Integrand size = 179, antiderivative size = 34 \[ \int \frac {-8 x^2+\left (4 x^2-4 x^5\right ) \log (x)+\left (8 x+\left (-4 x+4 x^2\right ) \log (x)\right ) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-4 x \log (x) \log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )}{\left (9 x^4+6 x^6+x^8\right ) \log (x)+\left (12 x^3+4 x^5\right ) \log (x) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\left (-2 x^2-2 x^4\right ) \log (x) \log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )-4 x \log (x) \log ^3\left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\log (x) \log ^4\left (\frac {3 \log (5) \log ^2(x)}{x}\right )} \, dx=2+\frac {2}{4+x^2-\frac {\left (-x+\log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )\right )^2}{x^2}} \]
Time = 0.07 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.38 \[ \int \frac {-8 x^2+\left (4 x^2-4 x^5\right ) \log (x)+\left (8 x+\left (-4 x+4 x^2\right ) \log (x)\right ) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-4 x \log (x) \log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )}{\left (9 x^4+6 x^6+x^8\right ) \log (x)+\left (12 x^3+4 x^5\right ) \log (x) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\left (-2 x^2-2 x^4\right ) \log (x) \log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )-4 x \log (x) \log ^3\left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\log (x) \log ^4\left (\frac {3 \log (5) \log ^2(x)}{x}\right )} \, dx=-\frac {2 x^2}{-3 x^2-x^4-2 x \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )} \]
Integrate[(-8*x^2 + (4*x^2 - 4*x^5)*Log[x] + (8*x + (-4*x + 4*x^2)*Log[x]) *Log[(3*Log[5]*Log[x]^2)/x] - 4*x*Log[x]*Log[(3*Log[5]*Log[x]^2)/x]^2)/((9 *x^4 + 6*x^6 + x^8)*Log[x] + (12*x^3 + 4*x^5)*Log[x]*Log[(3*Log[5]*Log[x]^ 2)/x] + (-2*x^2 - 2*x^4)*Log[x]*Log[(3*Log[5]*Log[x]^2)/x]^2 - 4*x*Log[x]* Log[(3*Log[5]*Log[x]^2)/x]^3 + Log[x]*Log[(3*Log[5]*Log[x]^2)/x]^4),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.
| \(\displaystyle \int \frac {-8 x^2+\left (\left (4 x^2-4 x\right ) \log (x)+8 x\right ) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\left (4 x^2-4 x^5\right ) \log (x)-4 x \log (x) \log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )}{\left (4 x^5+12 x^3\right ) \log (x) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\left (-2 x^4-2 x^2\right ) \log (x) \log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\left (x^8+6 x^6+9 x^4\right ) \log (x)+\log (x) \log ^4\left (\frac {3 \log (5) \log ^2(x)}{x}\right )-4 x \log (x) \log ^3\left (\frac {3 \log (5) \log ^2(x)}{x}\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-8 x^2+\left (\left (4 x^2-4 x\right ) \log (x)+8 x\right ) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\left (4 x^2-4 x^5\right ) \log (x)-4 x \log (x) \log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )}{\log (x) \left (x^4+3 x^2+2 x \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-\log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {4 x}{x^4+3 x^2+2 x \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-\log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )}-\frac {4 x \left (2 x^4 \log (x)+3 x^2 \log (x)+2 x+x \log (x) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\log (x) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-2 \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-x \log (x)\right )}{\log (x) \left (x^4+3 x^2+2 x \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-\log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 4 \int \frac {x^2}{\left (x^4+3 x^2+2 \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right ) x-\log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )\right )^2}dx-8 \int \frac {x^2}{\log (x) \left (x^4+3 x^2+2 \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right ) x-\log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )\right )^2}dx-4 \int \frac {x \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )}{\left (x^4+3 x^2+2 \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right ) x-\log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )\right )^2}dx-4 \int \frac {x^2 \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )}{\left (x^4+3 x^2+2 \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right ) x-\log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )\right )^2}dx+8 \int \frac {x \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )}{\log (x) \left (x^4+3 x^2+2 \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right ) x-\log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )\right )^2}dx+4 \int \frac {x}{x^4+3 x^2+2 \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right ) x-\log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )}dx-8 \int \frac {x^5}{\left (x^4+3 x^2+2 \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right ) x-\log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )\right )^2}dx-12 \int \frac {x^3}{\left (x^4+3 x^2+2 \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right ) x-\log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )\right )^2}dx\) |
Int[(-8*x^2 + (4*x^2 - 4*x^5)*Log[x] + (8*x + (-4*x + 4*x^2)*Log[x])*Log[( 3*Log[5]*Log[x]^2)/x] - 4*x*Log[x]*Log[(3*Log[5]*Log[x]^2)/x]^2)/((9*x^4 + 6*x^6 + x^8)*Log[x] + (12*x^3 + 4*x^5)*Log[x]*Log[(3*Log[5]*Log[x]^2)/x] + (-2*x^2 - 2*x^4)*Log[x]*Log[(3*Log[5]*Log[x]^2)/x]^2 - 4*x*Log[x]*Log[(3 *Log[5]*Log[x]^2)/x]^3 + Log[x]*Log[(3*Log[5]*Log[x]^2)/x]^4),x]
3.24.100.3.1 Defintions of rubi rules used
Time = 235.50 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.41
| method | result | size |
| parallelrisch | \(\frac {2 x^{2}}{x^{4}+3 x^{2}+2 x \ln \left (\frac {3 \ln \left (5\right ) \ln \left (x \right )^{2}}{x}\right )-\ln \left (\frac {3 \ln \left (5\right ) \ln \left (x \right )^{2}}{x}\right )^{2}}\) | \(48\) |
| risch | \(\text {Expression too large to display}\) | \(1723\) |
int((-4*x*ln(x)*ln(3*ln(5)*ln(x)^2/x)^2+((4*x^2-4*x)*ln(x)+8*x)*ln(3*ln(5) *ln(x)^2/x)+(-4*x^5+4*x^2)*ln(x)-8*x^2)/(ln(x)*ln(3*ln(5)*ln(x)^2/x)^4-4*x *ln(x)*ln(3*ln(5)*ln(x)^2/x)^3+(-2*x^4-2*x^2)*ln(x)*ln(3*ln(5)*ln(x)^2/x)^ 2+(4*x^5+12*x^3)*ln(x)*ln(3*ln(5)*ln(x)^2/x)+(x^8+6*x^6+9*x^4)*ln(x)),x,me thod=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.38 \[ \int \frac {-8 x^2+\left (4 x^2-4 x^5\right ) \log (x)+\left (8 x+\left (-4 x+4 x^2\right ) \log (x)\right ) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-4 x \log (x) \log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )}{\left (9 x^4+6 x^6+x^8\right ) \log (x)+\left (12 x^3+4 x^5\right ) \log (x) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\left (-2 x^2-2 x^4\right ) \log (x) \log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )-4 x \log (x) \log ^3\left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\log (x) \log ^4\left (\frac {3 \log (5) \log ^2(x)}{x}\right )} \, dx=\frac {2 \, x^{2}}{x^{4} + 3 \, x^{2} + 2 \, x \log \left (\frac {3 \, \log \left (5\right ) \log \left (x\right )^{2}}{x}\right ) - \log \left (\frac {3 \, \log \left (5\right ) \log \left (x\right )^{2}}{x}\right )^{2}} \]
integrate((-4*x*log(x)*log(3*log(5)*log(x)^2/x)^2+((4*x^2-4*x)*log(x)+8*x) *log(3*log(5)*log(x)^2/x)+(-4*x^5+4*x^2)*log(x)-8*x^2)/(log(x)*log(3*log(5 )*log(x)^2/x)^4-4*x*log(x)*log(3*log(5)*log(x)^2/x)^3+(-2*x^4-2*x^2)*log(x )*log(3*log(5)*log(x)^2/x)^2+(4*x^5+12*x^3)*log(x)*log(3*log(5)*log(x)^2/x )+(x^8+6*x^6+9*x^4)*log(x)),x, algorithm=\
Time = 0.19 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.35 \[ \int \frac {-8 x^2+\left (4 x^2-4 x^5\right ) \log (x)+\left (8 x+\left (-4 x+4 x^2\right ) \log (x)\right ) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-4 x \log (x) \log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )}{\left (9 x^4+6 x^6+x^8\right ) \log (x)+\left (12 x^3+4 x^5\right ) \log (x) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\left (-2 x^2-2 x^4\right ) \log (x) \log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )-4 x \log (x) \log ^3\left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\log (x) \log ^4\left (\frac {3 \log (5) \log ^2(x)}{x}\right )} \, dx=- \frac {2 x^{2}}{- x^{4} - 3 x^{2} - 2 x \log {\left (\frac {3 \log {\left (5 \right )} \log {\left (x \right )}^{2}}{x} \right )} + \log {\left (\frac {3 \log {\left (5 \right )} \log {\left (x \right )}^{2}}{x} \right )}^{2}} \]
integrate((-4*x*ln(x)*ln(3*ln(5)*ln(x)**2/x)**2+((4*x**2-4*x)*ln(x)+8*x)*l n(3*ln(5)*ln(x)**2/x)+(-4*x**5+4*x**2)*ln(x)-8*x**2)/(ln(x)*ln(3*ln(5)*ln( x)**2/x)**4-4*x*ln(x)*ln(3*ln(5)*ln(x)**2/x)**3+(-2*x**4-2*x**2)*ln(x)*ln( 3*ln(5)*ln(x)**2/x)**2+(4*x**5+12*x**3)*ln(x)*ln(3*ln(5)*ln(x)**2/x)+(x**8 +6*x**6+9*x**4)*ln(x)),x)
Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (34) = 68\).
Time = 0.39 (sec) , antiderivative size = 91, normalized size of antiderivative = 2.68 \[ \int \frac {-8 x^2+\left (4 x^2-4 x^5\right ) \log (x)+\left (8 x+\left (-4 x+4 x^2\right ) \log (x)\right ) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-4 x \log (x) \log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )}{\left (9 x^4+6 x^6+x^8\right ) \log (x)+\left (12 x^3+4 x^5\right ) \log (x) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\left (-2 x^2-2 x^4\right ) \log (x) \log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )-4 x \log (x) \log ^3\left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\log (x) \log ^4\left (\frac {3 \log (5) \log ^2(x)}{x}\right )} \, dx=\frac {2 \, x^{2}}{x^{4} + 3 \, x^{2} + 2 \, x {\left (\log \left (3\right ) + \log \left (\log \left (5\right )\right )\right )} - \log \left (3\right )^{2} - 2 \, {\left (x - \log \left (3\right ) - \log \left (\log \left (5\right )\right )\right )} \log \left (x\right ) - \log \left (x\right )^{2} - 2 \, \log \left (3\right ) \log \left (\log \left (5\right )\right ) - \log \left (\log \left (5\right )\right )^{2} + 4 \, {\left (x - \log \left (3\right ) + \log \left (x\right ) - \log \left (\log \left (5\right )\right )\right )} \log \left (\log \left (x\right )\right ) - 4 \, \log \left (\log \left (x\right )\right )^{2}} \]
integrate((-4*x*log(x)*log(3*log(5)*log(x)^2/x)^2+((4*x^2-4*x)*log(x)+8*x) *log(3*log(5)*log(x)^2/x)+(-4*x^5+4*x^2)*log(x)-8*x^2)/(log(x)*log(3*log(5 )*log(x)^2/x)^4-4*x*log(x)*log(3*log(5)*log(x)^2/x)^3+(-2*x^4-2*x^2)*log(x )*log(3*log(5)*log(x)^2/x)^2+(4*x^5+12*x^3)*log(x)*log(3*log(5)*log(x)^2/x )+(x^8+6*x^6+9*x^4)*log(x)),x, algorithm=\
2*x^2/(x^4 + 3*x^2 + 2*x*(log(3) + log(log(5))) - log(3)^2 - 2*(x - log(3) - log(log(5)))*log(x) - log(x)^2 - 2*log(3)*log(log(5)) - log(log(5))^2 + 4*(x - log(3) + log(x) - log(log(5)))*log(log(x)) - 4*log(log(x))^2)
Time = 2.01 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.91 \[ \int \frac {-8 x^2+\left (4 x^2-4 x^5\right ) \log (x)+\left (8 x+\left (-4 x+4 x^2\right ) \log (x)\right ) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-4 x \log (x) \log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )}{\left (9 x^4+6 x^6+x^8\right ) \log (x)+\left (12 x^3+4 x^5\right ) \log (x) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\left (-2 x^2-2 x^4\right ) \log (x) \log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )-4 x \log (x) \log ^3\left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\log (x) \log ^4\left (\frac {3 \log (5) \log ^2(x)}{x}\right )} \, dx=\frac {2 \, x^{2}}{x^{4} + 3 \, x^{2} + 2 \, x \log \left (3 \, \log \left (5\right ) \log \left (x\right )^{2}\right ) - \log \left (3 \, \log \left (5\right ) \log \left (x\right )^{2}\right )^{2} - 2 \, x \log \left (x\right ) + 2 \, \log \left (3 \, \log \left (5\right ) \log \left (x\right )^{2}\right ) \log \left (x\right ) - \log \left (x\right )^{2}} \]
integrate((-4*x*log(x)*log(3*log(5)*log(x)^2/x)^2+((4*x^2-4*x)*log(x)+8*x) *log(3*log(5)*log(x)^2/x)+(-4*x^5+4*x^2)*log(x)-8*x^2)/(log(x)*log(3*log(5 )*log(x)^2/x)^4-4*x*log(x)*log(3*log(5)*log(x)^2/x)^3+(-2*x^4-2*x^2)*log(x )*log(3*log(5)*log(x)^2/x)^2+(4*x^5+12*x^3)*log(x)*log(3*log(5)*log(x)^2/x )+(x^8+6*x^6+9*x^4)*log(x)),x, algorithm=\
2*x^2/(x^4 + 3*x^2 + 2*x*log(3*log(5)*log(x)^2) - log(3*log(5)*log(x)^2)^2 - 2*x*log(x) + 2*log(3*log(5)*log(x)^2)*log(x) - log(x)^2)
Timed out. \[ \int \frac {-8 x^2+\left (4 x^2-4 x^5\right ) \log (x)+\left (8 x+\left (-4 x+4 x^2\right ) \log (x)\right ) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )-4 x \log (x) \log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )}{\left (9 x^4+6 x^6+x^8\right ) \log (x)+\left (12 x^3+4 x^5\right ) \log (x) \log \left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\left (-2 x^2-2 x^4\right ) \log (x) \log ^2\left (\frac {3 \log (5) \log ^2(x)}{x}\right )-4 x \log (x) \log ^3\left (\frac {3 \log (5) \log ^2(x)}{x}\right )+\log (x) \log ^4\left (\frac {3 \log (5) \log ^2(x)}{x}\right )} \, dx=\int \frac {\ln \left (x\right )\,\left (4\,x^2-4\,x^5\right )+\ln \left (\frac {3\,\ln \left (5\right )\,{\ln \left (x\right )}^2}{x}\right )\,\left (8\,x-\ln \left (x\right )\,\left (4\,x-4\,x^2\right )\right )-8\,x^2-4\,x\,{\ln \left (\frac {3\,\ln \left (5\right )\,{\ln \left (x\right )}^2}{x}\right )}^2\,\ln \left (x\right )}{\ln \left (x\right )\,{\ln \left (\frac {3\,\ln \left (5\right )\,{\ln \left (x\right )}^2}{x}\right )}^4-4\,x\,\ln \left (x\right )\,{\ln \left (\frac {3\,\ln \left (5\right )\,{\ln \left (x\right )}^2}{x}\right )}^3-\ln \left (x\right )\,\left (2\,x^4+2\,x^2\right )\,{\ln \left (\frac {3\,\ln \left (5\right )\,{\ln \left (x\right )}^2}{x}\right )}^2+\ln \left (x\right )\,\left (4\,x^5+12\,x^3\right )\,\ln \left (\frac {3\,\ln \left (5\right )\,{\ln \left (x\right )}^2}{x}\right )+\ln \left (x\right )\,\left (x^8+6\,x^6+9\,x^4\right )} \,d x \]
int((log(x)*(4*x^2 - 4*x^5) + log((3*log(5)*log(x)^2)/x)*(8*x - log(x)*(4* x - 4*x^2)) - 8*x^2 - 4*x*log((3*log(5)*log(x)^2)/x)^2*log(x))/(log(x)*(9* x^4 + 6*x^6 + x^8) + log((3*log(5)*log(x)^2)/x)^4*log(x) + log((3*log(5)*l og(x)^2)/x)*log(x)*(12*x^3 + 4*x^5) - 4*x*log((3*log(5)*log(x)^2)/x)^3*log (x) - log((3*log(5)*log(x)^2)/x)^2*log(x)*(2*x^2 + 2*x^4)),x)
int((log(x)*(4*x^2 - 4*x^5) + log((3*log(5)*log(x)^2)/x)*(8*x - log(x)*(4* x - 4*x^2)) - 8*x^2 - 4*x*log((3*log(5)*log(x)^2)/x)^2*log(x))/(log(x)*(9* x^4 + 6*x^6 + x^8) + log((3*log(5)*log(x)^2)/x)^4*log(x) + log((3*log(5)*l og(x)^2)/x)*log(x)*(12*x^3 + 4*x^5) - 4*x*log((3*log(5)*log(x)^2)/x)^3*log (x) - log((3*log(5)*log(x)^2)/x)^2*log(x)*(2*x^2 + 2*x^4)), x)