Integrand size = 76, antiderivative size = 25 \[ \int \frac {256-1280 x^2+(-256-2560 x) \log (x)-1280 \log ^2(x)}{\left (x^2+5 x^3+\left (x+10 x^2\right ) \log (x)+5 x \log ^2(x)\right ) \log ^2\left (\frac {x+5 x^2+5 x \log (x)}{2 x+2 \log (x)}\right )} \, dx=2+\log (2)+\frac {256}{\log \left (\frac {1}{2} \left (5 x+\frac {x}{x+\log (x)}\right )\right )} \]
Time = 0.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {256-1280 x^2+(-256-2560 x) \log (x)-1280 \log ^2(x)}{\left (x^2+5 x^3+\left (x+10 x^2\right ) \log (x)+5 x \log ^2(x)\right ) \log ^2\left (\frac {x+5 x^2+5 x \log (x)}{2 x+2 \log (x)}\right )} \, dx=\frac {256}{\log \left (\frac {x (1+5 x+5 \log (x))}{2 (x+\log (x))}\right )} \]
Integrate[(256 - 1280*x^2 + (-256 - 2560*x)*Log[x] - 1280*Log[x]^2)/((x^2 + 5*x^3 + (x + 10*x^2)*Log[x] + 5*x*Log[x]^2)*Log[(x + 5*x^2 + 5*x*Log[x]) /(2*x + 2*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.
| \(\displaystyle \int \frac {-1280 x^2-1280 \log ^2(x)+(-2560 x-256) \log (x)+256}{\left (5 x^3+x^2+\left (10 x^2+x\right ) \log (x)+5 x \log ^2(x)\right ) \log ^2\left (\frac {5 x^2+x+5 x \log (x)}{2 x+2 \log (x)}\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {256 \left (-5 x^2-5 \log ^2(x)-10 x \log (x)-\log (x)+1\right )}{\left (5 x^3+x^2+\left (10 x^2+x\right ) \log (x)+5 x \log ^2(x)\right ) \log ^2\left (\frac {5 x^2+x+5 x \log (x)}{2 (x+\log (x))}\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 256 \int \frac {-5 x^2-10 \log (x) x-5 \log ^2(x)-\log (x)+1}{\left (5 x^3+x^2+5 \log ^2(x) x+\left (10 x^2+x\right ) \log (x)\right ) \log ^2\left (\frac {5 x^2+5 \log (x) x+x}{2 (x+\log (x))}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 256 \int \left (-\frac {5 \log ^2(x)}{x \left (5 x^2+10 \log (x) x+x+5 \log ^2(x)+\log (x)\right ) \log ^2\left (\frac {x (5 x+5 \log (x)+1)}{2 (x+\log (x))}\right )}-\frac {\log (x)}{x \left (5 x^2+10 \log (x) x+x+5 \log ^2(x)+\log (x)\right ) \log ^2\left (\frac {x (5 x+5 \log (x)+1)}{2 (x+\log (x))}\right )}-\frac {10 \log (x)}{\left (5 x^2+10 \log (x) x+x+5 \log ^2(x)+\log (x)\right ) \log ^2\left (\frac {x (5 x+5 \log (x)+1)}{2 (x+\log (x))}\right )}-\frac {5 x}{\left (5 x^2+10 \log (x) x+x+5 \log ^2(x)+\log (x)\right ) \log ^2\left (\frac {x (5 x+5 \log (x)+1)}{2 (x+\log (x))}\right )}+\frac {1}{x \left (5 x^2+10 \log (x) x+x+5 \log ^2(x)+\log (x)\right ) \log ^2\left (\frac {x (5 x+5 \log (x)+1)}{2 (x+\log (x))}\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 256 \left (\int \frac {1}{x \left (5 x^2+10 \log (x) x+x+5 \log ^2(x)+\log (x)\right ) \log ^2\left (\frac {x (5 x+5 \log (x)+1)}{2 (x+\log (x))}\right )}dx-5 \int \frac {x}{\left (5 x^2+10 \log (x) x+x+5 \log ^2(x)+\log (x)\right ) \log ^2\left (\frac {x (5 x+5 \log (x)+1)}{2 (x+\log (x))}\right )}dx-10 \int \frac {\log (x)}{\left (5 x^2+10 \log (x) x+x+5 \log ^2(x)+\log (x)\right ) \log ^2\left (\frac {x (5 x+5 \log (x)+1)}{2 (x+\log (x))}\right )}dx-\int \frac {\log (x)}{x \left (5 x^2+10 \log (x) x+x+5 \log ^2(x)+\log (x)\right ) \log ^2\left (\frac {x (5 x+5 \log (x)+1)}{2 (x+\log (x))}\right )}dx-5 \int \frac {\log ^2(x)}{x \left (5 x^2+10 \log (x) x+x+5 \log ^2(x)+\log (x)\right ) \log ^2\left (\frac {x (5 x+5 \log (x)+1)}{2 (x+\log (x))}\right )}dx\right )\) |
Int[(256 - 1280*x^2 + (-256 - 2560*x)*Log[x] - 1280*Log[x]^2)/((x^2 + 5*x^ 3 + (x + 10*x^2)*Log[x] + 5*x*Log[x]^2)*Log[(x + 5*x^2 + 5*x*Log[x])/(2*x + 2*Log[x])]^2),x]
3.8.9.3.1 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]]
Time = 1.64 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96
| method | result | size |
| parallelrisch | \(\frac {256}{\ln \left (\frac {x \left (5 \ln \left (x \right )+5 x +1\right )}{2 x +2 \ln \left (x \right )}\right )}\) | \(24\) |
| default | \(\frac {512 i}{\pi \,\operatorname {csgn}\left (\frac {i}{x +\ln \left (x \right )}\right ) \operatorname {csgn}\left (i \left (\ln \left (x \right )+x +\frac {1}{5}\right )\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )+x +\frac {1}{5}\right )}{x +\ln \left (x \right )}\right )-\pi \,\operatorname {csgn}\left (\frac {i}{x +\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )+x +\frac {1}{5}\right )}{x +\ln \left (x \right )}\right )^{2}-\pi \,\operatorname {csgn}\left (i \left (\ln \left (x \right )+x +\frac {1}{5}\right )\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )+x +\frac {1}{5}\right )}{x +\ln \left (x \right )}\right )^{2}+\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )+x +\frac {1}{5}\right )}{x +\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i x \left (\ln \left (x \right )+x +\frac {1}{5}\right )}{x +\ln \left (x \right )}\right )-\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i x \left (\ln \left (x \right )+x +\frac {1}{5}\right )}{x +\ln \left (x \right )}\right )^{2}+\pi \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )+x +\frac {1}{5}\right )}{x +\ln \left (x \right )}\right )^{3}-\pi \,\operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )+x +\frac {1}{5}\right )}{x +\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i x \left (\ln \left (x \right )+x +\frac {1}{5}\right )}{x +\ln \left (x \right )}\right )^{2}+\pi \operatorname {csgn}\left (\frac {i x \left (\ln \left (x \right )+x +\frac {1}{5}\right )}{x +\ln \left (x \right )}\right )^{3}+2 i \ln \left (x \right )+2 i \ln \left (\ln \left (x \right )+x +\frac {1}{5}\right )-2 i \ln \left (x +\ln \left (x \right )\right )-2 i \ln \left (2\right )+2 i \ln \left (5\right )}\) | \(273\) |
| risch | \(\frac {512 i}{\pi \,\operatorname {csgn}\left (\frac {i}{x +\ln \left (x \right )}\right ) \operatorname {csgn}\left (i \left (\ln \left (x \right )+x +\frac {1}{5}\right )\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )+x +\frac {1}{5}\right )}{x +\ln \left (x \right )}\right )-\pi \,\operatorname {csgn}\left (\frac {i}{x +\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )+x +\frac {1}{5}\right )}{x +\ln \left (x \right )}\right )^{2}-\pi \,\operatorname {csgn}\left (i \left (\ln \left (x \right )+x +\frac {1}{5}\right )\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )+x +\frac {1}{5}\right )}{x +\ln \left (x \right )}\right )^{2}+\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )+x +\frac {1}{5}\right )}{x +\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i x \left (\ln \left (x \right )+x +\frac {1}{5}\right )}{x +\ln \left (x \right )}\right )-\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i x \left (\ln \left (x \right )+x +\frac {1}{5}\right )}{x +\ln \left (x \right )}\right )^{2}+\pi \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )+x +\frac {1}{5}\right )}{x +\ln \left (x \right )}\right )^{3}-\pi \,\operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )+x +\frac {1}{5}\right )}{x +\ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i x \left (\ln \left (x \right )+x +\frac {1}{5}\right )}{x +\ln \left (x \right )}\right )^{2}+\pi \operatorname {csgn}\left (\frac {i x \left (\ln \left (x \right )+x +\frac {1}{5}\right )}{x +\ln \left (x \right )}\right )^{3}+2 i \ln \left (x \right )+2 i \ln \left (\ln \left (x \right )+x +\frac {1}{5}\right )-2 i \ln \left (x +\ln \left (x \right )\right )-2 i \ln \left (2\right )+2 i \ln \left (5\right )}\) | \(273\) |
int((-1280*ln(x)^2+(-2560*x-256)*ln(x)-1280*x^2+256)/(5*x*ln(x)^2+(10*x^2+ x)*ln(x)+5*x^3+x^2)/ln((5*x*ln(x)+5*x^2+x)/(2*x+2*ln(x)))^2,x,method=_RETU RNVERBOSE)
Time = 0.34 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {256-1280 x^2+(-256-2560 x) \log (x)-1280 \log ^2(x)}{\left (x^2+5 x^3+\left (x+10 x^2\right ) \log (x)+5 x \log ^2(x)\right ) \log ^2\left (\frac {x+5 x^2+5 x \log (x)}{2 x+2 \log (x)}\right )} \, dx=\frac {256}{\log \left (\frac {5 \, x^{2} + 5 \, x \log \left (x\right ) + x}{2 \, {\left (x + \log \left (x\right )\right )}}\right )} \]
integrate((-1280*log(x)^2+(-2560*x-256)*log(x)-1280*x^2+256)/(5*x*log(x)^2 +(10*x^2+x)*log(x)+5*x^3+x^2)/log((5*x*log(x)+5*x^2+x)/(2*x+2*log(x)))^2,x , algorithm=\
Time = 0.16 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {256-1280 x^2+(-256-2560 x) \log (x)-1280 \log ^2(x)}{\left (x^2+5 x^3+\left (x+10 x^2\right ) \log (x)+5 x \log ^2(x)\right ) \log ^2\left (\frac {x+5 x^2+5 x \log (x)}{2 x+2 \log (x)}\right )} \, dx=\frac {256}{\log {\left (\frac {5 x^{2} + 5 x \log {\left (x \right )} + x}{2 x + 2 \log {\left (x \right )}} \right )}} \]
integrate((-1280*ln(x)**2+(-2560*x-256)*ln(x)-1280*x**2+256)/(5*x*ln(x)**2 +(10*x**2+x)*ln(x)+5*x**3+x**2)/ln((5*x*ln(x)+5*x**2+x)/(2*x+2*ln(x)))**2, x)
Time = 0.33 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {256-1280 x^2+(-256-2560 x) \log (x)-1280 \log ^2(x)}{\left (x^2+5 x^3+\left (x+10 x^2\right ) \log (x)+5 x \log ^2(x)\right ) \log ^2\left (\frac {x+5 x^2+5 x \log (x)}{2 x+2 \log (x)}\right )} \, dx=-\frac {256}{\log \left (2\right ) - \log \left (5 \, x + 5 \, \log \left (x\right ) + 1\right ) + \log \left (x + \log \left (x\right )\right ) - \log \left (x\right )} \]
integrate((-1280*log(x)^2+(-2560*x-256)*log(x)-1280*x^2+256)/(5*x*log(x)^2 +(10*x^2+x)*log(x)+5*x^3+x^2)/log((5*x*log(x)+5*x^2+x)/(2*x+2*log(x)))^2,x , algorithm=\
Time = 0.36 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {256-1280 x^2+(-256-2560 x) \log (x)-1280 \log ^2(x)}{\left (x^2+5 x^3+\left (x+10 x^2\right ) \log (x)+5 x \log ^2(x)\right ) \log ^2\left (\frac {x+5 x^2+5 x \log (x)}{2 x+2 \log (x)}\right )} \, dx=\frac {256}{\log \left (5 \, x + 5 \, \log \left (x\right ) + 1\right ) - \log \left (2 \, x + 2 \, \log \left (x\right )\right ) + \log \left (x\right )} \]
integrate((-1280*log(x)^2+(-2560*x-256)*log(x)-1280*x^2+256)/(5*x*log(x)^2 +(10*x^2+x)*log(x)+5*x^3+x^2)/log((5*x*log(x)+5*x^2+x)/(2*x+2*log(x)))^2,x , algorithm=\
Time = 12.88 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {256-1280 x^2+(-256-2560 x) \log (x)-1280 \log ^2(x)}{\left (x^2+5 x^3+\left (x+10 x^2\right ) \log (x)+5 x \log ^2(x)\right ) \log ^2\left (\frac {x+5 x^2+5 x \log (x)}{2 x+2 \log (x)}\right )} \, dx=\frac {256}{\ln \left (\frac {x+5\,x\,\ln \left (x\right )+5\,x^2}{2\,x+2\,\ln \left (x\right )}\right )} \]