Integrand size = 128, antiderivative size = 30 \[ \int \frac {-20-2 x+\left (-20 x-2 x^2\right ) \log (x)+\left (\left (-40 x-2 x^2\right ) \log \left (\frac {16}{60 x+3 x^2}\right )+\left (-40 x-2 x^2\right ) \log (x) \log \left (\frac {16}{60 x+3 x^2}\right )\right ) \log (1+x \log (x))}{\left (20 x+x^2\right ) \log \left (\frac {16}{60 x+3 x^2}\right )+\left (20 x^2+x^3\right ) \log (x) \log \left (\frac {16}{60 x+3 x^2}\right )} \, dx=\log \left (\log \left (\frac {4}{3 \left (5+\frac {x}{4}\right ) x}\right )\right )-\log ^2(1+x \log (x)) \]
Time = 0.09 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {-20-2 x+\left (-20 x-2 x^2\right ) \log (x)+\left (\left (-40 x-2 x^2\right ) \log \left (\frac {16}{60 x+3 x^2}\right )+\left (-40 x-2 x^2\right ) \log (x) \log \left (\frac {16}{60 x+3 x^2}\right )\right ) \log (1+x \log (x))}{\left (20 x+x^2\right ) \log \left (\frac {16}{60 x+3 x^2}\right )+\left (20 x^2+x^3\right ) \log (x) \log \left (\frac {16}{60 x+3 x^2}\right )} \, dx=-2 \left (\frac {1}{2} \log ^2(1+x \log (x))-\frac {1}{2} \log \left (\log \left (\frac {16}{60 x+3 x^2}\right )\right )\right ) \]
Integrate[(-20 - 2*x + (-20*x - 2*x^2)*Log[x] + ((-40*x - 2*x^2)*Log[16/(6 0*x + 3*x^2)] + (-40*x - 2*x^2)*Log[x]*Log[16/(60*x + 3*x^2)])*Log[1 + x*L og[x]])/((20*x + x^2)*Log[16/(60*x + 3*x^2)] + (20*x^2 + x^3)*Log[x]*Log[1 6/(60*x + 3*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.
| \(\displaystyle \int \frac {\left (-2 x^2-20 x\right ) \log (x)+\left (\left (-2 x^2-40 x\right ) \log \left (\frac {16}{3 x^2+60 x}\right )+\left (-2 x^2-40 x\right ) \log (x) \log \left (\frac {16}{3 x^2+60 x}\right )\right ) \log (x \log (x)+1)-2 x-20}{\left (x^2+20 x\right ) \log \left (\frac {16}{3 x^2+60 x}\right )+\left (x^3+20 x^2\right ) \log (x) \log \left (\frac {16}{3 x^2+60 x}\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (-2 x^2-20 x\right ) \log (x)+\left (\left (-2 x^2-40 x\right ) \log \left (\frac {16}{3 x^2+60 x}\right )+\left (-2 x^2-40 x\right ) \log (x) \log \left (\frac {16}{3 x^2+60 x}\right )\right ) \log (x \log (x)+1)-2 x-20}{x (x+20) (x \log (x)+1) \log \left (\frac {16}{x (3 x+60)}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 (-x-10)}{x (x+20) \log \left (\frac {16}{x (3 x+60)}\right )}-\frac {2 (\log (x)+1) \log (x \log (x)+1)}{x \log (x)+1}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \int \frac {-x-10}{x (x+20) \log \left (\frac {16}{x (3 x+60)}\right )}dx-\log ^2(x \log (x)+1)\) |
Int[(-20 - 2*x + (-20*x - 2*x^2)*Log[x] + ((-40*x - 2*x^2)*Log[16/(60*x + 3*x^2)] + (-40*x - 2*x^2)*Log[x]*Log[16/(60*x + 3*x^2)])*Log[1 + x*Log[x]] )/((20*x + x^2)*Log[16/(60*x + 3*x^2)] + (20*x^2 + x^3)*Log[x]*Log[16/(60* x + 3*x^2)]),x]
3.26.42.3.1 Defintions of rubi rules used
Time = 9.98 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83
| method | result | size |
| parallelrisch | \(-\ln \left (x \ln \left (x \right )+1\right )^{2}+\ln \left (\ln \left (\frac {16}{3 x \left (20+x \right )}\right )\right )\) | \(25\) |
| default | \(-\ln \left (x \ln \left (x \right )+1\right )^{2}+\ln \left (4 \ln \left (2\right )-\ln \left (3\right )+\ln \left (\frac {1}{x \left (20+x \right )}\right )\right )\) | \(33\) |
| risch | \(-\ln \left (x \ln \left (x \right )+1\right )^{2}+\ln \left (-\pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i}{20+x}\right ) \operatorname {csgn}\left (\frac {i}{x \left (20+x \right )}\right )+\pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i}{x \left (20+x \right )}\right )^{2}+\pi \,\operatorname {csgn}\left (\frac {i}{20+x}\right ) \operatorname {csgn}\left (\frac {i}{x \left (20+x \right )}\right )^{2}-\pi \operatorname {csgn}\left (\frac {i}{x \left (20+x \right )}\right )^{3}+2 i \ln \left (3\right )-8 i \ln \left (2\right )+2 i \ln \left (x \right )+2 i \ln \left (20+x \right )\right )\) | \(133\) |
int((((-2*x^2-40*x)*ln(16/(3*x^2+60*x))*ln(x)+(-2*x^2-40*x)*ln(16/(3*x^2+6 0*x)))*ln(x*ln(x)+1)+(-2*x^2-20*x)*ln(x)-2*x-20)/((x^3+20*x^2)*ln(16/(3*x^ 2+60*x))*ln(x)+(x^2+20*x)*ln(16/(3*x^2+60*x))),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int \frac {-20-2 x+\left (-20 x-2 x^2\right ) \log (x)+\left (\left (-40 x-2 x^2\right ) \log \left (\frac {16}{60 x+3 x^2}\right )+\left (-40 x-2 x^2\right ) \log (x) \log \left (\frac {16}{60 x+3 x^2}\right )\right ) \log (1+x \log (x))}{\left (20 x+x^2\right ) \log \left (\frac {16}{60 x+3 x^2}\right )+\left (20 x^2+x^3\right ) \log (x) \log \left (\frac {16}{60 x+3 x^2}\right )} \, dx=-\log \left (x \log \left (x\right ) + 1\right )^{2} + \log \left (\log \left (\frac {16}{3 \, {\left (x^{2} + 20 \, x\right )}}\right )\right ) \]
integrate((((-2*x^2-40*x)*log(16/(3*x^2+60*x))*log(x)+(-2*x^2-40*x)*log(16 /(3*x^2+60*x)))*log(x*log(x)+1)+(-2*x^2-20*x)*log(x)-2*x-20)/((x^3+20*x^2) *log(16/(3*x^2+60*x))*log(x)+(x^2+20*x)*log(16/(3*x^2+60*x))),x, algorithm =\
Time = 0.50 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73 \[ \int \frac {-20-2 x+\left (-20 x-2 x^2\right ) \log (x)+\left (\left (-40 x-2 x^2\right ) \log \left (\frac {16}{60 x+3 x^2}\right )+\left (-40 x-2 x^2\right ) \log (x) \log \left (\frac {16}{60 x+3 x^2}\right )\right ) \log (1+x \log (x))}{\left (20 x+x^2\right ) \log \left (\frac {16}{60 x+3 x^2}\right )+\left (20 x^2+x^3\right ) \log (x) \log \left (\frac {16}{60 x+3 x^2}\right )} \, dx=- \log {\left (x \log {\left (x \right )} + 1 \right )}^{2} + \log {\left (\log {\left (\frac {16}{3 x^{2} + 60 x} \right )} \right )} \]
integrate((((-2*x**2-40*x)*ln(16/(3*x**2+60*x))*ln(x)+(-2*x**2-40*x)*ln(16 /(3*x**2+60*x)))*ln(x*ln(x)+1)+(-2*x**2-20*x)*ln(x)-2*x-20)/((x**3+20*x**2 )*ln(16/(3*x**2+60*x))*ln(x)+(x**2+20*x)*ln(16/(3*x**2+60*x))),x)
\[ \int \frac {-20-2 x+\left (-20 x-2 x^2\right ) \log (x)+\left (\left (-40 x-2 x^2\right ) \log \left (\frac {16}{60 x+3 x^2}\right )+\left (-40 x-2 x^2\right ) \log (x) \log \left (\frac {16}{60 x+3 x^2}\right )\right ) \log (1+x \log (x))}{\left (20 x+x^2\right ) \log \left (\frac {16}{60 x+3 x^2}\right )+\left (20 x^2+x^3\right ) \log (x) \log \left (\frac {16}{60 x+3 x^2}\right )} \, dx=\int { -\frac {2 \, {\left ({\left ({\left (x^{2} + 20 \, x\right )} \log \left (x\right ) \log \left (\frac {16}{3 \, {\left (x^{2} + 20 \, x\right )}}\right ) + {\left (x^{2} + 20 \, x\right )} \log \left (\frac {16}{3 \, {\left (x^{2} + 20 \, x\right )}}\right )\right )} \log \left (x \log \left (x\right ) + 1\right ) + {\left (x^{2} + 10 \, x\right )} \log \left (x\right ) + x + 10\right )}}{{\left (x^{3} + 20 \, x^{2}\right )} \log \left (x\right ) \log \left (\frac {16}{3 \, {\left (x^{2} + 20 \, x\right )}}\right ) + {\left (x^{2} + 20 \, x\right )} \log \left (\frac {16}{3 \, {\left (x^{2} + 20 \, x\right )}}\right )} \,d x } \]
integrate((((-2*x^2-40*x)*log(16/(3*x^2+60*x))*log(x)+(-2*x^2-40*x)*log(16 /(3*x^2+60*x)))*log(x*log(x)+1)+(-2*x^2-20*x)*log(x)-2*x-20)/((x^3+20*x^2) *log(16/(3*x^2+60*x))*log(x)+(x^2+20*x)*log(16/(3*x^2+60*x))),x, algorithm =\
-2*integrate((((x^2 + 20*x)*log(x)*log(16/3/(x^2 + 20*x)) + (x^2 + 20*x)*l og(16/3/(x^2 + 20*x)))*log(x*log(x) + 1) + (x^2 + 10*x)*log(x) + x + 10)/( (x^3 + 20*x^2)*log(x)*log(16/3/(x^2 + 20*x)) + (x^2 + 20*x)*log(16/3/(x^2 + 20*x))), x)
Time = 0.33 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {-20-2 x+\left (-20 x-2 x^2\right ) \log (x)+\left (\left (-40 x-2 x^2\right ) \log \left (\frac {16}{60 x+3 x^2}\right )+\left (-40 x-2 x^2\right ) \log (x) \log \left (\frac {16}{60 x+3 x^2}\right )\right ) \log (1+x \log (x))}{\left (20 x+x^2\right ) \log \left (\frac {16}{60 x+3 x^2}\right )+\left (20 x^2+x^3\right ) \log (x) \log \left (\frac {16}{60 x+3 x^2}\right )} \, dx=-\log \left (x \log \left (x\right ) + 1\right )^{2} + \log \left (-4 \, \log \left (2\right ) + \log \left (3 \, x + 60\right ) + \log \left (x\right )\right ) \]
integrate((((-2*x^2-40*x)*log(16/(3*x^2+60*x))*log(x)+(-2*x^2-40*x)*log(16 /(3*x^2+60*x)))*log(x*log(x)+1)+(-2*x^2-20*x)*log(x)-2*x-20)/((x^3+20*x^2) *log(16/(3*x^2+60*x))*log(x)+(x^2+20*x)*log(16/(3*x^2+60*x))),x, algorithm =\
Time = 12.06 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90 \[ \int \frac {-20-2 x+\left (-20 x-2 x^2\right ) \log (x)+\left (\left (-40 x-2 x^2\right ) \log \left (\frac {16}{60 x+3 x^2}\right )+\left (-40 x-2 x^2\right ) \log (x) \log \left (\frac {16}{60 x+3 x^2}\right )\right ) \log (1+x \log (x))}{\left (20 x+x^2\right ) \log \left (\frac {16}{60 x+3 x^2}\right )+\left (20 x^2+x^3\right ) \log (x) \log \left (\frac {16}{60 x+3 x^2}\right )} \, dx=\ln \left (\ln \left (\frac {16}{3\,x^2+60\,x}\right )\right )-{\ln \left (x\,\ln \left (x\right )+1\right )}^2 \]