Integrand size = 95, antiderivative size = 231 \[ \int \frac {3-7 x^2-21 x^4-32 x^5+72 x^6+108 x^7+45 x^8+6 x^9}{1-9 x^2-4 x^3+37 x^4+30 x^5-75 x^6-90 x^7+48 x^8+104 x^9+54 x^{10}+12 x^{11}+x^{12}} \, dx=\frac {1}{2} \left (1+\sqrt {5}\right ) \arctan \left (\frac {1}{2} \left (-1+\sqrt {5}\right ) \left (3 x+x^2\right )\right )+\frac {1}{2} \left (-1+\sqrt {5}\right ) \arctan \left (\frac {1}{2} \left (1+\sqrt {5}\right ) \left (3 x+x^2\right )\right )+\frac {1}{2} \left (1+\sqrt {5}\right ) \arctan \left (\frac {1}{2} \left (4 x-2 \sqrt {5} x+x^2-\sqrt {5} x^2-9 x^3+9 \sqrt {5} x^3-6 x^4+6 \sqrt {5} x^4-x^5+\sqrt {5} x^5\right )\right )+\frac {1}{2} \left (-1+\sqrt {5}\right ) \arctan \left (\frac {1}{2} \left (-4 x-2 \sqrt {5} x-x^2-\sqrt {5} x^2+9 x^3+9 \sqrt {5} x^3+6 x^4+6 \sqrt {5} x^4+x^5+\sqrt {5} x^5\right )\right ) \] Output:
1/2*(5^(1/2)+1)*arctan(1/2*(5^(1/2)-1)*(x^2+3*x))+1/2*(5^(1/2)-1)*arctan(1 /2*(5^(1/2)+1)*(x^2+3*x))+1/2*(5^(1/2)+1)*arctan(2*x-x*5^(1/2)+1/2*x^2-1/2 *5^(1/2)*x^2-9/2*x^3+9/2*5^(1/2)*x^3-3*x^4+3*5^(1/2)*x^4-1/2*x^5+1/2*5^(1/ 2)*x^5)+1/2*(5^(1/2)-1)*arctan(-2*x-x*5^(1/2)-1/2*x^2-1/2*5^(1/2)*x^2+9/2* x^3+9/2*5^(1/2)*x^3+3*x^4+3*5^(1/2)*x^4+1/2*x^5+1/2*5^(1/2)*x^5)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.04 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.05 \[ \int \frac {3-7 x^2-21 x^4-32 x^5+72 x^6+108 x^7+45 x^8+6 x^9}{1-9 x^2-4 x^3+37 x^4+30 x^5-75 x^6-90 x^7+48 x^8+104 x^9+54 x^{10}+12 x^{11}+x^{12}} \, dx=\frac {1}{2} \text {RootSum}\left [1-9 \text {$\#$1}^2-4 \text {$\#$1}^3+37 \text {$\#$1}^4+30 \text {$\#$1}^5-75 \text {$\#$1}^6-90 \text {$\#$1}^7+48 \text {$\#$1}^8+104 \text {$\#$1}^9+54 \text {$\#$1}^{10}+12 \text {$\#$1}^{11}+\text {$\#$1}^{12}\&,\frac {3 \log (x-\text {$\#$1})-7 \log (x-\text {$\#$1}) \text {$\#$1}^2-21 \log (x-\text {$\#$1}) \text {$\#$1}^4-32 \log (x-\text {$\#$1}) \text {$\#$1}^5+72 \log (x-\text {$\#$1}) \text {$\#$1}^6+108 \log (x-\text {$\#$1}) \text {$\#$1}^7+45 \log (x-\text {$\#$1}) \text {$\#$1}^8+6 \log (x-\text {$\#$1}) \text {$\#$1}^9}{-9 \text {$\#$1}-6 \text {$\#$1}^2+74 \text {$\#$1}^3+75 \text {$\#$1}^4-225 \text {$\#$1}^5-315 \text {$\#$1}^6+192 \text {$\#$1}^7+468 \text {$\#$1}^8+270 \text {$\#$1}^9+66 \text {$\#$1}^{10}+6 \text {$\#$1}^{11}}\&\right ] \] Input:
Integrate[(3 - 7*x^2 - 21*x^4 - 32*x^5 + 72*x^6 + 108*x^7 + 45*x^8 + 6*x^9 )/(1 - 9*x^2 - 4*x^3 + 37*x^4 + 30*x^5 - 75*x^6 - 90*x^7 + 48*x^8 + 104*x^ 9 + 54*x^10 + 12*x^11 + x^12),x]
Output:
RootSum[1 - 9*#1^2 - 4*#1^3 + 37*#1^4 + 30*#1^5 - 75*#1^6 - 90*#1^7 + 48*# 1^8 + 104*#1^9 + 54*#1^10 + 12*#1^11 + #1^12 & , (3*Log[x - #1] - 7*Log[x - #1]*#1^2 - 21*Log[x - #1]*#1^4 - 32*Log[x - #1]*#1^5 + 72*Log[x - #1]*#1 ^6 + 108*Log[x - #1]*#1^7 + 45*Log[x - #1]*#1^8 + 6*Log[x - #1]*#1^9)/(-9* #1 - 6*#1^2 + 74*#1^3 + 75*#1^4 - 225*#1^5 - 315*#1^6 + 192*#1^7 + 468*#1^ 8 + 270*#1^9 + 66*#1^10 + 6*#1^11) & ]/2
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.
| \(\displaystyle \int \frac {6 x^9+45 x^8+108 x^7+72 x^6-32 x^5-21 x^4-7 x^2+3}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {6 x^9}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1}+\frac {45 x^8}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1}+\frac {108 x^7}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1}+\frac {72 x^6}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1}-\frac {32 x^5}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1}-\frac {21 x^4}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1}-\frac {7 x^2}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1}+\frac {3}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 \int \frac {1}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1}dx-7 \int \frac {x^2}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1}dx-21 \int \frac {x^4}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1}dx-32 \int \frac {x^5}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1}dx+72 \int \frac {x^6}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1}dx+108 \int \frac {x^7}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1}dx+45 \int \frac {x^8}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1}dx+6 \int \frac {x^9}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1}dx\) |
Input:
Int[(3 - 7*x^2 - 21*x^4 - 32*x^5 + 72*x^6 + 108*x^7 + 45*x^8 + 6*x^9)/(1 - 9*x^2 - 4*x^3 + 37*x^4 + 30*x^5 - 75*x^6 - 90*x^7 + 48*x^8 + 104*x^9 + 54 *x^10 + 12*x^11 + x^12),x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.12 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.14
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (x^{3}+\textit {\_R} x +3 x^{2}-1\right )\right )}{2}\) | \(33\) |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (x^{3}+\textit {\_R} x +3 x^{2}-1\right )\right )}{2}\) | \(33\) |
Input:
int((6*x^9+45*x^8+108*x^7+72*x^6-32*x^5-21*x^4-7*x^2+3)/(x^12+12*x^11+54*x ^10+104*x^9+48*x^8-90*x^7-75*x^6+30*x^5+37*x^4-4*x^3-9*x^2+1),x,method=_RE TURNVERBOSE)
Output:
1/2*sum(_R*ln(x^3+_R*x+3*x^2-1),_R=RootOf(_Z^4+3*_Z^2+1))
Time = 0.09 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.77 \[ \int \frac {3-7 x^2-21 x^4-32 x^5+72 x^6+108 x^7+45 x^8+6 x^9}{1-9 x^2-4 x^3+37 x^4+30 x^5-75 x^6-90 x^7+48 x^8+104 x^9+54 x^{10}+12 x^{11}+x^{12}} \, dx=\frac {1}{2} \, {\left (\sqrt {5} - 1\right )} \arctan \left (\frac {1}{2} \, x^{5} + 3 \, x^{4} + \frac {9}{2} \, x^{3} - \frac {1}{2} \, x^{2} + \frac {1}{2} \, \sqrt {5} {\left (x^{5} + 6 \, x^{4} + 9 \, x^{3} - x^{2} - 2 \, x\right )} - 2 \, x\right ) + \frac {1}{2} \, {\left (\sqrt {5} + 1\right )} \arctan \left (-\frac {1}{2} \, x^{5} - 3 \, x^{4} - \frac {9}{2} \, x^{3} + \frac {1}{2} \, x^{2} + \frac {1}{2} \, \sqrt {5} {\left (x^{5} + 6 \, x^{4} + 9 \, x^{3} - x^{2} - 2 \, x\right )} + 2 \, x\right ) + \frac {1}{2} \, {\left (\sqrt {5} - 1\right )} \arctan \left (\frac {1}{2} \, x^{2} + \frac {1}{2} \, \sqrt {5} {\left (x^{2} + 3 \, x\right )} + \frac {3}{2} \, x\right ) + \frac {1}{2} \, {\left (\sqrt {5} + 1\right )} \arctan \left (-\frac {1}{2} \, x^{2} + \frac {1}{2} \, \sqrt {5} {\left (x^{2} + 3 \, x\right )} - \frac {3}{2} \, x\right ) \] Input:
integrate((6*x^9+45*x^8+108*x^7+72*x^6-32*x^5-21*x^4-7*x^2+3)/(x^12+12*x^1 1+54*x^10+104*x^9+48*x^8-90*x^7-75*x^6+30*x^5+37*x^4-4*x^3-9*x^2+1),x, alg orithm="fricas")
Output:
1/2*(sqrt(5) - 1)*arctan(1/2*x^5 + 3*x^4 + 9/2*x^3 - 1/2*x^2 + 1/2*sqrt(5) *(x^5 + 6*x^4 + 9*x^3 - x^2 - 2*x) - 2*x) + 1/2*(sqrt(5) + 1)*arctan(-1/2* x^5 - 3*x^4 - 9/2*x^3 + 1/2*x^2 + 1/2*sqrt(5)*(x^5 + 6*x^4 + 9*x^3 - x^2 - 2*x) + 2*x) + 1/2*(sqrt(5) - 1)*arctan(1/2*x^2 + 1/2*sqrt(5)*(x^2 + 3*x) + 3/2*x) + 1/2*(sqrt(5) + 1)*arctan(-1/2*x^2 + 1/2*sqrt(5)*(x^2 + 3*x) - 3 /2*x)
Time = 0.23 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.94 \[ \int \frac {3-7 x^2-21 x^4-32 x^5+72 x^6+108 x^7+45 x^8+6 x^9}{1-9 x^2-4 x^3+37 x^4+30 x^5-75 x^6-90 x^7+48 x^8+104 x^9+54 x^{10}+12 x^{11}+x^{12}} \, dx=\left (- \frac {1}{4} + \frac {\sqrt {5}}{4}\right ) \left (2 \operatorname {atan}{\left (\frac {2 x^{2}}{-1 + \sqrt {5}} + \frac {6 x}{-1 + \sqrt {5}} \right )} + 2 \operatorname {atan}{\left (\frac {2 x^{5}}{-1 + \sqrt {5}} + \frac {12 x^{4}}{-1 + \sqrt {5}} + \frac {18 x^{3}}{-1 + \sqrt {5}} - \frac {2 x^{2}}{-1 + \sqrt {5}} + x \left (- \frac {3}{-1 + \sqrt {5}} - \frac {\sqrt {5}}{-1 + \sqrt {5}}\right ) \right )}\right ) + \left (\frac {1}{4} + \frac {\sqrt {5}}{4}\right ) \left (2 \operatorname {atan}{\left (\frac {2 x^{2}}{1 + \sqrt {5}} + \frac {6 x}{1 + \sqrt {5}} \right )} + 2 \operatorname {atan}{\left (\frac {2 x^{5}}{1 + \sqrt {5}} + \frac {12 x^{4}}{1 + \sqrt {5}} + \frac {18 x^{3}}{1 + \sqrt {5}} - \frac {2 x^{2}}{1 + \sqrt {5}} + x \left (- \frac {3}{1 + \sqrt {5}} + \frac {\sqrt {5}}{1 + \sqrt {5}}\right ) \right )}\right ) \] Input:
integrate((6*x**9+45*x**8+108*x**7+72*x**6-32*x**5-21*x**4-7*x**2+3)/(x**1 2+12*x**11+54*x**10+104*x**9+48*x**8-90*x**7-75*x**6+30*x**5+37*x**4-4*x** 3-9*x**2+1),x)
Output:
(-1/4 + sqrt(5)/4)*(2*atan(2*x**2/(-1 + sqrt(5)) + 6*x/(-1 + sqrt(5))) + 2 *atan(2*x**5/(-1 + sqrt(5)) + 12*x**4/(-1 + sqrt(5)) + 18*x**3/(-1 + sqrt( 5)) - 2*x**2/(-1 + sqrt(5)) + x*(-3/(-1 + sqrt(5)) - sqrt(5)/(-1 + sqrt(5) )))) + (1/4 + sqrt(5)/4)*(2*atan(2*x**2/(1 + sqrt(5)) + 6*x/(1 + sqrt(5))) + 2*atan(2*x**5/(1 + sqrt(5)) + 12*x**4/(1 + sqrt(5)) + 18*x**3/(1 + sqrt (5)) - 2*x**2/(1 + sqrt(5)) + x*(-3/(1 + sqrt(5)) + sqrt(5)/(1 + sqrt(5))) ))
\[ \int \frac {3-7 x^2-21 x^4-32 x^5+72 x^6+108 x^7+45 x^8+6 x^9}{1-9 x^2-4 x^3+37 x^4+30 x^5-75 x^6-90 x^7+48 x^8+104 x^9+54 x^{10}+12 x^{11}+x^{12}} \, dx=\int { \frac {6 \, x^{9} + 45 \, x^{8} + 108 \, x^{7} + 72 \, x^{6} - 32 \, x^{5} - 21 \, x^{4} - 7 \, x^{2} + 3}{x^{12} + 12 \, x^{11} + 54 \, x^{10} + 104 \, x^{9} + 48 \, x^{8} - 90 \, x^{7} - 75 \, x^{6} + 30 \, x^{5} + 37 \, x^{4} - 4 \, x^{3} - 9 \, x^{2} + 1} \,d x } \] Input:
integrate((6*x^9+45*x^8+108*x^7+72*x^6-32*x^5-21*x^4-7*x^2+3)/(x^12+12*x^1 1+54*x^10+104*x^9+48*x^8-90*x^7-75*x^6+30*x^5+37*x^4-4*x^3-9*x^2+1),x, alg orithm="maxima")
Output:
integrate((6*x^9 + 45*x^8 + 108*x^7 + 72*x^6 - 32*x^5 - 21*x^4 - 7*x^2 + 3 )/(x^12 + 12*x^11 + 54*x^10 + 104*x^9 + 48*x^8 - 90*x^7 - 75*x^6 + 30*x^5 + 37*x^4 - 4*x^3 - 9*x^2 + 1), x)
Time = 0.12 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.68 \[ \int \frac {3-7 x^2-21 x^4-32 x^5+72 x^6+108 x^7+45 x^8+6 x^9}{1-9 x^2-4 x^3+37 x^4+30 x^5-75 x^6-90 x^7+48 x^8+104 x^9+54 x^{10}+12 x^{11}+x^{12}} \, dx=\frac {1}{2} \, {\left (\sqrt {5} - 1\right )} {\left (\arctan \left (\frac {1}{2} \, x^{5} {\left (\sqrt {5} + 1\right )} + 3 \, x^{4} {\left (\sqrt {5} + 1\right )} + \frac {9}{2} \, x^{3} {\left (\sqrt {5} + 1\right )} - \frac {1}{2} \, x^{2} {\left (\sqrt {5} + 1\right )} - x {\left (\sqrt {5} + 2\right )}\right ) + \arctan \left (\frac {1}{2} \, x^{2} {\left (\sqrt {5} + 1\right )} + \frac {3}{2} \, x {\left (\sqrt {5} + 1\right )}\right )\right )} - \frac {1}{2} \, {\left (\sqrt {5} + 1\right )} {\left (\arctan \left (-\frac {1}{2} \, x^{5} {\left (\sqrt {5} - 1\right )} - 3 \, x^{4} {\left (\sqrt {5} - 1\right )} - \frac {9}{2} \, x^{3} {\left (\sqrt {5} - 1\right )} + \frac {1}{2} \, x^{2} {\left (\sqrt {5} - 1\right )} + x {\left (\sqrt {5} - 2\right )}\right ) + \arctan \left (-\frac {1}{2} \, x^{2} {\left (\sqrt {5} - 1\right )} - \frac {3}{2} \, x {\left (\sqrt {5} - 1\right )}\right )\right )} \] Input:
integrate((6*x^9+45*x^8+108*x^7+72*x^6-32*x^5-21*x^4-7*x^2+3)/(x^12+12*x^1 1+54*x^10+104*x^9+48*x^8-90*x^7-75*x^6+30*x^5+37*x^4-4*x^3-9*x^2+1),x, alg orithm="giac")
Output:
1/2*(sqrt(5) - 1)*(arctan(1/2*x^5*(sqrt(5) + 1) + 3*x^4*(sqrt(5) + 1) + 9/ 2*x^3*(sqrt(5) + 1) - 1/2*x^2*(sqrt(5) + 1) - x*(sqrt(5) + 2)) + arctan(1/ 2*x^2*(sqrt(5) + 1) + 3/2*x*(sqrt(5) + 1))) - 1/2*(sqrt(5) + 1)*(arctan(-1 /2*x^5*(sqrt(5) - 1) - 3*x^4*(sqrt(5) - 1) - 9/2*x^3*(sqrt(5) - 1) + 1/2*x ^2*(sqrt(5) - 1) + x*(sqrt(5) - 2)) + arctan(-1/2*x^2*(sqrt(5) - 1) - 3/2* x*(sqrt(5) - 1)))
Time = 0.28 (sec) , antiderivative size = 959, normalized size of antiderivative = 4.15 \[ \int \frac {3-7 x^2-21 x^4-32 x^5+72 x^6+108 x^7+45 x^8+6 x^9}{1-9 x^2-4 x^3+37 x^4+30 x^5-75 x^6-90 x^7+48 x^8+104 x^9+54 x^{10}+12 x^{11}+x^{12}} \, dx=\text {Too large to display} \] Input:
int((72*x^6 - 21*x^4 - 32*x^5 - 7*x^2 + 108*x^7 + 45*x^8 + 6*x^9 + 3)/(37* x^4 - 4*x^3 - 9*x^2 + 30*x^5 - 75*x^6 - 90*x^7 + 48*x^8 + 104*x^9 + 54*x^1 0 + 12*x^11 + x^12 + 1),x)
Output:
2*atanh((12107306250*(- 5^(1/2)/8 - 3/8)^(1/2))/(2127206250*x - 1145137500 *5^(1/2)*x + 782583750*5^(1/2) - 2347751250*5^(1/2)*x^2 - 782583750*5^(1/2 )*x^3 + 4264406250*x^2 + 1421468750*x^3 - 1421468750) - (36321918750*x^2*( - 5^(1/2)/8 - 3/8)^(1/2))/(2127206250*x - 1145137500*5^(1/2)*x + 782583750 *5^(1/2) - 2347751250*5^(1/2)*x^2 - 782583750*5^(1/2)*x^3 + 4264406250*x^2 + 1421468750*x^3 - 1421468750) - (12107306250*x^3*(- 5^(1/2)/8 - 3/8)^(1/ 2))/(2127206250*x - 1145137500*5^(1/2)*x + 782583750*5^(1/2) - 2347751250* 5^(1/2)*x^2 - 782583750*5^(1/2)*x^3 + 4264406250*x^2 + 1421468750*x^3 - 14 21468750) - (5562618750*5^(1/2)*(- 5^(1/2)/8 - 3/8)^(1/2))/(2127206250*x - 1145137500*5^(1/2)*x + 782583750*5^(1/2) - 2347751250*5^(1/2)*x^2 - 78258 3750*5^(1/2)*x^3 + 4264406250*x^2 + 1421468750*x^3 - 1421468750) + (284293 7500*x*(- 5^(1/2)/8 - 3/8)^(1/2))/(2127206250*x - 1145137500*5^(1/2)*x + 7 82583750*5^(1/2) - 2347751250*5^(1/2)*x^2 - 782583750*5^(1/2)*x^3 + 426440 6250*x^2 + 1421468750*x^3 - 1421468750) - (1565167500*5^(1/2)*x*(- 5^(1/2) /8 - 3/8)^(1/2))/(2127206250*x - 1145137500*5^(1/2)*x + 782583750*5^(1/2) - 2347751250*5^(1/2)*x^2 - 782583750*5^(1/2)*x^3 + 4264406250*x^2 + 142146 8750*x^3 - 1421468750) + (16687856250*5^(1/2)*x^2*(- 5^(1/2)/8 - 3/8)^(1/2 ))/(2127206250*x - 1145137500*5^(1/2)*x + 782583750*5^(1/2) - 2347751250*5 ^(1/2)*x^2 - 782583750*5^(1/2)*x^3 + 4264406250*x^2 + 1421468750*x^3 - 142 1468750) + (5562618750*5^(1/2)*x^3*(- 5^(1/2)/8 - 3/8)^(1/2))/(21272062...
\[ \int \frac {3-7 x^2-21 x^4-32 x^5+72 x^6+108 x^7+45 x^8+6 x^9}{1-9 x^2-4 x^3+37 x^4+30 x^5-75 x^6-90 x^7+48 x^8+104 x^9+54 x^{10}+12 x^{11}+x^{12}} \, dx=6 \left (\int \frac {x^{9}}{x^{12}+12 x^{11}+54 x^{10}+104 x^{9}+48 x^{8}-90 x^{7}-75 x^{6}+30 x^{5}+37 x^{4}-4 x^{3}-9 x^{2}+1}d x \right )+45 \left (\int \frac {x^{8}}{x^{12}+12 x^{11}+54 x^{10}+104 x^{9}+48 x^{8}-90 x^{7}-75 x^{6}+30 x^{5}+37 x^{4}-4 x^{3}-9 x^{2}+1}d x \right )+108 \left (\int \frac {x^{7}}{x^{12}+12 x^{11}+54 x^{10}+104 x^{9}+48 x^{8}-90 x^{7}-75 x^{6}+30 x^{5}+37 x^{4}-4 x^{3}-9 x^{2}+1}d x \right )+72 \left (\int \frac {x^{6}}{x^{12}+12 x^{11}+54 x^{10}+104 x^{9}+48 x^{8}-90 x^{7}-75 x^{6}+30 x^{5}+37 x^{4}-4 x^{3}-9 x^{2}+1}d x \right )-32 \left (\int \frac {x^{5}}{x^{12}+12 x^{11}+54 x^{10}+104 x^{9}+48 x^{8}-90 x^{7}-75 x^{6}+30 x^{5}+37 x^{4}-4 x^{3}-9 x^{2}+1}d x \right )-21 \left (\int \frac {x^{4}}{x^{12}+12 x^{11}+54 x^{10}+104 x^{9}+48 x^{8}-90 x^{7}-75 x^{6}+30 x^{5}+37 x^{4}-4 x^{3}-9 x^{2}+1}d x \right )-7 \left (\int \frac {x^{2}}{x^{12}+12 x^{11}+54 x^{10}+104 x^{9}+48 x^{8}-90 x^{7}-75 x^{6}+30 x^{5}+37 x^{4}-4 x^{3}-9 x^{2}+1}d x \right )+3 \left (\int \frac {1}{x^{12}+12 x^{11}+54 x^{10}+104 x^{9}+48 x^{8}-90 x^{7}-75 x^{6}+30 x^{5}+37 x^{4}-4 x^{3}-9 x^{2}+1}d x \right ) \] Input:
int((6*x^9+45*x^8+108*x^7+72*x^6-32*x^5-21*x^4-7*x^2+3)/(x^12+12*x^11+54*x ^10+104*x^9+48*x^8-90*x^7-75*x^6+30*x^5+37*x^4-4*x^3-9*x^2+1),x)
Output:
6*int(x**9/(x**12 + 12*x**11 + 54*x**10 + 104*x**9 + 48*x**8 - 90*x**7 - 7 5*x**6 + 30*x**5 + 37*x**4 - 4*x**3 - 9*x**2 + 1),x) + 45*int(x**8/(x**12 + 12*x**11 + 54*x**10 + 104*x**9 + 48*x**8 - 90*x**7 - 75*x**6 + 30*x**5 + 37*x**4 - 4*x**3 - 9*x**2 + 1),x) + 108*int(x**7/(x**12 + 12*x**11 + 54*x **10 + 104*x**9 + 48*x**8 - 90*x**7 - 75*x**6 + 30*x**5 + 37*x**4 - 4*x**3 - 9*x**2 + 1),x) + 72*int(x**6/(x**12 + 12*x**11 + 54*x**10 + 104*x**9 + 48*x**8 - 90*x**7 - 75*x**6 + 30*x**5 + 37*x**4 - 4*x**3 - 9*x**2 + 1),x) - 32*int(x**5/(x**12 + 12*x**11 + 54*x**10 + 104*x**9 + 48*x**8 - 90*x**7 - 75*x**6 + 30*x**5 + 37*x**4 - 4*x**3 - 9*x**2 + 1),x) - 21*int(x**4/(x** 12 + 12*x**11 + 54*x**10 + 104*x**9 + 48*x**8 - 90*x**7 - 75*x**6 + 30*x** 5 + 37*x**4 - 4*x**3 - 9*x**2 + 1),x) - 7*int(x**2/(x**12 + 12*x**11 + 54* x**10 + 104*x**9 + 48*x**8 - 90*x**7 - 75*x**6 + 30*x**5 + 37*x**4 - 4*x** 3 - 9*x**2 + 1),x) + 3*int(1/(x**12 + 12*x**11 + 54*x**10 + 104*x**9 + 48* x**8 - 90*x**7 - 75*x**6 + 30*x**5 + 37*x**4 - 4*x**3 - 9*x**2 + 1),x)