Integrand size = 93, antiderivative size = 27 \[ \int \frac {-6144 x^3+24 x^4+\left (-1572894 x^2+12288 x^3-24 x^4\right ) \log \left (262149-2048 x+4 x^2\right )+\left (-262149+2048 x-4 x^2\right ) \log ^2\left (262149-2048 x+4 x^2\right )}{\left (262149 x-2048 x^2+4 x^3\right ) \log ^2\left (262149-2048 x+4 x^2\right )} \, dx=e^3-\frac {3 x^2}{\log \left (5+4 (256-x)^2\right )}-\log (x) \]
Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {-6144 x^3+24 x^4+\left (-1572894 x^2+12288 x^3-24 x^4\right ) \log \left (262149-2048 x+4 x^2\right )+\left (-262149+2048 x-4 x^2\right ) \log ^2\left (262149-2048 x+4 x^2\right )}{\left (262149 x-2048 x^2+4 x^3\right ) \log ^2\left (262149-2048 x+4 x^2\right )} \, dx=-\log (x)-\frac {3 x^2}{\log \left (262149-2048 x+4 x^2\right )} \]
Integrate[(-6144*x^3 + 24*x^4 + (-1572894*x^2 + 12288*x^3 - 24*x^4)*Log[26 2149 - 2048*x + 4*x^2] + (-262149 + 2048*x - 4*x^2)*Log[262149 - 2048*x + 4*x^2]^2)/((262149*x - 2048*x^2 + 4*x^3)*Log[262149 - 2048*x + 4*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.
\(\displaystyle \int \frac {24 x^4-6144 x^3+\left (-4 x^2+2048 x-262149\right ) \log ^2\left (4 x^2-2048 x+262149\right )+\left (-24 x^4+12288 x^3-1572894 x^2\right ) \log \left (4 x^2-2048 x+262149\right )}{\left (4 x^3-2048 x^2+262149 x\right ) \log ^2\left (4 x^2-2048 x+262149\right )} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {24 x^4-6144 x^3+\left (-4 x^2+2048 x-262149\right ) \log ^2\left (4 x^2-2048 x+262149\right )+\left (-24 x^4+12288 x^3-1572894 x^2\right ) \log \left (4 x^2-2048 x+262149\right )}{x \left (4 x^2-2048 x+262149\right ) \log ^2\left (4 x^2-2048 x+262149\right )}dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \left (\frac {24 (x-256) x^2}{\left (4 x^2-2048 x+262149\right ) \log ^2\left (4 x^2-2048 x+262149\right )}-\frac {6 x}{\log \left (4 x^2-2048 x+262149\right )}-\frac {1}{x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 1536 \int \frac {1}{\log ^2\left (4 x^2-2048 x+262149\right )}dx-\frac {805321728 i \int \frac {1}{\left (-8 x+4 i \sqrt {5}+2048\right ) \log ^2\left (4 x^2-2048 x+262149\right )}dx}{\sqrt {5}}+6 \int \frac {x}{\log ^2\left (4 x^2-2048 x+262149\right )}dx+\frac {1572834}{5} \left (5-512 i \sqrt {5}\right ) \int \frac {1}{\left (8 x-4 i \sqrt {5}-2048\right ) \log ^2\left (4 x^2-2048 x+262149\right )}dx+\frac {1572834}{5} \left (5+512 i \sqrt {5}\right ) \int \frac {1}{\left (8 x+4 i \sqrt {5}-2048\right ) \log ^2\left (4 x^2-2048 x+262149\right )}dx-\frac {805321728 i \int \frac {1}{\left (8 x+4 i \sqrt {5}-2048\right ) \log ^2\left (4 x^2-2048 x+262149\right )}dx}{\sqrt {5}}-6 \int \frac {x}{\log \left (4 x^2-2048 x+262149\right )}dx-\log (x)\) |
Int[(-6144*x^3 + 24*x^4 + (-1572894*x^2 + 12288*x^3 - 24*x^4)*Log[262149 - 2048*x + 4*x^2] + (-262149 + 2048*x - 4*x^2)*Log[262149 - 2048*x + 4*x^2] ^2)/((262149*x - 2048*x^2 + 4*x^3)*Log[262149 - 2048*x + 4*x^2]^2),x]
3.16.93.3.1 Defintions of rubi rules used
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.42 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89
method | result | size |
norman | \(-\frac {3 x^{2}}{\ln \left (4 x^{2}-2048 x +262149\right )}-\ln \left (x \right )\) | \(24\) |
risch | \(-\frac {3 x^{2}}{\ln \left (4 x^{2}-2048 x +262149\right )}-\ln \left (x \right )\) | \(24\) |
parallelrisch | \(-\frac {16 \ln \left (x \right ) \ln \left (4 x^{2}-2048 x +262149\right )+48 x^{2}}{16 \ln \left (4 x^{2}-2048 x +262149\right )}\) | \(37\) |
int(((-4*x^2+2048*x-262149)*ln(4*x^2-2048*x+262149)^2+(-24*x^4+12288*x^3-1 572894*x^2)*ln(4*x^2-2048*x+262149)+24*x^4-6144*x^3)/(4*x^3-2048*x^2+26214 9*x)/ln(4*x^2-2048*x+262149)^2,x,method=_RETURNVERBOSE)
Time = 0.23 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.30 \[ \int \frac {-6144 x^3+24 x^4+\left (-1572894 x^2+12288 x^3-24 x^4\right ) \log \left (262149-2048 x+4 x^2\right )+\left (-262149+2048 x-4 x^2\right ) \log ^2\left (262149-2048 x+4 x^2\right )}{\left (262149 x-2048 x^2+4 x^3\right ) \log ^2\left (262149-2048 x+4 x^2\right )} \, dx=-\frac {3 \, x^{2} + \log \left (4 \, x^{2} - 2048 \, x + 262149\right ) \log \left (x\right )}{\log \left (4 \, x^{2} - 2048 \, x + 262149\right )} \]
integrate(((-4*x^2+2048*x-262149)*log(4*x^2-2048*x+262149)^2+(-24*x^4+1228 8*x^3-1572894*x^2)*log(4*x^2-2048*x+262149)+24*x^4-6144*x^3)/(4*x^3-2048*x ^2+262149*x)/log(4*x^2-2048*x+262149)^2,x, algorithm=\
Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {-6144 x^3+24 x^4+\left (-1572894 x^2+12288 x^3-24 x^4\right ) \log \left (262149-2048 x+4 x^2\right )+\left (-262149+2048 x-4 x^2\right ) \log ^2\left (262149-2048 x+4 x^2\right )}{\left (262149 x-2048 x^2+4 x^3\right ) \log ^2\left (262149-2048 x+4 x^2\right )} \, dx=- \frac {3 x^{2}}{\log {\left (4 x^{2} - 2048 x + 262149 \right )}} - \log {\left (x \right )} \]
integrate(((-4*x**2+2048*x-262149)*ln(4*x**2-2048*x+262149)**2+(-24*x**4+1 2288*x**3-1572894*x**2)*ln(4*x**2-2048*x+262149)+24*x**4-6144*x**3)/(4*x** 3-2048*x**2+262149*x)/ln(4*x**2-2048*x+262149)**2,x)
Time = 0.22 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {-6144 x^3+24 x^4+\left (-1572894 x^2+12288 x^3-24 x^4\right ) \log \left (262149-2048 x+4 x^2\right )+\left (-262149+2048 x-4 x^2\right ) \log ^2\left (262149-2048 x+4 x^2\right )}{\left (262149 x-2048 x^2+4 x^3\right ) \log ^2\left (262149-2048 x+4 x^2\right )} \, dx=-\frac {3 \, x^{2}}{\log \left (4 \, x^{2} - 2048 \, x + 262149\right )} - \log \left (x\right ) \]
integrate(((-4*x^2+2048*x-262149)*log(4*x^2-2048*x+262149)^2+(-24*x^4+1228 8*x^3-1572894*x^2)*log(4*x^2-2048*x+262149)+24*x^4-6144*x^3)/(4*x^3-2048*x ^2+262149*x)/log(4*x^2-2048*x+262149)^2,x, algorithm=\
Time = 0.31 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {-6144 x^3+24 x^4+\left (-1572894 x^2+12288 x^3-24 x^4\right ) \log \left (262149-2048 x+4 x^2\right )+\left (-262149+2048 x-4 x^2\right ) \log ^2\left (262149-2048 x+4 x^2\right )}{\left (262149 x-2048 x^2+4 x^3\right ) \log ^2\left (262149-2048 x+4 x^2\right )} \, dx=-\frac {3 \, x^{2}}{\log \left (4 \, x^{2} - 2048 \, x + 262149\right )} - \log \left (x\right ) \]
integrate(((-4*x^2+2048*x-262149)*log(4*x^2-2048*x+262149)^2+(-24*x^4+1228 8*x^3-1572894*x^2)*log(4*x^2-2048*x+262149)+24*x^4-6144*x^3)/(4*x^3-2048*x ^2+262149*x)/log(4*x^2-2048*x+262149)^2,x, algorithm=\
Time = 13.35 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.67 \[ \int \frac {-6144 x^3+24 x^4+\left (-1572894 x^2+12288 x^3-24 x^4\right ) \log \left (262149-2048 x+4 x^2\right )+\left (-262149+2048 x-4 x^2\right ) \log ^2\left (262149-2048 x+4 x^2\right )}{\left (262149 x-2048 x^2+4 x^3\right ) \log ^2\left (262149-2048 x+4 x^2\right )} \, dx=768\,x-\ln \left (x\right )-\frac {960}{x-256}-\frac {3\,x^2-\frac {3\,x\,\ln \left (4\,x^2-2048\,x+262149\right )\,\left (4\,x^2-2048\,x+262149\right )}{4\,\left (x-256\right )}}{\ln \left (4\,x^2-2048\,x+262149\right )}-3\,x^2 \]
int(-(log(4*x^2 - 2048*x + 262149)^2*(4*x^2 - 2048*x + 262149) + log(4*x^2 - 2048*x + 262149)*(1572894*x^2 - 12288*x^3 + 24*x^4) + 6144*x^3 - 24*x^4 )/(log(4*x^2 - 2048*x + 262149)^2*(262149*x - 2048*x^2 + 4*x^3)),x)