Integrand size = 168, antiderivative size = 29 \[ \int \frac {-32 \log (3)+32 \log (3) \log \left (\frac {x}{3}\right )+(-8+16 x) \log (3) \log ^2\left (\frac {x}{3}\right )-8 \log (3) \log ^2\left (\frac {x}{3}\right ) \log (x)}{\left (-4 x \log \left (\frac {x}{3}\right )-x^2 \log ^2\left (\frac {x}{3}\right )+x \log ^2\left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {16 x^2+8 x^3 \log \left (\frac {x}{3}\right )+x^4 \log ^2\left (\frac {x}{3}\right )+\left (-8 x^2 \log \left (\frac {x}{3}\right )-2 x^3 \log ^2\left (\frac {x}{3}\right )\right ) \log (x)+x^2 \log ^2\left (\frac {x}{3}\right ) \log ^2(x)}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx=\frac {4 \log (3)}{\log \left (x^2 \left (x+\frac {4}{\log \left (\frac {x}{3}\right )}-\log (x)\right )^2\right )} \] Output:
4*ln(3)/ln(x^2*(4/ln(1/3*x)-ln(x)+x)^2)
\[ \int \frac {-32 \log (3)+32 \log (3) \log \left (\frac {x}{3}\right )+(-8+16 x) \log (3) \log ^2\left (\frac {x}{3}\right )-8 \log (3) \log ^2\left (\frac {x}{3}\right ) \log (x)}{\left (-4 x \log \left (\frac {x}{3}\right )-x^2 \log ^2\left (\frac {x}{3}\right )+x \log ^2\left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {16 x^2+8 x^3 \log \left (\frac {x}{3}\right )+x^4 \log ^2\left (\frac {x}{3}\right )+\left (-8 x^2 \log \left (\frac {x}{3}\right )-2 x^3 \log ^2\left (\frac {x}{3}\right )\right ) \log (x)+x^2 \log ^2\left (\frac {x}{3}\right ) \log ^2(x)}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx=\int \frac {-32 \log (3)+32 \log (3) \log \left (\frac {x}{3}\right )+(-8+16 x) \log (3) \log ^2\left (\frac {x}{3}\right )-8 \log (3) \log ^2\left (\frac {x}{3}\right ) \log (x)}{\left (-4 x \log \left (\frac {x}{3}\right )-x^2 \log ^2\left (\frac {x}{3}\right )+x \log ^2\left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {16 x^2+8 x^3 \log \left (\frac {x}{3}\right )+x^4 \log ^2\left (\frac {x}{3}\right )+\left (-8 x^2 \log \left (\frac {x}{3}\right )-2 x^3 \log ^2\left (\frac {x}{3}\right )\right ) \log (x)+x^2 \log ^2\left (\frac {x}{3}\right ) \log ^2(x)}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx \] Input:
Integrate[(-32*Log[3] + 32*Log[3]*Log[x/3] + (-8 + 16*x)*Log[3]*Log[x/3]^2 - 8*Log[3]*Log[x/3]^2*Log[x])/((-4*x*Log[x/3] - x^2*Log[x/3]^2 + x*Log[x/ 3]^2*Log[x])*Log[(16*x^2 + 8*x^3*Log[x/3] + x^4*Log[x/3]^2 + (-8*x^2*Log[x /3] - 2*x^3*Log[x/3]^2)*Log[x] + x^2*Log[x/3]^2*Log[x]^2)/Log[x/3]^2]^2),x ]
Output:
Integrate[(-32*Log[3] + 32*Log[3]*Log[x/3] + (-8 + 16*x)*Log[3]*Log[x/3]^2 - 8*Log[3]*Log[x/3]^2*Log[x])/((-4*x*Log[x/3] - x^2*Log[x/3]^2 + x*Log[x/ 3]^2*Log[x])*Log[(16*x^2 + 8*x^3*Log[x/3] + x^4*Log[x/3]^2 + (-8*x^2*Log[x /3] - 2*x^3*Log[x/3]^2)*Log[x] + x^2*Log[x/3]^2*Log[x]^2)/Log[x/3]^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.
| \(\displaystyle \int \frac {-8 \log (3) \log (x) \log ^2\left (\frac {x}{3}\right )+(16 x-8) \log (3) \log ^2\left (\frac {x}{3}\right )+32 \log (3) \log \left (\frac {x}{3}\right )-32 \log (3)}{\left (-x^2 \log ^2\left (\frac {x}{3}\right )+x \log (x) \log ^2\left (\frac {x}{3}\right )-4 x \log \left (\frac {x}{3}\right )\right ) \log ^2\left (\frac {x^4 \log ^2\left (\frac {x}{3}\right )+8 x^3 \log \left (\frac {x}{3}\right )+16 x^2+x^2 \log ^2\left (\frac {x}{3}\right ) \log ^2(x)+\left (-2 x^3 \log ^2\left (\frac {x}{3}\right )-8 x^2 \log \left (\frac {x}{3}\right )\right ) \log (x)}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {8 \log (3) \left ((-2 x+\log (x)+1) \log ^2\left (\frac {x}{3}\right )-4 \log (x)+4 (1+\log (3))\right )}{x \log \left (\frac {x}{3}\right ) \left (\log \left (\frac {x}{3}\right ) (x-\log (x))+4\right ) \log ^2\left (\frac {x^2 \left (\log \left (\frac {x}{3}\right ) (x-\log (x))+4\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 8 \log (3) \int \frac {(-2 x+\log (x)+1) \log ^2\left (\frac {x}{3}\right )-4 \log (x)+\log (81)+4}{x \log \left (\frac {x}{3}\right ) \left (\log \left (\frac {x}{3}\right ) (x-\log (x))+4\right ) \log ^2\left (\frac {x^2 \left (\log \left (\frac {x}{3}\right ) (x-\log (x))+4\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right )}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle 8 \log (3) \int \frac {(-2 x+\log (x)+1) \log ^2\left (\frac {x}{3}\right )-4 \log (x)+4 (1+\log (3))}{x \log \left (\frac {x}{3}\right ) \left (\log \left (\frac {x}{3}\right ) (x-\log (x))+4\right ) \log ^2\left (\frac {x^2 \left (\log \left (\frac {x}{3}\right ) (x-\log (x))+4\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 8 \log (3) \int \left (\frac {\log (x) \log \left (\frac {x}{3}\right )}{x \left (x \log \left (\frac {x}{3}\right )-\log (x) \log \left (\frac {x}{3}\right )+4\right ) \log ^2\left (\frac {x^2 \left (\log \left (\frac {x}{3}\right ) (x-\log (x))+4\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right )}+\frac {\log \left (\frac {x}{3}\right )}{x \left (x \log \left (\frac {x}{3}\right )-\log (x) \log \left (\frac {x}{3}\right )+4\right ) \log ^2\left (\frac {x^2 \left (\log \left (\frac {x}{3}\right ) (x-\log (x))+4\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right )}-\frac {2 \log \left (\frac {x}{3}\right )}{\left (x \log \left (\frac {x}{3}\right )-\log (x) \log \left (\frac {x}{3}\right )+4\right ) \log ^2\left (\frac {x^2 \left (\log \left (\frac {x}{3}\right ) (x-\log (x))+4\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right )}-\frac {4 \log (x)}{x \left (x \log \left (\frac {x}{3}\right )-\log (x) \log \left (\frac {x}{3}\right )+4\right ) \log ^2\left (\frac {x^2 \left (\log \left (\frac {x}{3}\right ) (x-\log (x))+4\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right ) \log \left (\frac {x}{3}\right )}+\frac {4 (1+\log (3))}{x \left (x \log \left (\frac {x}{3}\right )-\log (x) \log \left (\frac {x}{3}\right )+4\right ) \log ^2\left (\frac {x^2 \left (\log \left (\frac {x}{3}\right ) (x-\log (x))+4\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right ) \log \left (\frac {x}{3}\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 8 \log (3) \left (4 (1+\log (3)) \int \frac {1}{x \log \left (\frac {x}{3}\right ) \left (x \log \left (\frac {x}{3}\right )-\log (x) \log \left (\frac {x}{3}\right )+4\right ) \log ^2\left (\frac {x^2 \left (\log \left (\frac {x}{3}\right ) (x-\log (x))+4\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right )}dx-2 \int \frac {\log \left (\frac {x}{3}\right )}{\left (x \log \left (\frac {x}{3}\right )-\log (x) \log \left (\frac {x}{3}\right )+4\right ) \log ^2\left (\frac {x^2 \left (\log \left (\frac {x}{3}\right ) (x-\log (x))+4\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right )}dx+\int \frac {\log \left (\frac {x}{3}\right )}{x \left (x \log \left (\frac {x}{3}\right )-\log (x) \log \left (\frac {x}{3}\right )+4\right ) \log ^2\left (\frac {x^2 \left (\log \left (\frac {x}{3}\right ) (x-\log (x))+4\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right )}dx-4 \int \frac {\log (x)}{x \log \left (\frac {x}{3}\right ) \left (x \log \left (\frac {x}{3}\right )-\log (x) \log \left (\frac {x}{3}\right )+4\right ) \log ^2\left (\frac {x^2 \left (\log \left (\frac {x}{3}\right ) (x-\log (x))+4\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right )}dx+\int \frac {\log \left (\frac {x}{3}\right ) \log (x)}{x \left (x \log \left (\frac {x}{3}\right )-\log (x) \log \left (\frac {x}{3}\right )+4\right ) \log ^2\left (\frac {x^2 \left (\log \left (\frac {x}{3}\right ) (x-\log (x))+4\right )^2}{\log ^2\left (\frac {x}{3}\right )}\right )}dx\right )\) |
Input:
Int[(-32*Log[3] + 32*Log[3]*Log[x/3] + (-8 + 16*x)*Log[3]*Log[x/3]^2 - 8*L og[3]*Log[x/3]^2*Log[x])/((-4*x*Log[x/3] - x^2*Log[x/3]^2 + x*Log[x/3]^2*L og[x])*Log[(16*x^2 + 8*x^3*Log[x/3] + x^4*Log[x/3]^2 + (-8*x^2*Log[x/3] - 2*x^3*Log[x/3]^2)*Log[x] + x^2*Log[x/3]^2*Log[x]^2)/Log[x/3]^2]^2),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(66\) vs. \(2(27)=54\).
Time = 40.41 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.31
| method | result | size |
| parallelrisch | \(\frac {4 \ln \left (3\right )}{\ln \left (\frac {x^{2} \left (\ln \left (\frac {x}{3}\right )^{2} \ln \left (x \right )^{2}-2 x \ln \left (\frac {x}{3}\right )^{2} \ln \left (x \right )+x^{2} \ln \left (\frac {x}{3}\right )^{2}-8 \ln \left (\frac {x}{3}\right ) \ln \left (x \right )+8 x \ln \left (\frac {x}{3}\right )+16\right )}{\ln \left (\frac {x}{3}\right )^{2}}\right )}\) | \(67\) |
| default | \(\frac {4 \ln \left (3\right )}{\ln \left (\frac {x^{2} \left (\ln \left (3\right )^{2} \ln \left (x \right )^{2}-2 \ln \left (x \right )^{3} \ln \left (3\right )+\ln \left (x \right )^{4}-2 x \ln \left (x \right ) \ln \left (3\right )^{2}+4 \ln \left (3\right ) x \ln \left (x \right )^{2}-2 x \ln \left (x \right )^{3}+x^{2} \ln \left (3\right )^{2}-2 \ln \left (3\right ) \ln \left (x \right ) x^{2}+x^{2} \ln \left (x \right )^{2}+8 \ln \left (x \right ) \ln \left (3\right )-8 \ln \left (x \right )^{2}-8 x \ln \left (3\right )+8 x \ln \left (x \right )+16\right )}{\left (-\ln \left (3\right )+\ln \left (x \right )\right )^{2}}\right )}\) | \(116\) |
| risch | \(\text {Expression too large to display}\) | \(1002\) |
Input:
int((-8*ln(3)*ln(1/3*x)^2*ln(x)+(16*x-8)*ln(3)*ln(1/3*x)^2+32*ln(3)*ln(1/3 *x)-32*ln(3))/(x*ln(1/3*x)^2*ln(x)-x^2*ln(1/3*x)^2-4*x*ln(1/3*x))/ln((x^2* ln(1/3*x)^2*ln(x)^2+(-2*x^3*ln(1/3*x)^2-8*x^2*ln(1/3*x))*ln(x)+x^4*ln(1/3* x)^2+8*x^3*ln(1/3*x)+16*x^2)/ln(1/3*x)^2)^2,x,method=_RETURNVERBOSE)
Output:
4*ln(3)/ln(x^2*(ln(1/3*x)^2*ln(x)^2-2*x*ln(1/3*x)^2*ln(x)+x^2*ln(1/3*x)^2- 8*ln(1/3*x)*ln(x)+8*x*ln(1/3*x)+16)/ln(1/3*x)^2)
Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (27) = 54\).
Time = 0.10 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.34 \[ \int \frac {-32 \log (3)+32 \log (3) \log \left (\frac {x}{3}\right )+(-8+16 x) \log (3) \log ^2\left (\frac {x}{3}\right )-8 \log (3) \log ^2\left (\frac {x}{3}\right ) \log (x)}{\left (-4 x \log \left (\frac {x}{3}\right )-x^2 \log ^2\left (\frac {x}{3}\right )+x \log ^2\left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {16 x^2+8 x^3 \log \left (\frac {x}{3}\right )+x^4 \log ^2\left (\frac {x}{3}\right )+\left (-8 x^2 \log \left (\frac {x}{3}\right )-2 x^3 \log ^2\left (\frac {x}{3}\right )\right ) \log (x)+x^2 \log ^2\left (\frac {x}{3}\right ) \log ^2(x)}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx=\frac {4 \, \log \left (3\right )}{\log \left (\frac {x^{2} \log \left (\frac {1}{3} \, x\right )^{4} - 2 \, {\left (x^{3} - x^{2} \log \left (3\right )\right )} \log \left (\frac {1}{3} \, x\right )^{3} + {\left (x^{4} - 2 \, x^{3} \log \left (3\right ) + x^{2} \log \left (3\right )^{2} - 8 \, x^{2}\right )} \log \left (\frac {1}{3} \, x\right )^{2} + 16 \, x^{2} + 8 \, {\left (x^{3} - x^{2} \log \left (3\right )\right )} \log \left (\frac {1}{3} \, x\right )}{\log \left (\frac {1}{3} \, x\right )^{2}}\right )} \] Input:
integrate((-8*log(3)*log(1/3*x)^2*log(x)+(16*x-8)*log(3)*log(1/3*x)^2+32*l og(3)*log(1/3*x)-32*log(3))/(x*log(1/3*x)^2*log(x)-x^2*log(1/3*x)^2-4*x*lo g(1/3*x))/log((x^2*log(1/3*x)^2*log(x)^2+(-2*x^3*log(1/3*x)^2-8*x^2*log(1/ 3*x))*log(x)+x^4*log(1/3*x)^2+8*x^3*log(1/3*x)+16*x^2)/log(1/3*x)^2)^2,x, algorithm="fricas")
Output:
4*log(3)/log((x^2*log(1/3*x)^4 - 2*(x^3 - x^2*log(3))*log(1/3*x)^3 + (x^4 - 2*x^3*log(3) + x^2*log(3)^2 - 8*x^2)*log(1/3*x)^2 + 16*x^2 + 8*(x^3 - x^ 2*log(3))*log(1/3*x))/log(1/3*x)^2)
Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (22) = 44\).
Time = 0.59 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.10 \[ \int \frac {-32 \log (3)+32 \log (3) \log \left (\frac {x}{3}\right )+(-8+16 x) \log (3) \log ^2\left (\frac {x}{3}\right )-8 \log (3) \log ^2\left (\frac {x}{3}\right ) \log (x)}{\left (-4 x \log \left (\frac {x}{3}\right )-x^2 \log ^2\left (\frac {x}{3}\right )+x \log ^2\left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {16 x^2+8 x^3 \log \left (\frac {x}{3}\right )+x^4 \log ^2\left (\frac {x}{3}\right )+\left (-8 x^2 \log \left (\frac {x}{3}\right )-2 x^3 \log ^2\left (\frac {x}{3}\right )\right ) \log (x)+x^2 \log ^2\left (\frac {x}{3}\right ) \log ^2(x)}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx=\frac {4 \log {\left (3 \right )}}{\log {\left (\frac {x^{4} \left (\log {\left (x \right )} - \log {\left (3 \right )}\right )^{2} + 8 x^{3} \left (\log {\left (x \right )} - \log {\left (3 \right )}\right ) + x^{2} \left (\log {\left (x \right )} - \log {\left (3 \right )}\right )^{2} \log {\left (x \right )}^{2} + 16 x^{2} + \left (- 2 x^{3} \left (\log {\left (x \right )} - \log {\left (3 \right )}\right )^{2} - 8 x^{2} \left (\log {\left (x \right )} - \log {\left (3 \right )}\right )\right ) \log {\left (x \right )}}{\left (\log {\left (x \right )} - \log {\left (3 \right )}\right )^{2}} \right )}} \] Input:
integrate((-8*ln(3)*ln(1/3*x)**2*ln(x)+(16*x-8)*ln(3)*ln(1/3*x)**2+32*ln(3 )*ln(1/3*x)-32*ln(3))/(x*ln(1/3*x)**2*ln(x)-x**2*ln(1/3*x)**2-4*x*ln(1/3*x ))/ln((x**2*ln(1/3*x)**2*ln(x)**2+(-2*x**3*ln(1/3*x)**2-8*x**2*ln(1/3*x))* ln(x)+x**4*ln(1/3*x)**2+8*x**3*ln(1/3*x)+16*x**2)/ln(1/3*x)**2)**2,x)
Output:
4*log(3)/log((x**4*(log(x) - log(3))**2 + 8*x**3*(log(x) - log(3)) + x**2* (log(x) - log(3))**2*log(x)**2 + 16*x**2 + (-2*x**3*(log(x) - log(3))**2 - 8*x**2*(log(x) - log(3)))*log(x))/(log(x) - log(3))**2)
Time = 0.18 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.31 \[ \int \frac {-32 \log (3)+32 \log (3) \log \left (\frac {x}{3}\right )+(-8+16 x) \log (3) \log ^2\left (\frac {x}{3}\right )-8 \log (3) \log ^2\left (\frac {x}{3}\right ) \log (x)}{\left (-4 x \log \left (\frac {x}{3}\right )-x^2 \log ^2\left (\frac {x}{3}\right )+x \log ^2\left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {16 x^2+8 x^3 \log \left (\frac {x}{3}\right )+x^4 \log ^2\left (\frac {x}{3}\right )+\left (-8 x^2 \log \left (\frac {x}{3}\right )-2 x^3 \log ^2\left (\frac {x}{3}\right )\right ) \log (x)+x^2 \log ^2\left (\frac {x}{3}\right ) \log ^2(x)}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx=\frac {2 \, \log \left (3\right )}{\log \left (x \log \left (3\right ) - {\left (x + \log \left (3\right )\right )} \log \left (x\right ) + \log \left (x\right )^{2} - 4\right ) + \log \left (x\right ) - \log \left (-\log \left (3\right ) + \log \left (x\right )\right )} \] Input:
integrate((-8*log(3)*log(1/3*x)^2*log(x)+(16*x-8)*log(3)*log(1/3*x)^2+32*l og(3)*log(1/3*x)-32*log(3))/(x*log(1/3*x)^2*log(x)-x^2*log(1/3*x)^2-4*x*lo g(1/3*x))/log((x^2*log(1/3*x)^2*log(x)^2+(-2*x^3*log(1/3*x)^2-8*x^2*log(1/ 3*x))*log(x)+x^4*log(1/3*x)^2+8*x^3*log(1/3*x)+16*x^2)/log(1/3*x)^2)^2,x, algorithm="maxima")
Output:
2*log(3)/(log(x*log(3) - (x + log(3))*log(x) + log(x)^2 - 4) + log(x) - lo g(-log(3) + log(x)))
Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (27) = 54\).
Time = 4.98 (sec) , antiderivative size = 125, normalized size of antiderivative = 4.31 \[ \int \frac {-32 \log (3)+32 \log (3) \log \left (\frac {x}{3}\right )+(-8+16 x) \log (3) \log ^2\left (\frac {x}{3}\right )-8 \log (3) \log ^2\left (\frac {x}{3}\right ) \log (x)}{\left (-4 x \log \left (\frac {x}{3}\right )-x^2 \log ^2\left (\frac {x}{3}\right )+x \log ^2\left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {16 x^2+8 x^3 \log \left (\frac {x}{3}\right )+x^4 \log ^2\left (\frac {x}{3}\right )+\left (-8 x^2 \log \left (\frac {x}{3}\right )-2 x^3 \log ^2\left (\frac {x}{3}\right )\right ) \log (x)+x^2 \log ^2\left (\frac {x}{3}\right ) \log ^2(x)}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx=\frac {4 \, \log \left (3\right )}{\log \left (x^{2} \log \left (3\right )^{2} - 2 \, x^{2} \log \left (3\right ) \log \left (x\right ) - 2 \, x \log \left (3\right )^{2} \log \left (x\right ) + x^{2} \log \left (x\right )^{2} + 4 \, x \log \left (3\right ) \log \left (x\right )^{2} + \log \left (3\right )^{2} \log \left (x\right )^{2} - 2 \, x \log \left (x\right )^{3} - 2 \, \log \left (3\right ) \log \left (x\right )^{3} + \log \left (x\right )^{4} - 8 \, x \log \left (3\right ) + 8 \, x \log \left (x\right ) + 8 \, \log \left (3\right ) \log \left (x\right ) - 8 \, \log \left (x\right )^{2} + 16\right ) - \log \left (\log \left (3\right )^{2} - 2 \, \log \left (3\right ) \log \left (x\right ) + \log \left (x\right )^{2}\right ) + 2 \, \log \left (x\right )} \] Input:
integrate((-8*log(3)*log(1/3*x)^2*log(x)+(16*x-8)*log(3)*log(1/3*x)^2+32*l og(3)*log(1/3*x)-32*log(3))/(x*log(1/3*x)^2*log(x)-x^2*log(1/3*x)^2-4*x*lo g(1/3*x))/log((x^2*log(1/3*x)^2*log(x)^2+(-2*x^3*log(1/3*x)^2-8*x^2*log(1/ 3*x))*log(x)+x^4*log(1/3*x)^2+8*x^3*log(1/3*x)+16*x^2)/log(1/3*x)^2)^2,x, algorithm="giac")
Output:
4*log(3)/(log(x^2*log(3)^2 - 2*x^2*log(3)*log(x) - 2*x*log(3)^2*log(x) + x ^2*log(x)^2 + 4*x*log(3)*log(x)^2 + log(3)^2*log(x)^2 - 2*x*log(x)^3 - 2*l og(3)*log(x)^3 + log(x)^4 - 8*x*log(3) + 8*x*log(x) + 8*log(3)*log(x) - 8* log(x)^2 + 16) - log(log(3)^2 - 2*log(3)*log(x) + log(x)^2) + 2*log(x))
Time = 4.07 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.69 \[ \int \frac {-32 \log (3)+32 \log (3) \log \left (\frac {x}{3}\right )+(-8+16 x) \log (3) \log ^2\left (\frac {x}{3}\right )-8 \log (3) \log ^2\left (\frac {x}{3}\right ) \log (x)}{\left (-4 x \log \left (\frac {x}{3}\right )-x^2 \log ^2\left (\frac {x}{3}\right )+x \log ^2\left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {16 x^2+8 x^3 \log \left (\frac {x}{3}\right )+x^4 \log ^2\left (\frac {x}{3}\right )+\left (-8 x^2 \log \left (\frac {x}{3}\right )-2 x^3 \log ^2\left (\frac {x}{3}\right )\right ) \log (x)+x^2 \log ^2\left (\frac {x}{3}\right ) \log ^2(x)}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx=\frac {4\,\ln \left (3\right )}{\ln \left (\frac {8\,x^3\,\ln \left (\frac {x}{3}\right )+16\,x^2+x^4\,{\ln \left (\frac {x}{3}\right )}^2-\ln \left (x\right )\,\left (2\,x^3\,{\ln \left (\frac {x}{3}\right )}^2+8\,x^2\,\ln \left (\frac {x}{3}\right )\right )+x^2\,{\ln \left (\frac {x}{3}\right )}^2\,{\ln \left (x\right )}^2}{{\ln \left (\frac {x}{3}\right )}^2}\right )} \] Input:
int((32*log(3) - 32*log(x/3)*log(3) + 8*log(x/3)^2*log(3)*log(x) - log(x/3 )^2*log(3)*(16*x - 8))/(log((8*x^3*log(x/3) + 16*x^2 + x^4*log(x/3)^2 - lo g(x)*(8*x^2*log(x/3) + 2*x^3*log(x/3)^2) + x^2*log(x/3)^2*log(x)^2)/log(x/ 3)^2)^2*(4*x*log(x/3) + x^2*log(x/3)^2 - x*log(x/3)^2*log(x))),x)
Output:
(4*log(3))/log((8*x^3*log(x/3) + 16*x^2 + x^4*log(x/3)^2 - log(x)*(8*x^2*l og(x/3) + 2*x^3*log(x/3)^2) + x^2*log(x/3)^2*log(x)^2)/log(x/3)^2)
Time = 0.18 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.66 \[ \int \frac {-32 \log (3)+32 \log (3) \log \left (\frac {x}{3}\right )+(-8+16 x) \log (3) \log ^2\left (\frac {x}{3}\right )-8 \log (3) \log ^2\left (\frac {x}{3}\right ) \log (x)}{\left (-4 x \log \left (\frac {x}{3}\right )-x^2 \log ^2\left (\frac {x}{3}\right )+x \log ^2\left (\frac {x}{3}\right ) \log (x)\right ) \log ^2\left (\frac {16 x^2+8 x^3 \log \left (\frac {x}{3}\right )+x^4 \log ^2\left (\frac {x}{3}\right )+\left (-8 x^2 \log \left (\frac {x}{3}\right )-2 x^3 \log ^2\left (\frac {x}{3}\right )\right ) \log (x)+x^2 \log ^2\left (\frac {x}{3}\right ) \log ^2(x)}{\log ^2\left (\frac {x}{3}\right )}\right )} \, dx=\frac {4 \,\mathrm {log}\left (3\right )}{\mathrm {log}\left (\frac {\mathrm {log}\left (\frac {x}{3}\right )^{2} \mathrm {log}\left (x \right )^{2} x^{2}-2 \mathrm {log}\left (\frac {x}{3}\right )^{2} \mathrm {log}\left (x \right ) x^{3}+\mathrm {log}\left (\frac {x}{3}\right )^{2} x^{4}-8 \,\mathrm {log}\left (\frac {x}{3}\right ) \mathrm {log}\left (x \right ) x^{2}+8 \,\mathrm {log}\left (\frac {x}{3}\right ) x^{3}+16 x^{2}}{\mathrm {log}\left (\frac {x}{3}\right )^{2}}\right )} \] Input:
int((-8*log(3)*log(1/3*x)^2*log(x)+(16*x-8)*log(3)*log(1/3*x)^2+32*log(3)* log(1/3*x)-32*log(3))/(x*log(1/3*x)^2*log(x)-x^2*log(1/3*x)^2-4*x*log(1/3* x))/log((x^2*log(1/3*x)^2*log(x)^2+(-2*x^3*log(1/3*x)^2-8*x^2*log(1/3*x))* log(x)+x^4*log(1/3*x)^2+8*x^3*log(1/3*x)+16*x^2)/log(1/3*x)^2)^2,x)
Output:
(4*log(3))/log((log(x/3)**2*log(x)**2*x**2 - 2*log(x/3)**2*log(x)*x**3 + l og(x/3)**2*x**4 - 8*log(x/3)*log(x)*x**2 + 8*log(x/3)*x**3 + 16*x**2)/log( x/3)**2)