Integrand size = 157, antiderivative size = 33 \[ \int \frac {-2 x \log (x)+2 \log (x) \log ^2\left (x^2\right )+\left ((-x-x \log (x)) \log \left (x^2\right )+(2+2 \log (x)) \log ^2\left (x^2\right )\right ) \log \left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )+\left (-5 x \log \left (x^2\right )+10 \log ^2\left (x^2\right )\right ) \log ^2\left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )}{\left (-5 x \log \left (x^2\right )+10 \log ^2\left (x^2\right )\right ) \log ^2\left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )} \, dx=x+\frac {1}{5} \left (3+\frac {x \log (x)}{\log \left (\frac {2-\frac {x}{\log \left (x^2\right )}}{5 x}\right )}\right ) \]
Time = 0.12 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int \frac {-2 x \log (x)+2 \log (x) \log ^2\left (x^2\right )+\left ((-x-x \log (x)) \log \left (x^2\right )+(2+2 \log (x)) \log ^2\left (x^2\right )\right ) \log \left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )+\left (-5 x \log \left (x^2\right )+10 \log ^2\left (x^2\right )\right ) \log ^2\left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )}{\left (-5 x \log \left (x^2\right )+10 \log ^2\left (x^2\right )\right ) \log ^2\left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )} \, dx=-\frac {1}{5} x \left (-5-\frac {\log (x)}{\log \left (\frac {2}{5 x}-\frac {1}{5 \log \left (x^2\right )}\right )}\right ) \]
Integrate[(-2*x*Log[x] + 2*Log[x]*Log[x^2]^2 + ((-x - x*Log[x])*Log[x^2] + (2 + 2*Log[x])*Log[x^2]^2)*Log[(-x + 2*Log[x^2])/(5*x*Log[x^2])] + (-5*x* Log[x^2] + 10*Log[x^2]^2)*Log[(-x + 2*Log[x^2])/(5*x*Log[x^2])]^2)/((-5*x* Log[x^2] + 10*Log[x^2]^2)*Log[(-x + 2*Log[x^2])/(5*x*Log[x^2])]^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.
| \(\displaystyle \int \frac {2 \log (x) \log ^2\left (x^2\right )+\left (10 \log ^2\left (x^2\right )-5 x \log \left (x^2\right )\right ) \log ^2\left (\frac {2 \log \left (x^2\right )-x}{5 x \log \left (x^2\right )}\right )+\left ((2 \log (x)+2) \log ^2\left (x^2\right )+(x (-\log (x))-x) \log \left (x^2\right )\right ) \log \left (\frac {2 \log \left (x^2\right )-x}{5 x \log \left (x^2\right )}\right )-2 x \log (x)}{\left (10 \log ^2\left (x^2\right )-5 x \log \left (x^2\right )\right ) \log ^2\left (\frac {2 \log \left (x^2\right )-x}{5 x \log \left (x^2\right )}\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-2 \log (x) \log ^2\left (x^2\right )-\left (10 \log ^2\left (x^2\right )-5 x \log \left (x^2\right )\right ) \log ^2\left (\frac {2 \log \left (x^2\right )-x}{5 x \log \left (x^2\right )}\right )-\left ((2 \log (x)+2) \log ^2\left (x^2\right )+(x (-\log (x))-x) \log \left (x^2\right )\right ) \log \left (\frac {2 \log \left (x^2\right )-x}{5 x \log \left (x^2\right )}\right )+2 x \log (x)}{5 \left (x-2 \log \left (x^2\right )\right ) \log \left (x^2\right ) \log ^2\left (\frac {2 \log \left (x^2\right )-x}{5 x \log \left (x^2\right )}\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \int \frac {-2 \log (x) \log ^2\left (x^2\right )+5 \left (x \log \left (x^2\right )-2 \log ^2\left (x^2\right )\right ) \log ^2\left (-\frac {x-2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )+2 x \log (x)+\left ((\log (x) x+x) \log \left (x^2\right )-2 (\log (x)+1) \log ^2\left (x^2\right )\right ) \log \left (-\frac {x-2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )}{\left (x-2 \log \left (x^2\right )\right ) \log \left (x^2\right ) \log ^2\left (-\frac {x-2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (\frac {\log (x)+1}{\log \left (\frac {2}{5 x}-\frac {1}{5 \log \left (x^2\right )}\right )}+\frac {2 \log (x) \left (x-\log ^2\left (x^2\right )\right )}{\left (x-2 \log \left (x^2\right )\right ) \log \left (x^2\right ) \log ^2\left (\frac {2}{5 x}-\frac {1}{5 \log \left (x^2\right )}\right )}+5\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} \left (2 \int \frac {x \log (x)}{\left (x-2 \log \left (x^2\right )\right ) \log \left (x^2\right ) \log ^2\left (\frac {2}{5 x}-\frac {1}{5 \log \left (x^2\right )}\right )}dx-2 \int \frac {\log (x) \log \left (x^2\right )}{\left (x-2 \log \left (x^2\right )\right ) \log ^2\left (\frac {2}{5 x}-\frac {1}{5 \log \left (x^2\right )}\right )}dx+\int \frac {1}{\log \left (\frac {2}{5 x}-\frac {1}{5 \log \left (x^2\right )}\right )}dx+\int \frac {\log (x)}{\log \left (\frac {2}{5 x}-\frac {1}{5 \log \left (x^2\right )}\right )}dx+5 x\right )\) |
Int[(-2*x*Log[x] + 2*Log[x]*Log[x^2]^2 + ((-x - x*Log[x])*Log[x^2] + (2 + 2*Log[x])*Log[x^2]^2)*Log[(-x + 2*Log[x^2])/(5*x*Log[x^2])] + (-5*x*Log[x^ 2] + 10*Log[x^2]^2)*Log[(-x + 2*Log[x^2])/(5*x*Log[x^2])]^2)/((-5*x*Log[x^ 2] + 10*Log[x^2]^2)*Log[(-x + 2*Log[x^2])/(5*x*Log[x^2])]^2),x]
3.12.21.3.1 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]]
Leaf count of result is larger than twice the leaf count of optimal. \(111\) vs. \(2(27)=54\).
Time = 7.30 (sec) , antiderivative size = 112, normalized size of antiderivative = 3.39
| method | result | size |
| parallelrisch | \(\frac {16 x \ln \left (x \right )+80 \ln \left (\frac {2 \ln \left (x^{2}\right )-x}{5 x \ln \left (x^{2}\right )}\right ) x -160 \ln \left (x \right ) \ln \left (\frac {2 \ln \left (x^{2}\right )-x}{5 x \ln \left (x^{2}\right )}\right )+80 \ln \left (\frac {2 \ln \left (x^{2}\right )-x}{5 x \ln \left (x^{2}\right )}\right ) \ln \left (x^{2}\right )}{80 \ln \left (\frac {2 \ln \left (x^{2}\right )-x}{5 x \ln \left (x^{2}\right )}\right )}\) | \(112\) |
| risch | \(\text {Expression too large to display}\) | \(1531\) |
int(((10*ln(x^2)^2-5*x*ln(x^2))*ln(1/5*(2*ln(x^2)-x)/x/ln(x^2))^2+((2*ln(x )+2)*ln(x^2)^2+(-x*ln(x)-x)*ln(x^2))*ln(1/5*(2*ln(x^2)-x)/x/ln(x^2))+2*ln( x)*ln(x^2)^2-2*x*ln(x))/(10*ln(x^2)^2-5*x*ln(x^2))/ln(1/5*(2*ln(x^2)-x)/x/ ln(x^2))^2,x,method=_RETURNVERBOSE)
1/80*(16*x*ln(x)+80*ln(1/5*(2*ln(x^2)-x)/x/ln(x^2))*x-160*ln(x)*ln(1/5*(2* ln(x^2)-x)/x/ln(x^2))+80*ln(1/5*(2*ln(x^2)-x)/x/ln(x^2))*ln(x^2))/ln(1/5*( 2*ln(x^2)-x)/x/ln(x^2))
Time = 0.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.33 \[ \int \frac {-2 x \log (x)+2 \log (x) \log ^2\left (x^2\right )+\left ((-x-x \log (x)) \log \left (x^2\right )+(2+2 \log (x)) \log ^2\left (x^2\right )\right ) \log \left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )+\left (-5 x \log \left (x^2\right )+10 \log ^2\left (x^2\right )\right ) \log ^2\left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )}{\left (-5 x \log \left (x^2\right )+10 \log ^2\left (x^2\right )\right ) \log ^2\left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )} \, dx=\frac {x \log \left (x\right ) + 5 \, x \log \left (-\frac {x - 4 \, \log \left (x\right )}{10 \, x \log \left (x\right )}\right )}{5 \, \log \left (-\frac {x - 4 \, \log \left (x\right )}{10 \, x \log \left (x\right )}\right )} \]
integrate(((10*log(x^2)^2-5*x*log(x^2))*log(1/5*(2*log(x^2)-x)/x/log(x^2)) ^2+((2*log(x)+2)*log(x^2)^2+(-x*log(x)-x)*log(x^2))*log(1/5*(2*log(x^2)-x) /x/log(x^2))+2*log(x)*log(x^2)^2-2*x*log(x))/(10*log(x^2)^2-5*x*log(x^2))/ log(1/5*(2*log(x^2)-x)/x/log(x^2))^2,x, algorithm=\
1/5*(x*log(x) + 5*x*log(-1/10*(x - 4*log(x))/(x*log(x))))/log(-1/10*(x - 4 *log(x))/(x*log(x)))
Time = 0.12 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \frac {-2 x \log (x)+2 \log (x) \log ^2\left (x^2\right )+\left ((-x-x \log (x)) \log \left (x^2\right )+(2+2 \log (x)) \log ^2\left (x^2\right )\right ) \log \left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )+\left (-5 x \log \left (x^2\right )+10 \log ^2\left (x^2\right )\right ) \log ^2\left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )}{\left (-5 x \log \left (x^2\right )+10 \log ^2\left (x^2\right )\right ) \log ^2\left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )} \, dx=\frac {x \log {\left (x \right )}}{5 \log {\left (\frac {- \frac {x}{5} + \frac {4 \log {\left (x \right )}}{5}}{2 x \log {\left (x \right )}} \right )}} + x \]
integrate(((10*ln(x**2)**2-5*x*ln(x**2))*ln(1/5*(2*ln(x**2)-x)/x/ln(x**2)) **2+((2*ln(x)+2)*ln(x**2)**2+(-x*ln(x)-x)*ln(x**2))*ln(1/5*(2*ln(x**2)-x)/ x/ln(x**2))+2*ln(x)*ln(x**2)**2-2*x*ln(x))/(10*ln(x**2)**2-5*x*ln(x**2))/l n(1/5*(2*ln(x**2)-x)/x/ln(x**2))**2,x)
Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (26) = 52\).
Time = 0.34 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.73 \[ \int \frac {-2 x \log (x)+2 \log (x) \log ^2\left (x^2\right )+\left ((-x-x \log (x)) \log \left (x^2\right )+(2+2 \log (x)) \log ^2\left (x^2\right )\right ) \log \left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )+\left (-5 x \log \left (x^2\right )+10 \log ^2\left (x^2\right )\right ) \log ^2\left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )}{\left (-5 x \log \left (x^2\right )+10 \log ^2\left (x^2\right )\right ) \log ^2\left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )} \, dx=\frac {5 \, x {\left (\log \left (5\right ) + \log \left (2\right )\right )} + 4 \, x \log \left (x\right ) - 5 \, x \log \left (-x + 4 \, \log \left (x\right )\right ) + 5 \, x \log \left (\log \left (x\right )\right )}{5 \, {\left (\log \left (5\right ) + \log \left (2\right ) + \log \left (x\right ) - \log \left (-x + 4 \, \log \left (x\right )\right ) + \log \left (\log \left (x\right )\right )\right )}} \]
integrate(((10*log(x^2)^2-5*x*log(x^2))*log(1/5*(2*log(x^2)-x)/x/log(x^2)) ^2+((2*log(x)+2)*log(x^2)^2+(-x*log(x)-x)*log(x^2))*log(1/5*(2*log(x^2)-x) /x/log(x^2))+2*log(x)*log(x^2)^2-2*x*log(x))/(10*log(x^2)^2-5*x*log(x^2))/ log(1/5*(2*log(x^2)-x)/x/log(x^2))^2,x, algorithm=\
1/5*(5*x*(log(5) + log(2)) + 4*x*log(x) - 5*x*log(-x + 4*log(x)) + 5*x*log (log(x)))/(log(5) + log(2) + log(x) - log(-x + 4*log(x)) + log(log(x)))
Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (26) = 52\).
Time = 0.72 (sec) , antiderivative size = 218, normalized size of antiderivative = 6.61 \[ \int \frac {-2 x \log (x)+2 \log (x) \log ^2\left (x^2\right )+\left ((-x-x \log (x)) \log \left (x^2\right )+(2+2 \log (x)) \log ^2\left (x^2\right )\right ) \log \left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )+\left (-5 x \log \left (x^2\right )+10 \log ^2\left (x^2\right )\right ) \log ^2\left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )}{\left (-5 x \log \left (x^2\right )+10 \log ^2\left (x^2\right )\right ) \log ^2\left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )} \, dx=x - \frac {4 \, x^{2} \log \left (x^{2}\right ) \log \left (x\right )^{2} - 8 \, x \log \left (x^{2}\right )^{2} \log \left (x\right )^{2} - x^{3} \log \left (x^{2}\right ) + 2 \, x^{2} \log \left (x^{2}\right )^{2}}{10 \, {\left (x \log \left (x^{2}\right )^{2} \log \left (x\right ) - 4 \, \log \left (x^{2}\right )^{2} \log \left (x\right )^{2} - x \log \left (x^{2}\right )^{2} \log \left (-x + 2 \, \log \left (x^{2}\right )\right ) + 4 \, \log \left (x^{2}\right )^{2} \log \left (x\right ) \log \left (-x + 2 \, \log \left (x^{2}\right )\right ) + x \log \left (x^{2}\right )^{2} \log \left (5 \, \log \left (x^{2}\right )\right ) - 4 \, \log \left (x^{2}\right )^{2} \log \left (x\right ) \log \left (5 \, \log \left (x^{2}\right )\right ) - x^{2} \log \left (x\right ) + 4 \, x \log \left (x\right )^{2} + x^{2} \log \left (-x + 2 \, \log \left (x^{2}\right )\right ) - 4 \, x \log \left (x\right ) \log \left (-x + 2 \, \log \left (x^{2}\right )\right ) - x^{2} \log \left (5 \, \log \left (x^{2}\right )\right ) + 4 \, x \log \left (x\right ) \log \left (5 \, \log \left (x^{2}\right )\right )\right )}} \]
integrate(((10*log(x^2)^2-5*x*log(x^2))*log(1/5*(2*log(x^2)-x)/x/log(x^2)) ^2+((2*log(x)+2)*log(x^2)^2+(-x*log(x)-x)*log(x^2))*log(1/5*(2*log(x^2)-x) /x/log(x^2))+2*log(x)*log(x^2)^2-2*x*log(x))/(10*log(x^2)^2-5*x*log(x^2))/ log(1/5*(2*log(x^2)-x)/x/log(x^2))^2,x, algorithm=\
x - 1/10*(4*x^2*log(x^2)*log(x)^2 - 8*x*log(x^2)^2*log(x)^2 - x^3*log(x^2) + 2*x^2*log(x^2)^2)/(x*log(x^2)^2*log(x) - 4*log(x^2)^2*log(x)^2 - x*log( x^2)^2*log(-x + 2*log(x^2)) + 4*log(x^2)^2*log(x)*log(-x + 2*log(x^2)) + x *log(x^2)^2*log(5*log(x^2)) - 4*log(x^2)^2*log(x)*log(5*log(x^2)) - x^2*lo g(x) + 4*x*log(x)^2 + x^2*log(-x + 2*log(x^2)) - 4*x*log(x)*log(-x + 2*log (x^2)) - x^2*log(5*log(x^2)) + 4*x*log(x)*log(5*log(x^2)))
Time = 10.43 (sec) , antiderivative size = 296, normalized size of antiderivative = 8.97 \[ \int \frac {-2 x \log (x)+2 \log (x) \log ^2\left (x^2\right )+\left ((-x-x \log (x)) \log \left (x^2\right )+(2+2 \log (x)) \log ^2\left (x^2\right )\right ) \log \left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )+\left (-5 x \log \left (x^2\right )+10 \log ^2\left (x^2\right )\right ) \log ^2\left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )}{\left (-5 x \log \left (x^2\right )+10 \log ^2\left (x^2\right )\right ) \log ^2\left (\frac {-x+2 \log \left (x^2\right )}{5 x \log \left (x^2\right )}\right )} \, dx=\frac {4\,x}{5}+\frac {\frac {x\,\ln \left (x\right )}{5}-\frac {x\,\ln \left (-\frac {\frac {x}{5}-\frac {2\,\ln \left (x^2\right )}{5}}{x\,\ln \left (x^2\right )}\right )\,\ln \left (x^2\right )\,\left (\ln \left (x\right )+1\right )\,\left (x-2\,\ln \left (x^2\right )\right )}{10\,\left (4\,\ln \left (x\right )\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )-x+4\,{\ln \left (x\right )}^2+{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^2\right )}}{\ln \left (-\frac {\frac {x}{5}-\frac {2\,\ln \left (x^2\right )}{5}}{x\,\ln \left (x^2\right )}\right )}-\frac {x\,\ln \left (x\right )}{5}+\frac {x^2}{20}-\frac {\frac {2\,x^5\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )-32\,x^4\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )+16\,x^4\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^2-x^5\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^2+64\,x^4-20\,x^5+x^6}{20\,\left (16\,x^2-x^3\right )}+\frac {\ln \left (x\right )\,\left (16\,x^4\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )-x^5\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )\right )}{10\,\left (16\,x^2-x^3\right )}}{4\,\ln \left (x\right )\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )-x+4\,{\ln \left (x\right )}^2+{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^2} \]
int((log(-(x/5 - (2*log(x^2))/5)/(x*log(x^2)))^2*(5*x*log(x^2) - 10*log(x^ 2)^2) + 2*x*log(x) + log(-(x/5 - (2*log(x^2))/5)/(x*log(x^2)))*(log(x^2)*( x + x*log(x)) - log(x^2)^2*(2*log(x) + 2)) - 2*log(x^2)^2*log(x))/(log(-(x /5 - (2*log(x^2))/5)/(x*log(x^2)))^2*(5*x*log(x^2) - 10*log(x^2)^2)),x)
(4*x)/5 + ((x*log(x))/5 - (x*log(-(x/5 - (2*log(x^2))/5)/(x*log(x^2)))*log (x^2)*(log(x) + 1)*(x - 2*log(x^2)))/(10*(4*log(x)*(log(x^2) - 2*log(x)) - x + 4*log(x)^2 + (log(x^2) - 2*log(x))^2)))/log(-(x/5 - (2*log(x^2))/5)/( x*log(x^2))) - (x*log(x))/5 + x^2/20 - ((2*x^5*(log(x^2) - 2*log(x)) - 32* x^4*(log(x^2) - 2*log(x)) + 16*x^4*(log(x^2) - 2*log(x))^2 - x^5*(log(x^2) - 2*log(x))^2 + 64*x^4 - 20*x^5 + x^6)/(20*(16*x^2 - x^3)) + (log(x)*(16* x^4*(log(x^2) - 2*log(x)) - x^5*(log(x^2) - 2*log(x))))/(10*(16*x^2 - x^3) ))/(4*log(x)*(log(x^2) - 2*log(x)) - x + 4*log(x)^2 + (log(x^2) - 2*log(x) )^2)