*x^5+6238*x^4+160*x ^3+x^2),x, algorithm="maxima")
Output:
-x/(81*x^3 - 80*x^2 - (x^2 - 1)*log(x) - x)
Time = 0.12 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \frac {1-81 x^2+162 x^3+\left (-1-x^2\right ) \log (x)}{x^2+160 x^3+6238 x^4-12960 x^5+6561 x^6+\left (-2 x-160 x^2+164 x^3+160 x^4-162 x^5\right ) \log (x)+\left (1-2 x^2+x^4\right ) \log ^2(x)} \, dx=-\frac {x}{81 \, x^{3} - x^{2} \log \left (x\right ) - 80 \, x^{2} - x + \log \left (x\right )} \] Input:
integrate(((-x^2-1)*log(x)+162*x^3-81*x^2+1)/((x^4-2*x^2+1)*log(x)^2+(-162 *x^5+160*x^4+164*x^3-160*x^2-2*x)*log(x)+6561*x^6-12960*x^5+6238*x^4+160*x ^3+x^2),x, algorithm="giac")
Output:
-x/(81*x^3 - x^2*log(x) - 80*x^2 - x + log(x))
Timed out. \[ \int \frac {1-81 x^2+162 x^3+\left (-1-x^2\right ) \log (x)}{x^2+160 x^3+6238 x^4-12960 x^5+6561 x^6+\left (-2 x-160 x^2+164 x^3+160 x^4-162 x^5\right ) \log (x)+\left (1-2 x^2+x^4\right ) \log ^2(x)} \, dx=\int -\frac {\ln \left (x\right )\,\left (x^2+1\right )+81\,x^2-162\,x^3-1}{{\ln \left (x\right )}^2\,\left (x^4-2\,x^2+1\right )-\ln \left (x\right )\,\left (162\,x^5-160\,x^4-164\,x^3+160\,x^2+2\,x\right )+x^2+160\,x^3+6238\,x^4-12960\,x^5+6561\,x^6} \,d x \] Input:
int(-(log(x)*(x^2 + 1) + 81*x^2 - 162*x^3 - 1)/(log(x)^2*(x^4 - 2*x^2 + 1) - log(x)*(2*x + 160*x^2 - 164*x^3 - 160*x^4 + 162*x^5) + x^2 + 160*x^3 + 6238*x^4 - 12960*x^5 + 6561*x^6),x)
Output:
int(-(log(x)*(x^2 + 1) + 81*x^2 - 162*x^3 - 1)/(log(x)^2*(x^4 - 2*x^2 + 1) - log(x)*(2*x + 160*x^2 - 164*x^3 - 160*x^4 + 162*x^5) + x^2 + 160*x^3 + 6238*x^4 - 12960*x^5 + 6561*x^6), x)
Time = 0.17 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {1-81 x^2+162 x^3+\left (-1-x^2\right ) \log (x)}{x^2+160 x^3+6238 x^4-12960 x^5+6561 x^6+\left (-2 x-160 x^2+164 x^3+160 x^4-162 x^5\right ) \log (x)+\left (1-2 x^2+x^4\right ) \log ^2(x)} \, dx=\frac {x}{\mathrm {log}\left (x \right ) x^{2}-\mathrm {log}\left (x \right )-81 x^{3}+80 x^{2}+x} \] Input:
int(((-x^2-1)*log(x)+162*x^3-81*x^2+1)/((x^4-2*x^2+1)*log(x)^2+(-162*x^5+1 60*x^4+164*x^3-160*x^2-2*x)*log(x)+6561*x^6-12960*x^5+6238*x^4+160*x^3+x^2 ),x)
Output:
x/(log(x)*x**2 - log(x) - 81*x**3 + 80*x**2 + x)