Integrand size = 133, antiderivative size = 27 \[ \int \frac {4+4 x-x^2+\left (4+4 x-x^2\right ) \log \left (\frac {4+4 x-x^2}{x}\right )+\log (x) \left (6 x^2-x^3+\left (-4+5 x^2-x^3\right ) \log \left (\frac {4+4 x-x^2}{x}\right )\right ) \log (\log (x))}{\log (x) \left (-4 x-4 x^2+x^3+\left (-4 x-4 x^2+x^3\right ) \log \left (\frac {4+4 x-x^2}{x}\right )\right ) \log (\log (x))} \, dx=-x+\log \left (\frac {5 \left (x+x \log \left (4+\frac {4}{x}-x\right )\right )}{\log (\log (x))}\right ) \] Output:
ln(5*(ln(4-x+4/x)*x+x)/ln(ln(x)))-x
Time = 0.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {4+4 x-x^2+\left (4+4 x-x^2\right ) \log \left (\frac {4+4 x-x^2}{x}\right )+\log (x) \left (6 x^2-x^3+\left (-4+5 x^2-x^3\right ) \log \left (\frac {4+4 x-x^2}{x}\right )\right ) \log (\log (x))}{\log (x) \left (-4 x-4 x^2+x^3+\left (-4 x-4 x^2+x^3\right ) \log \left (\frac {4+4 x-x^2}{x}\right )\right ) \log (\log (x))} \, dx=-x+\log (x)+\log \left (1+\log \left (4+\frac {4}{x}-x\right )\right )-\log (\log (\log (x))) \] Input:
Integrate[(4 + 4*x - x^2 + (4 + 4*x - x^2)*Log[(4 + 4*x - x^2)/x] + Log[x] *(6*x^2 - x^3 + (-4 + 5*x^2 - x^3)*Log[(4 + 4*x - x^2)/x])*Log[Log[x]])/(L og[x]*(-4*x - 4*x^2 + x^3 + (-4*x - 4*x^2 + x^3)*Log[(4 + 4*x - x^2)/x])*L og[Log[x]]),x]
Output:
-x + Log[x] + Log[1 + Log[4 + 4/x - x]] - Log[Log[Log[x]]]
Time = 2.82 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {7292, 7279, 2009}
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.
\(\displaystyle \int \frac {-x^2+\left (-x^2+4 x+4\right ) \log \left (\frac {-x^2+4 x+4}{x}\right )+\log (x) \left (-x^3+6 x^2+\left (-x^3+5 x^2-4\right ) \log \left (\frac {-x^2+4 x+4}{x}\right )\right ) \log (\log (x))+4 x+4}{\log (x) \left (x^3-4 x^2+\left (x^3-4 x^2-4 x\right ) \log \left (\frac {-x^2+4 x+4}{x}\right )-4 x\right ) \log (\log (x))} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {x^2-\left (-x^2+4 x+4\right ) \log \left (\frac {-x^2+4 x+4}{x}\right )-\log (x) \left (-x^3+6 x^2+\left (-x^3+5 x^2-4\right ) \log \left (\frac {-x^2+4 x+4}{x}\right )\right ) \log (\log (x))-4 x-4}{x \left (-x^2+4 x+4\right ) \left (\log \left (-x+\frac {4}{x}+4\right )+1\right ) \log (x) \log (\log (x))}dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (\frac {-x^3+x^3 \left (-\log \left (-x+\frac {4}{x}+4\right )\right )+6 x^2+5 x^2 \log \left (-x+\frac {4}{x}+4\right )-4 \log \left (-x+\frac {4}{x}+4\right )}{x \left (x^2-4 x-4\right ) \left (\log \left (-x+\frac {4}{x}+4\right )+1\right )}-\frac {1}{x \log (x) \log (\log (x))}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -x+\log (x)+\log \left (\log \left (-x+\frac {4}{x}+4\right )+1\right )-\log (\log (\log (x)))\) |
Input:
Int[(4 + 4*x - x^2 + (4 + 4*x - x^2)*Log[(4 + 4*x - x^2)/x] + Log[x]*(6*x^ 2 - x^3 + (-4 + 5*x^2 - x^3)*Log[(4 + 4*x - x^2)/x])*Log[Log[x]])/(Log[x]* (-4*x - 4*x^2 + x^3 + (-4*x - 4*x^2 + x^3)*Log[(4 + 4*x - x^2)/x])*Log[Log [x]]),x]
Output:
-x + Log[x] + Log[1 + Log[4 + 4/x - x]] - Log[Log[Log[x]]]
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 27.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11
method | result | size |
default | \(-x +\ln \left (x \right )+\ln \left (\ln \left (-\frac {x^{2}-4 x -4}{x}\right )+1\right )-\ln \left (\ln \left (\ln \left (x \right )\right )\right )\) | \(30\) |
parallelrisch | \(-1+\ln \left (x \right )-\ln \left (\ln \left (\ln \left (x \right )\right )\right )+\ln \left (\ln \left (-\frac {x^{2}-4 x -4}{x}\right )+1\right )-x\) | \(31\) |
parts | \(-x +\ln \left (x \right )+\ln \left (\ln \left (\frac {-x^{2}+4 x +4}{x}\right )+1\right )-\ln \left (\ln \left (\ln \left (x \right )\right )\right )\) | \(31\) |
risch | \(\ln \left (x \right )-x +\ln \left (\ln \left (x^{2}-4 x -4\right )+\frac {i \left (-2 \pi {\operatorname {csgn}\left (\frac {i \left (x^{2}-4 x -4\right )}{x}\right )}^{2}-\pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (i \left (x^{2}-4 x -4\right )\right ) \operatorname {csgn}\left (\frac {i \left (x^{2}-4 x -4\right )}{x}\right )+\pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}-4 x -4\right )}{x}\right )}^{2}+\pi \,\operatorname {csgn}\left (i \left (x^{2}-4 x -4\right )\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}-4 x -4\right )}{x}\right )}^{2}+\pi {\operatorname {csgn}\left (\frac {i \left (x^{2}-4 x -4\right )}{x}\right )}^{3}+2 \pi +2 i \ln \left (x \right )-2 i\right )}{2}\right )-\ln \left (\ln \left (\ln \left (x \right )\right )\right )\) | \(171\) |
Input:
int((((-x^3+5*x^2-4)*ln((-x^2+4*x+4)/x)-x^3+6*x^2)*ln(x)*ln(ln(x))+(-x^2+4 *x+4)*ln((-x^2+4*x+4)/x)-x^2+4*x+4)/((x^3-4*x^2-4*x)*ln((-x^2+4*x+4)/x)+x^ 3-4*x^2-4*x)/ln(x)/ln(ln(x)),x,method=_RETURNVERBOSE)
Output:
-x+ln(x)+ln(ln(-(x^2-4*x-4)/x)+1)-ln(ln(ln(x)))
Time = 0.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {4+4 x-x^2+\left (4+4 x-x^2\right ) \log \left (\frac {4+4 x-x^2}{x}\right )+\log (x) \left (6 x^2-x^3+\left (-4+5 x^2-x^3\right ) \log \left (\frac {4+4 x-x^2}{x}\right )\right ) \log (\log (x))}{\log (x) \left (-4 x-4 x^2+x^3+\left (-4 x-4 x^2+x^3\right ) \log \left (\frac {4+4 x-x^2}{x}\right )\right ) \log (\log (x))} \, dx=-x + \log \left (x\right ) + \log \left (\log \left (-\frac {x^{2} - 4 \, x - 4}{x}\right ) + 1\right ) - \log \left (\log \left (\log \left (x\right )\right )\right ) \] Input:
integrate((((-x^3+5*x^2-4)*log((-x^2+4*x+4)/x)-x^3+6*x^2)*log(x)*log(log(x ))+(-x^2+4*x+4)*log((-x^2+4*x+4)/x)-x^2+4*x+4)/((x^3-4*x^2-4*x)*log((-x^2+ 4*x+4)/x)+x^3-4*x^2-4*x)/log(x)/log(log(x)),x, algorithm="fricas")
Output:
-x + log(x) + log(log(-(x^2 - 4*x - 4)/x) + 1) - log(log(log(x)))
Time = 0.31 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {4+4 x-x^2+\left (4+4 x-x^2\right ) \log \left (\frac {4+4 x-x^2}{x}\right )+\log (x) \left (6 x^2-x^3+\left (-4+5 x^2-x^3\right ) \log \left (\frac {4+4 x-x^2}{x}\right )\right ) \log (\log (x))}{\log (x) \left (-4 x-4 x^2+x^3+\left (-4 x-4 x^2+x^3\right ) \log \left (\frac {4+4 x-x^2}{x}\right )\right ) \log (\log (x))} \, dx=- x + \log {\left (x \right )} + \log {\left (\log {\left (\frac {- x^{2} + 4 x + 4}{x} \right )} + 1 \right )} - \log {\left (\log {\left (\log {\left (x \right )} \right )} \right )} \] Input:
integrate((((-x**3+5*x**2-4)*ln((-x**2+4*x+4)/x)-x**3+6*x**2)*ln(x)*ln(ln( x))+(-x**2+4*x+4)*ln((-x**2+4*x+4)/x)-x**2+4*x+4)/((x**3-4*x**2-4*x)*ln((- x**2+4*x+4)/x)+x**3-4*x**2-4*x)/ln(x)/ln(ln(x)),x)
Output:
-x + log(x) + log(log((-x**2 + 4*x + 4)/x) + 1) - log(log(log(x)))
Time = 0.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int \frac {4+4 x-x^2+\left (4+4 x-x^2\right ) \log \left (\frac {4+4 x-x^2}{x}\right )+\log (x) \left (6 x^2-x^3+\left (-4+5 x^2-x^3\right ) \log \left (\frac {4+4 x-x^2}{x}\right )\right ) \log (\log (x))}{\log (x) \left (-4 x-4 x^2+x^3+\left (-4 x-4 x^2+x^3\right ) \log \left (\frac {4+4 x-x^2}{x}\right )\right ) \log (\log (x))} \, dx=-x + \log \left (x\right ) + \log \left (\log \left (-x^{2} + 4 \, x + 4\right ) - \log \left (x\right ) + 1\right ) - \log \left (\log \left (\log \left (x\right )\right )\right ) \] Input:
integrate((((-x^3+5*x^2-4)*log((-x^2+4*x+4)/x)-x^3+6*x^2)*log(x)*log(log(x ))+(-x^2+4*x+4)*log((-x^2+4*x+4)/x)-x^2+4*x+4)/((x^3-4*x^2-4*x)*log((-x^2+ 4*x+4)/x)+x^3-4*x^2-4*x)/log(x)/log(log(x)),x, algorithm="maxima")
Output:
-x + log(x) + log(log(-x^2 + 4*x + 4) - log(x) + 1) - log(log(log(x)))
Time = 0.21 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int \frac {4+4 x-x^2+\left (4+4 x-x^2\right ) \log \left (\frac {4+4 x-x^2}{x}\right )+\log (x) \left (6 x^2-x^3+\left (-4+5 x^2-x^3\right ) \log \left (\frac {4+4 x-x^2}{x}\right )\right ) \log (\log (x))}{\log (x) \left (-4 x-4 x^2+x^3+\left (-4 x-4 x^2+x^3\right ) \log \left (\frac {4+4 x-x^2}{x}\right )\right ) \log (\log (x))} \, dx=-x + \log \left (x\right ) + \log \left (\log \left (-x^{2} + 4 \, x + 4\right ) - \log \left (x\right ) + 1\right ) - \log \left (\log \left (\log \left (x\right )\right )\right ) \] Input:
integrate((((-x^3+5*x^2-4)*log((-x^2+4*x+4)/x)-x^3+6*x^2)*log(x)*log(log(x ))+(-x^2+4*x+4)*log((-x^2+4*x+4)/x)-x^2+4*x+4)/((x^3-4*x^2-4*x)*log((-x^2+ 4*x+4)/x)+x^3-4*x^2-4*x)/log(x)/log(log(x)),x, algorithm="giac")
Output:
-x + log(x) + log(log(-x^2 + 4*x + 4) - log(x) + 1) - log(log(log(x)))
Time = 2.97 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int \frac {4+4 x-x^2+\left (4+4 x-x^2\right ) \log \left (\frac {4+4 x-x^2}{x}\right )+\log (x) \left (6 x^2-x^3+\left (-4+5 x^2-x^3\right ) \log \left (\frac {4+4 x-x^2}{x}\right )\right ) \log (\log (x))}{\log (x) \left (-4 x-4 x^2+x^3+\left (-4 x-4 x^2+x^3\right ) \log \left (\frac {4+4 x-x^2}{x}\right )\right ) \log (\log (x))} \, dx=\ln \left (\ln \left (\frac {-x^2+4\,x+4}{x}\right )+1\right )-\ln \left (\ln \left (\ln \left (x\right )\right )\right )-x+\ln \left (x\right ) \] Input:
int(-(4*x - x^2 + log((4*x - x^2 + 4)/x)*(4*x - x^2 + 4) - log(log(x))*log (x)*(x^3 - 6*x^2 + log((4*x - x^2 + 4)/x)*(x^3 - 5*x^2 + 4)) + 4)/(log(log (x))*log(x)*(4*x + log((4*x - x^2 + 4)/x)*(4*x + 4*x^2 - x^3) + 4*x^2 - x^ 3)),x)
Output:
log(log((4*x - x^2 + 4)/x) + 1) - log(log(log(x))) - x + log(x)
Time = 0.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int \frac {4+4 x-x^2+\left (4+4 x-x^2\right ) \log \left (\frac {4+4 x-x^2}{x}\right )+\log (x) \left (6 x^2-x^3+\left (-4+5 x^2-x^3\right ) \log \left (\frac {4+4 x-x^2}{x}\right )\right ) \log (\log (x))}{\log (x) \left (-4 x-4 x^2+x^3+\left (-4 x-4 x^2+x^3\right ) \log \left (\frac {4+4 x-x^2}{x}\right )\right ) \log (\log (x))} \, dx=-\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (x \right )\right )\right )+\mathrm {log}\left (\mathrm {log}\left (\frac {-x^{2}+4 x +4}{x}\right )+1\right )+\mathrm {log}\left (x \right )-x \] Input:
int((((-x^3+5*x^2-4)*log((-x^2+4*x+4)/x)-x^3+6*x^2)*log(x)*log(log(x))+(-x ^2+4*x+4)*log((-x^2+4*x+4)/x)-x^2+4*x+4)/((x^3-4*x^2-4*x)*log((-x^2+4*x+4) /x)+x^3-4*x^2-4*x)/log(x)/log(log(x)),x)
Output:
- log(log(log(x))) + log(log(( - x**2 + 4*x + 4)/x) + 1) + log(x) - x