Integrand size = 101, antiderivative size = 125 \[ \int \frac {4 \sqrt {3}-12 \sqrt {3} x^2+4 x^3+33 \sqrt {3} x^4+24 x^5}{4-8 \sqrt {3} x+12 x^2+12 \sqrt {3} x^3-27 x^4-16 \sqrt {3} x^5+27 x^6+24 \sqrt {3} x^7+16 x^8} \, dx=\arctan \left (\frac {1}{9} \left (6+\sqrt {3} x+12 x^2\right )\right )+\arctan \left (\frac {813+827 \sqrt {3} x+399 x^2+1437 \sqrt {3} x^3+2952 x^4+1008 \sqrt {3} x^5}{1764}\right )+\arctan \left (\frac {1}{188} \left (-306+368 \sqrt {3} x+483 x^2-543 \sqrt {3} x^3-349 x^4+911 \sqrt {3} x^5+1656 x^6+336 \sqrt {3} x^7\right )\right ) \] Output:
arctan(2/3+1/9*x*3^(1/2)+4/3*x^2)+arctan(271/588+827/1764*x*3^(1/2)+19/84* x^2+479/588*3^(1/2)*x^3+82/49*x^4+4/7*3^(1/2)*x^5)+arctan(-153/94+92/47*x* 3^(1/2)+483/188*x^2-543/188*3^(1/2)*x^3-349/188*x^4+911/188*3^(1/2)*x^5+41 4/47*x^6+84/47*3^(1/2)*x^7)
Result contains complex when optimal does not.
Time = 0.07 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.70 \[ \int \frac {4 \sqrt {3}-12 \sqrt {3} x^2+4 x^3+33 \sqrt {3} x^4+24 x^5}{4-8 \sqrt {3} x+12 x^2+12 \sqrt {3} x^3-27 x^4-16 \sqrt {3} x^5+27 x^6+24 \sqrt {3} x^7+16 x^8} \, dx=-\frac {1}{2} i \log \left (2 i-(2+2 i) \sqrt {3} x-3 i x^2+3 \sqrt {3} x^3+4 x^4\right )+\frac {1}{2} i \log \left (-2 i-(2-2 i) \sqrt {3} x+3 i x^2+3 \sqrt {3} x^3+4 x^4\right ) \] Input:
Integrate[(4*Sqrt[3] - 12*Sqrt[3]*x^2 + 4*x^3 + 33*Sqrt[3]*x^4 + 24*x^5)/( 4 - 8*Sqrt[3]*x + 12*x^2 + 12*Sqrt[3]*x^3 - 27*x^4 - 16*Sqrt[3]*x^5 + 27*x ^6 + 24*Sqrt[3]*x^7 + 16*x^8),x]
Output:
(-1/2*I)*Log[2*I - (2 + 2*I)*Sqrt[3]*x - (3*I)*x^2 + 3*Sqrt[3]*x^3 + 4*x^4 ] + (I/2)*Log[-2*I - (2 - 2*I)*Sqrt[3]*x + (3*I)*x^2 + 3*Sqrt[3]*x^3 + 4*x ^4]
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.
| \(\displaystyle \int \frac {24 x^5+33 \sqrt {3} x^4+4 x^3-12 \sqrt {3} x^2+4 \sqrt {3}}{16 x^8+24 \sqrt {3} x^7+27 x^6-16 \sqrt {3} x^5-27 x^4+12 \sqrt {3} x^3+12 x^2-8 \sqrt {3} x+4} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {24 x^5}{16 x^8+24 \sqrt {3} x^7+27 x^6-16 \sqrt {3} x^5-27 x^4+12 \sqrt {3} x^3+12 x^2-8 \sqrt {3} x+4}+\frac {33 \sqrt {3} x^4}{16 x^8+24 \sqrt {3} x^7+27 x^6-16 \sqrt {3} x^5-27 x^4+12 \sqrt {3} x^3+12 x^2-8 \sqrt {3} x+4}+\frac {4 x^3}{16 x^8+24 \sqrt {3} x^7+27 x^6-16 \sqrt {3} x^5-27 x^4+12 \sqrt {3} x^3+12 x^2-8 \sqrt {3} x+4}-\frac {12 \sqrt {3} x^2}{16 x^8+24 \sqrt {3} x^7+27 x^6-16 \sqrt {3} x^5-27 x^4+12 \sqrt {3} x^3+12 x^2-8 \sqrt {3} x+4}+\frac {4 \sqrt {3}}{16 x^8+24 \sqrt {3} x^7+27 x^6-16 \sqrt {3} x^5-27 x^4+12 \sqrt {3} x^3+12 x^2-8 \sqrt {3} x+4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 4 \sqrt {3} \int \frac {1}{16 x^8+24 \sqrt {3} x^7+27 x^6-16 \sqrt {3} x^5-27 x^4+12 \sqrt {3} x^3+12 x^2-8 \sqrt {3} x+4}dx-12 \sqrt {3} \int \frac {x^2}{16 x^8+24 \sqrt {3} x^7+27 x^6-16 \sqrt {3} x^5-27 x^4+12 \sqrt {3} x^3+12 x^2-8 \sqrt {3} x+4}dx+4 \int \frac {x^3}{16 x^8+24 \sqrt {3} x^7+27 x^6-16 \sqrt {3} x^5-27 x^4+12 \sqrt {3} x^3+12 x^2-8 \sqrt {3} x+4}dx+33 \sqrt {3} \int \frac {x^4}{16 x^8+24 \sqrt {3} x^7+27 x^6-16 \sqrt {3} x^5-27 x^4+12 \sqrt {3} x^3+12 x^2-8 \sqrt {3} x+4}dx+24 \int \frac {x^5}{16 x^8+24 \sqrt {3} x^7+27 x^6-16 \sqrt {3} x^5-27 x^4+12 \sqrt {3} x^3+12 x^2-8 \sqrt {3} x+4}dx\) |
Input:
Int[(4*Sqrt[3] - 12*Sqrt[3]*x^2 + 4*x^3 + 33*Sqrt[3]*x^4 + 24*x^5)/(4 - 8* Sqrt[3]*x + 12*x^2 + 12*Sqrt[3]*x^3 - 27*x^4 - 16*Sqrt[3]*x^5 + 27*x^6 + 2 4*Sqrt[3]*x^7 + 16*x^8),x]
Output:
$Aborted
Result contains complex when optimal does not.
Time = 0.50 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.61
| method | result | size |
| parallelrisch | \(-\frac {i \ln \left (\frac {3 \sqrt {3}\, x^{3}}{4}+x^{4}-\frac {i \sqrt {3}\, x}{2}-\frac {3 i x^{2}}{4}-\frac {\sqrt {3}\, x}{2}+\frac {i}{2}\right )}{2}+\frac {i \ln \left (\frac {3 \sqrt {3}\, x^{3}}{4}+x^{4}+\frac {i \sqrt {3}\, x}{2}+\frac {3 i x^{2}}{4}-\frac {\sqrt {3}\, x}{2}-\frac {i}{2}\right )}{2}\) | \(76\) |
| risch | \(\arctan \left (\frac {2}{3}+\frac {\sqrt {3}\, x}{9}+\frac {4 x^{2}}{3}\right )+\arctan \left (\frac {271}{588}+\frac {827 \sqrt {3}\, x}{1764}+\frac {19 x^{2}}{84}+\frac {479 \sqrt {3}\, x^{3}}{588}+\frac {82 x^{4}}{49}+\frac {4 \sqrt {3}\, x^{5}}{7}\right )+\arctan \left (-\frac {153}{94}+\frac {92 \sqrt {3}\, x}{47}+\frac {483 x^{2}}{188}-\frac {543 \sqrt {3}\, x^{3}}{188}-\frac {349 x^{4}}{188}+\frac {911 \sqrt {3}\, x^{5}}{188}+\frac {414 x^{6}}{47}+\frac {84 \sqrt {3}\, x^{7}}{47}\right )\) | \(99\) |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (4-8 \sqrt {3}\, \textit {\_Z} +12 \textit {\_Z}^{2}+12 \sqrt {3}\, \textit {\_Z}^{3}-27 \textit {\_Z}^{4}-16 \sqrt {3}\, \textit {\_Z}^{5}+27 \textit {\_Z}^{6}+24 \sqrt {3}\, \textit {\_Z}^{7}+16 \textit {\_Z}^{8}\right )}{\sum }\frac {\left (24 \textit {\_R}^{5}+4 \textit {\_R}^{3}+\sqrt {3}\, \left (33 \textit {\_R}^{4}-12 \textit {\_R}^{2}+4\right )\right ) \ln \left (x -\textit {\_R} \right )}{64 \textit {\_R}^{7}+81 \textit {\_R}^{5}-54 \textit {\_R}^{3}+12 \textit {\_R} +2 \sqrt {3}\, \left (42 \textit {\_R}^{6}-20 \textit {\_R}^{4}+9 \textit {\_R}^{2}-2\right )}\right )}{2}\) | \(136\) |
Input:
int((4*3^(1/2)-12*3^(1/2)*x^2+4*x^3+33*3^(1/2)*x^4+24*x^5)/(4-8*3^(1/2)*x+ 12*x^2+12*3^(1/2)*x^3-27*x^4-16*3^(1/2)*x^5+27*x^6+24*3^(1/2)*x^7+16*x^8), x,method=_RETURNVERBOSE)
Output:
-1/2*I*ln(3/4*3^(1/2)*x^3+x^4-1/2*I*3^(1/2)*x-3/4*I*x^2-1/2*3^(1/2)*x+1/2* I)+1/2*I*ln(3/4*3^(1/2)*x^3+x^4+1/2*I*3^(1/2)*x+3/4*I*x^2-1/2*3^(1/2)*x-1/ 2*I)
Time = 0.08 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.71 \[ \int \frac {4 \sqrt {3}-12 \sqrt {3} x^2+4 x^3+33 \sqrt {3} x^4+24 x^5}{4-8 \sqrt {3} x+12 x^2+12 \sqrt {3} x^3-27 x^4-16 \sqrt {3} x^5+27 x^6+24 \sqrt {3} x^7+16 x^8} \, dx=\arctan \left (\frac {414}{47} \, x^{6} - \frac {349}{188} \, x^{4} + \frac {483}{188} \, x^{2} + \frac {1}{188} \, \sqrt {3} {\left (336 \, x^{7} + 911 \, x^{5} - 543 \, x^{3} + 368 \, x\right )} - \frac {153}{94}\right ) + \arctan \left (\frac {82}{49} \, x^{4} + \frac {19}{84} \, x^{2} + \frac {1}{1764} \, \sqrt {3} {\left (1008 \, x^{5} + 1437 \, x^{3} + 827 \, x\right )} + \frac {271}{588}\right ) + \arctan \left (\frac {4}{3} \, x^{2} + \frac {1}{9} \, \sqrt {3} x + \frac {2}{3}\right ) \] Input:
integrate((4*3^(1/2)-12*3^(1/2)*x^2+4*x^3+33*3^(1/2)*x^4+24*x^5)/(4-8*3^(1 /2)*x+12*x^2+12*3^(1/2)*x^3-27*x^4-16*3^(1/2)*x^5+27*x^6+24*3^(1/2)*x^7+16 *x^8),x, algorithm="fricas")
Output:
arctan(414/47*x^6 - 349/188*x^4 + 483/188*x^2 + 1/188*sqrt(3)*(336*x^7 + 9 11*x^5 - 543*x^3 + 368*x) - 153/94) + arctan(82/49*x^4 + 19/84*x^2 + 1/176 4*sqrt(3)*(1008*x^5 + 1437*x^3 + 827*x) + 271/588) + arctan(4/3*x^2 + 1/9* sqrt(3)*x + 2/3)
Time = 1.92 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.14 \[ \int \frac {4 \sqrt {3}-12 \sqrt {3} x^2+4 x^3+33 \sqrt {3} x^4+24 x^5}{4-8 \sqrt {3} x+12 x^2+12 \sqrt {3} x^3-27 x^4-16 \sqrt {3} x^5+27 x^6+24 \sqrt {3} x^7+16 x^8} \, dx=\operatorname {atan}{\left (\frac {4 x^{2}}{3} + \frac {\sqrt {3} x}{9} + \frac {2}{3} \right )} + \operatorname {atan}{\left (\frac {4 \sqrt {3} x^{5}}{7} + \frac {82 x^{4}}{49} + \frac {479 \sqrt {3} x^{3}}{588} + \frac {19 x^{2}}{84} + \frac {827 \sqrt {3} x}{1764} + \frac {271}{588} \right )} + \operatorname {atan}{\left (\frac {84 \sqrt {3} x^{7}}{47} + \frac {414 x^{6}}{47} + \frac {911 \sqrt {3} x^{5}}{188} - \frac {349 x^{4}}{188} - \frac {543 \sqrt {3} x^{3}}{188} + \frac {483 x^{2}}{188} + \frac {92 \sqrt {3} x}{47} - \frac {153}{94} \right )} \] Input:
integrate((4*3**(1/2)-12*3**(1/2)*x**2+4*x**3+33*3**(1/2)*x**4+24*x**5)/(4 -8*3**(1/2)*x+12*x**2+12*3**(1/2)*x**3-27*x**4-16*3**(1/2)*x**5+27*x**6+24 *3**(1/2)*x**7+16*x**8),x)
Output:
atan(4*x**2/3 + sqrt(3)*x/9 + 2/3) + atan(4*sqrt(3)*x**5/7 + 82*x**4/49 + 479*sqrt(3)*x**3/588 + 19*x**2/84 + 827*sqrt(3)*x/1764 + 271/588) + atan(8 4*sqrt(3)*x**7/47 + 414*x**6/47 + 911*sqrt(3)*x**5/188 - 349*x**4/188 - 54 3*sqrt(3)*x**3/188 + 483*x**2/188 + 92*sqrt(3)*x/47 - 153/94)
\[ \int \frac {4 \sqrt {3}-12 \sqrt {3} x^2+4 x^3+33 \sqrt {3} x^4+24 x^5}{4-8 \sqrt {3} x+12 x^2+12 \sqrt {3} x^3-27 x^4-16 \sqrt {3} x^5+27 x^6+24 \sqrt {3} x^7+16 x^8} \, dx=\int { \frac {24 \, x^{5} + 33 \, \sqrt {3} x^{4} + 4 \, x^{3} - 12 \, \sqrt {3} x^{2} + 4 \, \sqrt {3}}{16 \, x^{8} + 24 \, \sqrt {3} x^{7} + 27 \, x^{6} - 16 \, \sqrt {3} x^{5} - 27 \, x^{4} + 12 \, \sqrt {3} x^{3} + 12 \, x^{2} - 8 \, \sqrt {3} x + 4} \,d x } \] Input:
integrate((4*3^(1/2)-12*3^(1/2)*x^2+4*x^3+33*3^(1/2)*x^4+24*x^5)/(4-8*3^(1 /2)*x+12*x^2+12*3^(1/2)*x^3-27*x^4-16*3^(1/2)*x^5+27*x^6+24*3^(1/2)*x^7+16 *x^8),x, algorithm="maxima")
Output:
integrate((24*x^5 + 33*sqrt(3)*x^4 + 4*x^3 - 12*sqrt(3)*x^2 + 4*sqrt(3))/( 16*x^8 + 24*sqrt(3)*x^7 + 27*x^6 - 16*sqrt(3)*x^5 - 27*x^4 + 12*sqrt(3)*x^ 3 + 12*x^2 - 8*sqrt(3)*x + 4), x)
Time = 0.16 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.90 \[ \int \frac {4 \sqrt {3}-12 \sqrt {3} x^2+4 x^3+33 \sqrt {3} x^4+24 x^5}{4-8 \sqrt {3} x+12 x^2+12 \sqrt {3} x^3-27 x^4-16 \sqrt {3} x^5+27 x^6+24 \sqrt {3} x^7+16 x^8} \, dx=-\arctan \left (-\frac {84}{47} \, \sqrt {3} x^{7} - \frac {414}{47} \, x^{6} - \frac {911}{188} \, \sqrt {3} x^{5} + \frac {349}{188} \, x^{4} + \frac {543}{188} \, \sqrt {3} x^{3} - \frac {483}{188} \, x^{2} - \frac {92}{47} \, \sqrt {3} x + \frac {153}{94}\right ) + \arctan \left (\frac {1}{1764} \, \sqrt {3} {\left (1008 \, x^{5} + 984 \, \sqrt {3} x^{4} + 1437 \, x^{3} + 133 \, \sqrt {3} x^{2} + 827 \, x + 271 \, \sqrt {3}\right )}\right ) + \arctan \left (\frac {1}{9} \, \sqrt {3} {\left (4 \, \sqrt {3} x^{2} + x + 2 \, \sqrt {3}\right )}\right ) \] Input:
integrate((4*3^(1/2)-12*3^(1/2)*x^2+4*x^3+33*3^(1/2)*x^4+24*x^5)/(4-8*3^(1 /2)*x+12*x^2+12*3^(1/2)*x^3-27*x^4-16*3^(1/2)*x^5+27*x^6+24*3^(1/2)*x^7+16 *x^8),x, algorithm="giac")
Output:
-arctan(-84/47*sqrt(3)*x^7 - 414/47*x^6 - 911/188*sqrt(3)*x^5 + 349/188*x^ 4 + 543/188*sqrt(3)*x^3 - 483/188*x^2 - 92/47*sqrt(3)*x + 153/94) + arctan (1/1764*sqrt(3)*(1008*x^5 + 984*sqrt(3)*x^4 + 1437*x^3 + 133*sqrt(3)*x^2 + 827*x + 271*sqrt(3))) + arctan(1/9*sqrt(3)*(4*sqrt(3)*x^2 + x + 2*sqrt(3) ))
Time = 11.54 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.94 \[ \int \frac {4 \sqrt {3}-12 \sqrt {3} x^2+4 x^3+33 \sqrt {3} x^4+24 x^5}{4-8 \sqrt {3} x+12 x^2+12 \sqrt {3} x^3-27 x^4-16 \sqrt {3} x^5+27 x^6+24 \sqrt {3} x^7+16 x^8} \, dx=\sum _{k=1}^8\ln \left (\frac {{\left (4\,{\mathrm {root}\left (z^8+z^6+\frac {3\,z^4}{8}+\frac {z^2}{16}+\frac {1}{256},z,k\right )}^2+1\right )}^3\,\left (210727320\,x-125336697\,\sqrt {3}\,\mathrm {root}\left (z^8+z^6+\frac {3\,z^4}{8}+\frac {z^2}{16}+\frac {1}{256},z,k\right )+\mathrm {root}\left (z^8+z^6+\frac {3\,z^4}{8}+\frac {z^2}{16}+\frac {1}{256},z,k\right )\,x\,614708847-135190001\,\sqrt {3}\right )\,343}{4294967296}\right )\,\mathrm {root}\left (z^8+z^6+\frac {3\,z^4}{8}+\frac {z^2}{16}+\frac {1}{256},z,k\right ) \] Input:
int((4*3^(1/2) - 12*3^(1/2)*x^2 + 33*3^(1/2)*x^4 + 4*x^3 + 24*x^5)/(12*3^( 1/2)*x^3 - 8*3^(1/2)*x - 16*3^(1/2)*x^5 + 24*3^(1/2)*x^7 + 12*x^2 - 27*x^4 + 27*x^6 + 16*x^8 + 4),x)
Output:
symsum(log((343*(4*root(z^8 + z^6 + (3*z^4)/8 + z^2/16 + 1/256, z, k)^2 + 1)^3*(210727320*x - 125336697*3^(1/2)*root(z^8 + z^6 + (3*z^4)/8 + z^2/16 + 1/256, z, k) + 614708847*root(z^8 + z^6 + (3*z^4)/8 + z^2/16 + 1/256, z, k)*x - 135190001*3^(1/2)))/4294967296)*root(z^8 + z^6 + (3*z^4)/8 + z^2/1 6 + 1/256, z, k), k, 1, 8)
\[ \int \frac {4 \sqrt {3}-12 \sqrt {3} x^2+4 x^3+33 \sqrt {3} x^4+24 x^5}{4-8 \sqrt {3} x+12 x^2+12 \sqrt {3} x^3-27 x^4-16 \sqrt {3} x^5+27 x^6+24 \sqrt {3} x^7+16 x^8} \, dx=-48 \sqrt {3}\, \left (\int \frac {x^{12}}{256 x^{16}-864 x^{14}+2169 x^{12}-3570 x^{10}+3809 x^{8}-1632 x^{6}+504 x^{4}-96 x^{2}+16}d x \right )+987 \sqrt {3}\, \left (\int \frac {x^{10}}{256 x^{16}-864 x^{14}+2169 x^{12}-3570 x^{10}+3809 x^{8}-1632 x^{6}+504 x^{4}-96 x^{2}+16}d x \right )-1375 \sqrt {3}\, \left (\int \frac {x^{8}}{256 x^{16}-864 x^{14}+2169 x^{12}-3570 x^{10}+3809 x^{8}-1632 x^{6}+504 x^{4}-96 x^{2}+16}d x \right )+972 \sqrt {3}\, \left (\int \frac {x^{6}}{256 x^{16}-864 x^{14}+2169 x^{12}-3570 x^{10}+3809 x^{8}-1632 x^{6}+504 x^{4}-96 x^{2}+16}d x \right )-88 \sqrt {3}\, \left (\int \frac {x^{4}}{256 x^{16}-864 x^{14}+2169 x^{12}-3570 x^{10}+3809 x^{8}-1632 x^{6}+504 x^{4}-96 x^{2}+16}d x \right )+16 \sqrt {3}\, \left (\int \frac {1}{256 x^{16}-864 x^{14}+2169 x^{12}-3570 x^{10}+3809 x^{8}-1632 x^{6}+504 x^{4}-96 x^{2}+16}d x \right )+384 \left (\int \frac {x^{13}}{256 x^{16}-864 x^{14}+2169 x^{12}-3570 x^{10}+3809 x^{8}-1632 x^{6}+504 x^{4}-96 x^{2}+16}d x \right )-1664 \left (\int \frac {x^{11}}{256 x^{16}-864 x^{14}+2169 x^{12}-3570 x^{10}+3809 x^{8}-1632 x^{6}+504 x^{4}-96 x^{2}+16}d x \right )+1908 \left (\int \frac {x^{9}}{256 x^{16}-864 x^{14}+2169 x^{12}-3570 x^{10}+3809 x^{8}-1632 x^{6}+504 x^{4}-96 x^{2}+16}d x \right )-1872 \left (\int \frac {x^{7}}{256 x^{16}-864 x^{14}+2169 x^{12}-3570 x^{10}+3809 x^{8}-1632 x^{6}+504 x^{4}-96 x^{2}+16}d x \right )+1560 \left (\int \frac {x^{5}}{256 x^{16}-864 x^{14}+2169 x^{12}-3570 x^{10}+3809 x^{8}-1632 x^{6}+504 x^{4}-96 x^{2}+16}d x \right )-416 \left (\int \frac {x^{3}}{256 x^{16}-864 x^{14}+2169 x^{12}-3570 x^{10}+3809 x^{8}-1632 x^{6}+504 x^{4}-96 x^{2}+16}d x \right )+96 \left (\int \frac {x}{256 x^{16}-864 x^{14}+2169 x^{12}-3570 x^{10}+3809 x^{8}-1632 x^{6}+504 x^{4}-96 x^{2}+16}d x \right ) \] Input:
int((4*3^(1/2)-12*3^(1/2)*x^2+4*x^3+33*3^(1/2)*x^4+24*x^5)/(4-8*3^(1/2)*x+ 12*x^2+12*3^(1/2)*x^3-27*x^4-16*3^(1/2)*x^5+27*x^6+24*3^(1/2)*x^7+16*x^8), x)
Output:
- 48*sqrt(3)*int(x**12/(256*x**16 - 864*x**14 + 2169*x**12 - 3570*x**10 + 3809*x**8 - 1632*x**6 + 504*x**4 - 96*x**2 + 16),x) + 987*sqrt(3)*int(x** 10/(256*x**16 - 864*x**14 + 2169*x**12 - 3570*x**10 + 3809*x**8 - 1632*x** 6 + 504*x**4 - 96*x**2 + 16),x) - 1375*sqrt(3)*int(x**8/(256*x**16 - 864*x **14 + 2169*x**12 - 3570*x**10 + 3809*x**8 - 1632*x**6 + 504*x**4 - 96*x** 2 + 16),x) + 972*sqrt(3)*int(x**6/(256*x**16 - 864*x**14 + 2169*x**12 - 35 70*x**10 + 3809*x**8 - 1632*x**6 + 504*x**4 - 96*x**2 + 16),x) - 88*sqrt(3 )*int(x**4/(256*x**16 - 864*x**14 + 2169*x**12 - 3570*x**10 + 3809*x**8 - 1632*x**6 + 504*x**4 - 96*x**2 + 16),x) + 16*sqrt(3)*int(1/(256*x**16 - 86 4*x**14 + 2169*x**12 - 3570*x**10 + 3809*x**8 - 1632*x**6 + 504*x**4 - 96* x**2 + 16),x) + 384*int(x**13/(256*x**16 - 864*x**14 + 2169*x**12 - 3570*x **10 + 3809*x**8 - 1632*x**6 + 504*x**4 - 96*x**2 + 16),x) - 1664*int(x**1 1/(256*x**16 - 864*x**14 + 2169*x**12 - 3570*x**10 + 3809*x**8 - 1632*x**6 + 504*x**4 - 96*x**2 + 16),x) + 1908*int(x**9/(256*x**16 - 864*x**14 + 21 69*x**12 - 3570*x**10 + 3809*x**8 - 1632*x**6 + 504*x**4 - 96*x**2 + 16),x ) - 1872*int(x**7/(256*x**16 - 864*x**14 + 2169*x**12 - 3570*x**10 + 3809* x**8 - 1632*x**6 + 504*x**4 - 96*x**2 + 16),x) + 1560*int(x**5/(256*x**16 - 864*x**14 + 2169*x**12 - 3570*x**10 + 3809*x**8 - 1632*x**6 + 504*x**4 - 96*x**2 + 16),x) - 416*int(x**3/(256*x**16 - 864*x**14 + 2169*x**12 - 357 0*x**10 + 3809*x**8 - 1632*x**6 + 504*x**4 - 96*x**2 + 16),x) + 96*int(...