Integrand size = 113, antiderivative size = 364 \[ \int \frac {1-5 x-5 x^2+37 x^3+31 x^4-118 x^5-129 x^6+133 x^7+249 x^8+137 x^9+33 x^{10}+3 x^{11}}{1-9 x^2-4 x^3+37 x^4+30 x^5-75 x^6-90 x^7+48 x^8+104 x^9+54 x^{10}+12 x^{11}+x^{12}} \, dx=\frac {\arctan \left (\frac {1}{2} \left (-1+\sqrt {5}\right ) \left (3 x+x^2\right )\right )}{\sqrt {5}}+\frac {\arctan \left (\frac {1}{2} \left (1+\sqrt {5}\right ) \left (3 x+x^2\right )\right )}{\sqrt {5}}+\frac {\arctan \left (\frac {1}{2} \left (4 x-2 \sqrt {5} x+x^2-\sqrt {5} x^2-9 x^3+9 \sqrt {5} x^3-6 x^4+6 \sqrt {5} x^4-x^5+\sqrt {5} x^5\right )\right )}{\sqrt {5}}+\frac {\arctan \left (\frac {1}{2} \left (-4 x-2 \sqrt {5} x-x^2-\sqrt {5} x^2+9 x^3+9 \sqrt {5} x^3+6 x^4+6 \sqrt {5} x^4+x^5+\sqrt {5} x^5\right )\right )}{\sqrt {5}}+\frac {\log \left (2-9 x^2-\sqrt {5} x^2-4 x^3+18 x^4+12 x^5+2 x^6\right )}{4 \sqrt {5}}-\frac {\log \left (2-9 x^2+\sqrt {5} x^2-4 x^3+18 x^4+12 x^5+2 x^6\right )}{4 \sqrt {5}}+\frac {1}{4} \log \left (1-9 x^2-4 x^3+37 x^4+30 x^5-75 x^6-90 x^7+48 x^8+104 x^9+54 x^{10}+12 x^{11}+x^{12}\right ) \] Output:
1/5*arctan(1/2*(5^(1/2)-1)*(x^2+3*x))*5^(1/2)+1/5*arctan(1/2*(5^(1/2)+1)*( x^2+3*x))*5^(1/2)+1/5*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)*5^(1/2)+1/5* 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)*5^(1/2)+1/20*ln(2-9*x^2-5^(1/2)*x ^2-4*x^3+18*x^4+12*x^5+2*x^6)*5^(1/2)-1/20*ln(2-9*x^2+5^(1/2)*x^2-4*x^3+18 *x^4+12*x^5+2*x^6)*5^(1/2)+1/4*ln(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)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.04 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.80 \[ \int \frac {1-5 x-5 x^2+37 x^3+31 x^4-118 x^5-129 x^6+133 x^7+249 x^8+137 x^9+33 x^{10}+3 x^{11}}{1-9 x^2-4 x^3+37 x^4+30 x^5-75 x^6-90 x^7+48 x^8+104 x^9+54 x^{10}+12 x^{11}+x^{12}} \, dx=\frac {1}{2} \text {RootSum}\left [1-9 \text {$\#$1}^2-4 \text {$\#$1}^3+37 \text {$\#$1}^4+30 \text {$\#$1}^5-75 \text {$\#$1}^6-90 \text {$\#$1}^7+48 \text {$\#$1}^8+104 \text {$\#$1}^9+54 \text {$\#$1}^{10}+12 \text {$\#$1}^{11}+\text {$\#$1}^{12}\&,\frac {\log (x-\text {$\#$1})-5 \log (x-\text {$\#$1}) \text {$\#$1}-5 \log (x-\text {$\#$1}) \text {$\#$1}^2+37 \log (x-\text {$\#$1}) \text {$\#$1}^3+31 \log (x-\text {$\#$1}) \text {$\#$1}^4-118 \log (x-\text {$\#$1}) \text {$\#$1}^5-129 \log (x-\text {$\#$1}) \text {$\#$1}^6+133 \log (x-\text {$\#$1}) \text {$\#$1}^7+249 \log (x-\text {$\#$1}) \text {$\#$1}^8+137 \log (x-\text {$\#$1}) \text {$\#$1}^9+33 \log (x-\text {$\#$1}) \text {$\#$1}^{10}+3 \log (x-\text {$\#$1}) \text {$\#$1}^{11}}{-9 \text {$\#$1}-6 \text {$\#$1}^2+74 \text {$\#$1}^3+75 \text {$\#$1}^4-225 \text {$\#$1}^5-315 \text {$\#$1}^6+192 \text {$\#$1}^7+468 \text {$\#$1}^8+270 \text {$\#$1}^9+66 \text {$\#$1}^{10}+6 \text {$\#$1}^{11}}\&\right ] \] Input:
Integrate[(1 - 5*x - 5*x^2 + 37*x^3 + 31*x^4 - 118*x^5 - 129*x^6 + 133*x^7 + 249*x^8 + 137*x^9 + 33*x^10 + 3*x^11)/(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 & , (Log[x - #1] - 5*Log[x - #1]*#1 - 5*Log[x - #1]*#1^2 + 37*Log[x - #1]*#1^3 + 31*Log[x - #1]*#1^4 - 118*Log[x - #1]*#1^5 - 129*Log[x - #1]*#1^6 + 133*Log[x - #1]*#1^7 + 249*L og[x - #1]*#1^8 + 137*Log[x - #1]*#1^9 + 33*Log[x - #1]*#1^10 + 3*Log[x - #1]*#1^11)/(-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.
| \(\displaystyle \int \frac {3 x^{11}+33 x^{10}+137 x^9+249 x^8+133 x^7-129 x^6-118 x^5+31 x^4+37 x^3-5 x^2-5 x+1}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1} \, dx\) |
| \(\Big \downarrow \) 2525 |
| \(\displaystyle \frac {1}{12} \int \frac {6 \left (4 x^9+30 x^8+74 x^7+57 x^6-11 x^5-13 x^4-4 x^2-x+2\right )}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1}dx+\frac {1}{4} \log \left (x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int \frac {4 x^9+30 x^8+74 x^7+57 x^6-11 x^5-13 x^4-4 x^2-x+2}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1}dx+\frac {1}{4} \log \left (x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1\right )\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {4 x^9}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1}+\frac {30 x^8}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1}+\frac {74 x^7}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1}+\frac {57 x^6}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1}-\frac {11 x^5}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1}-\frac {13 x^4}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1}-\frac {4 x^2}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1}-\frac {x}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1}+\frac {2}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1}\right )dx+\frac {1}{4} \log \left (x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (2 \int \frac {1}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1}dx-\int \frac {x}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1}dx-4 \int \frac {x^2}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1}dx-13 \int \frac {x^4}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1}dx-11 \int \frac {x^5}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1}dx+57 \int \frac {x^6}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1}dx+74 \int \frac {x^7}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1}dx+30 \int \frac {x^8}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1}dx+4 \int \frac {x^9}{x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1}dx\right )+\frac {1}{4} \log \left (x^{12}+12 x^{11}+54 x^{10}+104 x^9+48 x^8-90 x^7-75 x^6+30 x^5+37 x^4-4 x^3-9 x^2+1\right )\) |
Input:
Int[(1 - 5*x - 5*x^2 + 37*x^3 + 31*x^4 - 118*x^5 - 129*x^6 + 133*x^7 + 249 *x^8 + 137*x^9 + 33*x^10 + 3*x^11)/(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 = 0.10 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.16
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{3}+9 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )}{\sum }\textit {\_R} \ln \left (x^{3}+3 x^{2}+\left (5 \textit {\_R}^{3}-10 \textit {\_R}^{2}+9 \textit {\_R} -3\right ) x -1\right )\right )}{2}\) | \(57\) |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{3}+9 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )}{\sum }\textit {\_R} \ln \left (x^{3}+3 x^{2}+\left (5 \textit {\_R}^{3}-10 \textit {\_R}^{2}+9 \textit {\_R} -3\right ) x -1\right )\right )}{2}\) | \(57\) |
Input:
int((3*x^11+33*x^10+137*x^9+249*x^8+133*x^7-129*x^6-118*x^5+31*x^4+37*x^3- 5*x^2-5*x+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,method=_RETURNVERBOSE)
Output:
1/2*sum(_R*ln(x^3+3*x^2+(5*_R^3-10*_R^2+9*_R-3)*x-1),_R=RootOf(5*_Z^4-10*_ Z^3+9*_Z^2-4*_Z+1))
Time = 0.08 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.70 \[ \int \frac {1-5 x-5 x^2+37 x^3+31 x^4-118 x^5-129 x^6+133 x^7+249 x^8+137 x^9+33 x^{10}+3 x^{11}}{1-9 x^2-4 x^3+37 x^4+30 x^5-75 x^6-90 x^7+48 x^8+104 x^9+54 x^{10}+12 x^{11}+x^{12}} \, dx=-\frac {1}{20} \, {\left (\sqrt {5} - 5\right )} \log \left (2 \, x^{6} + 12 \, x^{5} + 18 \, x^{4} - 4 \, x^{3} + \sqrt {5} x^{2} - 9 \, x^{2} + 2\right ) + \frac {1}{20} \, {\left (\sqrt {5} + 5\right )} \log \left (2 \, x^{6} + 12 \, x^{5} + 18 \, x^{4} - 4 \, x^{3} - \sqrt {5} x^{2} - 9 \, x^{2} + 2\right ) + \frac {1}{5} \, \sqrt {5} \arctan \left (\frac {1}{2} \, x^{5} + 3 \, x^{4} + \frac {9}{2} \, x^{3} - \frac {1}{2} \, x^{2} + \frac {1}{2} \, \sqrt {5} {\left (x^{5} + 6 \, x^{4} + 9 \, x^{3} - x^{2} - 2 \, x\right )} - 2 \, x\right ) + \frac {1}{5} \, \sqrt {5} \arctan \left (-\frac {1}{2} \, x^{5} - 3 \, x^{4} - \frac {9}{2} \, x^{3} + \frac {1}{2} \, x^{2} + \frac {1}{2} \, \sqrt {5} {\left (x^{5} + 6 \, x^{4} + 9 \, x^{3} - x^{2} - 2 \, x\right )} + 2 \, x\right ) + \frac {1}{5} \, \sqrt {5} \arctan \left (\frac {1}{2} \, x^{2} + \frac {1}{2} \, \sqrt {5} {\left (x^{2} + 3 \, x\right )} + \frac {3}{2} \, x\right ) + \frac {1}{5} \, \sqrt {5} \arctan \left (-\frac {1}{2} \, x^{2} + \frac {1}{2} \, \sqrt {5} {\left (x^{2} + 3 \, x\right )} - \frac {3}{2} \, x\right ) \] Input:
integrate((3*x^11+33*x^10+137*x^9+249*x^8+133*x^7-129*x^6-118*x^5+31*x^4+3 7*x^3-5*x^2-5*x+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, algorithm="fricas")
Output:
-1/20*(sqrt(5) - 5)*log(2*x^6 + 12*x^5 + 18*x^4 - 4*x^3 + sqrt(5)*x^2 - 9* x^2 + 2) + 1/20*(sqrt(5) + 5)*log(2*x^6 + 12*x^5 + 18*x^4 - 4*x^3 - sqrt(5 )*x^2 - 9*x^2 + 2) + 1/5*sqrt(5)*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) + 1/5*sqrt(5)*arc tan(-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) + 1/5*sqrt(5)*arctan(1/2*x^2 + 1/2*sqrt(5)*(x^2 + 3* x) + 3/2*x) + 1/5*sqrt(5)*arctan(-1/2*x^2 + 1/2*sqrt(5)*(x^2 + 3*x) - 3/2* x)
Time = 0.20 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.84 \[ \int \frac {1-5 x-5 x^2+37 x^3+31 x^4-118 x^5-129 x^6+133 x^7+249 x^8+137 x^9+33 x^{10}+3 x^{11}}{1-9 x^2-4 x^3+37 x^4+30 x^5-75 x^6-90 x^7+48 x^8+104 x^9+54 x^{10}+12 x^{11}+x^{12}} \, dx=\frac {\sqrt {5} \cdot \left (2 \operatorname {atan}{\left (\frac {2 x^{2}}{-1 + \sqrt {5}} + \frac {6 x}{-1 + \sqrt {5}} \right )} + 2 \operatorname {atan}{\left (\frac {2 x^{5}}{-1 + \sqrt {5}} + \frac {12 x^{4}}{-1 + \sqrt {5}} + \frac {18 x^{3}}{-1 + \sqrt {5}} - \frac {2 x^{2}}{-1 + \sqrt {5}} + x \left (- \frac {3}{-1 + \sqrt {5}} - \frac {\sqrt {5}}{-1 + \sqrt {5}}\right ) \right )}\right )}{10} + \frac {\sqrt {5} \cdot \left (2 \operatorname {atan}{\left (\frac {2 x^{2}}{1 + \sqrt {5}} + \frac {6 x}{1 + \sqrt {5}} \right )} + 2 \operatorname {atan}{\left (\frac {2 x^{5}}{1 + \sqrt {5}} + \frac {12 x^{4}}{1 + \sqrt {5}} + \frac {18 x^{3}}{1 + \sqrt {5}} - \frac {2 x^{2}}{1 + \sqrt {5}} + x \left (- \frac {3}{1 + \sqrt {5}} + \frac {\sqrt {5}}{1 + \sqrt {5}}\right ) \right )}\right )}{10} + \left (\frac {\sqrt {5}}{20} + \frac {1}{4}\right ) \log {\left (x^{6} + 6 x^{5} + 9 x^{4} - 2 x^{3} + x^{2} \left (- \frac {9}{2} - \frac {\sqrt {5}}{2}\right ) + 1 \right )} + \left (\frac {1}{4} - \frac {\sqrt {5}}{20}\right ) \log {\left (x^{6} + 6 x^{5} + 9 x^{4} - 2 x^{3} + x^{2} \left (- \frac {9}{2} + \frac {\sqrt {5}}{2}\right ) + 1 \right )} \] Input:
integrate((3*x**11+33*x**10+137*x**9+249*x**8+133*x**7-129*x**6-118*x**5+3 1*x**4+37*x**3-5*x**2-5*x+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)
Output:
sqrt(5)*(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)))))/10 + s qrt(5)*(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)))))/10 + (sqrt(5)/2 0 + 1/4)*log(x**6 + 6*x**5 + 9*x**4 - 2*x**3 + x**2*(-9/2 - sqrt(5)/2) + 1 ) + (1/4 - sqrt(5)/20)*log(x**6 + 6*x**5 + 9*x**4 - 2*x**3 + x**2*(-9/2 + sqrt(5)/2) + 1)
\[ \int \frac {1-5 x-5 x^2+37 x^3+31 x^4-118 x^5-129 x^6+133 x^7+249 x^8+137 x^9+33 x^{10}+3 x^{11}}{1-9 x^2-4 x^3+37 x^4+30 x^5-75 x^6-90 x^7+48 x^8+104 x^9+54 x^{10}+12 x^{11}+x^{12}} \, dx=\int { \frac {3 \, x^{11} + 33 \, x^{10} + 137 \, x^{9} + 249 \, x^{8} + 133 \, x^{7} - 129 \, x^{6} - 118 \, x^{5} + 31 \, x^{4} + 37 \, x^{3} - 5 \, x^{2} - 5 \, x + 1}{x^{12} + 12 \, x^{11} + 54 \, x^{10} + 104 \, x^{9} + 48 \, x^{8} - 90 \, x^{7} - 75 \, x^{6} + 30 \, x^{5} + 37 \, x^{4} - 4 \, x^{3} - 9 \, x^{2} + 1} \,d x } \] Input:
integrate((3*x^11+33*x^10+137*x^9+249*x^8+133*x^7-129*x^6-118*x^5+31*x^4+3 7*x^3-5*x^2-5*x+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, algorithm="maxima")
Output:
integrate((3*x^11 + 33*x^10 + 137*x^9 + 249*x^8 + 133*x^7 - 129*x^6 - 118* x^5 + 31*x^4 + 37*x^3 - 5*x^2 - 5*x + 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)
Time = 0.13 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.75 \[ \int \frac {1-5 x-5 x^2+37 x^3+31 x^4-118 x^5-129 x^6+133 x^7+249 x^8+137 x^9+33 x^{10}+3 x^{11}}{1-9 x^2-4 x^3+37 x^4+30 x^5-75 x^6-90 x^7+48 x^8+104 x^9+54 x^{10}+12 x^{11}+x^{12}} \, dx=-\frac {1}{5} \, \sqrt {5} {\left (\arctan \left (-\frac {1}{2} \, x^{5} {\left (\sqrt {5} + 1\right )} - 3 \, x^{4} {\left (\sqrt {5} + 1\right )} - \frac {9}{2} \, x^{3} {\left (\sqrt {5} + 1\right )} + \frac {1}{2} \, x^{2} {\left (\sqrt {5} + 1\right )} + x {\left (\sqrt {5} + 2\right )}\right ) + \arctan \left (-\frac {1}{2} \, x^{2} {\left (\sqrt {5} + 1\right )} - \frac {3}{2} \, x {\left (\sqrt {5} + 1\right )}\right )\right )} + \frac {1}{5} \, \sqrt {5} {\left (\arctan \left (\frac {1}{2} \, x^{5} {\left (\sqrt {5} - 1\right )} + 3 \, x^{4} {\left (\sqrt {5} - 1\right )} + \frac {9}{2} \, x^{3} {\left (\sqrt {5} - 1\right )} - \frac {1}{2} \, x^{2} {\left (\sqrt {5} - 1\right )} - x {\left (\sqrt {5} - 2\right )}\right ) + \arctan \left (\frac {1}{2} \, x^{2} {\left (\sqrt {5} - 1\right )} + \frac {3}{2} \, x {\left (\sqrt {5} - 1\right )}\right )\right )} - \frac {1}{20} \, \sqrt {5} \log \left (4 \, {\left (x^{3} + 3 \, x^{2} - 1\right )}^{2} + {\left (\sqrt {5} x + x\right )}^{2}\right ) + \frac {1}{20} \, \sqrt {5} \log \left (4 \, {\left (x^{3} + 3 \, x^{2} - 1\right )}^{2} + {\left (\sqrt {5} x - x\right )}^{2}\right ) + \frac {1}{4} \, \log \left (x^{12} + 12 \, x^{11} + 54 \, x^{10} + 104 \, x^{9} + 48 \, x^{8} - 90 \, x^{7} - 75 \, x^{6} + 30 \, x^{5} + 37 \, x^{4} - 4 \, x^{3} - 9 \, x^{2} + 1\right ) \] Input:
integrate((3*x^11+33*x^10+137*x^9+249*x^8+133*x^7-129*x^6-118*x^5+31*x^4+3 7*x^3-5*x^2-5*x+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, algorithm="giac")
Output:
-1/5*sqrt(5)*(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))) + 1/5*sqrt(5)*(arctan(1/2*x^5*(sq rt(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))) - 1/20*sqrt(5)*log(4*(x^3 + 3*x^2 - 1)^2 + (sqrt(5)*x + x)^2) + 1/2 0*sqrt(5)*log(4*(x^3 + 3*x^2 - 1)^2 + (sqrt(5)*x - x)^2) + 1/4*log(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)
Time = 10.42 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.36 \[ \int \frac {1-5 x-5 x^2+37 x^3+31 x^4-118 x^5-129 x^6+133 x^7+249 x^8+137 x^9+33 x^{10}+3 x^{11}}{1-9 x^2-4 x^3+37 x^4+30 x^5-75 x^6-90 x^7+48 x^8+104 x^9+54 x^{10}+12 x^{11}+x^{12}} \, dx=-\ln \left (3\,x^2+x^3-1-\frac {x\,1{}\mathrm {i}}{2}-\frac {\sqrt {5}\,x\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{4}+\sqrt {5}\,\left (\frac {1}{20}+\frac {1}{10}{}\mathrm {i}\right )\right )+\ln \left (3\,x^2+x^3-1-\frac {x\,1{}\mathrm {i}}{2}+\frac {\sqrt {5}\,x\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{4}+\sqrt {5}\,\left (\frac {1}{20}+\frac {1}{10}{}\mathrm {i}\right )\right )+\ln \left (3\,x^2+x^3-1+\frac {x\,1{}\mathrm {i}}{2}-\frac {\sqrt {5}\,x\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{4}+\sqrt {5}\,\left (\frac {1}{20}-\frac {1}{10}{}\mathrm {i}\right )\right )-\ln \left (3\,x^2+x^3-1+\frac {x\,1{}\mathrm {i}}{2}+\frac {\sqrt {5}\,x\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{4}+\sqrt {5}\,\left (\frac {1}{20}-\frac {1}{10}{}\mathrm {i}\right )\right ) \] Input:
int((37*x^3 - 5*x^2 - 5*x + 31*x^4 - 118*x^5 - 129*x^6 + 133*x^7 + 249*x^8 + 137*x^9 + 33*x^10 + 3*x^11 + 1)/(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^10 + 12*x^11 + x^12 + 1),x)
Output:
log((5^(1/2)*x*1i)/2 - (x*1i)/2 + 3*x^2 + x^3 - 1)*(5^(1/2)*(1/20 + 1i/10) + 1/4) - log(3*x^2 - (5^(1/2)*x*1i)/2 - (x*1i)/2 + x^3 - 1)*(5^(1/2)*(1/2 0 + 1i/10) - 1/4) + log((x*1i)/2 - (5^(1/2)*x*1i)/2 + 3*x^2 + x^3 - 1)*(5^ (1/2)*(1/20 - 1i/10) + 1/4) - log((x*1i)/2 + (5^(1/2)*x*1i)/2 + 3*x^2 + x^ 3 - 1)*(5^(1/2)*(1/20 - 1i/10) - 1/4)
\[ \int \frac {1-5 x-5 x^2+37 x^3+31 x^4-118 x^5-129 x^6+133 x^7+249 x^8+137 x^9+33 x^{10}+3 x^{11}}{1-9 x^2-4 x^3+37 x^4+30 x^5-75 x^6-90 x^7+48 x^8+104 x^9+54 x^{10}+12 x^{11}+x^{12}} \, dx=2 \left (\int \frac {x^{9}}{x^{12}+12 x^{11}+54 x^{10}+104 x^{9}+48 x^{8}-90 x^{7}-75 x^{6}+30 x^{5}+37 x^{4}-4 x^{3}-9 x^{2}+1}d x \right )+15 \left (\int \frac {x^{8}}{x^{12}+12 x^{11}+54 x^{10}+104 x^{9}+48 x^{8}-90 x^{7}-75 x^{6}+30 x^{5}+37 x^{4}-4 x^{3}-9 x^{2}+1}d x \right )+37 \left (\int \frac {x^{7}}{x^{12}+12 x^{11}+54 x^{10}+104 x^{9}+48 x^{8}-90 x^{7}-75 x^{6}+30 x^{5}+37 x^{4}-4 x^{3}-9 x^{2}+1}d x \right )+\frac {57 \left (\int \frac {x^{6}}{x^{12}+12 x^{11}+54 x^{10}+104 x^{9}+48 x^{8}-90 x^{7}-75 x^{6}+30 x^{5}+37 x^{4}-4 x^{3}-9 x^{2}+1}d x \right )}{2}-\frac {11 \left (\int \frac {x^{5}}{x^{12}+12 x^{11}+54 x^{10}+104 x^{9}+48 x^{8}-90 x^{7}-75 x^{6}+30 x^{5}+37 x^{4}-4 x^{3}-9 x^{2}+1}d x \right )}{2}-\frac {13 \left (\int \frac {x^{4}}{x^{12}+12 x^{11}+54 x^{10}+104 x^{9}+48 x^{8}-90 x^{7}-75 x^{6}+30 x^{5}+37 x^{4}-4 x^{3}-9 x^{2}+1}d x \right )}{2}-2 \left (\int \frac {x^{2}}{x^{12}+12 x^{11}+54 x^{10}+104 x^{9}+48 x^{8}-90 x^{7}-75 x^{6}+30 x^{5}+37 x^{4}-4 x^{3}-9 x^{2}+1}d x \right )-\frac {\left (\int \frac {x}{x^{12}+12 x^{11}+54 x^{10}+104 x^{9}+48 x^{8}-90 x^{7}-75 x^{6}+30 x^{5}+37 x^{4}-4 x^{3}-9 x^{2}+1}d x \right )}{2}+\int \frac {1}{x^{12}+12 x^{11}+54 x^{10}+104 x^{9}+48 x^{8}-90 x^{7}-75 x^{6}+30 x^{5}+37 x^{4}-4 x^{3}-9 x^{2}+1}d x +\frac {\mathrm {log}\left (x^{12}+12 x^{11}+54 x^{10}+104 x^{9}+48 x^{8}-90 x^{7}-75 x^{6}+30 x^{5}+37 x^{4}-4 x^{3}-9 x^{2}+1\right )}{4} \] Input:
int((3*x^11+33*x^10+137*x^9+249*x^8+133*x^7-129*x^6-118*x^5+31*x^4+37*x^3- 5*x^2-5*x+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)
Output:
(8*int(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),x) + 60*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) + 148*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) + 114*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 ) - 22*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) - 26*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) - 8*int(x**2/(x**12 + 12*x**11 + 5 4*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) - 2*int(x/(x**12 + 12*x**11 + 54*x**10 + 104*x**9 + 4 8*x**8 - 90*x**7 - 75*x**6 + 30*x**5 + 37*x**4 - 4*x**3 - 9*x**2 + 1),x) + 4*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) + log(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))/4