Integrand size = 66, antiderivative size = 19 \[ \int \frac {\left (2 x^3+4 x^2 \log (4)+2 x \log ^2(4)\right ) \log (\log (x) \log (\log (4)))+\left (4 x^3+6 x^2 \log (4)+2 x \log ^2(4)\right ) \log (x) \log ^2(\log (x) \log (\log (4)))}{\log (x)} \, dx=x^2 (x+\log (4))^2 \log ^2(\log (x) \log (\log (4))) \] Output:
ln(ln(x)*ln(2*ln(2)))^2*(x+2*ln(2))^2*x^2
Time = 0.10 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {\left (2 x^3+4 x^2 \log (4)+2 x \log ^2(4)\right ) \log (\log (x) \log (\log (4)))+\left (4 x^3+6 x^2 \log (4)+2 x \log ^2(4)\right ) \log (x) \log ^2(\log (x) \log (\log (4)))}{\log (x)} \, dx=x^2 (x+\log (4))^2 \log ^2(\log (x) \log (\log (4))) \] Input:
Integrate[((2*x^3 + 4*x^2*Log[4] + 2*x*Log[4]^2)*Log[Log[x]*Log[Log[4]]] + (4*x^3 + 6*x^2*Log[4] + 2*x*Log[4]^2)*Log[x]*Log[Log[x]*Log[Log[4]]]^2)/L og[x],x]
Output:
x^2*(x + Log[4])^2*Log[Log[x]*Log[Log[4]]]^2
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.
\(\displaystyle \int \frac {\left (4 x^3+6 x^2 \log (4)+2 x \log ^2(4)\right ) \log (x) \log ^2(\log (\log (4)) \log (x))+\left (2 x^3+4 x^2 \log (4)+2 x \log ^2(4)\right ) \log (\log (\log (4)) \log (x))}{\log (x)} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {2 x (x+\log (4)) \log (\log (\log (4)) \log (x)) (x+(2 x+\log (4)) \log (x) \log (\log (\log (4)) \log (x))+\log (4))}{\log (x)}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int \frac {x (x+\log (4)) \log (\log (x) \log (\log (4))) (x+(2 x+\log (4)) \log (x) \log (\log (x) \log (\log (4)))+\log (4))}{\log (x)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle 2 \int \left (\frac {x \log (\log (x) \log (\log (4))) (x+\log (4))^2}{\log (x)}+x (2 x+\log (4)) \log ^2(\log (x) \log (\log (4))) (x+\log (4))\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (2 \int x^3 \log ^2(\log (x) \log (\log (4)))dx+\int \frac {x^3 \log (\log (x) \log (\log (4)))}{\log (x)}dx+3 \log (4) \int x^2 \log ^2(\log (x) \log (\log (4)))dx+2 \log (4) \int \frac {x^2 \log (\log (x) \log (\log (4)))}{\log (x)}dx+\log ^2(4) \int \frac {x \log (\log (x) \log (\log (4)))}{\log (x)}dx+\log ^2(4) \int x \log ^2(\log (x) \log (\log (4)))dx\right )\) |
Input:
Int[((2*x^3 + 4*x^2*Log[4] + 2*x*Log[4]^2)*Log[Log[x]*Log[Log[4]]] + (4*x^ 3 + 6*x^2*Log[4] + 2*x*Log[4]^2)*Log[x]*Log[Log[x]*Log[Log[4]]]^2)/Log[x], x]
Output:
$Aborted
Time = 1.86 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.79
method | result | size |
risch | \(\left (4 x^{2} \ln \left (2\right )^{2}+4 x^{3} \ln \left (2\right )+x^{4}\right ) \ln \left (\ln \left (x \right ) \left (\ln \left (2\right )+\ln \left (\ln \left (2\right )\right )\right )\right )^{2}\) | \(34\) |
parallelrisch | \(4 \ln \left (2\right )^{2} x^{2} \ln \left (\ln \left (x \right ) \ln \left (2 \ln \left (2\right )\right )\right )^{2}+4 \ln \left (2\right ) x^{3} \ln \left (\ln \left (x \right ) \ln \left (2 \ln \left (2\right )\right )\right )^{2}+\ln \left (\ln \left (x \right ) \ln \left (2 \ln \left (2\right )\right )\right )^{2} x^{4}\) | \(55\) |
Input:
int(((8*x*ln(2)^2+12*x^2*ln(2)+4*x^3)*ln(x)*ln(ln(x)*ln(2*ln(2)))^2+(8*x*l n(2)^2+8*x^2*ln(2)+2*x^3)*ln(ln(x)*ln(2*ln(2))))/ln(x),x,method=_RETURNVER BOSE)
Output:
(4*x^2*ln(2)^2+4*x^3*ln(2)+x^4)*ln(ln(x)*(ln(2)+ln(ln(2))))^2
Time = 0.08 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.68 \[ \int \frac {\left (2 x^3+4 x^2 \log (4)+2 x \log ^2(4)\right ) \log (\log (x) \log (\log (4)))+\left (4 x^3+6 x^2 \log (4)+2 x \log ^2(4)\right ) \log (x) \log ^2(\log (x) \log (\log (4)))}{\log (x)} \, dx={\left (x^{4} + 4 \, x^{3} \log \left (2\right ) + 4 \, x^{2} \log \left (2\right )^{2}\right )} \log \left (\log \left (x\right ) \log \left (2 \, \log \left (2\right )\right )\right )^{2} \] Input:
integrate(((8*x*log(2)^2+12*x^2*log(2)+4*x^3)*log(x)*log(log(x)*log(2*log( 2)))^2+(8*x*log(2)^2+8*x^2*log(2)+2*x^3)*log(log(x)*log(2*log(2))))/log(x) ,x, algorithm="fricas")
Output:
(x^4 + 4*x^3*log(2) + 4*x^2*log(2)^2)*log(log(x)*log(2*log(2)))^2
Time = 0.16 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.79 \[ \int \frac {\left (2 x^3+4 x^2 \log (4)+2 x \log ^2(4)\right ) \log (\log (x) \log (\log (4)))+\left (4 x^3+6 x^2 \log (4)+2 x \log ^2(4)\right ) \log (x) \log ^2(\log (x) \log (\log (4)))}{\log (x)} \, dx=\left (x^{4} + 4 x^{3} \log {\left (2 \right )} + 4 x^{2} \log {\left (2 \right )}^{2}\right ) \log {\left (\log {\left (x \right )} \log {\left (2 \log {\left (2 \right )} \right )} \right )}^{2} \] Input:
integrate(((8*x*ln(2)**2+12*x**2*ln(2)+4*x**3)*ln(x)*ln(ln(x)*ln(2*ln(2))) **2+(8*x*ln(2)**2+8*x**2*ln(2)+2*x**3)*ln(ln(x)*ln(2*ln(2))))/ln(x),x)
Output:
(x**4 + 4*x**3*log(2) + 4*x**2*log(2)**2)*log(log(x)*log(2*log(2)))**2
Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (23) = 46\).
Time = 0.21 (sec) , antiderivative size = 121, normalized size of antiderivative = 6.37 \[ \int \frac {\left (2 x^3+4 x^2 \log (4)+2 x \log ^2(4)\right ) \log (\log (x) \log (\log (4)))+\left (4 x^3+6 x^2 \log (4)+2 x \log ^2(4)\right ) \log (x) \log ^2(\log (x) \log (\log (4)))}{\log (x)} \, dx=x^{4} \log \left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )^{2} + 4 \, x^{3} \log \left (2\right ) \log \left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )^{2} + 4 \, x^{2} \log \left (2\right )^{2} \log \left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )^{2} + {\left (x^{4} + 4 \, x^{3} \log \left (2\right ) + 4 \, x^{2} \log \left (2\right )^{2}\right )} \log \left (\log \left (x\right )\right )^{2} + 2 \, {\left (x^{4} \log \left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right ) + 4 \, x^{3} \log \left (2\right ) \log \left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right ) + 4 \, x^{2} \log \left (2\right )^{2} \log \left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )\right )} \log \left (\log \left (x\right )\right ) \] Input:
integrate(((8*x*log(2)^2+12*x^2*log(2)+4*x^3)*log(x)*log(log(x)*log(2*log( 2)))^2+(8*x*log(2)^2+8*x^2*log(2)+2*x^3)*log(log(x)*log(2*log(2))))/log(x) ,x, algorithm="maxima")
Output:
x^4*log(log(2) + log(log(2)))^2 + 4*x^3*log(2)*log(log(2) + log(log(2)))^2 + 4*x^2*log(2)^2*log(log(2) + log(log(2)))^2 + (x^4 + 4*x^3*log(2) + 4*x^ 2*log(2)^2)*log(log(x))^2 + 2*(x^4*log(log(2) + log(log(2))) + 4*x^3*log(2 )*log(log(2) + log(log(2))) + 4*x^2*log(2)^2*log(log(2) + log(log(2))))*lo g(log(x))
Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (23) = 46\).
Time = 0.18 (sec) , antiderivative size = 134, normalized size of antiderivative = 7.05 \[ \int \frac {\left (2 x^3+4 x^2 \log (4)+2 x \log ^2(4)\right ) \log (\log (x) \log (\log (4)))+\left (4 x^3+6 x^2 \log (4)+2 x \log ^2(4)\right ) \log (x) \log ^2(\log (x) \log (\log (4)))}{\log (x)} \, dx=x^{4} \log \left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )^{2} + 4 \, x^{3} \log \left (2\right ) \log \left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )^{2} + 4 \, x^{2} \log \left (2\right )^{2} \log \left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )^{2} + 2 \, x^{4} \log \left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right ) \log \left (\log \left (x\right )\right ) + 8 \, x^{3} \log \left (2\right ) \log \left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right ) \log \left (\log \left (x\right )\right ) + 8 \, x^{2} \log \left (2\right )^{2} \log \left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right ) \log \left (\log \left (x\right )\right ) + x^{4} \log \left (\log \left (x\right )\right )^{2} + 4 \, x^{3} \log \left (2\right ) \log \left (\log \left (x\right )\right )^{2} + 4 \, x^{2} \log \left (2\right )^{2} \log \left (\log \left (x\right )\right )^{2} \] Input:
integrate(((8*x*log(2)^2+12*x^2*log(2)+4*x^3)*log(x)*log(log(x)*log(2*log( 2)))^2+(8*x*log(2)^2+8*x^2*log(2)+2*x^3)*log(log(x)*log(2*log(2))))/log(x) ,x, algorithm="giac")
Output:
x^4*log(log(2) + log(log(2)))^2 + 4*x^3*log(2)*log(log(2) + log(log(2)))^2 + 4*x^2*log(2)^2*log(log(2) + log(log(2)))^2 + 2*x^4*log(log(2) + log(log (2)))*log(log(x)) + 8*x^3*log(2)*log(log(2) + log(log(2)))*log(log(x)) + 8 *x^2*log(2)^2*log(log(2) + log(log(2)))*log(log(x)) + x^4*log(log(x))^2 + 4*x^3*log(2)*log(log(x))^2 + 4*x^2*log(2)^2*log(log(x))^2
Time = 3.11 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.58 \[ \int \frac {\left (2 x^3+4 x^2 \log (4)+2 x \log ^2(4)\right ) \log (\log (x) \log (\log (4)))+\left (4 x^3+6 x^2 \log (4)+2 x \log ^2(4)\right ) \log (x) \log ^2(\log (x) \log (\log (4)))}{\log (x)} \, dx={\ln \left (\ln \left (\ln \left (4\right )\right )\,\ln \left (x\right )\right )}^2\,\left (x^4+4\,\ln \left (2\right )\,x^3+4\,{\ln \left (2\right )}^2\,x^2\right ) \] Input:
int((log(log(2*log(2))*log(x))*(8*x*log(2)^2 + 8*x^2*log(2) + 2*x^3) + log (x)*log(log(2*log(2))*log(x))^2*(8*x*log(2)^2 + 12*x^2*log(2) + 4*x^3))/lo g(x),x)
Output:
log(log(log(4))*log(x))^2*(4*x^2*log(2)^2 + 4*x^3*log(2) + x^4)
Time = 0.15 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.58 \[ \int \frac {\left (2 x^3+4 x^2 \log (4)+2 x \log ^2(4)\right ) \log (\log (x) \log (\log (4)))+\left (4 x^3+6 x^2 \log (4)+2 x \log ^2(4)\right ) \log (x) \log ^2(\log (x) \log (\log (4)))}{\log (x)} \, dx=\mathrm {log}\left (\mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right ) \mathrm {log}\left (x \right )\right )^{2} x^{2} \left (4 \mathrm {log}\left (2\right )^{2}+4 \,\mathrm {log}\left (2\right ) x +x^{2}\right ) \] Input:
int(((8*x*log(2)^2+12*x^2*log(2)+4*x^3)*log(x)*log(log(x)*log(2*log(2)))^2 +(8*x*log(2)^2+8*x^2*log(2)+2*x^3)*log(log(x)*log(2*log(2))))/log(x),x)
Output:
log(log(2*log(2))*log(x))**2*x**2*(4*log(2)**2 + 4*log(2)*x + x**2)