Integrand size = 118, antiderivative size = 31 \[ \int \frac {-24 x^2+3 x \log (4)+\left (-12 x^2+3 x \log (4)\right ) \log \left (\frac {x^4}{16 x^2-8 x \log (4)+\log ^2(4)}\right )}{-36 x+9 \log (4)+(-24 x+6 \log (4)) \log \left (\frac {x^4}{16 x^2-8 x \log (4)+\log ^2(4)}\right )+(-4 x+\log (4)) \log ^2\left (\frac {x^4}{16 x^2-8 x \log (4)+\log ^2(4)}\right )} \, dx=3 \left (5+\frac {x^2}{2 \left (3+\log \left (\frac {x^2}{\left (4-\frac {\log (4)}{x}\right )^2}\right )\right )}\right ) \]
Time = 0.37 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \[ \int \frac {-24 x^2+3 x \log (4)+\left (-12 x^2+3 x \log (4)\right ) \log \left (\frac {x^4}{16 x^2-8 x \log (4)+\log ^2(4)}\right )}{-36 x+9 \log (4)+(-24 x+6 \log (4)) \log \left (\frac {x^4}{16 x^2-8 x \log (4)+\log ^2(4)}\right )+(-4 x+\log (4)) \log ^2\left (\frac {x^4}{16 x^2-8 x \log (4)+\log ^2(4)}\right )} \, dx=\frac {3 x^2}{2 \left (3+\log \left (\frac {x^4}{(-4 x+\log (4))^2}\right )\right )} \]
Integrate[(-24*x^2 + 3*x*Log[4] + (-12*x^2 + 3*x*Log[4])*Log[x^4/(16*x^2 - 8*x*Log[4] + Log[4]^2)])/(-36*x + 9*Log[4] + (-24*x + 6*Log[4])*Log[x^4/( 16*x^2 - 8*x*Log[4] + Log[4]^2)] + (-4*x + Log[4])*Log[x^4/(16*x^2 - 8*x*L og[4] + Log[4]^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.
\(\displaystyle \int \frac {-24 x^2+\left (3 x \log (4)-12 x^2\right ) \log \left (\frac {x^4}{16 x^2-8 x \log (4)+\log ^2(4)}\right )+3 x \log (4)}{(\log (4)-4 x) \log ^2\left (\frac {x^4}{16 x^2-8 x \log (4)+\log ^2(4)}\right )+(6 \log (4)-24 x) \log \left (\frac {x^4}{16 x^2-8 x \log (4)+\log ^2(4)}\right )-36 x+9 \log (4)} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {3 x \left ((4 x-\log (4)) \log \left (\frac {x^4}{(\log (4)-4 x)^2}\right )+8 x-\log (4)\right )}{(4 x-\log (4)) \left (\log \left (\frac {x^4}{(\log (4)-4 x)^2}\right )+3\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 3 \int \frac {x \left (8 x+(4 x-\log (4)) \log \left (\frac {x^4}{(4 x-\log (4))^2}\right )-\log (4)\right )}{(4 x-\log (4)) \left (\log \left (\frac {x^4}{(4 x-\log (4))^2}\right )+3\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle 3 \int \left (\frac {x}{\log \left (\frac {x^4}{(\log (4)-4 x)^2}\right )+3}-\frac {2 x (2 x-\log (4))}{(4 x-\log (4)) \left (\log \left (\frac {x^4}{(\log (4)-4 x)^2}\right )+3\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \left (\frac {1}{4} \log ^2(4) \int \frac {1}{(4 x-\log (4)) \left (\log \left (\frac {x^4}{(\log (4)-4 x)^2}\right )+3\right )^2}dx+\frac {1}{4} \log (4) \int \frac {1}{\left (\log \left (\frac {x^4}{(\log (4)-4 x)^2}\right )+3\right )^2}dx-\int \frac {x}{\left (\log \left (\frac {x^4}{(\log (4)-4 x)^2}\right )+3\right )^2}dx+\int \frac {x}{\log \left (\frac {x^4}{(\log (4)-4 x)^2}\right )+3}dx\right )\) |
Int[(-24*x^2 + 3*x*Log[4] + (-12*x^2 + 3*x*Log[4])*Log[x^4/(16*x^2 - 8*x*L og[4] + Log[4]^2)])/(-36*x + 9*Log[4] + (-24*x + 6*Log[4])*Log[x^4/(16*x^2 - 8*x*Log[4] + Log[4]^2)] + (-4*x + Log[4])*Log[x^4/(16*x^2 - 8*x*Log[4] + Log[4]^2)]^2),x]
3.9.96.3.1 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]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 5.54 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06
method | result | size |
parallelrisch | \(\frac {3 x^{2}}{2 \left (\ln \left (\frac {x^{4}}{4 \ln \left (2\right )^{2}-16 x \ln \left (2\right )+16 x^{2}}\right )+3\right )}\) | \(33\) |
norman | \(\frac {3 x^{2}}{2 \left (\ln \left (\frac {x^{4}}{4 \ln \left (2\right )^{2}-16 x \ln \left (2\right )+16 x^{2}}\right )+3\right )}\) | \(34\) |
risch | \(\frac {3 x^{2}}{2 \left (\ln \left (\frac {x^{4}}{4 \ln \left (2\right )^{2}-16 x \ln \left (2\right )+16 x^{2}}\right )+3\right )}\) | \(34\) |
int(((6*x*ln(2)-12*x^2)*ln(x^4/(4*ln(2)^2-16*x*ln(2)+16*x^2))+6*x*ln(2)-24 *x^2)/((2*ln(2)-4*x)*ln(x^4/(4*ln(2)^2-16*x*ln(2)+16*x^2))^2+(12*ln(2)-24* x)*ln(x^4/(4*ln(2)^2-16*x*ln(2)+16*x^2))+18*ln(2)-36*x),x,method=_RETURNVE RBOSE)
Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {-24 x^2+3 x \log (4)+\left (-12 x^2+3 x \log (4)\right ) \log \left (\frac {x^4}{16 x^2-8 x \log (4)+\log ^2(4)}\right )}{-36 x+9 \log (4)+(-24 x+6 \log (4)) \log \left (\frac {x^4}{16 x^2-8 x \log (4)+\log ^2(4)}\right )+(-4 x+\log (4)) \log ^2\left (\frac {x^4}{16 x^2-8 x \log (4)+\log ^2(4)}\right )} \, dx=\frac {3 \, x^{2}}{2 \, {\left (\log \left (\frac {x^{4}}{4 \, {\left (4 \, x^{2} - 4 \, x \log \left (2\right ) + \log \left (2\right )^{2}\right )}}\right ) + 3\right )}} \]
integrate(((6*x*log(2)-12*x^2)*log(x^4/(4*log(2)^2-16*x*log(2)+16*x^2))+6* x*log(2)-24*x^2)/((2*log(2)-4*x)*log(x^4/(4*log(2)^2-16*x*log(2)+16*x^2))^ 2+(12*log(2)-24*x)*log(x^4/(4*log(2)^2-16*x*log(2)+16*x^2))+18*log(2)-36*x ),x, algorithm=\
Time = 0.10 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {-24 x^2+3 x \log (4)+\left (-12 x^2+3 x \log (4)\right ) \log \left (\frac {x^4}{16 x^2-8 x \log (4)+\log ^2(4)}\right )}{-36 x+9 \log (4)+(-24 x+6 \log (4)) \log \left (\frac {x^4}{16 x^2-8 x \log (4)+\log ^2(4)}\right )+(-4 x+\log (4)) \log ^2\left (\frac {x^4}{16 x^2-8 x \log (4)+\log ^2(4)}\right )} \, dx=\frac {3 x^{2}}{2 \log {\left (\frac {x^{4}}{16 x^{2} - 16 x \log {\left (2 \right )} + 4 \log {\left (2 \right )}^{2}} \right )} + 6} \]
integrate(((6*x*ln(2)-12*x**2)*ln(x**4/(4*ln(2)**2-16*x*ln(2)+16*x**2))+6* x*ln(2)-24*x**2)/((2*ln(2)-4*x)*ln(x**4/(4*ln(2)**2-16*x*ln(2)+16*x**2))** 2+(12*ln(2)-24*x)*ln(x**4/(4*ln(2)**2-16*x*ln(2)+16*x**2))+18*ln(2)-36*x), x)
Time = 0.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \[ \int \frac {-24 x^2+3 x \log (4)+\left (-12 x^2+3 x \log (4)\right ) \log \left (\frac {x^4}{16 x^2-8 x \log (4)+\log ^2(4)}\right )}{-36 x+9 \log (4)+(-24 x+6 \log (4)) \log \left (\frac {x^4}{16 x^2-8 x \log (4)+\log ^2(4)}\right )+(-4 x+\log (4)) \log ^2\left (\frac {x^4}{16 x^2-8 x \log (4)+\log ^2(4)}\right )} \, dx=-\frac {3 \, x^{2}}{2 \, {\left (2 \, \log \left (2\right ) + 2 \, \log \left (2 \, x - \log \left (2\right )\right ) - 4 \, \log \left (x\right ) - 3\right )}} \]
integrate(((6*x*log(2)-12*x^2)*log(x^4/(4*log(2)^2-16*x*log(2)+16*x^2))+6* x*log(2)-24*x^2)/((2*log(2)-4*x)*log(x^4/(4*log(2)^2-16*x*log(2)+16*x^2))^ 2+(12*log(2)-24*x)*log(x^4/(4*log(2)^2-16*x*log(2)+16*x^2))+18*log(2)-36*x ),x, algorithm=\
Time = 0.44 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13 \[ \int \frac {-24 x^2+3 x \log (4)+\left (-12 x^2+3 x \log (4)\right ) \log \left (\frac {x^4}{16 x^2-8 x \log (4)+\log ^2(4)}\right )}{-36 x+9 \log (4)+(-24 x+6 \log (4)) \log \left (\frac {x^4}{16 x^2-8 x \log (4)+\log ^2(4)}\right )+(-4 x+\log (4)) \log ^2\left (\frac {x^4}{16 x^2-8 x \log (4)+\log ^2(4)}\right )} \, dx=-\frac {3 \, x^{2}}{2 \, {\left (2 \, \log \left (2\right ) - \log \left (x^{4}\right ) + \log \left (4 \, x^{2} - 4 \, x \log \left (2\right ) + \log \left (2\right )^{2}\right ) - 3\right )}} \]
integrate(((6*x*log(2)-12*x^2)*log(x^4/(4*log(2)^2-16*x*log(2)+16*x^2))+6* x*log(2)-24*x^2)/((2*log(2)-4*x)*log(x^4/(4*log(2)^2-16*x*log(2)+16*x^2))^ 2+(12*log(2)-24*x)*log(x^4/(4*log(2)^2-16*x*log(2)+16*x^2))+18*log(2)-36*x ),x, algorithm=\
Timed out. \[ \int \frac {-24 x^2+3 x \log (4)+\left (-12 x^2+3 x \log (4)\right ) \log \left (\frac {x^4}{16 x^2-8 x \log (4)+\log ^2(4)}\right )}{-36 x+9 \log (4)+(-24 x+6 \log (4)) \log \left (\frac {x^4}{16 x^2-8 x \log (4)+\log ^2(4)}\right )+(-4 x+\log (4)) \log ^2\left (\frac {x^4}{16 x^2-8 x \log (4)+\log ^2(4)}\right )} \, dx=-\int \frac {\ln \left (\frac {x^4}{16\,x^2-16\,\ln \left (2\right )\,x+4\,{\ln \left (2\right )}^2}\right )\,\left (6\,x\,\ln \left (2\right )-12\,x^2\right )+6\,x\,\ln \left (2\right )-24\,x^2}{\left (4\,x-2\,\ln \left (2\right )\right )\,{\ln \left (\frac {x^4}{16\,x^2-16\,\ln \left (2\right )\,x+4\,{\ln \left (2\right )}^2}\right )}^2+\left (24\,x-12\,\ln \left (2\right )\right )\,\ln \left (\frac {x^4}{16\,x^2-16\,\ln \left (2\right )\,x+4\,{\ln \left (2\right )}^2}\right )+36\,x-18\,\ln \left (2\right )} \,d x \]
int(-(log(x^4/(4*log(2)^2 - 16*x*log(2) + 16*x^2))*(6*x*log(2) - 12*x^2) + 6*x*log(2) - 24*x^2)/(36*x - 18*log(2) + log(x^4/(4*log(2)^2 - 16*x*log(2 ) + 16*x^2))*(24*x - 12*log(2)) + log(x^4/(4*log(2)^2 - 16*x*log(2) + 16*x ^2))^2*(4*x - 2*log(2))),x)