Integrand size = 415, antiderivative size = 40 \[ \int \frac {300 x^3+300 x^4+\left (-3600 x^2-4500 x^3-900 x^4\right ) \log \left (\frac {1+x}{3}\right )+\left (14400 x+21600 x^2+8100 x^3+900 x^4\right ) \log ^2\left (\frac {1+x}{3}\right )+\left (-19200-33600 x-18000 x^2-3900 x^3-300 x^4\right ) \log ^3\left (\frac {1+x}{3}\right )+\log ^2\left (x^2\right ) \left (-30 x^4+\left (120 x^3+30 x^4\right ) \log \left (\frac {1+x}{3}\right )+\left (120 x^2+120 x^3\right ) \log ^2\left (\frac {1+x}{3}\right )+\left (-480 x-600 x^2-120 x^3\right ) \log ^3\left (\frac {1+x}{3}\right )\right )+\log ^3\left (x^2\right ) \left (-6 x^4 \log \left (\frac {1+x}{3}\right )+\left (24 x^2+24 x^3\right ) \log ^3\left (\frac {1+x}{3}\right )\right )+\log \left (x^2\right ) \left (\left (60 x^3+60 x^4\right ) \log \left (\frac {1+x}{3}\right )+\left (-480 x^2-600 x^3-120 x^4\right ) \log ^2\left (\frac {1+x}{3}\right )+\left (960 x+1440 x^2+540 x^3+60 x^4\right ) \log ^3\left (\frac {1+x}{3}\right )\right )}{\log ^3\left (x^2\right ) \left (-5 x^4-5 x^5+\left (60 x^3+75 x^4+15 x^5\right ) \log \left (\frac {1+x}{3}\right )+\left (-240 x^2-360 x^3-135 x^4-15 x^5\right ) \log ^2\left (\frac {1+x}{3}\right )+\left (320 x+560 x^2+300 x^3+65 x^4+5 x^5\right ) \log ^3\left (\frac {1+x}{3}\right )\right )} \, dx=3 \left (1+\frac {1}{5} \left (\frac {5}{\log \left (x^2\right )}+\frac {x}{-4-x+\frac {x}{\log \left (\frac {1+x}{3}\right )}}\right )^2\right ) \] Output:
3+3/5*(5/ln(x^2)+x/(x/ln(1/3*x+1/3)-4-x))^2
Leaf count is larger than twice the leaf count of optimal. \(104\) vs. \(2(40)=80\).
Time = 0.19 (sec) , antiderivative size = 104, normalized size of antiderivative = 2.60 \[ \int \frac {300 x^3+300 x^4+\left (-3600 x^2-4500 x^3-900 x^4\right ) \log \left (\frac {1+x}{3}\right )+\left (14400 x+21600 x^2+8100 x^3+900 x^4\right ) \log ^2\left (\frac {1+x}{3}\right )+\left (-19200-33600 x-18000 x^2-3900 x^3-300 x^4\right ) \log ^3\left (\frac {1+x}{3}\right )+\log ^2\left (x^2\right ) \left (-30 x^4+\left (120 x^3+30 x^4\right ) \log \left (\frac {1+x}{3}\right )+\left (120 x^2+120 x^3\right ) \log ^2\left (\frac {1+x}{3}\right )+\left (-480 x-600 x^2-120 x^3\right ) \log ^3\left (\frac {1+x}{3}\right )\right )+\log ^3\left (x^2\right ) \left (-6 x^4 \log \left (\frac {1+x}{3}\right )+\left (24 x^2+24 x^3\right ) \log ^3\left (\frac {1+x}{3}\right )\right )+\log \left (x^2\right ) \left (\left (60 x^3+60 x^4\right ) \log \left (\frac {1+x}{3}\right )+\left (-480 x^2-600 x^3-120 x^4\right ) \log ^2\left (\frac {1+x}{3}\right )+\left (960 x+1440 x^2+540 x^3+60 x^4\right ) \log ^3\left (\frac {1+x}{3}\right )\right )}{\log ^3\left (x^2\right ) \left (-5 x^4-5 x^5+\left (60 x^3+75 x^4+15 x^5\right ) \log \left (\frac {1+x}{3}\right )+\left (-240 x^2-360 x^3-135 x^4-15 x^5\right ) \log ^2\left (\frac {1+x}{3}\right )+\left (320 x+560 x^2+300 x^3+65 x^4+5 x^5\right ) \log ^3\left (\frac {1+x}{3}\right )\right )} \, dx=-\frac {6}{5} \left (-\frac {25}{2 \log ^2\left (x^2\right )}+\frac {5 x \log \left (\frac {1+x}{3}\right )}{\log \left (x^2\right ) \left (-x+(4+x) \log \left (\frac {1+x}{3}\right )\right )}+\frac {x^2-2 x (4+x) \log \left (\frac {1+x}{3}\right )+8 (2+x) \log ^2\left (\frac {1+x}{3}\right )}{2 \left (x-(4+x) \log \left (\frac {1+x}{3}\right )\right )^2}\right ) \] Input:
Integrate[(300*x^3 + 300*x^4 + (-3600*x^2 - 4500*x^3 - 900*x^4)*Log[(1 + x )/3] + (14400*x + 21600*x^2 + 8100*x^3 + 900*x^4)*Log[(1 + x)/3]^2 + (-192 00 - 33600*x - 18000*x^2 - 3900*x^3 - 300*x^4)*Log[(1 + x)/3]^3 + Log[x^2] ^2*(-30*x^4 + (120*x^3 + 30*x^4)*Log[(1 + x)/3] + (120*x^2 + 120*x^3)*Log[ (1 + x)/3]^2 + (-480*x - 600*x^2 - 120*x^3)*Log[(1 + x)/3]^3) + Log[x^2]^3 *(-6*x^4*Log[(1 + x)/3] + (24*x^2 + 24*x^3)*Log[(1 + x)/3]^3) + Log[x^2]*( (60*x^3 + 60*x^4)*Log[(1 + x)/3] + (-480*x^2 - 600*x^3 - 120*x^4)*Log[(1 + x)/3]^2 + (960*x + 1440*x^2 + 540*x^3 + 60*x^4)*Log[(1 + x)/3]^3))/(Log[x ^2]^3*(-5*x^4 - 5*x^5 + (60*x^3 + 75*x^4 + 15*x^5)*Log[(1 + x)/3] + (-240* x^2 - 360*x^3 - 135*x^4 - 15*x^5)*Log[(1 + x)/3]^2 + (320*x + 560*x^2 + 30 0*x^3 + 65*x^4 + 5*x^5)*Log[(1 + x)/3]^3)),x]
Output:
(-6*(-25/(2*Log[x^2]^2) + (5*x*Log[(1 + x)/3])/(Log[x^2]*(-x + (4 + x)*Log [(1 + x)/3])) + (x^2 - 2*x*(4 + x)*Log[(1 + x)/3] + 8*(2 + x)*Log[(1 + x)/ 3]^2)/(2*(x - (4 + x)*Log[(1 + x)/3])^2)))/5
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.
| \(\displaystyle \int \frac {300 x^4+300 x^3+\left (-300 x^4-3900 x^3-18000 x^2-33600 x-19200\right ) \log ^3\left (\frac {x+1}{3}\right )+\log ^3\left (x^2\right ) \left (\left (24 x^3+24 x^2\right ) \log ^3\left (\frac {x+1}{3}\right )-6 x^4 \log \left (\frac {x+1}{3}\right )\right )+\left (900 x^4+8100 x^3+21600 x^2+14400 x\right ) \log ^2\left (\frac {x+1}{3}\right )+\log ^2\left (x^2\right ) \left (-30 x^4+\left (30 x^4+120 x^3\right ) \log \left (\frac {x+1}{3}\right )+\left (-120 x^3-600 x^2-480 x\right ) \log ^3\left (\frac {x+1}{3}\right )+\left (120 x^3+120 x^2\right ) \log ^2\left (\frac {x+1}{3}\right )\right )+\log \left (x^2\right ) \left (\left (60 x^4+60 x^3\right ) \log \left (\frac {x+1}{3}\right )+\left (60 x^4+540 x^3+1440 x^2+960 x\right ) \log ^3\left (\frac {x+1}{3}\right )+\left (-120 x^4-600 x^3-480 x^2\right ) \log ^2\left (\frac {x+1}{3}\right )\right )+\left (-900 x^4-4500 x^3-3600 x^2\right ) \log \left (\frac {x+1}{3}\right )}{\log ^3\left (x^2\right ) \left (-5 x^5-5 x^4+\left (15 x^5+75 x^4+60 x^3\right ) \log \left (\frac {x+1}{3}\right )+\left (5 x^5+65 x^4+300 x^3+560 x^2+320 x\right ) \log ^3\left (\frac {x+1}{3}\right )+\left (-15 x^5-135 x^4-360 x^3-240 x^2\right ) \log ^2\left (\frac {x+1}{3}\right )\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {6 \left (x^2 \log \left (\frac {x+1}{3}\right ) \left (x^2-4 (x+1) \log ^2\left (\frac {x+1}{3}\right )\right ) \log ^3\left (x^2\right )-10 x (x+1) \log \left (\frac {x+1}{3}\right ) \left (x-(x+4) \log \left (\frac {x+1}{3}\right )\right )^2 \log \left (x^2\right )-5 x \left (-x^3-4 \left (x^2+5 x+4\right ) \log ^3\left (\frac {x+1}{3}\right )+(x+4) x^2 \log \left (\frac {x+1}{3}\right )+4 (x+1) x \log ^2\left (\frac {x+1}{3}\right )\right ) \log ^2\left (x^2\right )+50 (x+1) \left ((x+4) \log \left (\frac {x+1}{3}\right )-x\right )^3\right )}{5 x (x+1) \log ^3\left (x^2\right ) \left (x-(x+4) \log \left (\frac {x+1}{3}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {6}{5} \int -\frac {-x^2 \log \left (\frac {x+1}{3}\right ) \left (x^2-4 (x+1) \log ^2\left (\frac {x+1}{3}\right )\right ) \log ^3\left (x^2\right )-5 x \left (x^3-(x+4) \log \left (\frac {x+1}{3}\right ) x^2-4 (x+1) \log ^2\left (\frac {x+1}{3}\right ) x+4 \left (x^2+5 x+4\right ) \log ^3\left (\frac {x+1}{3}\right )\right ) \log ^2\left (x^2\right )+10 x (x+1) \log \left (\frac {x+1}{3}\right ) \left (x-(x+4) \log \left (\frac {x+1}{3}\right )\right )^2 \log \left (x^2\right )+50 (x+1) \left (x-(x+4) \log \left (\frac {x+1}{3}\right )\right )^3}{x (x+1) \log ^3\left (x^2\right ) \left (x-(x+4) \log \left (\frac {x+1}{3}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {6}{5} \int \frac {-x^2 \log \left (\frac {x+1}{3}\right ) \left (x^2-4 (x+1) \log ^2\left (\frac {x+1}{3}\right )\right ) \log ^3\left (x^2\right )-5 x \left (x^3-(x+4) \log \left (\frac {x+1}{3}\right ) x^2-4 (x+1) \log ^2\left (\frac {x+1}{3}\right ) x+4 \left (x^2+5 x+4\right ) \log ^3\left (\frac {x+1}{3}\right )\right ) \log ^2\left (x^2\right )+10 x (x+1) \log \left (\frac {x+1}{3}\right ) \left (x-(x+4) \log \left (\frac {x+1}{3}\right )\right )^2 \log \left (x^2\right )+50 (x+1) \left (x-(x+4) \log \left (\frac {x+1}{3}\right )\right )^3}{x (x+1) \log ^3\left (x^2\right ) \left (x-(x+4) \log \left (\frac {x+1}{3}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {6}{5} \int \left (\frac {\left (x^2+4 x+12\right ) x^4}{(x+1) (x+4)^3 \left (\log \left (\frac {x+1}{3}\right ) x-x+4 \log \left (\frac {x+1}{3}\right )\right )^3}+\frac {\left (\log \left (x^2\right ) x^3-5 x^3-4 \log \left (x^2\right ) x^2-40 x^2+4 \log \left (x^2\right ) x-140 x-240\right ) x^2}{(x+1) (x+4)^3 \log \left (x^2\right ) \left (\log \left (\frac {x+1}{3}\right ) x-x+4 \log \left (\frac {x+1}{3}\right )\right )^2}-\frac {2 \left (5 x^2+6 \log ^2\left (x^2\right ) x-20 \log \left (x^2\right ) x+40 x-80 \log \left (x^2\right )+80\right ) x}{(x+4)^3 \log ^2\left (x^2\right ) \left (\log \left (\frac {x+1}{3}\right ) x-x+4 \log \left (\frac {x+1}{3}\right )\right )}-\frac {2 \left (5 \log \left (x^2\right ) x^3-25 x^3+2 \log ^3\left (x^2\right ) x^2-10 \log ^2\left (x^2\right ) x^2+40 \log \left (x^2\right ) x^2-300 x^2-40 \log ^2\left (x^2\right ) x+80 \log \left (x^2\right ) x-1200 x-1600\right )}{(x+4)^3 \log ^3\left (x^2\right ) x}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle -\frac {6}{5} \int \left (\frac {\left (x^2+4 x+12\right ) x^4}{(x+1) (x+4)^3 \left (\log \left (\frac {x+1}{3}\right ) x-x+4 \log \left (\frac {x+1}{3}\right )\right )^3}+\frac {\left (\log \left (x^2\right ) x^3-5 x^3-4 \log \left (x^2\right ) x^2-40 x^2+4 \log \left (x^2\right ) x-140 x-240\right ) x^2}{(x+1) (x+4)^3 \log \left (x^2\right ) \left (\log \left (\frac {x+1}{3}\right ) x-x+4 \log \left (\frac {x+1}{3}\right )\right )^2}-\frac {2 \left (5 x^2+6 \log ^2\left (x^2\right ) x-20 \log \left (x^2\right ) x+40 x-80 \log \left (x^2\right )+80\right ) x}{(x+4)^3 \log ^2\left (x^2\right ) \left (\log \left (\frac {x+1}{3}\right ) x-x+4 \log \left (\frac {x+1}{3}\right )\right )}-\frac {2 \left (5 \log \left (x^2\right ) x^3-25 x^3+2 \log ^3\left (x^2\right ) x^2-10 \log ^2\left (x^2\right ) x^2+40 \log \left (x^2\right ) x^2-300 x^2-40 \log ^2\left (x^2\right ) x+80 \log \left (x^2\right ) x-1200 x-1600\right )}{(x+4)^3 \log ^3\left (x^2\right ) x}\right )dx\) |
Input:
Int[(300*x^3 + 300*x^4 + (-3600*x^2 - 4500*x^3 - 900*x^4)*Log[(1 + x)/3] + (14400*x + 21600*x^2 + 8100*x^3 + 900*x^4)*Log[(1 + x)/3]^2 + (-19200 - 3 3600*x - 18000*x^2 - 3900*x^3 - 300*x^4)*Log[(1 + x)/3]^3 + Log[x^2]^2*(-3 0*x^4 + (120*x^3 + 30*x^4)*Log[(1 + x)/3] + (120*x^2 + 120*x^3)*Log[(1 + x )/3]^2 + (-480*x - 600*x^2 - 120*x^3)*Log[(1 + x)/3]^3) + Log[x^2]^3*(-6*x ^4*Log[(1 + x)/3] + (24*x^2 + 24*x^3)*Log[(1 + x)/3]^3) + Log[x^2]*((60*x^ 3 + 60*x^4)*Log[(1 + x)/3] + (-480*x^2 - 600*x^3 - 120*x^4)*Log[(1 + x)/3] ^2 + (960*x + 1440*x^2 + 540*x^3 + 60*x^4)*Log[(1 + x)/3]^3))/(Log[x^2]^3* (-5*x^4 - 5*x^5 + (60*x^3 + 75*x^4 + 15*x^5)*Log[(1 + x)/3] + (-240*x^2 - 360*x^3 - 135*x^4 - 15*x^5)*Log[(1 + x)/3]^2 + (320*x + 560*x^2 + 300*x^3 + 65*x^4 + 5*x^5)*Log[(1 + x)/3]^3)),x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.11 (sec) , antiderivative size = 892, normalized size of antiderivative = 22.30
\[\text {Expression too large to display}\]
Input:
int((((24*x^3+24*x^2)*ln(1/3*x+1/3)^3-6*x^4*ln(1/3*x+1/3))*ln(x^2)^3+((-12 0*x^3-600*x^2-480*x)*ln(1/3*x+1/3)^3+(120*x^3+120*x^2)*ln(1/3*x+1/3)^2+(30 *x^4+120*x^3)*ln(1/3*x+1/3)-30*x^4)*ln(x^2)^2+((60*x^4+540*x^3+1440*x^2+96 0*x)*ln(1/3*x+1/3)^3+(-120*x^4-600*x^3-480*x^2)*ln(1/3*x+1/3)^2+(60*x^4+60 *x^3)*ln(1/3*x+1/3))*ln(x^2)+(-300*x^4-3900*x^3-18000*x^2-33600*x-19200)*l n(1/3*x+1/3)^3+(900*x^4+8100*x^3+21600*x^2+14400*x)*ln(1/3*x+1/3)^2+(-900* x^4-4500*x^3-3600*x^2)*ln(1/3*x+1/3)+300*x^4+300*x^3)/((5*x^5+65*x^4+300*x ^3+560*x^2+320*x)*ln(1/3*x+1/3)^3+(-15*x^5-135*x^4-360*x^3-240*x^2)*ln(1/3 *x+1/3)^2+(15*x^5+75*x^4+60*x^3)*ln(1/3*x+1/3)-5*x^5-5*x^4)/ln(x^2)^3,x)
Output:
-12/5*(-400-200*x+32*x*ln(x)^2+80*x*ln(x)+20*x^2*ln(x)+64*ln(x)^2-25*x^2-2 *Pi^2*x*csgn(I*x^2)^6-4*Pi^2*csgn(I*x^2)^6-5*I*Pi*x^2*csgn(I*x)^2*csgn(I*x ^2)+10*I*Pi*x^2*csgn(I*x)*csgn(I*x^2)^2-20*I*Pi*x*csgn(I*x)^2*csgn(I*x^2)+ 40*I*Pi*x*csgn(I*x)*csgn(I*x^2)^2-32*I*ln(x)*Pi*csgn(I*x)^2*csgn(I*x^2)+64 *I*ln(x)*Pi*csgn(I*x)*csgn(I*x^2)^2-16*I*Pi*x*csgn(I*x^2)^3*ln(x)-16*I*Pi* x*csgn(I*x)^2*csgn(I*x^2)*ln(x)+32*I*Pi*x*csgn(I*x)*csgn(I*x^2)^2*ln(x)-4* Pi^2*csgn(I*x)^4*csgn(I*x^2)^2+16*Pi^2*csgn(I*x)^3*csgn(I*x^2)^3-24*Pi^2*c sgn(I*x)^2*csgn(I*x^2)^4+16*Pi^2*csgn(I*x)*csgn(I*x^2)^5-2*Pi^2*x*csgn(I*x )^4*csgn(I*x^2)^2+8*Pi^2*x*csgn(I*x)^3*csgn(I*x^2)^3-12*Pi^2*x*csgn(I*x)^2 *csgn(I*x^2)^4+8*Pi^2*x*csgn(I*x)*csgn(I*x^2)^5-5*I*Pi*x^2*csgn(I*x^2)^3-2 0*I*Pi*x*csgn(I*x^2)^3-32*I*ln(x)*Pi*csgn(I*x^2)^3)/(4+x)^2/(-I*Pi*csgn(I* x^2)^3+2*I*Pi*csgn(I*x^2)^2*csgn(I*x)-I*Pi*csgn(I*x)^2*csgn(I*x^2)+4*ln(x) )^2+3/5*(2*Pi*x^4*csgn(I*x)^2*csgn(I*x^2)*ln(1/3*x+1/3)-4*Pi*x^4*csgn(I*x) *csgn(I*x^2)^2*ln(1/3*x+1/3)+2*Pi*x^4*csgn(I*x^2)^3*ln(1/3*x+1/3)-csgn(I*x )^2*csgn(I*x^2)*x^4*Pi+2*csgn(I*x)*csgn(I*x^2)^2*x^4*Pi-csgn(I*x^2)^3*x^4* Pi+8*Pi*x^3*csgn(I*x)^2*csgn(I*x^2)*ln(1/3*x+1/3)-16*Pi*x^3*csgn(I*x)*csgn (I*x^2)^2*ln(1/3*x+1/3)+8*Pi*x^3*csgn(I*x^2)^3*ln(1/3*x+1/3)+8*I*x^4*ln(x) *ln(1/3*x+1/3)-20*I*x^4*ln(1/3*x+1/3)-4*I*x^4*ln(x)+32*I*x^3*ln(x)*ln(1/3* x+1/3)-320*I*x^2*ln(1/3*x+1/3)-160*I*x^3*ln(1/3*x+1/3)+20*I*x^4+80*I*x^3)/ (x*ln(1/3*x+1/3)-x+4*ln(1/3*x+1/3))^2/(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*...
Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (33) = 66\).
Time = 0.08 (sec) , antiderivative size = 158, normalized size of antiderivative = 3.95 \[ \int \frac {300 x^3+300 x^4+\left (-3600 x^2-4500 x^3-900 x^4\right ) \log \left (\frac {1+x}{3}\right )+\left (14400 x+21600 x^2+8100 x^3+900 x^4\right ) \log ^2\left (\frac {1+x}{3}\right )+\left (-19200-33600 x-18000 x^2-3900 x^3-300 x^4\right ) \log ^3\left (\frac {1+x}{3}\right )+\log ^2\left (x^2\right ) \left (-30 x^4+\left (120 x^3+30 x^4\right ) \log \left (\frac {1+x}{3}\right )+\left (120 x^2+120 x^3\right ) \log ^2\left (\frac {1+x}{3}\right )+\left (-480 x-600 x^2-120 x^3\right ) \log ^3\left (\frac {1+x}{3}\right )\right )+\log ^3\left (x^2\right ) \left (-6 x^4 \log \left (\frac {1+x}{3}\right )+\left (24 x^2+24 x^3\right ) \log ^3\left (\frac {1+x}{3}\right )\right )+\log \left (x^2\right ) \left (\left (60 x^3+60 x^4\right ) \log \left (\frac {1+x}{3}\right )+\left (-480 x^2-600 x^3-120 x^4\right ) \log ^2\left (\frac {1+x}{3}\right )+\left (960 x+1440 x^2+540 x^3+60 x^4\right ) \log ^3\left (\frac {1+x}{3}\right )\right )}{\log ^3\left (x^2\right ) \left (-5 x^4-5 x^5+\left (60 x^3+75 x^4+15 x^5\right ) \log \left (\frac {1+x}{3}\right )+\left (-240 x^2-360 x^3-135 x^4-15 x^5\right ) \log ^2\left (\frac {1+x}{3}\right )+\left (320 x+560 x^2+300 x^3+65 x^4+5 x^5\right ) \log ^3\left (\frac {1+x}{3}\right )\right )} \, dx=-\frac {3 \, {\left ({\left (8 \, {\left (x + 2\right )} \log \left (\frac {1}{3} \, x + \frac {1}{3}\right )^{2} + x^{2} - 2 \, {\left (x^{2} + 4 \, x\right )} \log \left (\frac {1}{3} \, x + \frac {1}{3}\right )\right )} \log \left (x^{2}\right )^{2} - 25 \, {\left (x^{2} + 8 \, x + 16\right )} \log \left (\frac {1}{3} \, x + \frac {1}{3}\right )^{2} - 25 \, x^{2} - 10 \, {\left (x^{2} \log \left (\frac {1}{3} \, x + \frac {1}{3}\right ) - {\left (x^{2} + 4 \, x\right )} \log \left (\frac {1}{3} \, x + \frac {1}{3}\right )^{2}\right )} \log \left (x^{2}\right ) + 50 \, {\left (x^{2} + 4 \, x\right )} \log \left (\frac {1}{3} \, x + \frac {1}{3}\right )\right )}}{5 \, {\left ({\left (x^{2} + 8 \, x + 16\right )} \log \left (\frac {1}{3} \, x + \frac {1}{3}\right )^{2} + x^{2} - 2 \, {\left (x^{2} + 4 \, x\right )} \log \left (\frac {1}{3} \, x + \frac {1}{3}\right )\right )} \log \left (x^{2}\right )^{2}} \] Input:
integrate((((24*x^3+24*x^2)*log(1/3*x+1/3)^3-6*x^4*log(1/3*x+1/3))*log(x^2 )^3+((-120*x^3-600*x^2-480*x)*log(1/3*x+1/3)^3+(120*x^3+120*x^2)*log(1/3*x +1/3)^2+(30*x^4+120*x^3)*log(1/3*x+1/3)-30*x^4)*log(x^2)^2+((60*x^4+540*x^ 3+1440*x^2+960*x)*log(1/3*x+1/3)^3+(-120*x^4-600*x^3-480*x^2)*log(1/3*x+1/ 3)^2+(60*x^4+60*x^3)*log(1/3*x+1/3))*log(x^2)+(-300*x^4-3900*x^3-18000*x^2 -33600*x-19200)*log(1/3*x+1/3)^3+(900*x^4+8100*x^3+21600*x^2+14400*x)*log( 1/3*x+1/3)^2+(-900*x^4-4500*x^3-3600*x^2)*log(1/3*x+1/3)+300*x^4+300*x^3)/ ((5*x^5+65*x^4+300*x^3+560*x^2+320*x)*log(1/3*x+1/3)^3+(-15*x^5-135*x^4-36 0*x^3-240*x^2)*log(1/3*x+1/3)^2+(15*x^5+75*x^4+60*x^3)*log(1/3*x+1/3)-5*x^ 5-5*x^4)/log(x^2)^3,x, algorithm="fricas")
Output:
-3/5*((8*(x + 2)*log(1/3*x + 1/3)^2 + x^2 - 2*(x^2 + 4*x)*log(1/3*x + 1/3) )*log(x^2)^2 - 25*(x^2 + 8*x + 16)*log(1/3*x + 1/3)^2 - 25*x^2 - 10*(x^2*l og(1/3*x + 1/3) - (x^2 + 4*x)*log(1/3*x + 1/3)^2)*log(x^2) + 50*(x^2 + 4*x )*log(1/3*x + 1/3))/(((x^2 + 8*x + 16)*log(1/3*x + 1/3)^2 + x^2 - 2*(x^2 + 4*x)*log(1/3*x + 1/3))*log(x^2)^2)
Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (27) = 54\).
Time = 0.39 (sec) , antiderivative size = 240, normalized size of antiderivative = 6.00 \[ \int \frac {300 x^3+300 x^4+\left (-3600 x^2-4500 x^3-900 x^4\right ) \log \left (\frac {1+x}{3}\right )+\left (14400 x+21600 x^2+8100 x^3+900 x^4\right ) \log ^2\left (\frac {1+x}{3}\right )+\left (-19200-33600 x-18000 x^2-3900 x^3-300 x^4\right ) \log ^3\left (\frac {1+x}{3}\right )+\log ^2\left (x^2\right ) \left (-30 x^4+\left (120 x^3+30 x^4\right ) \log \left (\frac {1+x}{3}\right )+\left (120 x^2+120 x^3\right ) \log ^2\left (\frac {1+x}{3}\right )+\left (-480 x-600 x^2-120 x^3\right ) \log ^3\left (\frac {1+x}{3}\right )\right )+\log ^3\left (x^2\right ) \left (-6 x^4 \log \left (\frac {1+x}{3}\right )+\left (24 x^2+24 x^3\right ) \log ^3\left (\frac {1+x}{3}\right )\right )+\log \left (x^2\right ) \left (\left (60 x^3+60 x^4\right ) \log \left (\frac {1+x}{3}\right )+\left (-480 x^2-600 x^3-120 x^4\right ) \log ^2\left (\frac {1+x}{3}\right )+\left (960 x+1440 x^2+540 x^3+60 x^4\right ) \log ^3\left (\frac {1+x}{3}\right )\right )}{\log ^3\left (x^2\right ) \left (-5 x^4-5 x^5+\left (60 x^3+75 x^4+15 x^5\right ) \log \left (\frac {1+x}{3}\right )+\left (-240 x^2-360 x^3-135 x^4-15 x^5\right ) \log ^2\left (\frac {1+x}{3}\right )+\left (320 x+560 x^2+300 x^3+65 x^4+5 x^5\right ) \log ^3\left (\frac {1+x}{3}\right )\right )} \, dx=\frac {24 \left (- x - 2\right )}{5 x^{2} + 40 x + 80} + \frac {- 3 x^{4} \log {\left (x^{2} \right )} + 30 x^{4} + 120 x^{3} + \left (6 x^{4} \log {\left (x^{2} \right )} - 30 x^{4} + 24 x^{3} \log {\left (x^{2} \right )} - 240 x^{3} - 480 x^{2}\right ) \log {\left (\frac {x}{3} + \frac {1}{3} \right )}}{5 x^{4} \log {\left (x^{2} \right )} + 40 x^{3} \log {\left (x^{2} \right )} + 80 x^{2} \log {\left (x^{2} \right )} + \left (- 10 x^{4} \log {\left (x^{2} \right )} - 120 x^{3} \log {\left (x^{2} \right )} - 480 x^{2} \log {\left (x^{2} \right )} - 640 x \log {\left (x^{2} \right )}\right ) \log {\left (\frac {x}{3} + \frac {1}{3} \right )} + \left (5 x^{4} \log {\left (x^{2} \right )} + 80 x^{3} \log {\left (x^{2} \right )} + 480 x^{2} \log {\left (x^{2} \right )} + 1280 x \log {\left (x^{2} \right )} + 1280 \log {\left (x^{2} \right )}\right ) \log {\left (\frac {x}{3} + \frac {1}{3} \right )}^{2}} + \frac {- 6 x \log {\left (x^{2} \right )} + 15 x + 60}{\left (x + 4\right ) \log {\left (x^{2} \right )}^{2}} \] Input:
integrate((((24*x**3+24*x**2)*ln(1/3*x+1/3)**3-6*x**4*ln(1/3*x+1/3))*ln(x* *2)**3+((-120*x**3-600*x**2-480*x)*ln(1/3*x+1/3)**3+(120*x**3+120*x**2)*ln (1/3*x+1/3)**2+(30*x**4+120*x**3)*ln(1/3*x+1/3)-30*x**4)*ln(x**2)**2+((60* x**4+540*x**3+1440*x**2+960*x)*ln(1/3*x+1/3)**3+(-120*x**4-600*x**3-480*x* *2)*ln(1/3*x+1/3)**2+(60*x**4+60*x**3)*ln(1/3*x+1/3))*ln(x**2)+(-300*x**4- 3900*x**3-18000*x**2-33600*x-19200)*ln(1/3*x+1/3)**3+(900*x**4+8100*x**3+2 1600*x**2+14400*x)*ln(1/3*x+1/3)**2+(-900*x**4-4500*x**3-3600*x**2)*ln(1/3 *x+1/3)+300*x**4+300*x**3)/((5*x**5+65*x**4+300*x**3+560*x**2+320*x)*ln(1/ 3*x+1/3)**3+(-15*x**5-135*x**4-360*x**3-240*x**2)*ln(1/3*x+1/3)**2+(15*x** 5+75*x**4+60*x**3)*ln(1/3*x+1/3)-5*x**5-5*x**4)/ln(x**2)**3,x)
Output:
24*(-x - 2)/(5*x**2 + 40*x + 80) + (-3*x**4*log(x**2) + 30*x**4 + 120*x**3 + (6*x**4*log(x**2) - 30*x**4 + 24*x**3*log(x**2) - 240*x**3 - 480*x**2)* log(x/3 + 1/3))/(5*x**4*log(x**2) + 40*x**3*log(x**2) + 80*x**2*log(x**2) + (-10*x**4*log(x**2) - 120*x**3*log(x**2) - 480*x**2*log(x**2) - 640*x*lo g(x**2))*log(x/3 + 1/3) + (5*x**4*log(x**2) + 80*x**3*log(x**2) + 480*x**2 *log(x**2) + 1280*x*log(x**2) + 1280*log(x**2))*log(x/3 + 1/3)**2) + (-6*x *log(x**2) + 15*x + 60)/((x + 4)*log(x**2)**2)
Leaf count of result is larger than twice the leaf count of optimal. 290 vs. \(2 (33) = 66\).
Time = 0.31 (sec) , antiderivative size = 290, normalized size of antiderivative = 7.25 \[ \int \frac {300 x^3+300 x^4+\left (-3600 x^2-4500 x^3-900 x^4\right ) \log \left (\frac {1+x}{3}\right )+\left (14400 x+21600 x^2+8100 x^3+900 x^4\right ) \log ^2\left (\frac {1+x}{3}\right )+\left (-19200-33600 x-18000 x^2-3900 x^3-300 x^4\right ) \log ^3\left (\frac {1+x}{3}\right )+\log ^2\left (x^2\right ) \left (-30 x^4+\left (120 x^3+30 x^4\right ) \log \left (\frac {1+x}{3}\right )+\left (120 x^2+120 x^3\right ) \log ^2\left (\frac {1+x}{3}\right )+\left (-480 x-600 x^2-120 x^3\right ) \log ^3\left (\frac {1+x}{3}\right )\right )+\log ^3\left (x^2\right ) \left (-6 x^4 \log \left (\frac {1+x}{3}\right )+\left (24 x^2+24 x^3\right ) \log ^3\left (\frac {1+x}{3}\right )\right )+\log \left (x^2\right ) \left (\left (60 x^3+60 x^4\right ) \log \left (\frac {1+x}{3}\right )+\left (-480 x^2-600 x^3-120 x^4\right ) \log ^2\left (\frac {1+x}{3}\right )+\left (960 x+1440 x^2+540 x^3+60 x^4\right ) \log ^3\left (\frac {1+x}{3}\right )\right )}{\log ^3\left (x^2\right ) \left (-5 x^4-5 x^5+\left (60 x^3+75 x^4+15 x^5\right ) \log \left (\frac {1+x}{3}\right )+\left (-240 x^2-360 x^3-135 x^4-15 x^5\right ) \log ^2\left (\frac {1+x}{3}\right )+\left (320 x+560 x^2+300 x^3+65 x^4+5 x^5\right ) \log ^3\left (\frac {1+x}{3}\right )\right )} \, dx=\frac {3 \, {\left (25 \, {\left (\log \left (3\right )^{2} + 2 \, \log \left (3\right ) + 1\right )} x^{2} - {\left (32 \, {\left (x + 2\right )} \log \left (x\right )^{2} - 25 \, x^{2} + 20 \, {\left (x^{2} + 4 \, x\right )} \log \left (x\right ) - 200 \, x - 400\right )} \log \left (x + 1\right )^{2} - 4 \, {\left (x^{2} {\left (2 \, \log \left (3\right ) + 1\right )} + 8 \, {\left (\log \left (3\right )^{2} + \log \left (3\right )\right )} x + 16 \, \log \left (3\right )^{2}\right )} \log \left (x\right )^{2} + 200 \, {\left (\log \left (3\right )^{2} + \log \left (3\right )\right )} x + 400 \, \log \left (3\right )^{2} - 2 \, {\left (25 \, x^{2} {\left (\log \left (3\right ) + 1\right )} - 4 \, {\left (x^{2} + 4 \, x {\left (2 \, \log \left (3\right ) + 1\right )} + 16 \, \log \left (3\right )\right )} \log \left (x\right )^{2} + 100 \, x {\left (2 \, \log \left (3\right ) + 1\right )} - 10 \, {\left (x^{2} {\left (2 \, \log \left (3\right ) + 1\right )} + 8 \, x \log \left (3\right )\right )} \log \left (x\right ) + 400 \, \log \left (3\right )\right )} \log \left (x + 1\right ) - 20 \, {\left ({\left (\log \left (3\right )^{2} + \log \left (3\right )\right )} x^{2} + 4 \, x \log \left (3\right )^{2}\right )} \log \left (x\right )\right )}}{20 \, {\left ({\left (x^{2} + 8 \, x + 16\right )} \log \left (x + 1\right )^{2} \log \left (x\right )^{2} - 2 \, {\left (x^{2} {\left (\log \left (3\right ) + 1\right )} + 4 \, x {\left (2 \, \log \left (3\right ) + 1\right )} + 16 \, \log \left (3\right )\right )} \log \left (x + 1\right ) \log \left (x\right )^{2} + {\left ({\left (\log \left (3\right )^{2} + 2 \, \log \left (3\right ) + 1\right )} x^{2} + 8 \, {\left (\log \left (3\right )^{2} + \log \left (3\right )\right )} x + 16 \, \log \left (3\right )^{2}\right )} \log \left (x\right )^{2}\right )}} \] Input:
integrate((((24*x^3+24*x^2)*log(1/3*x+1/3)^3-6*x^4*log(1/3*x+1/3))*log(x^2 )^3+((-120*x^3-600*x^2-480*x)*log(1/3*x+1/3)^3+(120*x^3+120*x^2)*log(1/3*x +1/3)^2+(30*x^4+120*x^3)*log(1/3*x+1/3)-30*x^4)*log(x^2)^2+((60*x^4+540*x^ 3+1440*x^2+960*x)*log(1/3*x+1/3)^3+(-120*x^4-600*x^3-480*x^2)*log(1/3*x+1/ 3)^2+(60*x^4+60*x^3)*log(1/3*x+1/3))*log(x^2)+(-300*x^4-3900*x^3-18000*x^2 -33600*x-19200)*log(1/3*x+1/3)^3+(900*x^4+8100*x^3+21600*x^2+14400*x)*log( 1/3*x+1/3)^2+(-900*x^4-4500*x^3-3600*x^2)*log(1/3*x+1/3)+300*x^4+300*x^3)/ ((5*x^5+65*x^4+300*x^3+560*x^2+320*x)*log(1/3*x+1/3)^3+(-15*x^5-135*x^4-36 0*x^3-240*x^2)*log(1/3*x+1/3)^2+(15*x^5+75*x^4+60*x^3)*log(1/3*x+1/3)-5*x^ 5-5*x^4)/log(x^2)^3,x, algorithm="maxima")
Output:
3/20*(25*(log(3)^2 + 2*log(3) + 1)*x^2 - (32*(x + 2)*log(x)^2 - 25*x^2 + 2 0*(x^2 + 4*x)*log(x) - 200*x - 400)*log(x + 1)^2 - 4*(x^2*(2*log(3) + 1) + 8*(log(3)^2 + log(3))*x + 16*log(3)^2)*log(x)^2 + 200*(log(3)^2 + log(3)) *x + 400*log(3)^2 - 2*(25*x^2*(log(3) + 1) - 4*(x^2 + 4*x*(2*log(3) + 1) + 16*log(3))*log(x)^2 + 100*x*(2*log(3) + 1) - 10*(x^2*(2*log(3) + 1) + 8*x *log(3))*log(x) + 400*log(3))*log(x + 1) - 20*((log(3)^2 + log(3))*x^2 + 4 *x*log(3)^2)*log(x))/((x^2 + 8*x + 16)*log(x + 1)^2*log(x)^2 - 2*(x^2*(log (3) + 1) + 4*x*(2*log(3) + 1) + 16*log(3))*log(x + 1)*log(x)^2 + ((log(3)^ 2 + 2*log(3) + 1)*x^2 + 8*(log(3)^2 + log(3))*x + 16*log(3)^2)*log(x)^2)
Leaf count of result is larger than twice the leaf count of optimal. 485 vs. \(2 (33) = 66\).
Time = 0.52 (sec) , antiderivative size = 485, normalized size of antiderivative = 12.12 \[ \int \frac {300 x^3+300 x^4+\left (-3600 x^2-4500 x^3-900 x^4\right ) \log \left (\frac {1+x}{3}\right )+\left (14400 x+21600 x^2+8100 x^3+900 x^4\right ) \log ^2\left (\frac {1+x}{3}\right )+\left (-19200-33600 x-18000 x^2-3900 x^3-300 x^4\right ) \log ^3\left (\frac {1+x}{3}\right )+\log ^2\left (x^2\right ) \left (-30 x^4+\left (120 x^3+30 x^4\right ) \log \left (\frac {1+x}{3}\right )+\left (120 x^2+120 x^3\right ) \log ^2\left (\frac {1+x}{3}\right )+\left (-480 x-600 x^2-120 x^3\right ) \log ^3\left (\frac {1+x}{3}\right )\right )+\log ^3\left (x^2\right ) \left (-6 x^4 \log \left (\frac {1+x}{3}\right )+\left (24 x^2+24 x^3\right ) \log ^3\left (\frac {1+x}{3}\right )\right )+\log \left (x^2\right ) \left (\left (60 x^3+60 x^4\right ) \log \left (\frac {1+x}{3}\right )+\left (-480 x^2-600 x^3-120 x^4\right ) \log ^2\left (\frac {1+x}{3}\right )+\left (960 x+1440 x^2+540 x^3+60 x^4\right ) \log ^3\left (\frac {1+x}{3}\right )\right )}{\log ^3\left (x^2\right ) \left (-5 x^4-5 x^5+\left (60 x^3+75 x^4+15 x^5\right ) \log \left (\frac {1+x}{3}\right )+\left (-240 x^2-360 x^3-135 x^4-15 x^5\right ) \log ^2\left (\frac {1+x}{3}\right )+\left (320 x+560 x^2+300 x^3+65 x^4+5 x^5\right ) \log ^3\left (\frac {1+x}{3}\right )\right )} \, dx =\text {Too large to display} \] Input:
integrate((((24*x^3+24*x^2)*log(1/3*x+1/3)^3-6*x^4*log(1/3*x+1/3))*log(x^2 )^3+((-120*x^3-600*x^2-480*x)*log(1/3*x+1/3)^3+(120*x^3+120*x^2)*log(1/3*x +1/3)^2+(30*x^4+120*x^3)*log(1/3*x+1/3)-30*x^4)*log(x^2)^2+((60*x^4+540*x^ 3+1440*x^2+960*x)*log(1/3*x+1/3)^3+(-120*x^4-600*x^3-480*x^2)*log(1/3*x+1/ 3)^2+(60*x^4+60*x^3)*log(1/3*x+1/3))*log(x^2)+(-300*x^4-3900*x^3-18000*x^2 -33600*x-19200)*log(1/3*x+1/3)^3+(900*x^4+8100*x^3+21600*x^2+14400*x)*log( 1/3*x+1/3)^2+(-900*x^4-4500*x^3-3600*x^2)*log(1/3*x+1/3)+300*x^4+300*x^3)/ ((5*x^5+65*x^4+300*x^3+560*x^2+320*x)*log(1/3*x+1/3)^3+(-15*x^5-135*x^4-36 0*x^3-240*x^2)*log(1/3*x+1/3)^2+(15*x^5+75*x^4+60*x^3)*log(1/3*x+1/3)-5*x^ 5-5*x^4)/log(x^2)^3,x, algorithm="giac")
Output:
-3/5*(2*x^4*log(3)*log(x^2) - 2*x^4*log(x^2)*log(x + 1) - 10*x^4*log(3) + x^4*log(x^2) + 8*x^3*log(3)*log(x^2) + 10*x^4*log(x + 1) - 8*x^3*log(x^2)* log(x + 1) - 10*x^4 - 80*x^3*log(3) + 80*x^3*log(x + 1) - 40*x^3 - 160*x^2 *log(3) + 160*x^2*log(x + 1))/(x^4*log(3)^2*log(x^2) - 2*x^4*log(3)*log(x^ 2)*log(x + 1) + x^4*log(x^2)*log(x + 1)^2 + 2*x^4*log(3)*log(x^2) + 16*x^3 *log(3)^2*log(x^2) - 2*x^4*log(x^2)*log(x + 1) - 32*x^3*log(3)*log(x^2)*lo g(x + 1) + 16*x^3*log(x^2)*log(x + 1)^2 + x^4*log(x^2) + 24*x^3*log(3)*log (x^2) + 96*x^2*log(3)^2*log(x^2) - 24*x^3*log(x^2)*log(x + 1) - 192*x^2*lo g(3)*log(x^2)*log(x + 1) + 96*x^2*log(x^2)*log(x + 1)^2 + 8*x^3*log(x^2) + 96*x^2*log(3)*log(x^2) + 256*x*log(3)^2*log(x^2) - 96*x^2*log(x^2)*log(x + 1) - 512*x*log(3)*log(x^2)*log(x + 1) + 256*x*log(x^2)*log(x + 1)^2 + 16 *x^2*log(x^2) + 128*x*log(3)*log(x^2) + 256*log(3)^2*log(x^2) - 128*x*log( x^2)*log(x + 1) - 512*log(3)*log(x^2)*log(x + 1) + 256*log(x^2)*log(x + 1) ^2) - 3*(2*x*log(x^2) - 5*x - 20)/(x*log(x^2)^2 + 4*log(x^2)^2) - 24/5*(x + 2)/(x^2 + 8*x + 16)
Time = 2.03 (sec) , antiderivative size = 366, normalized size of antiderivative = 9.15 \[ \int \frac {300 x^3+300 x^4+\left (-3600 x^2-4500 x^3-900 x^4\right ) \log \left (\frac {1+x}{3}\right )+\left (14400 x+21600 x^2+8100 x^3+900 x^4\right ) \log ^2\left (\frac {1+x}{3}\right )+\left (-19200-33600 x-18000 x^2-3900 x^3-300 x^4\right ) \log ^3\left (\frac {1+x}{3}\right )+\log ^2\left (x^2\right ) \left (-30 x^4+\left (120 x^3+30 x^4\right ) \log \left (\frac {1+x}{3}\right )+\left (120 x^2+120 x^3\right ) \log ^2\left (\frac {1+x}{3}\right )+\left (-480 x-600 x^2-120 x^3\right ) \log ^3\left (\frac {1+x}{3}\right )\right )+\log ^3\left (x^2\right ) \left (-6 x^4 \log \left (\frac {1+x}{3}\right )+\left (24 x^2+24 x^3\right ) \log ^3\left (\frac {1+x}{3}\right )\right )+\log \left (x^2\right ) \left (\left (60 x^3+60 x^4\right ) \log \left (\frac {1+x}{3}\right )+\left (-480 x^2-600 x^3-120 x^4\right ) \log ^2\left (\frac {1+x}{3}\right )+\left (960 x+1440 x^2+540 x^3+60 x^4\right ) \log ^3\left (\frac {1+x}{3}\right )\right )}{\log ^3\left (x^2\right ) \left (-5 x^4-5 x^5+\left (60 x^3+75 x^4+15 x^5\right ) \log \left (\frac {1+x}{3}\right )+\left (-240 x^2-360 x^3-135 x^4-15 x^5\right ) \log ^2\left (\frac {1+x}{3}\right )+\left (320 x+560 x^2+300 x^3+65 x^4+5 x^5\right ) \log ^3\left (\frac {1+x}{3}\right )\right )} \, dx=\frac {\frac {6\,x\,{\ln \left (x^2\right )}^2}{{\left (x+4\right )}^2}-\frac {3\,x\,\ln \left (x^2\right )}{x+4}+15}{{\ln \left (x^2\right )}^2}-\frac {\frac {3\,x}{x+4}-\frac {6\,x\,\ln \left (x^2\right )}{{\left (x+4\right )}^2}+\frac {3\,x\,{\ln \left (x^2\right )}^2\,\left (x-4\right )}{{\left (x+4\right )}^3}}{\ln \left (x^2\right )}-\frac {\frac {84\,x}{5}+\frac {48}{5}}{x^2+8\,x+16}-\frac {\ln \left (x^2\right )\,\left (12\,x-3\,x^2\right )}{x^3+12\,x^2+48\,x+64}+\frac {6\,\left (-x^6\,{\ln \left (x^2\right )}^2+5\,x^6\,\ln \left (x^2\right )-5\,x^5\,{\ln \left (x^2\right )}^2+45\,x^5\,\ln \left (x^2\right )-16\,x^4\,{\ln \left (x^2\right )}^2+180\,x^4\,\ln \left (x^2\right )-12\,x^3\,{\ln \left (x^2\right )}^2+380\,x^3\,\ln \left (x^2\right )+240\,x^2\,\ln \left (x^2\right )\right )}{5\,{\ln \left (x^2\right )}^2\,\left (x-\ln \left (\frac {x}{3}+\frac {1}{3}\right )\,\left (x+4\right )\right )\,{\left (x+4\right )}^2\,\left (x^3+5\,x^2+16\,x+12\right )}+\frac {3\,\left (x^7\,{\ln \left (x^2\right )}^2+5\,x^6\,{\ln \left (x^2\right )}^2+16\,x^5\,{\ln \left (x^2\right )}^2+12\,x^4\,{\ln \left (x^2\right )}^2\right )}{5\,{\ln \left (x^2\right )}^2\,{\left (x+4\right )}^2\,\left ({\ln \left (\frac {x}{3}+\frac {1}{3}\right )}^2\,{\left (x+4\right )}^2+x^2-2\,x\,\ln \left (\frac {x}{3}+\frac {1}{3}\right )\,\left (x+4\right )\right )\,\left (x^3+5\,x^2+16\,x+12\right )} \] Input:
int(-(log(x/3 + 1/3)^2*(14400*x + 21600*x^2 + 8100*x^3 + 900*x^4) - log(x/ 3 + 1/3)*(3600*x^2 + 4500*x^3 + 900*x^4) - log(x/3 + 1/3)^3*(33600*x + 180 00*x^2 + 3900*x^3 + 300*x^4 + 19200) + log(x^2)^3*(log(x/3 + 1/3)^3*(24*x^ 2 + 24*x^3) - 6*x^4*log(x/3 + 1/3)) + log(x^2)*(log(x/3 + 1/3)^3*(960*x + 1440*x^2 + 540*x^3 + 60*x^4) - log(x/3 + 1/3)^2*(480*x^2 + 600*x^3 + 120*x ^4) + log(x/3 + 1/3)*(60*x^3 + 60*x^4)) - log(x^2)^2*(log(x/3 + 1/3)^3*(48 0*x + 600*x^2 + 120*x^3) - log(x/3 + 1/3)^2*(120*x^2 + 120*x^3) - log(x/3 + 1/3)*(120*x^3 + 30*x^4) + 30*x^4) + 300*x^3 + 300*x^4)/(log(x^2)^3*(log( x/3 + 1/3)^2*(240*x^2 + 360*x^3 + 135*x^4 + 15*x^5) - log(x/3 + 1/3)^3*(32 0*x + 560*x^2 + 300*x^3 + 65*x^4 + 5*x^5) - log(x/3 + 1/3)*(60*x^3 + 75*x^ 4 + 15*x^5) + 5*x^4 + 5*x^5)),x)
Output:
((6*x*log(x^2)^2)/(x + 4)^2 - (3*x*log(x^2))/(x + 4) + 15)/log(x^2)^2 - (( 3*x)/(x + 4) - (6*x*log(x^2))/(x + 4)^2 + (3*x*log(x^2)^2*(x - 4))/(x + 4) ^3)/log(x^2) - ((84*x)/5 + 48/5)/(8*x + x^2 + 16) - (log(x^2)*(12*x - 3*x^ 2))/(48*x + 12*x^2 + x^3 + 64) + (6*(240*x^2*log(x^2) + 380*x^3*log(x^2) + 180*x^4*log(x^2) + 45*x^5*log(x^2) + 5*x^6*log(x^2) - 12*x^3*log(x^2)^2 - 16*x^4*log(x^2)^2 - 5*x^5*log(x^2)^2 - x^6*log(x^2)^2))/(5*log(x^2)^2*(x - log(x/3 + 1/3)*(x + 4))*(x + 4)^2*(16*x + 5*x^2 + x^3 + 12)) + (3*(12*x^ 4*log(x^2)^2 + 16*x^5*log(x^2)^2 + 5*x^6*log(x^2)^2 + x^7*log(x^2)^2))/(5* log(x^2)^2*(x + 4)^2*(log(x/3 + 1/3)^2*(x + 4)^2 + x^2 - 2*x*log(x/3 + 1/3 )*(x + 4))*(16*x + 5*x^2 + x^3 + 12))
Time = 0.20 (sec) , antiderivative size = 192, normalized size of antiderivative = 4.80 \[ \int \frac {300 x^3+300 x^4+\left (-3600 x^2-4500 x^3-900 x^4\right ) \log \left (\frac {1+x}{3}\right )+\left (14400 x+21600 x^2+8100 x^3+900 x^4\right ) \log ^2\left (\frac {1+x}{3}\right )+\left (-19200-33600 x-18000 x^2-3900 x^3-300 x^4\right ) \log ^3\left (\frac {1+x}{3}\right )+\log ^2\left (x^2\right ) \left (-30 x^4+\left (120 x^3+30 x^4\right ) \log \left (\frac {1+x}{3}\right )+\left (120 x^2+120 x^3\right ) \log ^2\left (\frac {1+x}{3}\right )+\left (-480 x-600 x^2-120 x^3\right ) \log ^3\left (\frac {1+x}{3}\right )\right )+\log ^3\left (x^2\right ) \left (-6 x^4 \log \left (\frac {1+x}{3}\right )+\left (24 x^2+24 x^3\right ) \log ^3\left (\frac {1+x}{3}\right )\right )+\log \left (x^2\right ) \left (\left (60 x^3+60 x^4\right ) \log \left (\frac {1+x}{3}\right )+\left (-480 x^2-600 x^3-120 x^4\right ) \log ^2\left (\frac {1+x}{3}\right )+\left (960 x+1440 x^2+540 x^3+60 x^4\right ) \log ^3\left (\frac {1+x}{3}\right )\right )}{\log ^3\left (x^2\right ) \left (-5 x^4-5 x^5+\left (60 x^3+75 x^4+15 x^5\right ) \log \left (\frac {1+x}{3}\right )+\left (-240 x^2-360 x^3-135 x^4-15 x^5\right ) \log ^2\left (\frac {1+x}{3}\right )+\left (320 x+560 x^2+300 x^3+65 x^4+5 x^5\right ) \log ^3\left (\frac {1+x}{3}\right )\right )} \, dx=\frac {\frac {3 \mathrm {log}\left (x^{2}\right )^{2} \mathrm {log}\left (\frac {x}{3}+\frac {1}{3}\right )^{2} x^{2}}{5}-6 \,\mathrm {log}\left (x^{2}\right ) \mathrm {log}\left (\frac {x}{3}+\frac {1}{3}\right )^{2} x^{2}-24 \,\mathrm {log}\left (x^{2}\right ) \mathrm {log}\left (\frac {x}{3}+\frac {1}{3}\right )^{2} x +6 \,\mathrm {log}\left (x^{2}\right ) \mathrm {log}\left (\frac {x}{3}+\frac {1}{3}\right ) x^{2}+15 \mathrm {log}\left (\frac {x}{3}+\frac {1}{3}\right )^{2} x^{2}+120 \mathrm {log}\left (\frac {x}{3}+\frac {1}{3}\right )^{2} x +240 \mathrm {log}\left (\frac {x}{3}+\frac {1}{3}\right )^{2}-30 \,\mathrm {log}\left (\frac {x}{3}+\frac {1}{3}\right ) x^{2}-120 \,\mathrm {log}\left (\frac {x}{3}+\frac {1}{3}\right ) x +15 x^{2}}{\mathrm {log}\left (x^{2}\right )^{2} \left (\mathrm {log}\left (\frac {x}{3}+\frac {1}{3}\right )^{2} x^{2}+8 \mathrm {log}\left (\frac {x}{3}+\frac {1}{3}\right )^{2} x +16 \mathrm {log}\left (\frac {x}{3}+\frac {1}{3}\right )^{2}-2 \,\mathrm {log}\left (\frac {x}{3}+\frac {1}{3}\right ) x^{2}-8 \,\mathrm {log}\left (\frac {x}{3}+\frac {1}{3}\right ) x +x^{2}\right )} \] Input:
int((((24*x^3+24*x^2)*log(1/3*x+1/3)^3-6*x^4*log(1/3*x+1/3))*log(x^2)^3+(( -120*x^3-600*x^2-480*x)*log(1/3*x+1/3)^3+(120*x^3+120*x^2)*log(1/3*x+1/3)^ 2+(30*x^4+120*x^3)*log(1/3*x+1/3)-30*x^4)*log(x^2)^2+((60*x^4+540*x^3+1440 *x^2+960*x)*log(1/3*x+1/3)^3+(-120*x^4-600*x^3-480*x^2)*log(1/3*x+1/3)^2+( 60*x^4+60*x^3)*log(1/3*x+1/3))*log(x^2)+(-300*x^4-3900*x^3-18000*x^2-33600 *x-19200)*log(1/3*x+1/3)^3+(900*x^4+8100*x^3+21600*x^2+14400*x)*log(1/3*x+ 1/3)^2+(-900*x^4-4500*x^3-3600*x^2)*log(1/3*x+1/3)+300*x^4+300*x^3)/((5*x^ 5+65*x^4+300*x^3+560*x^2+320*x)*log(1/3*x+1/3)^3+(-15*x^5-135*x^4-360*x^3- 240*x^2)*log(1/3*x+1/3)^2+(15*x^5+75*x^4+60*x^3)*log(1/3*x+1/3)-5*x^5-5*x^ 4)/log(x^2)^3,x)
Output:
(3*(log(x**2)**2*log((x + 1)/3)**2*x**2 - 10*log(x**2)*log((x + 1)/3)**2*x **2 - 40*log(x**2)*log((x + 1)/3)**2*x + 10*log(x**2)*log((x + 1)/3)*x**2 + 25*log((x + 1)/3)**2*x**2 + 200*log((x + 1)/3)**2*x + 400*log((x + 1)/3) **2 - 50*log((x + 1)/3)*x**2 - 200*log((x + 1)/3)*x + 25*x**2))/(5*log(x** 2)**2*(log((x + 1)/3)**2*x**2 + 8*log((x + 1)/3)**2*x + 16*log((x + 1)/3)* *2 - 2*log((x + 1)/3)*x**2 - 8*log((x + 1)/3)*x + x**2))