Integrand size = 71, antiderivative size = 14 \[ \int \frac {2 x^2+8 x \log (\log (50))+\left (-4+x^2+8 x \log (\log (50))+16 \log ^2(\log (50))\right ) \log \left (-4+x^2+8 x \log (\log (50))+16 \log ^2(\log (50))\right )}{-4+x^2+8 x \log (\log (50))+16 \log ^2(\log (50))} \, dx=x \log \left (-4+(x+4 \log (\log (50)))^2\right ) \] Output:
x*ln((4*ln(ln(50))+x)^2-4)
Time = 0.04 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.50 \[ \int \frac {2 x^2+8 x \log (\log (50))+\left (-4+x^2+8 x \log (\log (50))+16 \log ^2(\log (50))\right ) \log \left (-4+x^2+8 x \log (\log (50))+16 \log ^2(\log (50))\right )}{-4+x^2+8 x \log (\log (50))+16 \log ^2(\log (50))} \, dx=x \log \left (-4+x^2+8 x \log (\log (50))+16 \log ^2(\log (50))\right ) \] Input:
Integrate[(2*x^2 + 8*x*Log[Log[50]] + (-4 + x^2 + 8*x*Log[Log[50]] + 16*Lo g[Log[50]]^2)*Log[-4 + x^2 + 8*x*Log[Log[50]] + 16*Log[Log[50]]^2])/(-4 + x^2 + 8*x*Log[Log[50]] + 16*Log[Log[50]]^2),x]
Output:
x*Log[-4 + x^2 + 8*x*Log[Log[50]] + 16*Log[Log[50]]^2]
Time = 0.42 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.71, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {7279, 2009}
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.
\(\displaystyle \int \frac {2 x^2+\left (x^2+8 x \log (\log (50))-4+16 \log ^2(\log (50))\right ) \log \left (x^2+8 x \log (\log (50))-4+16 \log ^2(\log (50))\right )+8 x \log (\log (50))}{x^2+8 x \log (\log (50))-4+16 \log ^2(\log (50))} \, dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (\log \left (x^2+8 x \log (\log (50))-4 \left (1-4 \log ^2(\log (50))\right )\right )+\frac {2 x (x+4 \log (\log (50)))}{(x-2+4 \log (\log (50))) (x+2+4 \log (\log (50)))}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x \log \left (x^2+8 x \log (\log (50))-4 \left (1-4 \log ^2(\log (50))\right )\right )\) |
Input:
Int[(2*x^2 + 8*x*Log[Log[50]] + (-4 + x^2 + 8*x*Log[Log[50]] + 16*Log[Log[ 50]]^2)*Log[-4 + x^2 + 8*x*Log[Log[50]] + 16*Log[Log[50]]^2])/(-4 + x^2 + 8*x*Log[Log[50]] + 16*Log[Log[50]]^2),x]
Output:
x*Log[x^2 + 8*x*Log[Log[50]] - 4*(1 - 4*Log[Log[50]]^2)]
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 2.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.57
method | result | size |
default | \(x \ln \left (16 \ln \left (\ln \left (50\right )\right )^{2}+8 x \ln \left (\ln \left (50\right )\right )+x^{2}-4\right )\) | \(22\) |
norman | \(x \ln \left (16 \ln \left (\ln \left (50\right )\right )^{2}+8 x \ln \left (\ln \left (50\right )\right )+x^{2}-4\right )\) | \(22\) |
parallelrisch | \(x \ln \left (16 \ln \left (\ln \left (50\right )\right )^{2}+8 x \ln \left (\ln \left (50\right )\right )+x^{2}-4\right )\) | \(22\) |
parts | \(x \ln \left (16 \ln \left (\ln \left (50\right )\right )^{2}+8 x \ln \left (\ln \left (50\right )\right )+x^{2}-4\right )\) | \(22\) |
risch | \(\ln \left (16 \ln \left (\ln \left (2\right )+2 \ln \left (5\right )\right )^{2}+8 x \ln \left (\ln \left (2\right )+2 \ln \left (5\right )\right )+x^{2}-4\right ) x\) | \(32\) |
Input:
int(((16*ln(ln(50))^2+8*x*ln(ln(50))+x^2-4)*ln(16*ln(ln(50))^2+8*x*ln(ln(5 0))+x^2-4)+8*x*ln(ln(50))+2*x^2)/(16*ln(ln(50))^2+8*x*ln(ln(50))+x^2-4),x, method=_RETURNVERBOSE)
Output:
x*ln(16*ln(ln(50))^2+8*x*ln(ln(50))+x^2-4)
Time = 0.08 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.50 \[ \int \frac {2 x^2+8 x \log (\log (50))+\left (-4+x^2+8 x \log (\log (50))+16 \log ^2(\log (50))\right ) \log \left (-4+x^2+8 x \log (\log (50))+16 \log ^2(\log (50))\right )}{-4+x^2+8 x \log (\log (50))+16 \log ^2(\log (50))} \, dx=x \log \left (x^{2} + 8 \, x \log \left (\log \left (50\right )\right ) + 16 \, \log \left (\log \left (50\right )\right )^{2} - 4\right ) \] Input:
integrate(((16*log(log(50))^2+8*x*log(log(50))+x^2-4)*log(16*log(log(50))^ 2+8*x*log(log(50))+x^2-4)+8*x*log(log(50))+2*x^2)/(16*log(log(50))^2+8*x*l og(log(50))+x^2-4),x, algorithm="fricas")
Output:
x*log(x^2 + 8*x*log(log(50)) + 16*log(log(50))^2 - 4)
Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.71 \[ \int \frac {2 x^2+8 x \log (\log (50))+\left (-4+x^2+8 x \log (\log (50))+16 \log ^2(\log (50))\right ) \log \left (-4+x^2+8 x \log (\log (50))+16 \log ^2(\log (50))\right )}{-4+x^2+8 x \log (\log (50))+16 \log ^2(\log (50))} \, dx=x \log {\left (x^{2} + 8 x \log {\left (\log {\left (50 \right )} \right )} - 4 + 16 \log {\left (\log {\left (50 \right )} \right )}^{2} \right )} \] Input:
integrate(((16*ln(ln(50))**2+8*x*ln(ln(50))+x**2-4)*ln(16*ln(ln(50))**2+8* x*ln(ln(50))+x**2-4)+8*x*ln(ln(50))+2*x**2)/(16*ln(ln(50))**2+8*x*ln(ln(50 ))+x**2-4),x)
Output:
x*log(x**2 + 8*x*log(log(50)) - 4 + 16*log(log(50))**2)
Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (14) = 28\).
Time = 0.16 (sec) , antiderivative size = 148, normalized size of antiderivative = 10.57 \[ \int \frac {2 x^2+8 x \log (\log (50))+\left (-4+x^2+8 x \log (\log (50))+16 \log ^2(\log (50))\right ) \log \left (-4+x^2+8 x \log (\log (50))+16 \log ^2(\log (50))\right )}{-4+x^2+8 x \log (\log (50))+16 \log ^2(\log (50))} \, dx={\left (x + 4 \, \log \left (2 \, \log \left (5\right ) + \log \left (2\right )\right ) + 2\right )} \log \left (x + 4 \, \log \left (2 \, \log \left (5\right ) + \log \left (2\right )\right ) + 2\right ) + {\left (x + 4 \, \log \left (2 \, \log \left (5\right ) + \log \left (2\right )\right ) - 2\right )} \log \left (x + 4 \, \log \left (2 \, \log \left (5\right ) + \log \left (2\right )\right ) - 2\right ) - 2 \, {\left (4 \, \log \left (\log \left (50\right )\right )^{2} + 4 \, \log \left (\log \left (50\right )\right ) + 1\right )} \log \left (x + 4 \, \log \left (\log \left (50\right )\right ) + 2\right ) + 2 \, {\left (4 \, \log \left (\log \left (50\right )\right )^{2} - 4 \, \log \left (\log \left (50\right )\right ) + 1\right )} \log \left (x + 4 \, \log \left (\log \left (50\right )\right ) - 2\right ) + 4 \, {\left ({\left (2 \, \log \left (\log \left (50\right )\right ) + 1\right )} \log \left (x + 4 \, \log \left (\log \left (50\right )\right ) + 2\right ) - {\left (2 \, \log \left (\log \left (50\right )\right ) - 1\right )} \log \left (x + 4 \, \log \left (\log \left (50\right )\right ) - 2\right )\right )} \log \left (\log \left (50\right )\right ) \] Input:
integrate(((16*log(log(50))^2+8*x*log(log(50))+x^2-4)*log(16*log(log(50))^ 2+8*x*log(log(50))+x^2-4)+8*x*log(log(50))+2*x^2)/(16*log(log(50))^2+8*x*l og(log(50))+x^2-4),x, algorithm="maxima")
Output:
(x + 4*log(2*log(5) + log(2)) + 2)*log(x + 4*log(2*log(5) + log(2)) + 2) + (x + 4*log(2*log(5) + log(2)) - 2)*log(x + 4*log(2*log(5) + log(2)) - 2) - 2*(4*log(log(50))^2 + 4*log(log(50)) + 1)*log(x + 4*log(log(50)) + 2) + 2*(4*log(log(50))^2 - 4*log(log(50)) + 1)*log(x + 4*log(log(50)) - 2) + 4* ((2*log(log(50)) + 1)*log(x + 4*log(log(50)) + 2) - (2*log(log(50)) - 1)*l og(x + 4*log(log(50)) - 2))*log(log(50))
Time = 0.13 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.50 \[ \int \frac {2 x^2+8 x \log (\log (50))+\left (-4+x^2+8 x \log (\log (50))+16 \log ^2(\log (50))\right ) \log \left (-4+x^2+8 x \log (\log (50))+16 \log ^2(\log (50))\right )}{-4+x^2+8 x \log (\log (50))+16 \log ^2(\log (50))} \, dx=x \log \left (x^{2} + 8 \, x \log \left (\log \left (50\right )\right ) + 16 \, \log \left (\log \left (50\right )\right )^{2} - 4\right ) \] Input:
integrate(((16*log(log(50))^2+8*x*log(log(50))+x^2-4)*log(16*log(log(50))^ 2+8*x*log(log(50))+x^2-4)+8*x*log(log(50))+2*x^2)/(16*log(log(50))^2+8*x*l og(log(50))+x^2-4),x, algorithm="giac")
Output:
x*log(x^2 + 8*x*log(log(50)) + 16*log(log(50))^2 - 4)
Time = 0.35 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.50 \[ \int \frac {2 x^2+8 x \log (\log (50))+\left (-4+x^2+8 x \log (\log (50))+16 \log ^2(\log (50))\right ) \log \left (-4+x^2+8 x \log (\log (50))+16 \log ^2(\log (50))\right )}{-4+x^2+8 x \log (\log (50))+16 \log ^2(\log (50))} \, dx=x\,\ln \left (x^2+8\,\ln \left (\ln \left (50\right )\right )\,x+16\,{\ln \left (\ln \left (50\right )\right )}^2-4\right ) \] Input:
int((8*x*log(log(50)) + log(16*log(log(50))^2 + 8*x*log(log(50)) + x^2 - 4 )*(16*log(log(50))^2 + 8*x*log(log(50)) + x^2 - 4) + 2*x^2)/(16*log(log(50 ))^2 + 8*x*log(log(50)) + x^2 - 4),x)
Output:
x*log(16*log(log(50))^2 + 8*x*log(log(50)) + x^2 - 4)
Time = 0.19 (sec) , antiderivative size = 175, normalized size of antiderivative = 12.50 \[ \int \frac {2 x^2+8 x \log (\log (50))+\left (-4+x^2+8 x \log (\log (50))+16 \log ^2(\log (50))\right ) \log \left (-4+x^2+8 x \log (\log (50))+16 \log ^2(\log (50))\right )}{-4+x^2+8 x \log (\log (50))+16 \log ^2(\log (50))} \, dx=8 \mathrm {log}\left (\mathrm {log}\left (50\right )\right )^{2} \mathrm {log}\left (16 \mathrm {log}\left (\mathrm {log}\left (50\right )\right )^{2}+8 \,\mathrm {log}\left (\mathrm {log}\left (50\right )\right ) x +x^{2}-4\right )-8 \mathrm {log}\left (\mathrm {log}\left (50\right )\right )^{2} \mathrm {log}\left (4 \,\mathrm {log}\left (\mathrm {log}\left (50\right )\right )+x -2\right )-8 \mathrm {log}\left (\mathrm {log}\left (50\right )\right )^{2} \mathrm {log}\left (4 \,\mathrm {log}\left (\mathrm {log}\left (50\right )\right )+x +2\right )+8 \,\mathrm {log}\left (\mathrm {log}\left (50\right )\right ) \mathrm {log}\left (16 \mathrm {log}\left (\mathrm {log}\left (50\right )\right )^{2}+8 \,\mathrm {log}\left (\mathrm {log}\left (50\right )\right ) x +x^{2}-4\right )-8 \,\mathrm {log}\left (\mathrm {log}\left (50\right )\right ) \mathrm {log}\left (4 \,\mathrm {log}\left (\mathrm {log}\left (50\right )\right )+x -2\right )-8 \,\mathrm {log}\left (\mathrm {log}\left (50\right )\right ) \mathrm {log}\left (4 \,\mathrm {log}\left (\mathrm {log}\left (50\right )\right )+x +2\right )+\mathrm {log}\left (16 \mathrm {log}\left (\mathrm {log}\left (50\right )\right )^{2}+8 \,\mathrm {log}\left (\mathrm {log}\left (50\right )\right ) x +x^{2}-4\right ) x +2 \,\mathrm {log}\left (16 \mathrm {log}\left (\mathrm {log}\left (50\right )\right )^{2}+8 \,\mathrm {log}\left (\mathrm {log}\left (50\right )\right ) x +x^{2}-4\right )-2 \,\mathrm {log}\left (4 \,\mathrm {log}\left (\mathrm {log}\left (50\right )\right )+x -2\right )-2 \,\mathrm {log}\left (4 \,\mathrm {log}\left (\mathrm {log}\left (50\right )\right )+x +2\right ) \] Input:
int(((16*log(log(50))^2+8*x*log(log(50))+x^2-4)*log(16*log(log(50))^2+8*x* log(log(50))+x^2-4)+8*x*log(log(50))+2*x^2)/(16*log(log(50))^2+8*x*log(log (50))+x^2-4),x)
Output:
8*log(log(50))**2*log(16*log(log(50))**2 + 8*log(log(50))*x + x**2 - 4) - 8*log(log(50))**2*log(4*log(log(50)) + x - 2) - 8*log(log(50))**2*log(4*lo g(log(50)) + x + 2) + 8*log(log(50))*log(16*log(log(50))**2 + 8*log(log(50 ))*x + x**2 - 4) - 8*log(log(50))*log(4*log(log(50)) + x - 2) - 8*log(log( 50))*log(4*log(log(50)) + x + 2) + log(16*log(log(50))**2 + 8*log(log(50)) *x + x**2 - 4)*x + 2*log(16*log(log(50))**2 + 8*log(log(50))*x + x**2 - 4) - 2*log(4*log(log(50)) + x - 2) - 2*log(4*log(log(50)) + x + 2)