Integrand size = 68, antiderivative size = 26 \[ \int \frac {8-4 x+\left (-20+4 x-x^2\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+4 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right )}{\left (20-20 x+5 x^2\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx=\frac {1}{5} x \left (-5+\frac {2 \left (-2 x+\log \left (\log \left (\log \left (x^2\right )\right )\right )\right )}{2-x}\right ) \]
Time = 0.15 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {8-4 x+\left (-20+4 x-x^2\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+4 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right )}{\left (20-20 x+5 x^2\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx=-\frac {-16-2 x+x^2+2 x \log \left (\log \left (\log \left (x^2\right )\right )\right )}{5 (-2+x)} \]
Integrate[(8 - 4*x + (-20 + 4*x - x^2)*Log[x^2]*Log[Log[x^2]] + 4*Log[x^2] *Log[Log[x^2]]*Log[Log[Log[x^2]]])/((20 - 20*x + 5*x^2)*Log[x^2]*Log[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.
\(\displaystyle \int \frac {\left (-x^2+4 x-20\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+4 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right )-4 x+8}{\left (5 x^2-20 x+20\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx\) |
\(\Big \downarrow \) 7277 |
\(\displaystyle 20 \int \frac {-4 x-\left (x^2-4 x+20\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+4 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right )+8}{100 (2-x)^2 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{5} \int \frac {-4 x-\left (x^2-4 x+20\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+4 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right )+8}{(2-x)^2 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{5} \int \left (\frac {-\log \left (x^2\right ) \log \left (\log \left (x^2\right )\right ) x^2+4 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right ) x-4 x-20 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+8}{(x-2)^2 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}+\frac {4 \log \left (\log \left (\log \left (x^2\right )\right )\right )}{(x-2)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{5} \left (-4 \int \frac {1}{(x-2) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}dx+4 \int \frac {\log \left (\log \left (\log \left (x^2\right )\right )\right )}{(x-2)^2}dx-x-\frac {16}{2-x}\right )\) |
Int[(8 - 4*x + (-20 + 4*x - x^2)*Log[x^2]*Log[Log[x^2]] + 4*Log[x^2]*Log[L og[x^2]]*Log[Log[Log[x^2]]])/((20 - 20*x + 5*x^2)*Log[x^2]*Log[Log[x^2]]), x]
3.2.49.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> Simp[1/(4^p*c^p) Int[u*(b + 2*c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n} , x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p] && !AlgebraicFu nctionQ[u, x]
Time = 2.23 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92
method | result | size |
parallelrisch | \(\frac {40-4 \ln \left (\ln \left (\ln \left (x^{2}\right )\right )\right ) x -2 x^{2}}{10 x -20}\) | \(24\) |
int((4*ln(x^2)*ln(ln(x^2))*ln(ln(ln(x^2)))+(-x^2+4*x-20)*ln(x^2)*ln(ln(x^2 ))-4*x+8)/(5*x^2-20*x+20)/ln(x^2)/ln(ln(x^2)),x,method=_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {8-4 x+\left (-20+4 x-x^2\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+4 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right )}{\left (20-20 x+5 x^2\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx=-\frac {x^{2} + 2 \, x \log \left (\log \left (\log \left (x^{2}\right )\right )\right ) - 2 \, x - 16}{5 \, {\left (x - 2\right )}} \]
integrate((4*log(x^2)*log(log(x^2))*log(log(log(x^2)))+(-x^2+4*x-20)*log(x ^2)*log(log(x^2))-4*x+8)/(5*x^2-20*x+20)/log(x^2)/log(log(x^2)),x, algorit hm=\
Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38 \[ \int \frac {8-4 x+\left (-20+4 x-x^2\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+4 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right )}{\left (20-20 x+5 x^2\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx=- \frac {x}{5} - \frac {2 \log {\left (\log {\left (\log {\left (x^{2} \right )} \right )} \right )}}{5} - \frac {4 \log {\left (\log {\left (\log {\left (x^{2} \right )} \right )} \right )}}{5 x - 10} + \frac {16}{5 x - 10} \]
integrate((4*ln(x**2)*ln(ln(x**2))*ln(ln(ln(x**2)))+(-x**2+4*x-20)*ln(x**2 )*ln(ln(x**2))-4*x+8)/(5*x**2-20*x+20)/ln(x**2)/ln(ln(x**2)),x)
Time = 0.32 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {8-4 x+\left (-20+4 x-x^2\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+4 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right )}{\left (20-20 x+5 x^2\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx=-\frac {x^{2} + 2 \, x \log \left (\log \left (2\right ) + \log \left (\log \left (x\right )\right )\right ) - 2 \, x - 16}{5 \, {\left (x - 2\right )}} \]
integrate((4*log(x^2)*log(log(x^2))*log(log(log(x^2)))+(-x^2+4*x-20)*log(x ^2)*log(log(x^2))-4*x+8)/(5*x^2-20*x+20)/log(x^2)/log(log(x^2)),x, algorit hm=\
Time = 0.36 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {8-4 x+\left (-20+4 x-x^2\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+4 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right )}{\left (20-20 x+5 x^2\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx=-\frac {1}{5} \, x - \frac {4 \, \log \left (\log \left (\log \left (x^{2}\right )\right )\right )}{5 \, {\left (x - 2\right )}} + \frac {16}{5 \, {\left (x - 2\right )}} - \frac {2}{5} \, \log \left (\log \left (\log \left (x^{2}\right )\right )\right ) \]
integrate((4*log(x^2)*log(log(x^2))*log(log(log(x^2)))+(-x^2+4*x-20)*log(x ^2)*log(log(x^2))-4*x+8)/(5*x^2-20*x+20)/log(x^2)/log(log(x^2)),x, algorit hm=\
Time = 8.85 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.77 \[ \int \frac {8-4 x+\left (-20+4 x-x^2\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )+4 \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right ) \log \left (\log \left (\log \left (x^2\right )\right )\right )}{\left (20-20 x+5 x^2\right ) \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )} \, dx=\frac {16}{5\,\left (x-2\right )}-\frac {2\,\ln \left (\ln \left (\ln \left (x^2\right )\right )\right )}{5}-\frac {x}{5}+\frac {\ln \left (\ln \left (\ln \left (x^2\right )\right )\right )\,\left (8\,x-4\,x^2\right )}{5\,x\,{\left (x-2\right )}^2} \]
int(-(4*x + log(x^2)*log(log(x^2))*(x^2 - 4*x + 20) - 4*log(log(log(x^2))) *log(x^2)*log(log(x^2)) - 8)/(log(x^2)*log(log(x^2))*(5*x^2 - 20*x + 20)), x)