Integrand size = 280, antiderivative size = 42 \[ \int \frac {32 x^4-16 x^5+2 x^6+\left (240 x^2-120 x^3-x^4+8 x^5-x^6\right ) \log \left (\frac {1}{5} \left (-15+x^2\right )\right )+\left (-15 x^2+x^4+\left (-240+120 x+x^2-8 x^3+x^4\right ) \log (3)\right ) \log ^2\left (\frac {1}{5} \left (-15+x^2\right )\right )}{-960 x^4+480 x^5+4 x^6-32 x^7+4 x^8+\left (2400 x^3-1080 x^4-40 x^5+72 x^6-8 x^7+\left (-1920 x^2+960 x^3+8 x^4-64 x^5+8 x^6\right ) \log (3)\right ) \log \left (\frac {1}{5} \left (-15+x^2\right )\right )+\left (-1500 x^2+600 x^3+40 x^4-40 x^5+4 x^6+\left (2400 x-1080 x^2-40 x^3+72 x^4-8 x^5\right ) \log (3)+\left (-960+480 x+4 x^2-32 x^3+4 x^4\right ) \log ^2(3)\right ) \log ^2\left (\frac {1}{5} \left (-15+x^2\right )\right )} \, dx=\frac {1}{4 \left (\frac {-x+\frac {x}{-4+x}+\log (3)}{x}+\frac {x}{\log \left (\left (-\frac {3}{x}+\frac {x}{5}\right ) x\right )}\right )} \] Output:
1/4/(x/ln((1/5*x-3/x)*x)+(x/(-4+x)+ln(3)-x)/x)
Time = 0.08 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.67 \[ \int \frac {32 x^4-16 x^5+2 x^6+\left (240 x^2-120 x^3-x^4+8 x^5-x^6\right ) \log \left (\frac {1}{5} \left (-15+x^2\right )\right )+\left (-15 x^2+x^4+\left (-240+120 x+x^2-8 x^3+x^4\right ) \log (3)\right ) \log ^2\left (\frac {1}{5} \left (-15+x^2\right )\right )}{-960 x^4+480 x^5+4 x^6-32 x^7+4 x^8+\left (2400 x^3-1080 x^4-40 x^5+72 x^6-8 x^7+\left (-1920 x^2+960 x^3+8 x^4-64 x^5+8 x^6\right ) \log (3)\right ) \log \left (\frac {1}{5} \left (-15+x^2\right )\right )+\left (-1500 x^2+600 x^3+40 x^4-40 x^5+4 x^6+\left (2400 x-1080 x^2-40 x^3+72 x^4-8 x^5\right ) \log (3)+\left (-960+480 x+4 x^2-32 x^3+4 x^4\right ) \log ^2(3)\right ) \log ^2\left (\frac {1}{5} \left (-15+x^2\right )\right )} \, dx=\frac {(-4+x) x^2+(x-4 \log (3)+x \log (3)) \log \left (-3+\frac {x^2}{5}\right )}{4 \left ((-4+x) x^2+\left (-x^2-4 \log (3)+x (5+\log (3))\right ) \log \left (-3+\frac {x^2}{5}\right )\right )} \] Input:
Integrate[(32*x^4 - 16*x^5 + 2*x^6 + (240*x^2 - 120*x^3 - x^4 + 8*x^5 - x^ 6)*Log[(-15 + x^2)/5] + (-15*x^2 + x^4 + (-240 + 120*x + x^2 - 8*x^3 + x^4 )*Log[3])*Log[(-15 + x^2)/5]^2)/(-960*x^4 + 480*x^5 + 4*x^6 - 32*x^7 + 4*x ^8 + (2400*x^3 - 1080*x^4 - 40*x^5 + 72*x^6 - 8*x^7 + (-1920*x^2 + 960*x^3 + 8*x^4 - 64*x^5 + 8*x^6)*Log[3])*Log[(-15 + x^2)/5] + (-1500*x^2 + 600*x ^3 + 40*x^4 - 40*x^5 + 4*x^6 + (2400*x - 1080*x^2 - 40*x^3 + 72*x^4 - 8*x^ 5)*Log[3] + (-960 + 480*x + 4*x^2 - 32*x^3 + 4*x^4)*Log[3]^2)*Log[(-15 + x ^2)/5]^2),x]
Output:
((-4 + x)*x^2 + (x - 4*Log[3] + x*Log[3])*Log[-3 + x^2/5])/(4*((-4 + x)*x^ 2 + (-x^2 - 4*Log[3] + x*(5 + Log[3]))*Log[-3 + x^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.
| \(\displaystyle \int \frac {2 x^6-16 x^5+32 x^4+\left (x^4-15 x^2+\left (x^4-8 x^3+x^2+120 x-240\right ) \log (3)\right ) \log ^2\left (\frac {1}{5} \left (x^2-15\right )\right )+\left (-x^6+8 x^5-x^4-120 x^3+240 x^2\right ) \log \left (\frac {1}{5} \left (x^2-15\right )\right )}{4 x^8-32 x^7+4 x^6+480 x^5-960 x^4+\left (4 x^6-40 x^5+40 x^4+600 x^3-1500 x^2+\left (4 x^4-32 x^3+4 x^2+480 x-960\right ) \log ^2(3)+\left (-8 x^5+72 x^4-40 x^3-1080 x^2+2400 x\right ) \log (3)\right ) \log ^2\left (\frac {1}{5} \left (x^2-15\right )\right )+\left (-8 x^7+72 x^6-40 x^5-1080 x^4+2400 x^3+\left (8 x^6-64 x^5+8 x^4+960 x^3-1920 x^2\right ) \log (3)\right ) \log \left (\frac {1}{5} \left (x^2-15\right )\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-2 (x-4)^2 x^4-\left (x^2-15\right ) \left (x^2 (1+\log (3))-8 x \log (3)+16 \log (3)\right ) \log ^2\left (\frac {x^2}{5}-3\right )+(x-4)^2 \left (x^2-15\right ) x^2 \log \left (\frac {x^2}{5}-3\right )}{4 \left (15-x^2\right ) \left ((x-4) x^2+\left (-x^2+x (5+\log (3))-4 \log (3)\right ) \log \left (\frac {x^2}{5}-3\right )\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \int -\frac {2 (4-x)^2 x^4+(4-x)^2 \left (15-x^2\right ) \log \left (\frac {x^2}{5}-3\right ) x^2-\left (15-x^2\right ) \left ((1+\log (3)) x^2-8 \log (3) x+16 \log (3)\right ) \log ^2\left (\frac {x^2}{5}-3\right )}{\left (15-x^2\right ) \left ((4-x) x^2+\left (x^2-(5+\log (3)) x+4 \log (3)\right ) \log \left (\frac {x^2}{5}-3\right )\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{4} \int \frac {2 (4-x)^2 x^4+(4-x)^2 \left (15-x^2\right ) \log \left (\frac {x^2}{5}-3\right ) x^2-\left (15-x^2\right ) \left ((1+\log (3)) x^2-8 \log (3) x+16 \log (3)\right ) \log ^2\left (\frac {x^2}{5}-3\right )}{\left (15-x^2\right ) \left ((4-x) x^2+\left (x^2-(5+\log (3)) x+\log (81)\right ) \log \left (\frac {x^2}{5}-3\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {1}{4} \int \left (\frac {(4-x)^2 \left (-x^5+12 \left (1+\frac {\log (3)}{6}\right ) x^4-25 \left (1+\frac {4 \log (3)}{5}\right ) x^3-100 \left (1-\frac {1}{50} \log (3) (19+\log (3))\right ) x^2+300 \left (1-\frac {1}{75} \log (3) (-40+\log (81))\right ) x-240 \log (3) \left (1-\frac {(-15+\log (9)) \log (81)}{60 \log (3)}\right )\right ) x^4}{\left (15-x^2\right ) \left (x^2-(5+\log (3)) x+\log (81)\right )^2 \left (-x^3+\log \left (\frac {x^2}{5}-3\right ) x^2+4 x^2-5 \left (1+\frac {\log (3)}{5}\right ) \log \left (\frac {x^2}{5}-3\right ) x+\log (81) \log \left (\frac {x^2}{5}-3\right )\right )^2}+\frac {(4-x) \left (-x^3+(11+\log (27)) x^2-(20+20 \log (3)+\log (81)) x+4 \log (531441)\right ) x^2}{\left (x^2-(5+\log (3)) x+\log (81)\right )^2 \left (-x^3+\log \left (\frac {x^2}{5}-3\right ) x^2+4 x^2-5 \left (1+\frac {\log (3)}{5}\right ) \log \left (\frac {x^2}{5}-3\right ) x+\log (81) \log \left (\frac {x^2}{5}-3\right )\right )}+\frac {-\left ((1+\log (3)) x^2\right )+8 \log (3) x-16 \log (3)}{\left (x^2-(5+\log (3)) x+\log (81)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{4} \int \left (\frac {(4-x)^2 \left (-x^5+12 \left (1+\frac {\log (3)}{6}\right ) x^4-25 \left (1+\frac {4 \log (3)}{5}\right ) x^3-100 \left (1-\frac {1}{50} \log (3) (19+\log (3))\right ) x^2+300 \left (1+\frac {8 \log (3)}{15}-\frac {\log ^2(81)}{300}\right ) x-240 \log (3) \left (1-\frac {(-15+\log (9)) \log (81)}{60 \log (3)}\right )\right ) x^4}{\left (15-x^2\right ) \left (x^2-(5+\log (3)) x+\log (81)\right )^2 \left (-x^3+\log \left (\frac {x^2}{5}-3\right ) x^2+4 x^2-5 \left (1+\frac {\log (3)}{5}\right ) \log \left (\frac {x^2}{5}-3\right ) x+\log (81) \log \left (\frac {x^2}{5}-3\right )\right )^2}+\frac {(4-x) \left (-x^3+(11+\log (27)) x^2-(20+20 \log (3)+\log (81)) x+4 \log (531441)\right ) x^2}{\left (x^2-(5+\log (3)) x+\log (81)\right )^2 \left (-x^3+\log \left (\frac {x^2}{5}-3\right ) x^2+4 x^2-5 \left (1+\frac {\log (3)}{5}\right ) \log \left (\frac {x^2}{5}-3\right ) x+\log (81) \log \left (\frac {x^2}{5}-3\right )\right )}+\frac {-\left ((1+\log (3)) x^2\right )+8 \log (3) x-16 \log (3)}{\left (x^2-(5+\log (3)) x+\log (81)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle -\frac {1}{4} \int \left (\frac {(4-x)^2 \left (-x^5+12 \left (1+\frac {\log (3)}{6}\right ) x^4-25 \left (1+\frac {4 \log (3)}{5}\right ) x^3-100 \left (1-\frac {1}{50} \log (3) (19+\log (3))\right ) x^2+300 \left (1+\frac {8 \log (3)}{15}-\frac {\log ^2(81)}{300}\right ) x-240 \log (3) \left (1-\frac {(-15+\log (9)) \log (81)}{60 \log (3)}\right )\right ) x^4}{\left (15-x^2\right ) \left (x^2-(5+\log (3)) x+\log (81)\right )^2 \left (-x^3+\log \left (\frac {x^2}{5}-3\right ) x^2+4 x^2-5 \left (1+\frac {\log (3)}{5}\right ) \log \left (\frac {x^2}{5}-3\right ) x+\log (81) \log \left (\frac {x^2}{5}-3\right )\right )^2}+\frac {(4-x) \left (-x^3+(11+\log (27)) x^2-(20+20 \log (3)+\log (81)) x+4 \log (531441)\right ) x^2}{\left (x^2-(5+\log (3)) x+\log (81)\right )^2 \left (-x^3+\log \left (\frac {x^2}{5}-3\right ) x^2+4 x^2-5 \left (1+\frac {\log (3)}{5}\right ) \log \left (\frac {x^2}{5}-3\right ) x+\log (81) \log \left (\frac {x^2}{5}-3\right )\right )}+\frac {-\left ((1+\log (3)) x^2\right )+8 \log (3) x-16 \log (3)}{\left (x^2-(5+\log (3)) x+\log (81)\right )^2}\right )dx\) |
Input:
Int[(32*x^4 - 16*x^5 + 2*x^6 + (240*x^2 - 120*x^3 - x^4 + 8*x^5 - x^6)*Log [(-15 + x^2)/5] + (-15*x^2 + x^4 + (-240 + 120*x + x^2 - 8*x^3 + x^4)*Log[ 3])*Log[(-15 + x^2)/5]^2)/(-960*x^4 + 480*x^5 + 4*x^6 - 32*x^7 + 4*x^8 + ( 2400*x^3 - 1080*x^4 - 40*x^5 + 72*x^6 - 8*x^7 + (-1920*x^2 + 960*x^3 + 8*x ^4 - 64*x^5 + 8*x^6)*Log[3])*Log[(-15 + x^2)/5] + (-1500*x^2 + 600*x^3 + 4 0*x^4 - 40*x^5 + 4*x^6 + (2400*x - 1080*x^2 - 40*x^3 + 72*x^4 - 8*x^5)*Log [3] + (-960 + 480*x + 4*x^2 - 32*x^3 + 4*x^4)*Log[3]^2)*Log[(-15 + x^2)/5] ^2),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(104\) vs. \(2(38)=76\).
Time = 5.49 (sec) , antiderivative size = 105, normalized size of antiderivative = 2.50
| method | result | size |
| parallelrisch | \(\frac {-4 \ln \left (3\right ) \ln \left (\frac {x^{2}}{5}-3\right )+\ln \left (\frac {x^{2}}{5}-3\right ) x +x^{3}-4 x^{2}+x \ln \left (3\right ) \ln \left (\frac {x^{2}}{5}-3\right )}{4 x \ln \left (3\right ) \ln \left (\frac {x^{2}}{5}-3\right )-4 x^{2} \ln \left (\frac {x^{2}}{5}-3\right )+4 x^{3}-16 \ln \left (3\right ) \ln \left (\frac {x^{2}}{5}-3\right )+20 \ln \left (\frac {x^{2}}{5}-3\right ) x -16 x^{2}}\) | \(105\) |
| risch | \(\frac {x \ln \left (3\right )-4 \ln \left (3\right )+x}{4 x \ln \left (3\right )-4 x^{2}-16 \ln \left (3\right )+20 x}-\frac {x^{3} \left (x^{2}-8 x +16\right )}{4 \left (x \ln \left (3\right )-x^{2}-4 \ln \left (3\right )+5 x \right ) \left (x \ln \left (3\right ) \ln \left (\frac {x^{2}}{5}-3\right )-x^{2} \ln \left (\frac {x^{2}}{5}-3\right )+x^{3}-4 \ln \left (3\right ) \ln \left (\frac {x^{2}}{5}-3\right )+5 \ln \left (\frac {x^{2}}{5}-3\right ) x -4 x^{2}\right )}\) | \(124\) |
Input:
int((((x^4-8*x^3+x^2+120*x-240)*ln(3)+x^4-15*x^2)*ln(1/5*x^2-3)^2+(-x^6+8* x^5-x^4-120*x^3+240*x^2)*ln(1/5*x^2-3)+2*x^6-16*x^5+32*x^4)/(((4*x^4-32*x^ 3+4*x^2+480*x-960)*ln(3)^2+(-8*x^5+72*x^4-40*x^3-1080*x^2+2400*x)*ln(3)+4* x^6-40*x^5+40*x^4+600*x^3-1500*x^2)*ln(1/5*x^2-3)^2+((8*x^6-64*x^5+8*x^4+9 60*x^3-1920*x^2)*ln(3)-8*x^7+72*x^6-40*x^5-1080*x^4+2400*x^3)*ln(1/5*x^2-3 )+4*x^8-32*x^7+4*x^6+480*x^5-960*x^4),x,method=_RETURNVERBOSE)
Output:
1/4*(-4*ln(3)*ln(1/5*x^2-3)+ln(1/5*x^2-3)*x+x^3-4*x^2+x*ln(3)*ln(1/5*x^2-3 ))/(x*ln(3)*ln(1/5*x^2-3)-x^2*ln(1/5*x^2-3)+x^3-4*ln(3)*ln(1/5*x^2-3)+5*ln (1/5*x^2-3)*x-4*x^2)
Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.50 \[ \int \frac {32 x^4-16 x^5+2 x^6+\left (240 x^2-120 x^3-x^4+8 x^5-x^6\right ) \log \left (\frac {1}{5} \left (-15+x^2\right )\right )+\left (-15 x^2+x^4+\left (-240+120 x+x^2-8 x^3+x^4\right ) \log (3)\right ) \log ^2\left (\frac {1}{5} \left (-15+x^2\right )\right )}{-960 x^4+480 x^5+4 x^6-32 x^7+4 x^8+\left (2400 x^3-1080 x^4-40 x^5+72 x^6-8 x^7+\left (-1920 x^2+960 x^3+8 x^4-64 x^5+8 x^6\right ) \log (3)\right ) \log \left (\frac {1}{5} \left (-15+x^2\right )\right )+\left (-1500 x^2+600 x^3+40 x^4-40 x^5+4 x^6+\left (2400 x-1080 x^2-40 x^3+72 x^4-8 x^5\right ) \log (3)+\left (-960+480 x+4 x^2-32 x^3+4 x^4\right ) \log ^2(3)\right ) \log ^2\left (\frac {1}{5} \left (-15+x^2\right )\right )} \, dx=\frac {x^{3} - 4 \, x^{2} + {\left ({\left (x - 4\right )} \log \left (3\right ) + x\right )} \log \left (\frac {1}{5} \, x^{2} - 3\right )}{4 \, {\left (x^{3} - 4 \, x^{2} - {\left (x^{2} - {\left (x - 4\right )} \log \left (3\right ) - 5 \, x\right )} \log \left (\frac {1}{5} \, x^{2} - 3\right )\right )}} \] Input:
integrate((((x^4-8*x^3+x^2+120*x-240)*log(3)+x^4-15*x^2)*log(1/5*x^2-3)^2+ (-x^6+8*x^5-x^4-120*x^3+240*x^2)*log(1/5*x^2-3)+2*x^6-16*x^5+32*x^4)/(((4* x^4-32*x^3+4*x^2+480*x-960)*log(3)^2+(-8*x^5+72*x^4-40*x^3-1080*x^2+2400*x )*log(3)+4*x^6-40*x^5+40*x^4+600*x^3-1500*x^2)*log(1/5*x^2-3)^2+((8*x^6-64 *x^5+8*x^4+960*x^3-1920*x^2)*log(3)-8*x^7+72*x^6-40*x^5-1080*x^4+2400*x^3) *log(1/5*x^2-3)+4*x^8-32*x^7+4*x^6+480*x^5-960*x^4),x, algorithm="fricas")
Output:
1/4*(x^3 - 4*x^2 + ((x - 4)*log(3) + x)*log(1/5*x^2 - 3))/(x^3 - 4*x^2 - ( x^2 - (x - 4)*log(3) - 5*x)*log(1/5*x^2 - 3))
Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (31) = 62\).
Time = 0.79 (sec) , antiderivative size = 160, normalized size of antiderivative = 3.81 \[ \int \frac {32 x^4-16 x^5+2 x^6+\left (240 x^2-120 x^3-x^4+8 x^5-x^6\right ) \log \left (\frac {1}{5} \left (-15+x^2\right )\right )+\left (-15 x^2+x^4+\left (-240+120 x+x^2-8 x^3+x^4\right ) \log (3)\right ) \log ^2\left (\frac {1}{5} \left (-15+x^2\right )\right )}{-960 x^4+480 x^5+4 x^6-32 x^7+4 x^8+\left (2400 x^3-1080 x^4-40 x^5+72 x^6-8 x^7+\left (-1920 x^2+960 x^3+8 x^4-64 x^5+8 x^6\right ) \log (3)\right ) \log \left (\frac {1}{5} \left (-15+x^2\right )\right )+\left (-1500 x^2+600 x^3+40 x^4-40 x^5+4 x^6+\left (2400 x-1080 x^2-40 x^3+72 x^4-8 x^5\right ) \log (3)+\left (-960+480 x+4 x^2-32 x^3+4 x^4\right ) \log ^2(3)\right ) \log ^2\left (\frac {1}{5} \left (-15+x^2\right )\right )} \, dx=\frac {x \left (- \log {\left (3 \right )} - 1\right ) + 4 \log {\left (3 \right )}}{4 x^{2} + x \left (-20 - 4 \log {\left (3 \right )}\right ) + 16 \log {\left (3 \right )}} + \frac {- x^{5} + 8 x^{4} - 16 x^{3}}{- 4 x^{5} + 4 x^{4} \log {\left (3 \right )} + 36 x^{4} - 80 x^{3} - 32 x^{3} \log {\left (3 \right )} + 64 x^{2} \log {\left (3 \right )} + \left (4 x^{4} - 40 x^{3} - 8 x^{3} \log {\left (3 \right )} + 4 x^{2} \log {\left (3 \right )}^{2} + 72 x^{2} \log {\left (3 \right )} + 100 x^{2} - 160 x \log {\left (3 \right )} - 32 x \log {\left (3 \right )}^{2} + 64 \log {\left (3 \right )}^{2}\right ) \log {\left (\frac {x^{2}}{5} - 3 \right )}} \] Input:
integrate((((x**4-8*x**3+x**2+120*x-240)*ln(3)+x**4-15*x**2)*ln(1/5*x**2-3 )**2+(-x**6+8*x**5-x**4-120*x**3+240*x**2)*ln(1/5*x**2-3)+2*x**6-16*x**5+3 2*x**4)/(((4*x**4-32*x**3+4*x**2+480*x-960)*ln(3)**2+(-8*x**5+72*x**4-40*x **3-1080*x**2+2400*x)*ln(3)+4*x**6-40*x**5+40*x**4+600*x**3-1500*x**2)*ln( 1/5*x**2-3)**2+((8*x**6-64*x**5+8*x**4+960*x**3-1920*x**2)*ln(3)-8*x**7+72 *x**6-40*x**5-1080*x**4+2400*x**3)*ln(1/5*x**2-3)+4*x**8-32*x**7+4*x**6+48 0*x**5-960*x**4),x)
Output:
(x*(-log(3) - 1) + 4*log(3))/(4*x**2 + x*(-20 - 4*log(3)) + 16*log(3)) + ( -x**5 + 8*x**4 - 16*x**3)/(-4*x**5 + 4*x**4*log(3) + 36*x**4 - 80*x**3 - 3 2*x**3*log(3) + 64*x**2*log(3) + (4*x**4 - 40*x**3 - 8*x**3*log(3) + 4*x** 2*log(3)**2 + 72*x**2*log(3) + 100*x**2 - 160*x*log(3) - 32*x*log(3)**2 + 64*log(3)**2)*log(x**2/5 - 3))
Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (39) = 78\).
Time = 0.21 (sec) , antiderivative size = 102, normalized size of antiderivative = 2.43 \[ \int \frac {32 x^4-16 x^5+2 x^6+\left (240 x^2-120 x^3-x^4+8 x^5-x^6\right ) \log \left (\frac {1}{5} \left (-15+x^2\right )\right )+\left (-15 x^2+x^4+\left (-240+120 x+x^2-8 x^3+x^4\right ) \log (3)\right ) \log ^2\left (\frac {1}{5} \left (-15+x^2\right )\right )}{-960 x^4+480 x^5+4 x^6-32 x^7+4 x^8+\left (2400 x^3-1080 x^4-40 x^5+72 x^6-8 x^7+\left (-1920 x^2+960 x^3+8 x^4-64 x^5+8 x^6\right ) \log (3)\right ) \log \left (\frac {1}{5} \left (-15+x^2\right )\right )+\left (-1500 x^2+600 x^3+40 x^4-40 x^5+4 x^6+\left (2400 x-1080 x^2-40 x^3+72 x^4-8 x^5\right ) \log (3)+\left (-960+480 x+4 x^2-32 x^3+4 x^4\right ) \log ^2(3)\right ) \log ^2\left (\frac {1}{5} \left (-15+x^2\right )\right )} \, dx=\frac {x^{3} - {\left (\log \left (5\right ) \log \left (3\right ) + \log \left (5\right )\right )} x - 4 \, x^{2} + 4 \, \log \left (5\right ) \log \left (3\right ) + {\left (x {\left (\log \left (3\right ) + 1\right )} - 4 \, \log \left (3\right )\right )} \log \left (x^{2} - 15\right )}{4 \, {\left (x^{3} + x^{2} {\left (\log \left (5\right ) - 4\right )} - {\left (\log \left (5\right ) \log \left (3\right ) + 5 \, \log \left (5\right )\right )} x + 4 \, \log \left (5\right ) \log \left (3\right ) - {\left (x^{2} - x {\left (\log \left (3\right ) + 5\right )} + 4 \, \log \left (3\right )\right )} \log \left (x^{2} - 15\right )\right )}} \] Input:
integrate((((x^4-8*x^3+x^2+120*x-240)*log(3)+x^4-15*x^2)*log(1/5*x^2-3)^2+ (-x^6+8*x^5-x^4-120*x^3+240*x^2)*log(1/5*x^2-3)+2*x^6-16*x^5+32*x^4)/(((4* x^4-32*x^3+4*x^2+480*x-960)*log(3)^2+(-8*x^5+72*x^4-40*x^3-1080*x^2+2400*x )*log(3)+4*x^6-40*x^5+40*x^4+600*x^3-1500*x^2)*log(1/5*x^2-3)^2+((8*x^6-64 *x^5+8*x^4+960*x^3-1920*x^2)*log(3)-8*x^7+72*x^6-40*x^5-1080*x^4+2400*x^3) *log(1/5*x^2-3)+4*x^8-32*x^7+4*x^6+480*x^5-960*x^4),x, algorithm="maxima")
Output:
1/4*(x^3 - (log(5)*log(3) + log(5))*x - 4*x^2 + 4*log(5)*log(3) + (x*(log( 3) + 1) - 4*log(3))*log(x^2 - 15))/(x^3 + x^2*(log(5) - 4) - (log(5)*log(3 ) + 5*log(5))*x + 4*log(5)*log(3) - (x^2 - x*(log(3) + 5) + 4*log(3))*log( x^2 - 15))
Leaf count of result is larger than twice the leaf count of optimal. 266 vs. \(2 (39) = 78\).
Time = 0.68 (sec) , antiderivative size = 266, normalized size of antiderivative = 6.33 \[ \int \frac {32 x^4-16 x^5+2 x^6+\left (240 x^2-120 x^3-x^4+8 x^5-x^6\right ) \log \left (\frac {1}{5} \left (-15+x^2\right )\right )+\left (-15 x^2+x^4+\left (-240+120 x+x^2-8 x^3+x^4\right ) \log (3)\right ) \log ^2\left (\frac {1}{5} \left (-15+x^2\right )\right )}{-960 x^4+480 x^5+4 x^6-32 x^7+4 x^8+\left (2400 x^3-1080 x^4-40 x^5+72 x^6-8 x^7+\left (-1920 x^2+960 x^3+8 x^4-64 x^5+8 x^6\right ) \log (3)\right ) \log \left (\frac {1}{5} \left (-15+x^2\right )\right )+\left (-1500 x^2+600 x^3+40 x^4-40 x^5+4 x^6+\left (2400 x-1080 x^2-40 x^3+72 x^4-8 x^5\right ) \log (3)+\left (-960+480 x+4 x^2-32 x^3+4 x^4\right ) \log ^2(3)\right ) \log ^2\left (\frac {1}{5} \left (-15+x^2\right )\right )} \, dx=\frac {x^{5} - 8 \, x^{4} + 16 \, x^{3}}{4 \, {\left (x^{5} + x^{4} \log \left (5\right ) - x^{4} \log \left (3\right ) - 2 \, x^{3} \log \left (5\right ) \log \left (3\right ) + x^{2} \log \left (5\right ) \log \left (3\right )^{2} - x^{4} \log \left (x^{2} - 15\right ) + 2 \, x^{3} \log \left (3\right ) \log \left (x^{2} - 15\right ) - x^{2} \log \left (3\right )^{2} \log \left (x^{2} - 15\right ) - 9 \, x^{4} - 10 \, x^{3} \log \left (5\right ) + 8 \, x^{3} \log \left (3\right ) + 18 \, x^{2} \log \left (5\right ) \log \left (3\right ) - 8 \, x \log \left (5\right ) \log \left (3\right )^{2} + 10 \, x^{3} \log \left (x^{2} - 15\right ) - 18 \, x^{2} \log \left (3\right ) \log \left (x^{2} - 15\right ) + 8 \, x \log \left (3\right )^{2} \log \left (x^{2} - 15\right ) + 20 \, x^{3} + 25 \, x^{2} \log \left (5\right ) - 16 \, x^{2} \log \left (3\right ) - 40 \, x \log \left (5\right ) \log \left (3\right ) + 16 \, \log \left (5\right ) \log \left (3\right )^{2} - 25 \, x^{2} \log \left (x^{2} - 15\right ) + 40 \, x \log \left (3\right ) \log \left (x^{2} - 15\right ) - 16 \, \log \left (3\right )^{2} \log \left (x^{2} - 15\right )\right )}} - \frac {x \log \left (3\right ) + x - 4 \, \log \left (3\right )}{4 \, {\left (x^{2} - x \log \left (3\right ) - 5 \, x + 4 \, \log \left (3\right )\right )}} \] Input:
integrate((((x^4-8*x^3+x^2+120*x-240)*log(3)+x^4-15*x^2)*log(1/5*x^2-3)^2+ (-x^6+8*x^5-x^4-120*x^3+240*x^2)*log(1/5*x^2-3)+2*x^6-16*x^5+32*x^4)/(((4* x^4-32*x^3+4*x^2+480*x-960)*log(3)^2+(-8*x^5+72*x^4-40*x^3-1080*x^2+2400*x )*log(3)+4*x^6-40*x^5+40*x^4+600*x^3-1500*x^2)*log(1/5*x^2-3)^2+((8*x^6-64 *x^5+8*x^4+960*x^3-1920*x^2)*log(3)-8*x^7+72*x^6-40*x^5-1080*x^4+2400*x^3) *log(1/5*x^2-3)+4*x^8-32*x^7+4*x^6+480*x^5-960*x^4),x, algorithm="giac")
Output:
1/4*(x^5 - 8*x^4 + 16*x^3)/(x^5 + x^4*log(5) - x^4*log(3) - 2*x^3*log(5)*l og(3) + x^2*log(5)*log(3)^2 - x^4*log(x^2 - 15) + 2*x^3*log(3)*log(x^2 - 1 5) - x^2*log(3)^2*log(x^2 - 15) - 9*x^4 - 10*x^3*log(5) + 8*x^3*log(3) + 1 8*x^2*log(5)*log(3) - 8*x*log(5)*log(3)^2 + 10*x^3*log(x^2 - 15) - 18*x^2* log(3)*log(x^2 - 15) + 8*x*log(3)^2*log(x^2 - 15) + 20*x^3 + 25*x^2*log(5) - 16*x^2*log(3) - 40*x*log(5)*log(3) + 16*log(5)*log(3)^2 - 25*x^2*log(x^ 2 - 15) + 40*x*log(3)*log(x^2 - 15) - 16*log(3)^2*log(x^2 - 15)) - 1/4*(x* log(3) + x - 4*log(3))/(x^2 - x*log(3) - 5*x + 4*log(3))
Timed out. \[ \int \frac {32 x^4-16 x^5+2 x^6+\left (240 x^2-120 x^3-x^4+8 x^5-x^6\right ) \log \left (\frac {1}{5} \left (-15+x^2\right )\right )+\left (-15 x^2+x^4+\left (-240+120 x+x^2-8 x^3+x^4\right ) \log (3)\right ) \log ^2\left (\frac {1}{5} \left (-15+x^2\right )\right )}{-960 x^4+480 x^5+4 x^6-32 x^7+4 x^8+\left (2400 x^3-1080 x^4-40 x^5+72 x^6-8 x^7+\left (-1920 x^2+960 x^3+8 x^4-64 x^5+8 x^6\right ) \log (3)\right ) \log \left (\frac {1}{5} \left (-15+x^2\right )\right )+\left (-1500 x^2+600 x^3+40 x^4-40 x^5+4 x^6+\left (2400 x-1080 x^2-40 x^3+72 x^4-8 x^5\right ) \log (3)+\left (-960+480 x+4 x^2-32 x^3+4 x^4\right ) \log ^2(3)\right ) \log ^2\left (\frac {1}{5} \left (-15+x^2\right )\right )} \, dx=\int \frac {{\ln \left (\frac {x^2}{5}-3\right )}^2\,\left (x^4-15\,x^2+\ln \left (3\right )\,\left (x^4-8\,x^3+x^2+120\,x-240\right )\right )-\ln \left (\frac {x^2}{5}-3\right )\,\left (x^6-8\,x^5+x^4+120\,x^3-240\,x^2\right )+32\,x^4-16\,x^5+2\,x^6}{\ln \left (\frac {x^2}{5}-3\right )\,\left (\ln \left (3\right )\,\left (8\,x^6-64\,x^5+8\,x^4+960\,x^3-1920\,x^2\right )+2400\,x^3-1080\,x^4-40\,x^5+72\,x^6-8\,x^7\right )+{\ln \left (\frac {x^2}{5}-3\right )}^2\,\left ({\ln \left (3\right )}^2\,\left (4\,x^4-32\,x^3+4\,x^2+480\,x-960\right )-1500\,x^2+600\,x^3+40\,x^4-40\,x^5+4\,x^6-\ln \left (3\right )\,\left (8\,x^5-72\,x^4+40\,x^3+1080\,x^2-2400\,x\right )\right )-960\,x^4+480\,x^5+4\,x^6-32\,x^7+4\,x^8} \,d x \] Input:
int((log(x^2/5 - 3)^2*(x^4 - 15*x^2 + log(3)*(120*x + x^2 - 8*x^3 + x^4 - 240)) - log(x^2/5 - 3)*(120*x^3 - 240*x^2 + x^4 - 8*x^5 + x^6) + 32*x^4 - 16*x^5 + 2*x^6)/(log(x^2/5 - 3)*(log(3)*(960*x^3 - 1920*x^2 + 8*x^4 - 64*x ^5 + 8*x^6) + 2400*x^3 - 1080*x^4 - 40*x^5 + 72*x^6 - 8*x^7) + log(x^2/5 - 3)^2*(log(3)^2*(480*x + 4*x^2 - 32*x^3 + 4*x^4 - 960) - 1500*x^2 + 600*x^ 3 + 40*x^4 - 40*x^5 + 4*x^6 - log(3)*(1080*x^2 - 2400*x + 40*x^3 - 72*x^4 + 8*x^5)) - 960*x^4 + 480*x^5 + 4*x^6 - 32*x^7 + 4*x^8),x)
Output:
int((log(x^2/5 - 3)^2*(x^4 - 15*x^2 + log(3)*(120*x + x^2 - 8*x^3 + x^4 - 240)) - log(x^2/5 - 3)*(120*x^3 - 240*x^2 + x^4 - 8*x^5 + x^6) + 32*x^4 - 16*x^5 + 2*x^6)/(log(x^2/5 - 3)*(log(3)*(960*x^3 - 1920*x^2 + 8*x^4 - 64*x ^5 + 8*x^6) + 2400*x^3 - 1080*x^4 - 40*x^5 + 72*x^6 - 8*x^7) + log(x^2/5 - 3)^2*(log(3)^2*(480*x + 4*x^2 - 32*x^3 + 4*x^4 - 960) - 1500*x^2 + 600*x^ 3 + 40*x^4 - 40*x^5 + 4*x^6 - log(3)*(1080*x^2 - 2400*x + 40*x^3 - 72*x^4 + 8*x^5)) - 960*x^4 + 480*x^5 + 4*x^6 - 32*x^7 + 4*x^8), x)
Time = 0.19 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.79 \[ \int \frac {32 x^4-16 x^5+2 x^6+\left (240 x^2-120 x^3-x^4+8 x^5-x^6\right ) \log \left (\frac {1}{5} \left (-15+x^2\right )\right )+\left (-15 x^2+x^4+\left (-240+120 x+x^2-8 x^3+x^4\right ) \log (3)\right ) \log ^2\left (\frac {1}{5} \left (-15+x^2\right )\right )}{-960 x^4+480 x^5+4 x^6-32 x^7+4 x^8+\left (2400 x^3-1080 x^4-40 x^5+72 x^6-8 x^7+\left (-1920 x^2+960 x^3+8 x^4-64 x^5+8 x^6\right ) \log (3)\right ) \log \left (\frac {1}{5} \left (-15+x^2\right )\right )+\left (-1500 x^2+600 x^3+40 x^4-40 x^5+4 x^6+\left (2400 x-1080 x^2-40 x^3+72 x^4-8 x^5\right ) \log (3)+\left (-960+480 x+4 x^2-32 x^3+4 x^4\right ) \log ^2(3)\right ) \log ^2\left (\frac {1}{5} \left (-15+x^2\right )\right )} \, dx=\frac {\mathrm {log}\left (\frac {x^{2}}{5}-3\right ) x \left (x -4\right )}{4 \,\mathrm {log}\left (\frac {x^{2}}{5}-3\right ) \mathrm {log}\left (3\right ) x -16 \,\mathrm {log}\left (\frac {x^{2}}{5}-3\right ) \mathrm {log}\left (3\right )-4 \,\mathrm {log}\left (\frac {x^{2}}{5}-3\right ) x^{2}+20 \,\mathrm {log}\left (\frac {x^{2}}{5}-3\right ) x +4 x^{3}-16 x^{2}} \] Input:
int((((x^4-8*x^3+x^2+120*x-240)*log(3)+x^4-15*x^2)*log(1/5*x^2-3)^2+(-x^6+ 8*x^5-x^4-120*x^3+240*x^2)*log(1/5*x^2-3)+2*x^6-16*x^5+32*x^4)/(((4*x^4-32 *x^3+4*x^2+480*x-960)*log(3)^2+(-8*x^5+72*x^4-40*x^3-1080*x^2+2400*x)*log( 3)+4*x^6-40*x^5+40*x^4+600*x^3-1500*x^2)*log(1/5*x^2-3)^2+((8*x^6-64*x^5+8 *x^4+960*x^3-1920*x^2)*log(3)-8*x^7+72*x^6-40*x^5-1080*x^4+2400*x^3)*log(1 /5*x^2-3)+4*x^8-32*x^7+4*x^6+480*x^5-960*x^4),x)
Output:
(log((x**2 - 15)/5)*x*(x - 4))/(4*(log((x**2 - 15)/5)*log(3)*x - 4*log((x* *2 - 15)/5)*log(3) - log((x**2 - 15)/5)*x**2 + 5*log((x**2 - 15)/5)*x + x* *3 - 4*x**2))