Integrand size = 91, antiderivative size = 27 \[ \int \frac {1-81 x^2+162 x^3+\left (-1-x^2\right ) \log (x)}{x^2+160 x^3+6238 x^4-12960 x^5+6561 x^6+\left (-2 x-160 x^2+164 x^3+160 x^4-162 x^5\right ) \log (x)+\left (1-2 x^2+x^4\right ) \log ^2(x)} \, dx=\frac {x}{(1-x) \left (x+81 x^2-\log (x)-x \log (x)\right )} \]
Time = 0.23 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {1-81 x^2+162 x^3+\left (-1-x^2\right ) \log (x)}{x^2+160 x^3+6238 x^4-12960 x^5+6561 x^6+\left (-2 x-160 x^2+164 x^3+160 x^4-162 x^5\right ) \log (x)+\left (1-2 x^2+x^4\right ) \log ^2(x)} \, dx=\frac {x}{(-1+x) \left (-x-81 x^2+\log (x)+x \log (x)\right )} \]
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]
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 |
\(\displaystyle 161 \int \frac {1}{\left (81 x^2-\log (x) x+x-\log (x)\right )^2}dx+120 \int \frac {1}{(x-1) \left (81 x^2-\log (x) x+x-\log (x)\right )^2}dx+81 \int \frac {x}{\left (81 x^2-\log (x) x+x-\log (x)\right )^2}dx-40 \int \frac {1}{(x+1) \left (81 x^2-\log (x) x+x-\log (x)\right )^2}dx+\int \frac {1}{(x-1)^2 \left (81 x^2-\log (x) x+x-\log (x)\right )}dx+\frac {1}{2} \int \frac {1}{(x-1) \left (81 x^2-\log (x) x+x-\log (x)\right )}dx+\frac {1}{2} \int \frac {1}{(x+1) \left (81 x^2-\log (x) x+x-\log (x)\right )}dx\) |
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]
3.15.61.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.20 (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 | \(-\frac {x}{81 x^{3}-x^{2} \ln \left (x \right )-80 x^{2}-x +\ln \left (x \right )}\) | \(29\) |
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)
Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \frac {1-81 x^2+162 x^3+\left (-1-x^2\right ) \log (x)}{x^2+160 x^3+6238 x^4-12960 x^5+6561 x^6+\left (-2 x-160 x^2+164 x^3+160 x^4-162 x^5\right ) \log (x)+\left (1-2 x^2+x^4\right ) \log ^2(x)} \, dx=-\frac {x}{81 \, x^{3} - 80 \, x^{2} - {\left (x^{2} - 1\right )} \log \left (x\right ) - x} \]
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=\
Time = 0.11 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {1-81 x^2+162 x^3+\left (-1-x^2\right ) \log (x)}{x^2+160 x^3+6238 x^4-12960 x^5+6561 x^6+\left (-2 x-160 x^2+164 x^3+160 x^4-162 x^5\right ) \log (x)+\left (1-2 x^2+x^4\right ) \log ^2(x)} \, dx=\frac {x}{- 81 x^{3} + 80 x^{2} + x + \left (x^{2} - 1\right ) \log {\left (x \right )}} \]
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)
Time = 0.23 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \frac {1-81 x^2+162 x^3+\left (-1-x^2\right ) \log (x)}{x^2+160 x^3+6238 x^4-12960 x^5+6561 x^6+\left (-2 x-160 x^2+164 x^3+160 x^4-162 x^5\right ) \log (x)+\left (1-2 x^2+x^4\right ) \log ^2(x)} \, dx=-\frac {x}{81 \, x^{3} - 80 \, x^{2} - {\left (x^{2} - 1\right )} \log \left (x\right ) - x} \]
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=\
Time = 0.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \frac {1-81 x^2+162 x^3+\left (-1-x^2\right ) \log (x)}{x^2+160 x^3+6238 x^4-12960 x^5+6561 x^6+\left (-2 x-160 x^2+164 x^3+160 x^4-162 x^5\right ) \log (x)+\left (1-2 x^2+x^4\right ) \log ^2(x)} \, dx=-\frac {x}{81 \, x^{3} - x^{2} \log \left (x\right ) - 80 \, x^{2} - x + \log \left (x\right )} \]
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=\
Timed out. \[ \int \frac {1-81 x^2+162 x^3+\left (-1-x^2\right ) \log (x)}{x^2+160 x^3+6238 x^4-12960 x^5+6561 x^6+\left (-2 x-160 x^2+164 x^3+160 x^4-162 x^5\right ) \log (x)+\left (1-2 x^2+x^4\right ) \log ^2(x)} \, dx=\int -\frac {\ln \left (x\right )\,\left (x^2+1\right )+81\,x^2-162\,x^3-1}{{\ln \left (x\right )}^2\,\left (x^4-2\,x^2+1\right )-\ln \left (x\right )\,\left (162\,x^5-160\,x^4-164\,x^3+160\,x^2+2\,x\right )+x^2+160\,x^3+6238\,x^4-12960\,x^5+6561\,x^6} \,d x \]
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)