Integrand size = 244, antiderivative size = 29 \[ \int \frac {-2 x \log (x)+\left (3 x+x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (6+2 \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )+\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log \left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )}{\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log ^2\left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )} \, dx=\frac {x}{\log \left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \]
Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {-2 x \log (x)+\left (3 x+x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (6+2 \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )+\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log \left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )}{\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log ^2\left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )} \, dx=\frac {x}{\log \left (-5+\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (3+\log ^2(x)\right )}\right )} \]
Integrate[(-2*x*Log[x] + (3*x + x*Log[x]^2)*Log[3 + Log[x]^2] + (6 + 2*Log [x]^2)*Log[3 + Log[x]^2]^2 + ((-3*x - x*Log[x]^2)*Log[3 + Log[x]^2] + (-15 + 3*Log[5] + 3*Log[3/(4*x^2)] + (-5 + Log[5] + Log[3/(4*x^2)])*Log[x]^2)* Log[3 + Log[x]^2]^2)*Log[(-x + (-5 + Log[5] + Log[3/(4*x^2)])*Log[3 + Log[ x]^2])/Log[3 + Log[x]^2]])/(((-3*x - x*Log[x]^2)*Log[3 + Log[x]^2] + (-15 + 3*Log[5] + 3*Log[3/(4*x^2)] + (-5 + Log[5] + Log[3/(4*x^2)])*Log[x]^2)*L og[3 + Log[x]^2]^2)*Log[(-x + (-5 + Log[5] + Log[3/(4*x^2)])*Log[3 + Log[x ]^2])/Log[3 + Log[x]^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.
| \(\displaystyle \int \frac {\left (\left (\left (\log \left (\frac {3}{4 x^2}\right )-5+\log (5)\right ) \log ^2(x)+3 \log \left (\frac {3}{4 x^2}\right )-15+3 \log (5)\right ) \log ^2\left (\log ^2(x)+3\right )+\left (-3 x-x \log ^2(x)\right ) \log \left (\log ^2(x)+3\right )\right ) \log \left (\frac {\left (\log \left (\frac {3}{4 x^2}\right )-5+\log (5)\right ) \log \left (\log ^2(x)+3\right )-x}{\log \left (\log ^2(x)+3\right )}\right )+\left (2 \log ^2(x)+6\right ) \log ^2\left (\log ^2(x)+3\right )+\left (3 x+x \log ^2(x)\right ) \log \left (\log ^2(x)+3\right )-2 x \log (x)}{\left (\left (\left (\log \left (\frac {3}{4 x^2}\right )-5+\log (5)\right ) \log ^2(x)+3 \log \left (\frac {3}{4 x^2}\right )-15+3 \log (5)\right ) \log ^2\left (\log ^2(x)+3\right )+\left (-3 x-x \log ^2(x)\right ) \log \left (\log ^2(x)+3\right )\right ) \log ^2\left (\frac {\left (\log \left (\frac {3}{4 x^2}\right )-5+\log (5)\right ) \log \left (\log ^2(x)+3\right )-x}{\log \left (\log ^2(x)+3\right )}\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-\left (\left (\left (\log \left (\frac {3}{4 x^2}\right )-5+\log (5)\right ) \log ^2(x)+3 \log \left (\frac {3}{4 x^2}\right )-15+3 \log (5)\right ) \log ^2\left (\log ^2(x)+3\right )+\left (-3 x-x \log ^2(x)\right ) \log \left (\log ^2(x)+3\right )\right ) \log \left (\frac {\left (\log \left (\frac {3}{4 x^2}\right )-5+\log (5)\right ) \log \left (\log ^2(x)+3\right )-x}{\log \left (\log ^2(x)+3\right )}\right )-\left (\left (2 \log ^2(x)+6\right ) \log ^2\left (\log ^2(x)+3\right )\right )-\left (3 x+x \log ^2(x)\right ) \log \left (\log ^2(x)+3\right )+2 x \log (x)}{\left (\log ^2(x)+3\right ) \log \left (\log ^2(x)+3\right ) \left (-\log \left (\frac {15}{4 x^2}\right ) \log \left (\log ^2(x)+3\right )+x+5 \log \left (\log ^2(x)+3\right )\right ) \log ^2\left (\frac {\left (\log \left (\frac {3}{4 x^2}\right )-5+\log (5)\right ) \log \left (\log ^2(x)+3\right )-x}{\log \left (\log ^2(x)+3\right )}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 \log ^2\left (\log ^2(x)+3\right ) \log ^2(x)+x \log \left (\log ^2(x)+3\right ) \log ^2(x)+6 \log ^2\left (\log ^2(x)+3\right )+3 x \log \left (\log ^2(x)+3\right )-2 x \log (x)}{\left (\log ^2(x)+3\right ) \log \left (\log ^2(x)+3\right ) \left (\log \left (\frac {15}{4 x^2}\right ) \log \left (\log ^2(x)+3\right )-x-5 \log \left (\log ^2(x)+3\right )\right ) \log ^2\left (\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (\log ^2(x)+3\right )}-5\right )}+\frac {1}{\log \left (\log \left (\frac {15}{4 x^2}\right )-\frac {x}{\log \left (\log ^2(x)+3\right )}-5\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -3 \int \frac {x}{\left (\log ^2(x)+3\right ) \left (x-\log \left (\frac {15}{4 x^2}\right ) \log \left (\log ^2(x)+3\right )+5 \log \left (\log ^2(x)+3\right )\right ) \log ^2\left (-\frac {x}{\log \left (\log ^2(x)+3\right )}+\log \left (\frac {15}{4 x^2}\right )-5\right )}dx-\int \frac {x \log ^2(x)}{\left (\log ^2(x)+3\right ) \left (x-\log \left (\frac {15}{4 x^2}\right ) \log \left (\log ^2(x)+3\right )+5 \log \left (\log ^2(x)+3\right )\right ) \log ^2\left (-\frac {x}{\log \left (\log ^2(x)+3\right )}+\log \left (\frac {15}{4 x^2}\right )-5\right )}dx-2 \int \frac {x \log (x)}{\left (\log ^2(x)+3\right ) \log \left (\log ^2(x)+3\right ) \left (-x+\log \left (\frac {15}{4 x^2}\right ) \log \left (\log ^2(x)+3\right )-5 \log \left (\log ^2(x)+3\right )\right ) \log ^2\left (-\frac {x}{\log \left (\log ^2(x)+3\right )}+\log \left (\frac {15}{4 x^2}\right )-5\right )}dx+6 \int \frac {\log \left (\log ^2(x)+3\right )}{\left (\log ^2(x)+3\right ) \left (-x+\log \left (\frac {15}{4 x^2}\right ) \log \left (\log ^2(x)+3\right )-5 \log \left (\log ^2(x)+3\right )\right ) \log ^2\left (-\frac {x}{\log \left (\log ^2(x)+3\right )}+\log \left (\frac {15}{4 x^2}\right )-5\right )}dx+2 \int \frac {\log ^2(x) \log \left (\log ^2(x)+3\right )}{\left (\log ^2(x)+3\right ) \left (-x+\log \left (\frac {15}{4 x^2}\right ) \log \left (\log ^2(x)+3\right )-5 \log \left (\log ^2(x)+3\right )\right ) \log ^2\left (-\frac {x}{\log \left (\log ^2(x)+3\right )}+\log \left (\frac {15}{4 x^2}\right )-5\right )}dx+\int \frac {1}{\log \left (-\frac {x}{\log \left (\log ^2(x)+3\right )}+\log \left (\frac {15}{4 x^2}\right )-5\right )}dx\) |
Int[(-2*x*Log[x] + (3*x + x*Log[x]^2)*Log[3 + Log[x]^2] + (6 + 2*Log[x]^2) *Log[3 + Log[x]^2]^2 + ((-3*x - x*Log[x]^2)*Log[3 + Log[x]^2] + (-15 + 3*L og[5] + 3*Log[3/(4*x^2)] + (-5 + Log[5] + Log[3/(4*x^2)])*Log[x]^2)*Log[3 + Log[x]^2]^2)*Log[(-x + (-5 + Log[5] + Log[3/(4*x^2)])*Log[3 + Log[x]^2]) /Log[3 + Log[x]^2]])/(((-3*x - x*Log[x]^2)*Log[3 + Log[x]^2] + (-15 + 3*Lo g[5] + 3*Log[3/(4*x^2)] + (-5 + Log[5] + Log[3/(4*x^2)])*Log[x]^2)*Log[3 + Log[x]^2]^2)*Log[(-x + (-5 + Log[5] + Log[3/(4*x^2)])*Log[3 + Log[x]^2])/ Log[3 + Log[x]^2]]^2),x]
3.26.36.3.1 Defintions of rubi rules used
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.06 (sec) , antiderivative size = 1746, normalized size of antiderivative = 60.21
\[\text {Expression too large to display}\]
int(((((ln(3/4/x^2)+ln(5)-5)*ln(x)^2+3*ln(3/4/x^2)+3*ln(5)-15)*ln(ln(x)^2+ 3)^2+(-x*ln(x)^2-3*x)*ln(ln(x)^2+3))*ln(((ln(3/4/x^2)+ln(5)-5)*ln(ln(x)^2+ 3)-x)/ln(ln(x)^2+3))+(2*ln(x)^2+6)*ln(ln(x)^2+3)^2+(x*ln(x)^2+3*x)*ln(ln(x )^2+3)-2*x*ln(x))/(((ln(3/4/x^2)+ln(5)-5)*ln(x)^2+3*ln(3/4/x^2)+3*ln(5)-15 )*ln(ln(x)^2+3)^2+(-x*ln(x)^2-3*x)*ln(ln(x)^2+3))/ln(((ln(3/4/x^2)+ln(5)-5 )*ln(ln(x)^2+3)-x)/ln(ln(x)^2+3))^2,x)
2*I*x/(-Pi*csgn(I*ln(ln(x)^2+3)*Pi*csgn(I*x)^2*csgn(I*x^2)-2*I*ln(ln(x)^2+ 3)*Pi*csgn(I*x)*csgn(I*x^2)^2+I*ln(ln(x)^2+3)*Pi*csgn(I*x^2)^3-4*ln(ln(x)^ 2+3)*ln(x)+2*ln(ln(x)^2+3)*ln(5)+2*ln(ln(x)^2+3)*ln(3/4)-10*ln(ln(x)^2+3)- 2*x)*csgn(I/ln(ln(x)^2+3))*csgn(I/ln(ln(x)^2+3)*(-2*I*x-10*I*ln(ln(x)^2+3) -ln(ln(x)^2+3)*Pi*csgn(I*x)^2*csgn(I*x^2)+2*ln(ln(x)^2+3)*Pi*csgn(I*x)*csg n(I*x^2)^2-ln(ln(x)^2+3)*Pi*csgn(I*x^2)^3+2*I*ln(ln(x)^2+3)*ln(3/4)-4*I*ln (ln(x)^2+3)*ln(x)+2*I*ln(ln(x)^2+3)*ln(5)))+Pi*csgn(I*ln(ln(x)^2+3)*Pi*csg n(I*x)^2*csgn(I*x^2)-2*I*ln(ln(x)^2+3)*Pi*csgn(I*x)*csgn(I*x^2)^2+I*ln(ln( x)^2+3)*Pi*csgn(I*x^2)^3-4*ln(ln(x)^2+3)*ln(x)+2*ln(ln(x)^2+3)*ln(5)+2*ln( ln(x)^2+3)*ln(3/4)-10*ln(ln(x)^2+3)-2*x)*csgn(I/ln(ln(x)^2+3)*(-2*I*x-10*I *ln(ln(x)^2+3)-ln(ln(x)^2+3)*Pi*csgn(I*x)^2*csgn(I*x^2)+2*ln(ln(x)^2+3)*Pi *csgn(I*x)*csgn(I*x^2)^2-ln(ln(x)^2+3)*Pi*csgn(I*x^2)^3+2*I*ln(ln(x)^2+3)* ln(3/4)-4*I*ln(ln(x)^2+3)*ln(x)+2*I*ln(ln(x)^2+3)*ln(5)))^2-Pi*csgn(I/ln(l n(x)^2+3))*csgn(I/ln(ln(x)^2+3)*(-2*I*x-10*I*ln(ln(x)^2+3)-ln(ln(x)^2+3)*P i*csgn(I*x)^2*csgn(I*x^2)+2*ln(ln(x)^2+3)*Pi*csgn(I*x)*csgn(I*x^2)^2-ln(ln (x)^2+3)*Pi*csgn(I*x^2)^3+2*I*ln(ln(x)^2+3)*ln(3/4)-4*I*ln(ln(x)^2+3)*ln(x )+2*I*ln(ln(x)^2+3)*ln(5)))^2+Pi*csgn(I/ln(ln(x)^2+3)*(-2*I*x-10*I*ln(ln(x )^2+3)-ln(ln(x)^2+3)*Pi*csgn(I*x)^2*csgn(I*x^2)+2*ln(ln(x)^2+3)*Pi*csgn(I* x)*csgn(I*x^2)^2-ln(ln(x)^2+3)*Pi*csgn(I*x^2)^3+2*I*ln(ln(x)^2+3)*ln(3/4)- 4*I*ln(ln(x)^2+3)*ln(x)+2*I*ln(ln(x)^2+3)*ln(5)))^3-Pi*csgn(I/ln(ln(x)^...
Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (27) = 54\).
Time = 0.26 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.79 \[ \int \frac {-2 x \log (x)+\left (3 x+x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (6+2 \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )+\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log \left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )}{\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log ^2\left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )} \, dx=\frac {x}{\log \left (\frac {{\left (\log \left (5\right ) + \log \left (\frac {3}{4 \, x^{2}}\right ) - 5\right )} \log \left (\frac {1}{4} \, \log \left (\frac {3}{4}\right )^{2} - \frac {1}{2} \, \log \left (\frac {3}{4}\right ) \log \left (\frac {3}{4 \, x^{2}}\right ) + \frac {1}{4} \, \log \left (\frac {3}{4 \, x^{2}}\right )^{2} + 3\right ) - x}{\log \left (\frac {1}{4} \, \log \left (\frac {3}{4}\right )^{2} - \frac {1}{2} \, \log \left (\frac {3}{4}\right ) \log \left (\frac {3}{4 \, x^{2}}\right ) + \frac {1}{4} \, \log \left (\frac {3}{4 \, x^{2}}\right )^{2} + 3\right )}\right )} \]
integrate(((((log(3/4/x^2)+log(5)-5)*log(x)^2+3*log(3/4/x^2)+3*log(5)-15)* log(log(x)^2+3)^2+(-x*log(x)^2-3*x)*log(log(x)^2+3))*log(((log(3/4/x^2)+lo g(5)-5)*log(log(x)^2+3)-x)/log(log(x)^2+3))+(2*log(x)^2+6)*log(log(x)^2+3) ^2+(x*log(x)^2+3*x)*log(log(x)^2+3)-2*x*log(x))/(((log(3/4/x^2)+log(5)-5)* log(x)^2+3*log(3/4/x^2)+3*log(5)-15)*log(log(x)^2+3)^2+(-x*log(x)^2-3*x)*l og(log(x)^2+3))/log(((log(3/4/x^2)+log(5)-5)*log(log(x)^2+3)-x)/log(log(x) ^2+3))^2,x, algorithm=\
x/log(((log(5) + log(3/4/x^2) - 5)*log(1/4*log(3/4)^2 - 1/2*log(3/4)*log(3 /4/x^2) + 1/4*log(3/4/x^2)^2 + 3) - x)/log(1/4*log(3/4)^2 - 1/2*log(3/4)*l og(3/4/x^2) + 1/4*log(3/4/x^2)^2 + 3))
Time = 107.45 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.24 \[ \int \frac {-2 x \log (x)+\left (3 x+x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (6+2 \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )+\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log \left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )}{\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log ^2\left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )} \, dx=\frac {x}{\log {\left (\frac {- x + \left (- 2 \log {\left (x \right )} - 5 + \log {\left (\frac {3}{4} \right )} + \log {\left (5 \right )}\right ) \log {\left (\log {\left (x \right )}^{2} + 3 \right )}}{\log {\left (\log {\left (x \right )}^{2} + 3 \right )}} \right )}} \]
integrate(((((ln(3/4/x**2)+ln(5)-5)*ln(x)**2+3*ln(3/4/x**2)+3*ln(5)-15)*ln (ln(x)**2+3)**2+(-x*ln(x)**2-3*x)*ln(ln(x)**2+3))*ln(((ln(3/4/x**2)+ln(5)- 5)*ln(ln(x)**2+3)-x)/ln(ln(x)**2+3))+(2*ln(x)**2+6)*ln(ln(x)**2+3)**2+(x*l n(x)**2+3*x)*ln(ln(x)**2+3)-2*x*ln(x))/(((ln(3/4/x**2)+ln(5)-5)*ln(x)**2+3 *ln(3/4/x**2)+3*ln(5)-15)*ln(ln(x)**2+3)**2+(-x*ln(x)**2-3*x)*ln(ln(x)**2+ 3))/ln(((ln(3/4/x**2)+ln(5)-5)*ln(ln(x)**2+3)-x)/ln(ln(x)**2+3))**2,x)
Time = 0.39 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.69 \[ \int \frac {-2 x \log (x)+\left (3 x+x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (6+2 \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )+\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log \left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )}{\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log ^2\left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )} \, dx=\frac {x}{\log \left ({\left (\log \left (5\right ) + \log \left (3\right ) - 2 \, \log \left (2\right ) - 5\right )} \log \left (\log \left (x\right )^{2} + 3\right ) - 2 \, \log \left (\log \left (x\right )^{2} + 3\right ) \log \left (x\right ) - x\right ) - \log \left (\log \left (\log \left (x\right )^{2} + 3\right )\right )} \]
integrate(((((log(3/4/x^2)+log(5)-5)*log(x)^2+3*log(3/4/x^2)+3*log(5)-15)* log(log(x)^2+3)^2+(-x*log(x)^2-3*x)*log(log(x)^2+3))*log(((log(3/4/x^2)+lo g(5)-5)*log(log(x)^2+3)-x)/log(log(x)^2+3))+(2*log(x)^2+6)*log(log(x)^2+3) ^2+(x*log(x)^2+3*x)*log(log(x)^2+3)-2*x*log(x))/(((log(3/4/x^2)+log(5)-5)* log(x)^2+3*log(3/4/x^2)+3*log(5)-15)*log(log(x)^2+3)^2+(-x*log(x)^2-3*x)*l og(log(x)^2+3))/log(((log(3/4/x^2)+log(5)-5)*log(log(x)^2+3)-x)/log(log(x) ^2+3))^2,x, algorithm=\
x/(log((log(5) + log(3) - 2*log(2) - 5)*log(log(x)^2 + 3) - 2*log(log(x)^2 + 3)*log(x) - x) - log(log(log(x)^2 + 3)))
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (27) = 54\).
Time = 2.71 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.45 \[ \int \frac {-2 x \log (x)+\left (3 x+x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (6+2 \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )+\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log \left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )}{\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log ^2\left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )} \, dx=\frac {x}{\log \left (\log \left (5\right ) \log \left (\log \left (x\right )^{2} + 3\right ) + \log \left (3\right ) \log \left (\log \left (x\right )^{2} + 3\right ) - 2 \, \log \left (2\right ) \log \left (\log \left (x\right )^{2} + 3\right ) - 2 \, \log \left (\log \left (x\right )^{2} + 3\right ) \log \left (x\right ) - x - 5 \, \log \left (\log \left (x\right )^{2} + 3\right )\right ) - \log \left (\log \left (\log \left (x\right )^{2} + 3\right )\right )} \]
integrate(((((log(3/4/x^2)+log(5)-5)*log(x)^2+3*log(3/4/x^2)+3*log(5)-15)* log(log(x)^2+3)^2+(-x*log(x)^2-3*x)*log(log(x)^2+3))*log(((log(3/4/x^2)+lo g(5)-5)*log(log(x)^2+3)-x)/log(log(x)^2+3))+(2*log(x)^2+6)*log(log(x)^2+3) ^2+(x*log(x)^2+3*x)*log(log(x)^2+3)-2*x*log(x))/(((log(3/4/x^2)+log(5)-5)* log(x)^2+3*log(3/4/x^2)+3*log(5)-15)*log(log(x)^2+3)^2+(-x*log(x)^2-3*x)*l og(log(x)^2+3))/log(((log(3/4/x^2)+log(5)-5)*log(log(x)^2+3)-x)/log(log(x) ^2+3))^2,x, algorithm=\
x/(log(log(5)*log(log(x)^2 + 3) + log(3)*log(log(x)^2 + 3) - 2*log(2)*log( log(x)^2 + 3) - 2*log(log(x)^2 + 3)*log(x) - x - 5*log(log(x)^2 + 3)) - lo g(log(log(x)^2 + 3)))
Timed out. \[ \int \frac {-2 x \log (x)+\left (3 x+x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (6+2 \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )+\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log \left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )}{\left (\left (-3 x-x \log ^2(x)\right ) \log \left (3+\log ^2(x)\right )+\left (-15+3 \log (5)+3 \log \left (\frac {3}{4 x^2}\right )+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log ^2(x)\right ) \log ^2\left (3+\log ^2(x)\right )\right ) \log ^2\left (\frac {-x+\left (-5+\log (5)+\log \left (\frac {3}{4 x^2}\right )\right ) \log \left (3+\log ^2(x)\right )}{\log \left (3+\log ^2(x)\right )}\right )} \, dx=\int \frac {\ln \left (-\frac {x-\ln \left ({\ln \left (x\right )}^2+3\right )\,\left (\ln \left (5\right )+\ln \left (\frac {3}{4\,x^2}\right )-5\right )}{\ln \left ({\ln \left (x\right )}^2+3\right )}\right )\,\left ({\ln \left ({\ln \left (x\right )}^2+3\right )}^2\,\left (\left (\ln \left (5\right )+\ln \left (\frac {3}{4\,x^2}\right )-5\right )\,{\ln \left (x\right )}^2+3\,\ln \left (5\right )+3\,\ln \left (\frac {3}{4\,x^2}\right )-15\right )-\ln \left ({\ln \left (x\right )}^2+3\right )\,\left (x\,{\ln \left (x\right )}^2+3\,x\right )\right )-2\,x\,\ln \left (x\right )+\ln \left ({\ln \left (x\right )}^2+3\right )\,\left (x\,{\ln \left (x\right )}^2+3\,x\right )+{\ln \left ({\ln \left (x\right )}^2+3\right )}^2\,\left (2\,{\ln \left (x\right )}^2+6\right )}{{\ln \left (-\frac {x-\ln \left ({\ln \left (x\right )}^2+3\right )\,\left (\ln \left (5\right )+\ln \left (\frac {3}{4\,x^2}\right )-5\right )}{\ln \left ({\ln \left (x\right )}^2+3\right )}\right )}^2\,\left ({\ln \left ({\ln \left (x\right )}^2+3\right )}^2\,\left (\left (\ln \left (5\right )+\ln \left (\frac {3}{4\,x^2}\right )-5\right )\,{\ln \left (x\right )}^2+3\,\ln \left (5\right )+3\,\ln \left (\frac {3}{4\,x^2}\right )-15\right )-\ln \left ({\ln \left (x\right )}^2+3\right )\,\left (x\,{\ln \left (x\right )}^2+3\,x\right )\right )} \,d x \]
int((log(-(x - log(log(x)^2 + 3)*(log(5) + log(3/(4*x^2)) - 5))/log(log(x) ^2 + 3))*(log(log(x)^2 + 3)^2*(3*log(5) + 3*log(3/(4*x^2)) + log(x)^2*(log (5) + log(3/(4*x^2)) - 5) - 15) - log(log(x)^2 + 3)*(3*x + x*log(x)^2)) - 2*x*log(x) + log(log(x)^2 + 3)*(3*x + x*log(x)^2) + log(log(x)^2 + 3)^2*(2 *log(x)^2 + 6))/(log(-(x - log(log(x)^2 + 3)*(log(5) + log(3/(4*x^2)) - 5) )/log(log(x)^2 + 3))^2*(log(log(x)^2 + 3)^2*(3*log(5) + 3*log(3/(4*x^2)) + log(x)^2*(log(5) + log(3/(4*x^2)) - 5) - 15) - log(log(x)^2 + 3)*(3*x + x *log(x)^2))),x)
int((log(-(x - log(log(x)^2 + 3)*(log(5) + log(3/(4*x^2)) - 5))/log(log(x) ^2 + 3))*(log(log(x)^2 + 3)^2*(3*log(5) + 3*log(3/(4*x^2)) + log(x)^2*(log (5) + log(3/(4*x^2)) - 5) - 15) - log(log(x)^2 + 3)*(3*x + x*log(x)^2)) - 2*x*log(x) + log(log(x)^2 + 3)*(3*x + x*log(x)^2) + log(log(x)^2 + 3)^2*(2 *log(x)^2 + 6))/(log(-(x - log(log(x)^2 + 3)*(log(5) + log(3/(4*x^2)) - 5) )/log(log(x)^2 + 3))^2*(log(log(x)^2 + 3)^2*(3*log(5) + 3*log(3/(4*x^2)) + log(x)^2*(log(5) + log(3/(4*x^2)) - 5) - 15) - log(log(x)^2 + 3)*(3*x + x *log(x)^2))), x)