Integrand size = 316, antiderivative size = 31 \[ \int \frac {-x-x^2+x^3-2 x^4+\left (x^2-2 x^3\right ) \log (x)+\left (-x+2 x^3+\left (-1+2 x^2\right ) \log (x)\right ) \log \left (x^2+x \log (x)\right )+\left (1+2 x-x^2+3 x^3+\left (1-x+3 x^2\right ) \log (x)+\left (-3 x^2-3 x \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )+\left (-x^2-x \log (x)+(x+\log (x)) \log \left (x^2+x \log (x)\right )\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )}{x^3-x^5+\left (x^2-x^4\right ) \log (x)+\left (-x^2+x^4+\left (-x+x^3\right ) \log (x)\right ) \log \left (x^2+x \log (x)\right )+\left (2 x^4+2 x^3 \log (x)+\left (-2 x^3-2 x^2 \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )+\left (-x^3-x^2 \log (x)+\left (x^2+x \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )} \, dx=\frac {1}{2} \log \left (x \left (-x+x (x-\log (x-\log (x (x+\log (x)))))^2\right )\right ) \]
Time = 0.24 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.48 \[ \int \frac {-x-x^2+x^3-2 x^4+\left (x^2-2 x^3\right ) \log (x)+\left (-x+2 x^3+\left (-1+2 x^2\right ) \log (x)\right ) \log \left (x^2+x \log (x)\right )+\left (1+2 x-x^2+3 x^3+\left (1-x+3 x^2\right ) \log (x)+\left (-3 x^2-3 x \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )+\left (-x^2-x \log (x)+(x+\log (x)) \log \left (x^2+x \log (x)\right )\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )}{x^3-x^5+\left (x^2-x^4\right ) \log (x)+\left (-x^2+x^4+\left (-x+x^3\right ) \log (x)\right ) \log \left (x^2+x \log (x)\right )+\left (2 x^4+2 x^3 \log (x)+\left (-2 x^3-2 x^2 \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )+\left (-x^3-x^2 \log (x)+\left (x^2+x \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )} \, dx=\log (x)+\frac {1}{2} \log \left (1-x^2+2 x \log (x-\log (x (x+\log (x))))-\log ^2(x-\log (x (x+\log (x))))\right ) \]
Integrate[(-x - x^2 + x^3 - 2*x^4 + (x^2 - 2*x^3)*Log[x] + (-x + 2*x^3 + ( -1 + 2*x^2)*Log[x])*Log[x^2 + x*Log[x]] + (1 + 2*x - x^2 + 3*x^3 + (1 - x + 3*x^2)*Log[x] + (-3*x^2 - 3*x*Log[x])*Log[x^2 + x*Log[x]])*Log[x - Log[x ^2 + x*Log[x]]] + (-x^2 - x*Log[x] + (x + Log[x])*Log[x^2 + x*Log[x]])*Log [x - Log[x^2 + x*Log[x]]]^2)/(x^3 - x^5 + (x^2 - x^4)*Log[x] + (-x^2 + x^4 + (-x + x^3)*Log[x])*Log[x^2 + x*Log[x]] + (2*x^4 + 2*x^3*Log[x] + (-2*x^ 3 - 2*x^2*Log[x])*Log[x^2 + x*Log[x]])*Log[x - Log[x^2 + x*Log[x]]] + (-x^ 3 - x^2*Log[x] + (x^2 + x*Log[x])*Log[x^2 + x*Log[x]])*Log[x - Log[x^2 + x *Log[x]]]^2),x]
Time = 21.61 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.52, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {7292, 7293, 7239, 7293, 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.
| \(\displaystyle \int \frac {-2 x^4+x^3-x^2+\left (-x^2+(x+\log (x)) \log \left (x^2+x \log (x)\right )-x \log (x)\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )+\left (x^2-2 x^3\right ) \log (x)+\left (2 x^3+\left (2 x^2-1\right ) \log (x)-x\right ) \log \left (x^2+x \log (x)\right )+\left (3 x^3-x^2+\left (3 x^2-x+1\right ) \log (x)+\left (-3 x^2-3 x \log (x)\right ) \log \left (x^2+x \log (x)\right )+2 x+1\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )-x}{-x^5+x^3+\left (x^2-x^4\right ) \log (x)+\left (-x^3-x^2 \log (x)+\left (x^2+x \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )+\left (x^4+\left (x^3-x\right ) \log (x)-x^2\right ) \log \left (x^2+x \log (x)\right )+\left (2 x^4+2 x^3 \log (x)+\left (-2 x^3-2 x^2 \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-2 x^4+x^3-x^2+\left (-x^2+(x+\log (x)) \log \left (x^2+x \log (x)\right )-x \log (x)\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )+\left (x^2-2 x^3\right ) \log (x)+\left (2 x^3+\left (2 x^2-1\right ) \log (x)-x\right ) \log \left (x^2+x \log (x)\right )+\left (3 x^3-x^2+\left (3 x^2-x+1\right ) \log (x)+\left (-3 x^2-3 x \log (x)\right ) \log \left (x^2+x \log (x)\right )+2 x+1\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )-x}{x (x+\log (x)) (x-\log (x (x+\log (x)))) \left (-x^2-\log ^2(x-\log (x (x+\log (x))))+2 x \log (x-\log (x (x+\log (x))))+1\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 x^3}{(x+\log (x)) (x-\log (x (x+\log (x)))) (x-\log (x-\log (x (x+\log (x))))-1) (x-\log (x-\log (x (x+\log (x))))+1)}-\frac {x^2}{(x+\log (x)) (x-\log (x (x+\log (x)))) (x-\log (x-\log (x (x+\log (x))))-1) (x-\log (x-\log (x (x+\log (x))))+1)}+\frac {\left (2 x^2-1\right ) \log \left (x^2+x \log (x)\right )}{x (x-\log (x (x+\log (x)))) (x-\log (x-\log (x (x+\log (x))))+1) (-x+\log (x-\log (x (x+\log (x))))+1)}-\frac {\left (3 x^3-x^2+3 x^2 \log (x)-3 x^2 \log (x (x+\log (x)))+2 x-x \log (x)-3 x \log (x) \log (x (x+\log (x)))+\log (x)+1\right ) \log (x-\log (x (x+\log (x))))}{x (x+\log (x)) (x-\log (x (x+\log (x)))) (x-\log (x-\log (x (x+\log (x))))-1) (x-\log (x-\log (x (x+\log (x))))+1)}+\frac {\log ^2(x-\log (x (x+\log (x))))}{x (x-\log (x-\log (x (x+\log (x))))-1) (x-\log (x-\log (x (x+\log (x))))+1)}+\frac {(2 x-1) x \log (x)}{(x+\log (x)) (x-\log (x (x+\log (x)))) (x-\log (x-\log (x (x+\log (x))))-1) (x-\log (x-\log (x (x+\log (x))))+1)}+\frac {x}{(x+\log (x)) (x-\log (x (x+\log (x)))) (x-\log (x-\log (x (x+\log (x))))-1) (x-\log (x-\log (x (x+\log (x))))+1)}+\frac {1}{(x+\log (x)) (x-\log (x (x+\log (x)))) (x-\log (x-\log (x (x+\log (x))))-1) (x-\log (x-\log (x (x+\log (x))))+1)}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-2 x^4+x^3+3 x^3 \log (x-\log (x (x+\log (x))))-x^2-x^2 \log ^2(x-\log (x (x+\log (x))))+x \log (x (x+\log (x))) \left (2 x^2+\log ^2(x-\log (x (x+\log (x))))-3 x \log (x-\log (x (x+\log (x))))-1\right )-\log (x) \left ((2 x-1) x^2-\log (x (x+\log (x))) \left (2 x^2+\log ^2(x-\log (x (x+\log (x))))-3 x \log (x-\log (x (x+\log (x))))-1\right )+\left (-3 x^2+x-1\right ) \log (x-\log (x (x+\log (x))))+x \log ^2(x-\log (x (x+\log (x))))\right )-x^2 \log (x-\log (x (x+\log (x))))-x+2 x \log (x-\log (x (x+\log (x))))+\log (x-\log (x (x+\log (x))))}{x (x+\log (x)) (x-\log (x (x+\log (x)))) (x-\log (x-\log (x (x+\log (x))))+1) (-x+\log (x-\log (x (x+\log (x))))+1)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x^3-x^2+x^2 \log (x)-x^2 \log (x (x+\log (x)))+2 x-x \log (x)-x \log (x) \log (x (x+\log (x)))+\log (x)+1}{2 x (x+\log (x)) (x-\log (x (x+\log (x)))) (x-\log (x-\log (x (x+\log (x))))-1)}+\frac {x^3-x^2+x^2 \log (x)-x^2 \log (x (x+\log (x)))+2 x-x \log (x)-x \log (x) \log (x (x+\log (x)))+\log (x)+1}{2 x (x+\log (x)) (x-\log (x (x+\log (x)))) (x-\log (x-\log (x (x+\log (x))))+1)}+\frac {1}{x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \log (x)+\frac {1}{2} \log (x-\log (x-\log (x (x+\log (x))))+1)+\frac {1}{2} \log (-x+\log (x-\log (x (x+\log (x))))+1)\) |
Int[(-x - x^2 + x^3 - 2*x^4 + (x^2 - 2*x^3)*Log[x] + (-x + 2*x^3 + (-1 + 2 *x^2)*Log[x])*Log[x^2 + x*Log[x]] + (1 + 2*x - x^2 + 3*x^3 + (1 - x + 3*x^ 2)*Log[x] + (-3*x^2 - 3*x*Log[x])*Log[x^2 + x*Log[x]])*Log[x - Log[x^2 + x *Log[x]]] + (-x^2 - x*Log[x] + (x + Log[x])*Log[x^2 + x*Log[x]])*Log[x - L og[x^2 + x*Log[x]]]^2)/(x^3 - x^5 + (x^2 - x^4)*Log[x] + (-x^2 + x^4 + (-x + x^3)*Log[x])*Log[x^2 + x*Log[x]] + (2*x^4 + 2*x^3*Log[x] + (-2*x^3 - 2* x^2*Log[x])*Log[x^2 + x*Log[x]])*Log[x - Log[x^2 + x*Log[x]]] + (-x^3 - x^ 2*Log[x] + (x^2 + x*Log[x])*Log[x^2 + x*Log[x]])*Log[x - Log[x^2 + x*Log[x ]]]^2),x]
Log[x] + Log[1 + x - Log[x - Log[x*(x + Log[x])]]]/2 + Log[1 - x + Log[x - Log[x*(x + Log[x])]]]/2
3.16.78.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 130.12 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.42
| method | result | size |
| parallelrisch | \(\frac {\ln \left (x -\ln \left (x -\ln \left (\left (x +\ln \left (x \right )\right ) x \right )\right )-1\right )}{2}+\frac {\ln \left (x -\ln \left (x -\ln \left (\left (x +\ln \left (x \right )\right ) x \right )\right )+1\right )}{2}+\ln \left (x \right )\) | \(44\) |
| default | \(\ln \left (x \right )+\frac {\ln \left (x^{2}-2 x \ln \left (-\ln \left (x \right )-\ln \left (x +\ln \left (x \right )\right )+\frac {i \pi \,\operatorname {csgn}\left (i x \left (x +\ln \left (x \right )\right )\right ) \left (-\operatorname {csgn}\left (i x \left (x +\ln \left (x \right )\right )\right )+\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i x \left (x +\ln \left (x \right )\right )\right )+\operatorname {csgn}\left (i \left (x +\ln \left (x \right )\right )\right )\right )}{2}+x \right )+\ln \left (-\ln \left (x \right )-\ln \left (x +\ln \left (x \right )\right )+\frac {i \pi \,\operatorname {csgn}\left (i x \left (x +\ln \left (x \right )\right )\right ) \left (-\operatorname {csgn}\left (i x \left (x +\ln \left (x \right )\right )\right )+\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i x \left (x +\ln \left (x \right )\right )\right )+\operatorname {csgn}\left (i \left (x +\ln \left (x \right )\right )\right )\right )}{2}+x \right )^{2}-1\right )}{2}\) | \(145\) |
| risch | \(\ln \left (x \right )+\frac {\ln \left (x^{2}-2 x \ln \left (-\ln \left (x \right )-\ln \left (x +\ln \left (x \right )\right )+\frac {i \pi \,\operatorname {csgn}\left (i x \left (x +\ln \left (x \right )\right )\right ) \left (-\operatorname {csgn}\left (i x \left (x +\ln \left (x \right )\right )\right )+\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i x \left (x +\ln \left (x \right )\right )\right )+\operatorname {csgn}\left (i \left (x +\ln \left (x \right )\right )\right )\right )}{2}+x \right )+\ln \left (-\ln \left (x \right )-\ln \left (x +\ln \left (x \right )\right )+\frac {i \pi \,\operatorname {csgn}\left (i x \left (x +\ln \left (x \right )\right )\right ) \left (-\operatorname {csgn}\left (i x \left (x +\ln \left (x \right )\right )\right )+\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i x \left (x +\ln \left (x \right )\right )\right )+\operatorname {csgn}\left (i \left (x +\ln \left (x \right )\right )\right )\right )}{2}+x \right )^{2}-1\right )}{2}\) | \(145\) |
int((((x+ln(x))*ln(x*ln(x)+x^2)-x*ln(x)-x^2)*ln(-ln(x*ln(x)+x^2)+x)^2+((-3 *x*ln(x)-3*x^2)*ln(x*ln(x)+x^2)+(3*x^2-x+1)*ln(x)+3*x^3-x^2+2*x+1)*ln(-ln( x*ln(x)+x^2)+x)+((2*x^2-1)*ln(x)+2*x^3-x)*ln(x*ln(x)+x^2)+(-2*x^3+x^2)*ln( x)-2*x^4+x^3-x^2-x)/(((x*ln(x)+x^2)*ln(x*ln(x)+x^2)-x^2*ln(x)-x^3)*ln(-ln( x*ln(x)+x^2)+x)^2+((-2*x^2*ln(x)-2*x^3)*ln(x*ln(x)+x^2)+2*x^3*ln(x)+2*x^4) *ln(-ln(x*ln(x)+x^2)+x)+((x^3-x)*ln(x)+x^4-x^2)*ln(x*ln(x)+x^2)+(-x^4+x^2) *ln(x)-x^5+x^3),x,method=_RETURNVERBOSE)
Time = 0.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.42 \[ \int \frac {-x-x^2+x^3-2 x^4+\left (x^2-2 x^3\right ) \log (x)+\left (-x+2 x^3+\left (-1+2 x^2\right ) \log (x)\right ) \log \left (x^2+x \log (x)\right )+\left (1+2 x-x^2+3 x^3+\left (1-x+3 x^2\right ) \log (x)+\left (-3 x^2-3 x \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )+\left (-x^2-x \log (x)+(x+\log (x)) \log \left (x^2+x \log (x)\right )\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )}{x^3-x^5+\left (x^2-x^4\right ) \log (x)+\left (-x^2+x^4+\left (-x+x^3\right ) \log (x)\right ) \log \left (x^2+x \log (x)\right )+\left (2 x^4+2 x^3 \log (x)+\left (-2 x^3-2 x^2 \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )+\left (-x^3-x^2 \log (x)+\left (x^2+x \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )} \, dx=\frac {1}{2} \, \log \left (x^{2} - 2 \, x \log \left (x - \log \left (x^{2} + x \log \left (x\right )\right )\right ) + \log \left (x - \log \left (x^{2} + x \log \left (x\right )\right )\right )^{2} - 1\right ) + \log \left (x\right ) \]
integrate((((x+log(x))*log(x*log(x)+x^2)-x*log(x)-x^2)*log(-log(x*log(x)+x ^2)+x)^2+((-3*x*log(x)-3*x^2)*log(x*log(x)+x^2)+(3*x^2-x+1)*log(x)+3*x^3-x ^2+2*x+1)*log(-log(x*log(x)+x^2)+x)+((2*x^2-1)*log(x)+2*x^3-x)*log(x*log(x )+x^2)+(-2*x^3+x^2)*log(x)-2*x^4+x^3-x^2-x)/(((x*log(x)+x^2)*log(x*log(x)+ x^2)-x^2*log(x)-x^3)*log(-log(x*log(x)+x^2)+x)^2+((-2*x^2*log(x)-2*x^3)*lo g(x*log(x)+x^2)+2*x^3*log(x)+2*x^4)*log(-log(x*log(x)+x^2)+x)+((x^3-x)*log (x)+x^4-x^2)*log(x*log(x)+x^2)+(-x^4+x^2)*log(x)-x^5+x^3),x, algorithm=\
1/2*log(x^2 - 2*x*log(x - log(x^2 + x*log(x))) + log(x - log(x^2 + x*log(x )))^2 - 1) + log(x)
Time = 4.62 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35 \[ \int \frac {-x-x^2+x^3-2 x^4+\left (x^2-2 x^3\right ) \log (x)+\left (-x+2 x^3+\left (-1+2 x^2\right ) \log (x)\right ) \log \left (x^2+x \log (x)\right )+\left (1+2 x-x^2+3 x^3+\left (1-x+3 x^2\right ) \log (x)+\left (-3 x^2-3 x \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )+\left (-x^2-x \log (x)+(x+\log (x)) \log \left (x^2+x \log (x)\right )\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )}{x^3-x^5+\left (x^2-x^4\right ) \log (x)+\left (-x^2+x^4+\left (-x+x^3\right ) \log (x)\right ) \log \left (x^2+x \log (x)\right )+\left (2 x^4+2 x^3 \log (x)+\left (-2 x^3-2 x^2 \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )+\left (-x^3-x^2 \log (x)+\left (x^2+x \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )} \, dx=\log {\left (x \right )} + \frac {\log {\left (x^{2} - 2 x \log {\left (x - \log {\left (x^{2} + x \log {\left (x \right )} \right )} \right )} + \log {\left (x - \log {\left (x^{2} + x \log {\left (x \right )} \right )} \right )}^{2} - 1 \right )}}{2} \]
integrate((((x+ln(x))*ln(x*ln(x)+x**2)-x*ln(x)-x**2)*ln(-ln(x*ln(x)+x**2)+ x)**2+((-3*x*ln(x)-3*x**2)*ln(x*ln(x)+x**2)+(3*x**2-x+1)*ln(x)+3*x**3-x**2 +2*x+1)*ln(-ln(x*ln(x)+x**2)+x)+((2*x**2-1)*ln(x)+2*x**3-x)*ln(x*ln(x)+x** 2)+(-2*x**3+x**2)*ln(x)-2*x**4+x**3-x**2-x)/(((x*ln(x)+x**2)*ln(x*ln(x)+x* *2)-x**2*ln(x)-x**3)*ln(-ln(x*ln(x)+x**2)+x)**2+((-2*x**2*ln(x)-2*x**3)*ln (x*ln(x)+x**2)+2*x**3*ln(x)+2*x**4)*ln(-ln(x*ln(x)+x**2)+x)+((x**3-x)*ln(x )+x**4-x**2)*ln(x*ln(x)+x**2)+(-x**4+x**2)*ln(x)-x**5+x**3),x)
log(x) + log(x**2 - 2*x*log(x - log(x**2 + x*log(x))) + log(x - log(x**2 + x*log(x)))**2 - 1)/2
Time = 0.27 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.52 \[ \int \frac {-x-x^2+x^3-2 x^4+\left (x^2-2 x^3\right ) \log (x)+\left (-x+2 x^3+\left (-1+2 x^2\right ) \log (x)\right ) \log \left (x^2+x \log (x)\right )+\left (1+2 x-x^2+3 x^3+\left (1-x+3 x^2\right ) \log (x)+\left (-3 x^2-3 x \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )+\left (-x^2-x \log (x)+(x+\log (x)) \log \left (x^2+x \log (x)\right )\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )}{x^3-x^5+\left (x^2-x^4\right ) \log (x)+\left (-x^2+x^4+\left (-x+x^3\right ) \log (x)\right ) \log \left (x^2+x \log (x)\right )+\left (2 x^4+2 x^3 \log (x)+\left (-2 x^3-2 x^2 \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )+\left (-x^3-x^2 \log (x)+\left (x^2+x \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )} \, dx=\log \left (x\right ) + \frac {1}{2} \, \log \left (-x + \log \left (x - \log \left (x + \log \left (x\right )\right ) - \log \left (x\right )\right ) + 1\right ) + \frac {1}{2} \, \log \left (-x + \log \left (x - \log \left (x + \log \left (x\right )\right ) - \log \left (x\right )\right ) - 1\right ) \]
integrate((((x+log(x))*log(x*log(x)+x^2)-x*log(x)-x^2)*log(-log(x*log(x)+x ^2)+x)^2+((-3*x*log(x)-3*x^2)*log(x*log(x)+x^2)+(3*x^2-x+1)*log(x)+3*x^3-x ^2+2*x+1)*log(-log(x*log(x)+x^2)+x)+((2*x^2-1)*log(x)+2*x^3-x)*log(x*log(x )+x^2)+(-2*x^3+x^2)*log(x)-2*x^4+x^3-x^2-x)/(((x*log(x)+x^2)*log(x*log(x)+ x^2)-x^2*log(x)-x^3)*log(-log(x*log(x)+x^2)+x)^2+((-2*x^2*log(x)-2*x^3)*lo g(x*log(x)+x^2)+2*x^3*log(x)+2*x^4)*log(-log(x*log(x)+x^2)+x)+((x^3-x)*log (x)+x^4-x^2)*log(x*log(x)+x^2)+(-x^4+x^2)*log(x)-x^5+x^3),x, algorithm=\
log(x) + 1/2*log(-x + log(x - log(x + log(x)) - log(x)) + 1) + 1/2*log(-x + log(x - log(x + log(x)) - log(x)) - 1)
Time = 1.06 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.42 \[ \int \frac {-x-x^2+x^3-2 x^4+\left (x^2-2 x^3\right ) \log (x)+\left (-x+2 x^3+\left (-1+2 x^2\right ) \log (x)\right ) \log \left (x^2+x \log (x)\right )+\left (1+2 x-x^2+3 x^3+\left (1-x+3 x^2\right ) \log (x)+\left (-3 x^2-3 x \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )+\left (-x^2-x \log (x)+(x+\log (x)) \log \left (x^2+x \log (x)\right )\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )}{x^3-x^5+\left (x^2-x^4\right ) \log (x)+\left (-x^2+x^4+\left (-x+x^3\right ) \log (x)\right ) \log \left (x^2+x \log (x)\right )+\left (2 x^4+2 x^3 \log (x)+\left (-2 x^3-2 x^2 \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )+\left (-x^3-x^2 \log (x)+\left (x^2+x \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )} \, dx=\frac {1}{2} \, \log \left (x^{2} - 2 \, x \log \left (x - \log \left (x + \log \left (x\right )\right ) - \log \left (x\right )\right ) + \log \left (x - \log \left (x + \log \left (x\right )\right ) - \log \left (x\right )\right )^{2} - 1\right ) + \log \left (x\right ) \]
integrate((((x+log(x))*log(x*log(x)+x^2)-x*log(x)-x^2)*log(-log(x*log(x)+x ^2)+x)^2+((-3*x*log(x)-3*x^2)*log(x*log(x)+x^2)+(3*x^2-x+1)*log(x)+3*x^3-x ^2+2*x+1)*log(-log(x*log(x)+x^2)+x)+((2*x^2-1)*log(x)+2*x^3-x)*log(x*log(x )+x^2)+(-2*x^3+x^2)*log(x)-2*x^4+x^3-x^2-x)/(((x*log(x)+x^2)*log(x*log(x)+ x^2)-x^2*log(x)-x^3)*log(-log(x*log(x)+x^2)+x)^2+((-2*x^2*log(x)-2*x^3)*lo g(x*log(x)+x^2)+2*x^3*log(x)+2*x^4)*log(-log(x*log(x)+x^2)+x)+((x^3-x)*log (x)+x^4-x^2)*log(x*log(x)+x^2)+(-x^4+x^2)*log(x)-x^5+x^3),x, algorithm=\
1/2*log(x^2 - 2*x*log(x - log(x + log(x)) - log(x)) + log(x - log(x + log( x)) - log(x))^2 - 1) + log(x)
Time = 14.73 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.29 \[ \int \frac {-x-x^2+x^3-2 x^4+\left (x^2-2 x^3\right ) \log (x)+\left (-x+2 x^3+\left (-1+2 x^2\right ) \log (x)\right ) \log \left (x^2+x \log (x)\right )+\left (1+2 x-x^2+3 x^3+\left (1-x+3 x^2\right ) \log (x)+\left (-3 x^2-3 x \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )+\left (-x^2-x \log (x)+(x+\log (x)) \log \left (x^2+x \log (x)\right )\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )}{x^3-x^5+\left (x^2-x^4\right ) \log (x)+\left (-x^2+x^4+\left (-x+x^3\right ) \log (x)\right ) \log \left (x^2+x \log (x)\right )+\left (2 x^4+2 x^3 \log (x)+\left (-2 x^3-2 x^2 \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )+\left (-x^3-x^2 \log (x)+\left (x^2+x \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )} \, dx=\frac {\ln \left (x^2-2\,x\,\ln \left (x-\ln \left (x\,\left (x+\ln \left (x\right )\right )\right )\right )+{\ln \left (x-\ln \left (x\,\left (x+\ln \left (x\right )\right )\right )\right )}^2-1\right )}{2}+\ln \left (x\right ) \]
int((x - log(x*log(x) + x^2)*(2*x^3 - x + log(x)*(2*x^2 - 1)) + log(x - lo g(x*log(x) + x^2))^2*(x*log(x) - log(x*log(x) + x^2)*(x + log(x)) + x^2) - log(x)*(x^2 - 2*x^3) - log(x - log(x*log(x) + x^2))*(2*x - log(x*log(x) + x^2)*(3*x*log(x) + 3*x^2) + log(x)*(3*x^2 - x + 1) - x^2 + 3*x^3 + 1) + x ^2 - x^3 + 2*x^4)/(log(x*log(x) + x^2)*(log(x)*(x - x^3) + x^2 - x^4) - lo g(x - log(x*log(x) + x^2))*(2*x^3*log(x) - log(x*log(x) + x^2)*(2*x^2*log( x) + 2*x^3) + 2*x^4) - log(x)*(x^2 - x^4) + log(x - log(x*log(x) + x^2))^2 *(x^2*log(x) - log(x*log(x) + x^2)*(x*log(x) + x^2) + x^3) - x^3 + x^5),x)