Integrand size = 132, antiderivative size = 25 \[ \int \left (2 x^2-4 x \log (2)+2 \log ^2(2)+\left (3 x^2-4 x \log (2)+\log ^2(2)\right ) \log \left (\frac {x^2}{\log ^4(3)}\right )+(-12 x+12 \log (2)) \log \left (x^3\right ) \log \left (\frac {x^2}{\log ^4(3)}\right )+12 \log ^3\left (x^3\right ) \log \left (\frac {x^2}{\log ^4(3)}\right )+\log ^4\left (x^3\right ) \left (2+\log \left (\frac {x^2}{\log ^4(3)}\right )\right )+\log ^2\left (x^3\right ) \left (-4 x+4 \log (2)+(-4 x+2 \log (2)) \log \left (\frac {x^2}{\log ^4(3)}\right )\right )\right ) \, dx=x \left (-x+\log (2)+\log ^2\left (x^3\right )\right )^2 \log \left (\frac {x^2}{\log ^4(3)}\right ) \] Output:
(ln(2)-x+ln(x^3)^2)^2*ln(x^2/ln(3)^4)*x
Leaf count is larger than twice the leaf count of optimal. \(112\) vs. \(2(25)=50\).
Time = 0.19 (sec) , antiderivative size = 112, normalized size of antiderivative = 4.48 \[ \int \left (2 x^2-4 x \log (2)+2 \log ^2(2)+\left (3 x^2-4 x \log (2)+\log ^2(2)\right ) \log \left (\frac {x^2}{\log ^4(3)}\right )+(-12 x+12 \log (2)) \log \left (x^3\right ) \log \left (\frac {x^2}{\log ^4(3)}\right )+12 \log ^3\left (x^3\right ) \log \left (\frac {x^2}{\log ^4(3)}\right )+\log ^4\left (x^3\right ) \left (2+\log \left (\frac {x^2}{\log ^4(3)}\right )\right )+\log ^2\left (x^3\right ) \left (-4 x+4 \log (2)+(-4 x+2 \log (2)) \log \left (\frac {x^2}{\log ^4(3)}\right )\right )\right ) \, dx=\frac {1}{2} x \left (-4 \log ^2(2)+\log ^2(4)+\log \left (x^2\right ) \left (2 x^2+2 \log ^2(2)-x \log (16)+(-4 x+\log (16)) \log ^2\left (x^3\right )+2 \log ^4\left (x^3\right )\right )-8 x^2 \log (\log (3))-8 \log ^2(2) \log (\log (3))+x \log (65536) \log (\log (3))+(16 x-\log (65536)) \log ^2\left (x^3\right ) \log (\log (3))-8 \log ^4\left (x^3\right ) \log (\log (3))\right ) \] Input:
Integrate[2*x^2 - 4*x*Log[2] + 2*Log[2]^2 + (3*x^2 - 4*x*Log[2] + Log[2]^2 )*Log[x^2/Log[3]^4] + (-12*x + 12*Log[2])*Log[x^3]*Log[x^2/Log[3]^4] + 12* Log[x^3]^3*Log[x^2/Log[3]^4] + Log[x^3]^4*(2 + Log[x^2/Log[3]^4]) + Log[x^ 3]^2*(-4*x + 4*Log[2] + (-4*x + 2*Log[2])*Log[x^2/Log[3]^4]),x]
Output:
(x*(-4*Log[2]^2 + Log[4]^2 + Log[x^2]*(2*x^2 + 2*Log[2]^2 - x*Log[16] + (- 4*x + Log[16])*Log[x^3]^2 + 2*Log[x^3]^4) - 8*x^2*Log[Log[3]] - 8*Log[2]^2 *Log[Log[3]] + x*Log[65536]*Log[Log[3]] + (16*x - Log[65536])*Log[x^3]^2*L og[Log[3]] - 8*Log[x^3]^4*Log[Log[3]]))/2
Leaf count is larger than twice the leaf count of optimal. \(441\) vs. \(2(25)=50\).
Time = 0.97 (sec) , antiderivative size = 441, normalized size of antiderivative = 17.64, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.008, Rules used = {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.
| \(\displaystyle \int \left (2 x^2+\left (3 x^2-4 x \log (2)+\log ^2(2)\right ) \log \left (\frac {x^2}{\log ^4(3)}\right )+\left (\log \left (\frac {x^2}{\log ^4(3)}\right )+2\right ) \log ^4\left (x^3\right )+(12 \log (2)-12 x) \log \left (\frac {x^2}{\log ^4(3)}\right ) \log \left (x^3\right )+12 \log \left (\frac {x^2}{\log ^4(3)}\right ) \log ^3\left (x^3\right )+\left ((2 \log (2)-4 x) \log \left (\frac {x^2}{\log ^4(3)}\right )-4 x+4 \log (2)\right ) \log ^2\left (x^3\right )-4 x \log (2)+2 \log ^2(2)\right ) \, dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 x \log ^4\left (x^3\right )+24 x \log ^3\left (x^3\right )-216 x \log ^2\left (x^3\right )+\frac {1}{12} \log ^2(4096) \log (x) \log \left (x^3\right )+1296 x \log \left (x^3\right )-2 x \log (4096) \log \left (x^3\right )+24 x \log (2) \log \left (x^3\right )-1944 x \log \left (\frac {x^2}{\log ^4(3)}\right )-3 x \log (4096) \log \left (\frac {x^2}{\log ^4(3)}\right )-2 x^2 \log (2) \log \left (\frac {x^2}{\log ^4(3)}\right )+36 x \log (2) \log \left (\frac {x^2}{\log ^4(3)}\right )+1944 x \left (\log \left (\frac {x^2}{\log ^4(3)}\right )+2\right )+\frac {1}{8} \log ^2(4096) \log (x) \log \left (\frac {x^2}{\log ^4(3)}\right )+x \log ^2(2) \log \left (\frac {x^2}{\log ^4(3)}\right )+x \left (\log \left (\frac {x^2}{\log ^4(3)}\right )+2\right ) \log ^4\left (x^3\right )+6 x^2 \log \left (\frac {x^2}{\log ^4(3)}\right ) \log \left (x^3\right )-\frac {1}{24} (12 x-\log (4096))^2 \log \left (\frac {x^2}{\log ^4(3)}\right ) \log \left (x^3\right )+648 x \log \left (\frac {x^2}{\log ^4(3)}\right ) \log \left (x^3\right )-12 x \log (2) \log \left (\frac {x^2}{\log ^4(3)}\right ) \log \left (x^3\right )-648 x \left (\log \left (\frac {x^2}{\log ^4(3)}\right )+2\right ) \log \left (x^3\right )+x^3 \log \left (\frac {x^2}{\log ^4(3)}\right )+12 x \log \left (\frac {x^2}{\log ^4(3)}\right ) \log ^3\left (x^3\right )-12 x \left (\log \left (\frac {x^2}{\log ^4(3)}\right )+2\right ) \log ^3\left (x^3\right )-2 x^2 \log \left (\frac {x^2}{\log ^4(3)}\right ) \log ^2\left (x^3\right )-108 x \log \left (\frac {x^2}{\log ^4(3)}\right ) \log ^2\left (x^3\right )+2 x \log (2) \log \left (\frac {x^2}{\log ^4(3)}\right ) \log ^2\left (x^3\right )+108 x \left (\log \left (\frac {x^2}{\log ^4(3)}\right )+2\right ) \log ^2\left (x^3\right )-3888 x-\frac {1}{4} \log ^2(4096) \log ^2(x)+12 x \log (4096)-144 x \log (2)\) |
Input:
Int[2*x^2 - 4*x*Log[2] + 2*Log[2]^2 + (3*x^2 - 4*x*Log[2] + Log[2]^2)*Log[ x^2/Log[3]^4] + (-12*x + 12*Log[2])*Log[x^3]*Log[x^2/Log[3]^4] + 12*Log[x^ 3]^3*Log[x^2/Log[3]^4] + Log[x^3]^4*(2 + Log[x^2/Log[3]^4]) + Log[x^3]^2*( -4*x + 4*Log[2] + (-4*x + 2*Log[2])*Log[x^2/Log[3]^4]),x]
Output:
-3888*x - 144*x*Log[2] + 12*x*Log[4096] - (Log[4096]^2*Log[x]^2)/4 + 1296* x*Log[x^3] + 24*x*Log[2]*Log[x^3] - 2*x*Log[4096]*Log[x^3] + (Log[4096]^2* Log[x]*Log[x^3])/12 - 216*x*Log[x^3]^2 + 24*x*Log[x^3]^3 - 2*x*Log[x^3]^4 - 1944*x*Log[x^2/Log[3]^4] + x^3*Log[x^2/Log[3]^4] + 36*x*Log[2]*Log[x^2/L og[3]^4] - 2*x^2*Log[2]*Log[x^2/Log[3]^4] + x*Log[2]^2*Log[x^2/Log[3]^4] - 3*x*Log[4096]*Log[x^2/Log[3]^4] + (Log[4096]^2*Log[x]*Log[x^2/Log[3]^4])/ 8 + 648*x*Log[x^3]*Log[x^2/Log[3]^4] + 6*x^2*Log[x^3]*Log[x^2/Log[3]^4] - 12*x*Log[2]*Log[x^3]*Log[x^2/Log[3]^4] - ((12*x - Log[4096])^2*Log[x^3]*Lo g[x^2/Log[3]^4])/24 - 108*x*Log[x^3]^2*Log[x^2/Log[3]^4] - 2*x^2*Log[x^3]^ 2*Log[x^2/Log[3]^4] + 2*x*Log[2]*Log[x^3]^2*Log[x^2/Log[3]^4] + 12*x*Log[x ^3]^3*Log[x^2/Log[3]^4] + 1944*x*(2 + Log[x^2/Log[3]^4]) - 648*x*Log[x^3]* (2 + Log[x^2/Log[3]^4]) + 108*x*Log[x^3]^2*(2 + Log[x^2/Log[3]^4]) - 12*x* Log[x^3]^3*(2 + Log[x^2/Log[3]^4]) + x*Log[x^3]^4*(2 + Log[x^2/Log[3]^4])
Leaf count of result is larger than twice the leaf count of optimal. \(102\) vs. \(2(25)=50\).
Time = 12.64 (sec) , antiderivative size = 103, normalized size of antiderivative = 4.12
| method | result | size |
| parallelrisch | \(2 \ln \left (2\right ) x \ln \left (x^{3}\right )^{2} \ln \left (\frac {x^{2}}{\ln \left (3\right )^{4}}\right )+\ln \left (\frac {x^{2}}{\ln \left (3\right )^{4}}\right ) \ln \left (2\right )^{2} x -2 \ln \left (\frac {x^{2}}{\ln \left (3\right )^{4}}\right ) \ln \left (2\right ) x^{2}-2 x^{2} \ln \left (x^{3}\right )^{2} \ln \left (\frac {x^{2}}{\ln \left (3\right )^{4}}\right )+\ln \left (\frac {x^{2}}{\ln \left (3\right )^{4}}\right ) x \ln \left (x^{3}\right )^{4}+\ln \left (\frac {x^{2}}{\ln \left (3\right )^{4}}\right ) x^{3}\) | \(103\) |
| default | \(-4 x^{3} \ln \left (\ln \left (3\right )\right )+x^{3} \ln \left (x^{2}\right )+36 \ln \left (\ln \left (3\right )\right ) x^{2}+2 x \ln \left (2\right )^{2}+\ln \left (2\right )^{2} \left (x \ln \left (x^{2}\right )-2 x \right )+2 x \ln \left (2\right ) \ln \left (x^{2}\right ) \ln \left (x^{3}\right )^{2}+48 x \ln \left (2\right ) \ln \left (\ln \left (3\right )\right ) \ln \left (x^{3}\right )-8 x \ln \left (2\right ) \ln \left (\ln \left (3\right )\right ) \ln \left (x^{3}\right )^{2}-48 \ln \left (2\right ) \ln \left (\ln \left (3\right )\right ) \left (x \ln \left (x^{3}\right )-3 x \right )+8 \ln \left (\ln \left (3\right )\right ) \ln \left (2\right ) x^{2}+48 \ln \left (\ln \left (3\right )\right ) \left (\frac {x^{2} \ln \left (x^{3}\right )}{2}-\frac {3 x^{2}}{4}\right )+\ln \left (x^{3}\right )^{4} \ln \left (x^{2}\right ) x -4 \ln \left (\ln \left (3\right )\right ) x \ln \left (x^{3}\right )^{4}-2 x^{2} \ln \left (x^{2}\right ) \ln \left (2\right )-4 \ln \left (2\right )^{2} \ln \left (\ln \left (3\right )\right ) x -2 x^{2} \ln \left (x^{2}\right ) \ln \left (x^{3}\right )^{2}-144 x \ln \left (2\right ) \ln \left (\ln \left (3\right )\right )-24 x^{2} \ln \left (\ln \left (3\right )\right ) \ln \left (x^{3}\right )+8 x^{2} \ln \left (\ln \left (3\right )\right ) \ln \left (x^{3}\right )^{2}\) | \(230\) |
| parts | \(-4 x^{3} \ln \left (\ln \left (3\right )\right )+x^{3} \ln \left (x^{2}\right )+36 \ln \left (\ln \left (3\right )\right ) x^{2}+2 x \ln \left (2\right )^{2}+\ln \left (2\right )^{2} \left (x \ln \left (x^{2}\right )-2 x \right )+2 x \ln \left (2\right ) \ln \left (x^{2}\right ) \ln \left (x^{3}\right )^{2}+48 x \ln \left (2\right ) \ln \left (\ln \left (3\right )\right ) \ln \left (x^{3}\right )-8 x \ln \left (2\right ) \ln \left (\ln \left (3\right )\right ) \ln \left (x^{3}\right )^{2}-48 \ln \left (2\right ) \ln \left (\ln \left (3\right )\right ) \left (x \ln \left (x^{3}\right )-3 x \right )+8 \ln \left (\ln \left (3\right )\right ) \ln \left (2\right ) x^{2}+48 \ln \left (\ln \left (3\right )\right ) \left (\frac {x^{2} \ln \left (x^{3}\right )}{2}-\frac {3 x^{2}}{4}\right )+\ln \left (x^{3}\right )^{4} \ln \left (x^{2}\right ) x -4 \ln \left (\ln \left (3\right )\right ) x \ln \left (x^{3}\right )^{4}-2 x^{2} \ln \left (x^{2}\right ) \ln \left (2\right )-4 \ln \left (2\right )^{2} \ln \left (\ln \left (3\right )\right ) x -2 x^{2} \ln \left (x^{2}\right ) \ln \left (x^{3}\right )^{2}-144 x \ln \left (2\right ) \ln \left (\ln \left (3\right )\right )-24 x^{2} \ln \left (\ln \left (3\right )\right ) \ln \left (x^{3}\right )+8 x^{2} \ln \left (\ln \left (3\right )\right ) \ln \left (x^{3}\right )^{2}\) | \(230\) |
| risch | \(\text {Expression too large to display}\) | \(16210\) |
Input:
int((ln(x^2/ln(3)^4)+2)*ln(x^3)^4+12*ln(x^2/ln(3)^4)*ln(x^3)^3+((2*ln(2)-4 *x)*ln(x^2/ln(3)^4)+4*ln(2)-4*x)*ln(x^3)^2+(12*ln(2)-12*x)*ln(x^2/ln(3)^4) *ln(x^3)+(ln(2)^2-4*x*ln(2)+3*x^2)*ln(x^2/ln(3)^4)+2*ln(2)^2-4*x*ln(2)+2*x ^2,x,method=_RETURNVERBOSE)
Output:
2*ln(2)*x*ln(x^3)^2*ln(x^2/ln(3)^4)+ln(x^2/ln(3)^4)*ln(2)^2*x-2*ln(x^2/ln( 3)^4)*ln(2)*x^2-2*x^2*ln(x^3)^2*ln(x^2/ln(3)^4)+ln(x^2/ln(3)^4)*x*ln(x^3)^ 4+ln(x^2/ln(3)^4)*x^3
Leaf count of result is larger than twice the leaf count of optimal. 192 vs. \(2 (25) = 50\).
Time = 0.09 (sec) , antiderivative size = 192, normalized size of antiderivative = 7.68 \[ \int \left (2 x^2-4 x \log (2)+2 \log ^2(2)+\left (3 x^2-4 x \log (2)+\log ^2(2)\right ) \log \left (\frac {x^2}{\log ^4(3)}\right )+(-12 x+12 \log (2)) \log \left (x^3\right ) \log \left (\frac {x^2}{\log ^4(3)}\right )+12 \log ^3\left (x^3\right ) \log \left (\frac {x^2}{\log ^4(3)}\right )+\log ^4\left (x^3\right ) \left (2+\log \left (\frac {x^2}{\log ^4(3)}\right )\right )+\log ^2\left (x^3\right ) \left (-4 x+4 \log (2)+(-4 x+2 \log (2)) \log \left (\frac {x^2}{\log ^4(3)}\right )\right )\right ) \, dx=\frac {1}{16} \, x \log \left (\log \left (3\right )^{12}\right )^{4} \log \left (\frac {x^{2}}{\log \left (3\right )^{4}}\right ) + \frac {3}{4} \, x \log \left (\log \left (3\right )^{12}\right )^{3} \log \left (\frac {x^{2}}{\log \left (3\right )^{4}}\right )^{2} + \frac {81}{16} \, x \log \left (\frac {x^{2}}{\log \left (3\right )^{4}}\right )^{5} - \frac {9}{2} \, {\left (x^{2} - x \log \left (2\right )\right )} \log \left (\frac {x^{2}}{\log \left (3\right )^{4}}\right )^{3} + \frac {1}{8} \, {\left (27 \, x \log \left (\frac {x^{2}}{\log \left (3\right )^{4}}\right )^{3} - 4 \, {\left (x^{2} - x \log \left (2\right )\right )} \log \left (\frac {x^{2}}{\log \left (3\right )^{4}}\right )\right )} \log \left (\log \left (3\right )^{12}\right )^{2} + \frac {3}{4} \, {\left (9 \, x \log \left (\frac {x^{2}}{\log \left (3\right )^{4}}\right )^{4} - 4 \, {\left (x^{2} - x \log \left (2\right )\right )} \log \left (\frac {x^{2}}{\log \left (3\right )^{4}}\right )^{2}\right )} \log \left (\log \left (3\right )^{12}\right ) + {\left (x^{3} - 2 \, x^{2} \log \left (2\right ) + x \log \left (2\right )^{2}\right )} \log \left (\frac {x^{2}}{\log \left (3\right )^{4}}\right ) \] Input:
integrate((log(x^2/log(3)^4)+2)*log(x^3)^4+12*log(x^2/log(3)^4)*log(x^3)^3 +((2*log(2)-4*x)*log(x^2/log(3)^4)+4*log(2)-4*x)*log(x^3)^2+(12*log(2)-12* x)*log(x^2/log(3)^4)*log(x^3)+(log(2)^2-4*x*log(2)+3*x^2)*log(x^2/log(3)^4 )+2*log(2)^2-4*x*log(2)+2*x^2,x, algorithm="fricas")
Output:
1/16*x*log(log(3)^12)^4*log(x^2/log(3)^4) + 3/4*x*log(log(3)^12)^3*log(x^2 /log(3)^4)^2 + 81/16*x*log(x^2/log(3)^4)^5 - 9/2*(x^2 - x*log(2))*log(x^2/ log(3)^4)^3 + 1/8*(27*x*log(x^2/log(3)^4)^3 - 4*(x^2 - x*log(2))*log(x^2/l og(3)^4))*log(log(3)^12)^2 + 3/4*(9*x*log(x^2/log(3)^4)^4 - 4*(x^2 - x*log (2))*log(x^2/log(3)^4)^2)*log(log(3)^12) + (x^3 - 2*x^2*log(2) + x*log(2)^ 2)*log(x^2/log(3)^4)
Leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (24) = 48\).
Time = 0.16 (sec) , antiderivative size = 146, normalized size of antiderivative = 5.84 \[ \int \left (2 x^2-4 x \log (2)+2 \log ^2(2)+\left (3 x^2-4 x \log (2)+\log ^2(2)\right ) \log \left (\frac {x^2}{\log ^4(3)}\right )+(-12 x+12 \log (2)) \log \left (x^3\right ) \log \left (\frac {x^2}{\log ^4(3)}\right )+12 \log ^3\left (x^3\right ) \log \left (\frac {x^2}{\log ^4(3)}\right )+\log ^4\left (x^3\right ) \left (2+\log \left (\frac {x^2}{\log ^4(3)}\right )\right )+\log ^2\left (x^3\right ) \left (-4 x+4 \log (2)+(-4 x+2 \log (2)) \log \left (\frac {x^2}{\log ^4(3)}\right )\right )\right ) \, dx=- 4 x^{3} \log {\left (\log {\left (3 \right )} \right )} + 8 x^{2} \log {\left (2 \right )} \log {\left (\log {\left (3 \right )} \right )} + \frac {2 x \log {\left (x^{3} \right )}^{5}}{3} - 4 x \log {\left (x^{3} \right )}^{4} \log {\left (\log {\left (3 \right )} \right )} - 4 x \log {\left (2 \right )}^{2} \log {\left (\log {\left (3 \right )} \right )} + \left (- \frac {4 x^{2}}{3} + \frac {4 x \log {\left (2 \right )}}{3}\right ) \log {\left (x^{3} \right )}^{3} + \left (8 x^{2} \log {\left (\log {\left (3 \right )} \right )} - 8 x \log {\left (2 \right )} \log {\left (\log {\left (3 \right )} \right )}\right ) \log {\left (x^{3} \right )}^{2} + \left (\frac {2 x^{3}}{3} - \frac {4 x^{2} \log {\left (2 \right )}}{3} + \frac {2 x \log {\left (2 \right )}^{2}}{3}\right ) \log {\left (x^{3} \right )} \] Input:
integrate((ln(x**2/ln(3)**4)+2)*ln(x**3)**4+12*ln(x**2/ln(3)**4)*ln(x**3)* *3+((2*ln(2)-4*x)*ln(x**2/ln(3)**4)+4*ln(2)-4*x)*ln(x**3)**2+(12*ln(2)-12* x)*ln(x**2/ln(3)**4)*ln(x**3)+(ln(2)**2-4*x*ln(2)+3*x**2)*ln(x**2/ln(3)**4 )+2*ln(2)**2-4*x*ln(2)+2*x**2,x)
Output:
-4*x**3*log(log(3)) + 8*x**2*log(2)*log(log(3)) + 2*x*log(x**3)**5/3 - 4*x *log(x**3)**4*log(log(3)) - 4*x*log(2)**2*log(log(3)) + (-4*x**2/3 + 4*x*l og(2)/3)*log(x**3)**3 + (8*x**2*log(log(3)) - 8*x*log(2)*log(log(3)))*log( x**3)**2 + (2*x**3/3 - 4*x**2*log(2)/3 + 2*x*log(2)**2/3)*log(x**3)
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (25) = 50\).
Time = 0.11 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.84 \[ \int \left (2 x^2-4 x \log (2)+2 \log ^2(2)+\left (3 x^2-4 x \log (2)+\log ^2(2)\right ) \log \left (\frac {x^2}{\log ^4(3)}\right )+(-12 x+12 \log (2)) \log \left (x^3\right ) \log \left (\frac {x^2}{\log ^4(3)}\right )+12 \log ^3\left (x^3\right ) \log \left (\frac {x^2}{\log ^4(3)}\right )+\log ^4\left (x^3\right ) \left (2+\log \left (\frac {x^2}{\log ^4(3)}\right )\right )+\log ^2\left (x^3\right ) \left (-4 x+4 \log (2)+(-4 x+2 \log (2)) \log \left (\frac {x^2}{\log ^4(3)}\right )\right )\right ) \, dx=x \log \left (x^{3}\right )^{4} \log \left (\frac {x^{2}}{\log \left (3\right )^{4}}\right ) - 2 \, {\left (x^{2} - x \log \left (2\right )\right )} \log \left (x^{3}\right )^{2} \log \left (\frac {x^{2}}{\log \left (3\right )^{4}}\right ) + {\left (x^{3} - 2 \, x^{2} \log \left (2\right ) + x \log \left (2\right )^{2}\right )} \log \left (\frac {x^{2}}{\log \left (3\right )^{4}}\right ) \] Input:
integrate((log(x^2/log(3)^4)+2)*log(x^3)^4+12*log(x^2/log(3)^4)*log(x^3)^3 +((2*log(2)-4*x)*log(x^2/log(3)^4)+4*log(2)-4*x)*log(x^3)^2+(12*log(2)-12* x)*log(x^2/log(3)^4)*log(x^3)+(log(2)^2-4*x*log(2)+3*x^2)*log(x^2/log(3)^4 )+2*log(2)^2-4*x*log(2)+2*x^2,x, algorithm="maxima")
Output:
x*log(x^3)^4*log(x^2/log(3)^4) - 2*(x^2 - x*log(2))*log(x^3)^2*log(x^2/log (3)^4) + (x^3 - 2*x^2*log(2) + x*log(2)^2)*log(x^2/log(3)^4)
Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (25) = 50\).
Time = 0.14 (sec) , antiderivative size = 114, normalized size of antiderivative = 4.56 \[ \int \left (2 x^2-4 x \log (2)+2 \log ^2(2)+\left (3 x^2-4 x \log (2)+\log ^2(2)\right ) \log \left (\frac {x^2}{\log ^4(3)}\right )+(-12 x+12 \log (2)) \log \left (x^3\right ) \log \left (\frac {x^2}{\log ^4(3)}\right )+12 \log ^3\left (x^3\right ) \log \left (\frac {x^2}{\log ^4(3)}\right )+\log ^4\left (x^3\right ) \left (2+\log \left (\frac {x^2}{\log ^4(3)}\right )\right )+\log ^2\left (x^3\right ) \left (-4 x+4 \log (2)+(-4 x+2 \log (2)) \log \left (\frac {x^2}{\log ^4(3)}\right )\right )\right ) \, dx=-324 \, x {\left (\log \left (\log \left (3\right )\right ) + 2\right )} \log \left (x\right )^{4} + 162 \, x \log \left (x\right )^{5} + 648 \, x \log \left (x\right )^{4} - 36 \, {\left (x^{2} - x \log \left (2\right )\right )} \log \left (x\right )^{3} + 36 \, {\left (x^{2} {\left (2 \, \log \left (\log \left (3\right )\right ) + 1\right )} - 2 \, {\left (\log \left (2\right ) \log \left (\log \left (3\right )\right ) + \log \left (2\right )\right )} x\right )} \log \left (x\right )^{2} - 36 \, {\left (x^{2} - 2 \, x \log \left (2\right )\right )} \log \left (x\right )^{2} + {\left (x^{3} - 2 \, x^{2} \log \left (2\right ) + x \log \left (2\right )^{2}\right )} \log \left (\frac {x^{2}}{\log \left (3\right )^{4}}\right ) \] Input:
integrate((log(x^2/log(3)^4)+2)*log(x^3)^4+12*log(x^2/log(3)^4)*log(x^3)^3 +((2*log(2)-4*x)*log(x^2/log(3)^4)+4*log(2)-4*x)*log(x^3)^2+(12*log(2)-12* x)*log(x^2/log(3)^4)*log(x^3)+(log(2)^2-4*x*log(2)+3*x^2)*log(x^2/log(3)^4 )+2*log(2)^2-4*x*log(2)+2*x^2,x, algorithm="giac")
Output:
-324*x*(log(log(3)) + 2)*log(x)^4 + 162*x*log(x)^5 + 648*x*log(x)^4 - 36*( x^2 - x*log(2))*log(x)^3 + 36*(x^2*(2*log(log(3)) + 1) - 2*(log(2)*log(log (3)) + log(2))*x)*log(x)^2 - 36*(x^2 - 2*x*log(2))*log(x)^2 + (x^3 - 2*x^2 *log(2) + x*log(2)^2)*log(x^2/log(3)^4)
Timed out. \[ \int \left (2 x^2-4 x \log (2)+2 \log ^2(2)+\left (3 x^2-4 x \log (2)+\log ^2(2)\right ) \log \left (\frac {x^2}{\log ^4(3)}\right )+(-12 x+12 \log (2)) \log \left (x^3\right ) \log \left (\frac {x^2}{\log ^4(3)}\right )+12 \log ^3\left (x^3\right ) \log \left (\frac {x^2}{\log ^4(3)}\right )+\log ^4\left (x^3\right ) \left (2+\log \left (\frac {x^2}{\log ^4(3)}\right )\right )+\log ^2\left (x^3\right ) \left (-4 x+4 \log (2)+(-4 x+2 \log (2)) \log \left (\frac {x^2}{\log ^4(3)}\right )\right )\right ) \, dx=\int 12\,{\ln \left (x^3\right )}^3\,\ln \left (\frac {x^2}{{\ln \left (3\right )}^4}\right )-4\,x\,\ln \left (2\right )+\ln \left (\frac {x^2}{{\ln \left (3\right )}^4}\right )\,\left (3\,x^2-4\,\ln \left (2\right )\,x+{\ln \left (2\right )}^2\right )+2\,{\ln \left (2\right )}^2+{\ln \left (x^3\right )}^4\,\left (\ln \left (\frac {x^2}{{\ln \left (3\right )}^4}\right )+2\right )-{\ln \left (x^3\right )}^2\,\left (4\,x-4\,\ln \left (2\right )+\ln \left (\frac {x^2}{{\ln \left (3\right )}^4}\right )\,\left (4\,x-2\,\ln \left (2\right )\right )\right )+2\,x^2-\ln \left (x^3\right )\,\ln \left (\frac {x^2}{{\ln \left (3\right )}^4}\right )\,\left (12\,x-12\,\ln \left (2\right )\right ) \,d x \] Input:
int(12*log(x^3)^3*log(x^2/log(3)^4) - 4*x*log(2) + log(x^2/log(3)^4)*(log( 2)^2 - 4*x*log(2) + 3*x^2) + 2*log(2)^2 + log(x^3)^4*(log(x^2/log(3)^4) + 2) - log(x^3)^2*(4*x - 4*log(2) + log(x^2/log(3)^4)*(4*x - 2*log(2))) + 2* x^2 - log(x^3)*log(x^2/log(3)^4)*(12*x - 12*log(2)),x)
Output:
int(12*log(x^3)^3*log(x^2/log(3)^4) - 4*x*log(2) + log(x^2/log(3)^4)*(log( 2)^2 - 4*x*log(2) + 3*x^2) + 2*log(2)^2 + log(x^3)^4*(log(x^2/log(3)^4) + 2) - log(x^3)^2*(4*x - 4*log(2) + log(x^2/log(3)^4)*(4*x - 2*log(2))) + 2* x^2 - log(x^3)*log(x^2/log(3)^4)*(12*x - 12*log(2)), x)
Time = 0.16 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.96 \[ \int \left (2 x^2-4 x \log (2)+2 \log ^2(2)+\left (3 x^2-4 x \log (2)+\log ^2(2)\right ) \log \left (\frac {x^2}{\log ^4(3)}\right )+(-12 x+12 \log (2)) \log \left (x^3\right ) \log \left (\frac {x^2}{\log ^4(3)}\right )+12 \log ^3\left (x^3\right ) \log \left (\frac {x^2}{\log ^4(3)}\right )+\log ^4\left (x^3\right ) \left (2+\log \left (\frac {x^2}{\log ^4(3)}\right )\right )+\log ^2\left (x^3\right ) \left (-4 x+4 \log (2)+(-4 x+2 \log (2)) \log \left (\frac {x^2}{\log ^4(3)}\right )\right )\right ) \, dx=\mathrm {log}\left (\frac {x^{2}}{\mathrm {log}\left (3\right )^{4}}\right ) x \left (\mathrm {log}\left (x^{3}\right )^{4}+2 \mathrm {log}\left (x^{3}\right )^{2} \mathrm {log}\left (2\right )-2 \mathrm {log}\left (x^{3}\right )^{2} x +\mathrm {log}\left (2\right )^{2}-2 \,\mathrm {log}\left (2\right ) x +x^{2}\right ) \] Input:
int((log(x^2/log(3)^4)+2)*log(x^3)^4+12*log(x^2/log(3)^4)*log(x^3)^3+((2*l og(2)-4*x)*log(x^2/log(3)^4)+4*log(2)-4*x)*log(x^3)^2+(12*log(2)-12*x)*log (x^2/log(3)^4)*log(x^3)+(log(2)^2-4*x*log(2)+3*x^2)*log(x^2/log(3)^4)+2*lo g(2)^2-4*x*log(2)+2*x^2,x)
Output:
log(x**2/log(3)**4)*x*(log(x**3)**4 + 2*log(x**3)**2*log(2) - 2*log(x**3)* *2*x + log(2)**2 - 2*log(2)*x + x**2)