Integrand size = 80, antiderivative size = 484 \[ \int \frac {-8+24 x^4-272 x^8-252 x^{12}+244 x^{16}-296 x^{20}-16 x^{24}-12 x^{28}}{1-2 x^4+69 x^8-236 x^{12}-34 x^{16}-114 x^{20}+4 x^{24}-8 x^{28}+x^{32}} \, dx=-\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \arctan \left (\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} x-\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} x^2\right )-\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \arctan \left (\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} x^2\right )-\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \arctan \left (\frac {1}{2} \sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} x+\frac {1}{2} \sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} x^3\right )-\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \arctan \left (\frac {3}{2} \sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} x-\sqrt {-\frac {11}{8}+\frac {5 \sqrt {5}}{8}} x^3+\sqrt {-2+\sqrt {5}} x^5+\frac {1}{2} \sqrt {-2+\sqrt {5}} x^7\right )-\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \text {arctanh}\left (\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} x-\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} x^2\right )-\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \text {arctanh}\left (\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} x+\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} x^2\right )+\frac {1}{2} \sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \log \left (-1+\sqrt {5}-2 \sqrt {2 \left (-1+\sqrt {5}\right )} x+2 x^2+2 x^4\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \log \left (-1+\sqrt {5}+2 \sqrt {2 \left (-1+\sqrt {5}\right )} x+2 x^2+2 x^4\right ) \] Output:
1/2*(-2+2*5^(1/2))^(1/2)*arctan(-1/2*(2+2*5^(1/2))^(1/2)*x+1/2*(2+2*5^(1/2 ))^(1/2)*x^2)-1/2*(-2+2*5^(1/2))^(1/2)*arctan(1/2*(2+2*5^(1/2))^(1/2)*x+1/ 2*(2+2*5^(1/2))^(1/2)*x^2)-1/2*(2+2*5^(1/2))^(1/2)*arctan(1/4*(-2+2*5^(1/2 ))^(1/2)*x+1/4*(-2+2*5^(1/2))^(1/2)*x^3)-1/2*(2+2*5^(1/2))^(1/2)*arctan(3/ 4*(-2+2*5^(1/2))^(1/2)*x-1/4*(-22+10*5^(1/2))^(1/2)*x^3+(-2+5^(1/2))^(1/2) *x^5+1/2*(-2+5^(1/2))^(1/2)*x^7)+1/2*(2+2*5^(1/2))^(1/2)*arctanh(-1/2*(-2+ 2*5^(1/2))^(1/2)*x+1/2*(-2+2*5^(1/2))^(1/2)*x^2)-1/2*(2+2*5^(1/2))^(1/2)*a rctanh(1/2*(-2+2*5^(1/2))^(1/2)*x+1/2*(-2+2*5^(1/2))^(1/2)*x^2)+1/4*(-2+2* 5^(1/2))^(1/2)*ln(-1+5^(1/2)-2*(-2+2*5^(1/2))^(1/2)*x+2*x^2+2*x^4)-1/4*(-2 +2*5^(1/2))^(1/2)*ln(-1+5^(1/2)+2*(-2+2*5^(1/2))^(1/2)*x+2*x^2+2*x^4)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.12 (sec) , antiderivative size = 440, normalized size of antiderivative = 0.91 \[ \int \frac {-8+24 x^4-272 x^8-252 x^{12}+244 x^{16}-296 x^{20}-16 x^{24}-12 x^{28}}{1-2 x^4+69 x^8-236 x^{12}-34 x^{16}-114 x^{20}+4 x^{24}-8 x^{28}+x^{32}} \, dx=\frac {1}{2} \left (-\text {RootSum}\left [-1-\text {$\#$1}^2+2 \text {$\#$1}^3-4 \text {$\#$1}^5+6 \text {$\#$1}^6-4 \text {$\#$1}^7+\text {$\#$1}^8\&,\frac {2 \log (x-\text {$\#$1})+\log (x-\text {$\#$1}) \text {$\#$1}^2-2 \log (x-\text {$\#$1}) \text {$\#$1}^3+\log (x-\text {$\#$1}) \text {$\#$1}^4}{\text {$\#$1}-\text {$\#$1}^2-2 \text {$\#$1}^3+6 \text {$\#$1}^4-6 \text {$\#$1}^5+2 \text {$\#$1}^6}\&\right ]+\text {RootSum}\left [-1-\text {$\#$1}^2-2 \text {$\#$1}^3+4 \text {$\#$1}^5+6 \text {$\#$1}^6+4 \text {$\#$1}^7+\text {$\#$1}^8\&,\frac {2 \log (x-\text {$\#$1})+\log (x-\text {$\#$1}) \text {$\#$1}^2+2 \log (x-\text {$\#$1}) \text {$\#$1}^3+\log (x-\text {$\#$1}) \text {$\#$1}^4}{-\text {$\#$1}-\text {$\#$1}^2+2 \text {$\#$1}^3+6 \text {$\#$1}^4+6 \text {$\#$1}^5+2 \text {$\#$1}^6}\&\right ]-\text {RootSum}\left [1-2 \text {$\#$1}^2+\text {$\#$1}^4+16 \text {$\#$1}^6+2 \text {$\#$1}^8+2 \text {$\#$1}^{10}+4 \text {$\#$1}^{12}+4 \text {$\#$1}^{14}+\text {$\#$1}^{16}\&,\frac {2 \log (x-\text {$\#$1})-3 \log (x-\text {$\#$1}) \text {$\#$1}^2-2 \log (x-\text {$\#$1}) \text {$\#$1}^4-\log (x-\text {$\#$1}) \text {$\#$1}^6+20 \log (x-\text {$\#$1}) \text {$\#$1}^8+7 \log (x-\text {$\#$1}) \text {$\#$1}^{10}+3 \log (x-\text {$\#$1}) \text {$\#$1}^{12}}{-\text {$\#$1}+\text {$\#$1}^3+24 \text {$\#$1}^5+4 \text {$\#$1}^7+5 \text {$\#$1}^9+12 \text {$\#$1}^{11}+14 \text {$\#$1}^{13}+4 \text {$\#$1}^{15}}\&\right ]\right ) \] Input:
Integrate[(-8 + 24*x^4 - 272*x^8 - 252*x^12 + 244*x^16 - 296*x^20 - 16*x^2 4 - 12*x^28)/(1 - 2*x^4 + 69*x^8 - 236*x^12 - 34*x^16 - 114*x^20 + 4*x^24 - 8*x^28 + x^32),x]
Output:
(-RootSum[-1 - #1^2 + 2*#1^3 - 4*#1^5 + 6*#1^6 - 4*#1^7 + #1^8 & , (2*Log[ x - #1] + Log[x - #1]*#1^2 - 2*Log[x - #1]*#1^3 + Log[x - #1]*#1^4)/(#1 - #1^2 - 2*#1^3 + 6*#1^4 - 6*#1^5 + 2*#1^6) & ] + RootSum[-1 - #1^2 - 2*#1^3 + 4*#1^5 + 6*#1^6 + 4*#1^7 + #1^8 & , (2*Log[x - #1] + Log[x - #1]*#1^2 + 2*Log[x - #1]*#1^3 + Log[x - #1]*#1^4)/(-#1 - #1^2 + 2*#1^3 + 6*#1^4 + 6* #1^5 + 2*#1^6) & ] - RootSum[1 - 2*#1^2 + #1^4 + 16*#1^6 + 2*#1^8 + 2*#1^1 0 + 4*#1^12 + 4*#1^14 + #1^16 & , (2*Log[x - #1] - 3*Log[x - #1]*#1^2 - 2* Log[x - #1]*#1^4 - Log[x - #1]*#1^6 + 20*Log[x - #1]*#1^8 + 7*Log[x - #1]* #1^10 + 3*Log[x - #1]*#1^12)/(-#1 + #1^3 + 24*#1^5 + 4*#1^7 + 5*#1^9 + 12* #1^11 + 14*#1^13 + 4*#1^15) & ])/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.
| \(\displaystyle \int \frac {-12 x^{28}-16 x^{24}-296 x^{20}+244 x^{16}-252 x^{12}-272 x^8+24 x^4-8}{x^{32}-8 x^{28}+4 x^{24}-114 x^{20}-34 x^{16}-236 x^{12}+69 x^8-2 x^4+1} \, dx\) |
| \(\Big \downarrow \) 2460 |
| \(\displaystyle \int \left (\frac {-2 x^5+5 x^4-4 x^3+x^2-4 x+2}{x^8-4 x^7+6 x^6-4 x^5+2 x^3-x^2-1}+\frac {2 x^5+5 x^4+4 x^3+x^2+4 x+2}{x^8+4 x^7+6 x^6+4 x^5-2 x^3-x^2-1}-\frac {2 \left (3 x^{12}+7 x^{10}+20 x^8-x^6-2 x^4-3 x^2+2\right )}{x^{16}+4 x^{14}+4 x^{12}+2 x^{10}+2 x^8+16 x^6+x^4-2 x^2+1}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \int \frac {1}{x^8-4 x^7+6 x^6-4 x^5+2 x^3-x^2-1}dx-4 \int \frac {x}{x^8-4 x^7+6 x^6-4 x^5+2 x^3-x^2-1}dx+\int \frac {x^2}{x^8-4 x^7+6 x^6-4 x^5+2 x^3-x^2-1}dx-4 \int \frac {x^3}{x^8-4 x^7+6 x^6-4 x^5+2 x^3-x^2-1}dx+5 \int \frac {x^4}{x^8-4 x^7+6 x^6-4 x^5+2 x^3-x^2-1}dx-2 \int \frac {x^5}{x^8-4 x^7+6 x^6-4 x^5+2 x^3-x^2-1}dx+2 \int \frac {1}{x^8+4 x^7+6 x^6+4 x^5-2 x^3-x^2-1}dx+4 \int \frac {x}{x^8+4 x^7+6 x^6+4 x^5-2 x^3-x^2-1}dx+\int \frac {x^2}{x^8+4 x^7+6 x^6+4 x^5-2 x^3-x^2-1}dx+4 \int \frac {x^3}{x^8+4 x^7+6 x^6+4 x^5-2 x^3-x^2-1}dx+5 \int \frac {x^4}{x^8+4 x^7+6 x^6+4 x^5-2 x^3-x^2-1}dx+2 \int \frac {x^5}{x^8+4 x^7+6 x^6+4 x^5-2 x^3-x^2-1}dx-4 \int \frac {1}{x^{16}+4 x^{14}+4 x^{12}+2 x^{10}+2 x^8+16 x^6+x^4-2 x^2+1}dx+6 \int \frac {x^2}{x^{16}+4 x^{14}+4 x^{12}+2 x^{10}+2 x^8+16 x^6+x^4-2 x^2+1}dx+4 \int \frac {x^4}{x^{16}+4 x^{14}+4 x^{12}+2 x^{10}+2 x^8+16 x^6+x^4-2 x^2+1}dx+2 \int \frac {x^6}{x^{16}+4 x^{14}+4 x^{12}+2 x^{10}+2 x^8+16 x^6+x^4-2 x^2+1}dx-40 \int \frac {x^8}{x^{16}+4 x^{14}+4 x^{12}+2 x^{10}+2 x^8+16 x^6+x^4-2 x^2+1}dx-14 \int \frac {x^{10}}{x^{16}+4 x^{14}+4 x^{12}+2 x^{10}+2 x^8+16 x^6+x^4-2 x^2+1}dx-6 \int \frac {x^{12}}{x^{16}+4 x^{14}+4 x^{12}+2 x^{10}+2 x^8+16 x^6+x^4-2 x^2+1}dx\) |
Input:
Int[(-8 + 24*x^4 - 272*x^8 - 252*x^12 + 244*x^16 - 296*x^20 - 16*x^24 - 12 *x^28)/(1 - 2*x^4 + 69*x^8 - 236*x^12 - 34*x^16 - 114*x^20 + 4*x^24 - 8*x^ 28 + x^32),x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.30 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.14
| method | result | size |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-\textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (x^{4}-\textit {\_R}^{2}+2 \textit {\_R} x -x^{2}\right )\right )}{2}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+\textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (x^{4}+\textit {\_R}^{2}-2 \textit {\_R} x +x^{2}\right )\right )}{2}\) | \(70\) |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-\textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (x^{2}-\textit {\_R} +x \right )\right )}{2}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-\textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (x^{2}+\textit {\_R} -x \right )\right )}{2}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+\textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (x^{4}+\textit {\_R}^{2}-2 \textit {\_R} x +x^{2}\right )\right )}{2}\) | \(87\) |
Input:
int((-12*x^28-16*x^24-296*x^20+244*x^16-252*x^12-272*x^8+24*x^4-8)/(x^32-8 *x^28+4*x^24-114*x^20-34*x^16-236*x^12+69*x^8-2*x^4+1),x,method=_RETURNVER BOSE)
Output:
1/2*sum(_R*ln(x^4-_R^2+2*_R*x-x^2),_R=RootOf(_Z^4-_Z^2-1))+1/2*sum(_R*ln(x ^4+_R^2-2*_R*x+x^2),_R=RootOf(_Z^4+_Z^2-1))
Time = 0.11 (sec) , antiderivative size = 365, normalized size of antiderivative = 0.75 \[ \int \frac {-8+24 x^4-272 x^8-252 x^{12}+244 x^{16}-296 x^{20}-16 x^{24}-12 x^{28}}{1-2 x^4+69 x^8-236 x^{12}-34 x^{16}-114 x^{20}+4 x^{24}-8 x^{28}+x^{32}} \, dx=\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \arctan \left (-\frac {1}{4} \, {\left (3 \, x^{7} + 6 \, x^{5} + 4 \, x^{3} - \sqrt {5} {\left (x^{7} + 2 \, x^{5} + 2 \, x^{3} - 3 \, x\right )} - 3 \, x\right )} \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}\right ) - \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \arctan \left (-\frac {1}{4} \, {\left (x^{3} - \sqrt {5} {\left (x^{3} + x\right )} + x\right )} \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}\right ) - \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \arctan \left (\frac {1}{4} \, {\left (3 \, x^{7} - 6 \, x^{5} + 4 \, x^{3} + \sqrt {5} {\left (x^{7} - 2 \, x^{5} + 2 \, x^{3} + 3 \, x\right )} + 3 \, x\right )} \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}\right ) + \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \arctan \left (\frac {1}{4} \, {\left (x^{3} + \sqrt {5} {\left (x^{3} - x\right )} - x\right )} \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}\right ) - \frac {1}{2} \, \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \log \left (2 \, x^{4} + 2 \, x^{2} + 4 \, x \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} + \sqrt {5} - 1\right ) + \frac {1}{2} \, \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \log \left (2 \, x^{4} + 2 \, x^{2} - 4 \, x \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} + \sqrt {5} - 1\right ) + \frac {1}{2} \, \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \log \left (2 \, x^{4} - 2 \, x^{2} + 4 \, x \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} - \sqrt {5} - 1\right ) - \frac {1}{2} \, \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \log \left (2 \, x^{4} - 2 \, x^{2} - 4 \, x \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} - \sqrt {5} - 1\right ) \] Input:
integrate((-12*x^28-16*x^24-296*x^20+244*x^16-252*x^12-272*x^8+24*x^4-8)/( x^32-8*x^28+4*x^24-114*x^20-34*x^16-236*x^12+69*x^8-2*x^4+1),x, algorithm= "fricas")
Output:
sqrt(1/2*sqrt(5) + 1/2)*arctan(-1/4*(3*x^7 + 6*x^5 + 4*x^3 - sqrt(5)*(x^7 + 2*x^5 + 2*x^3 - 3*x) - 3*x)*sqrt(1/2*sqrt(5) + 1/2)) - sqrt(1/2*sqrt(5) + 1/2)*arctan(-1/4*(x^3 - sqrt(5)*(x^3 + x) + x)*sqrt(1/2*sqrt(5) + 1/2)) - sqrt(1/2*sqrt(5) - 1/2)*arctan(1/4*(3*x^7 - 6*x^5 + 4*x^3 + sqrt(5)*(x^7 - 2*x^5 + 2*x^3 + 3*x) + 3*x)*sqrt(1/2*sqrt(5) - 1/2)) + sqrt(1/2*sqrt(5) - 1/2)*arctan(1/4*(x^3 + sqrt(5)*(x^3 - x) - x)*sqrt(1/2*sqrt(5) - 1/2)) - 1/2*sqrt(1/2*sqrt(5) - 1/2)*log(2*x^4 + 2*x^2 + 4*x*sqrt(1/2*sqrt(5) - 1 /2) + sqrt(5) - 1) + 1/2*sqrt(1/2*sqrt(5) - 1/2)*log(2*x^4 + 2*x^2 - 4*x*s qrt(1/2*sqrt(5) - 1/2) + sqrt(5) - 1) + 1/2*sqrt(1/2*sqrt(5) + 1/2)*log(2* x^4 - 2*x^2 + 4*x*sqrt(1/2*sqrt(5) + 1/2) - sqrt(5) - 1) - 1/2*sqrt(1/2*sq rt(5) + 1/2)*log(2*x^4 - 2*x^2 - 4*x*sqrt(1/2*sqrt(5) + 1/2) - sqrt(5) - 1 )
Time = 0.71 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.21 \[ \int \frac {-8+24 x^4-272 x^8-252 x^{12}+244 x^{16}-296 x^{20}-16 x^{24}-12 x^{28}}{1-2 x^4+69 x^8-236 x^{12}-34 x^{16}-114 x^{20}+4 x^{24}-8 x^{28}+x^{32}} \, dx=- \operatorname {RootSum} {\left (16 t^{4} - 4 t^{2} - 1, \left ( t \mapsto t \log {\left (- 16 t^{4} + x^{4} + x^{2} \left (- 64 t^{6} + 8 t^{2}\right ) + x \left (- 256 t^{7} + 32 t^{3}\right ) + 1 \right )} \right )\right )} - \operatorname {RootSum} {\left (16 t^{4} + 4 t^{2} - 1, \left ( t \mapsto t \log {\left (- 16 t^{4} + x^{4} + x^{2} \left (- 64 t^{6} + 8 t^{2}\right ) + x \left (- 256 t^{7} + 32 t^{3}\right ) + 1 \right )} \right )\right )} \] Input:
integrate((-12*x**28-16*x**24-296*x**20+244*x**16-252*x**12-272*x**8+24*x* *4-8)/(x**32-8*x**28+4*x**24-114*x**20-34*x**16-236*x**12+69*x**8-2*x**4+1 ),x)
Output:
-RootSum(16*_t**4 - 4*_t**2 - 1, Lambda(_t, _t*log(-16*_t**4 + x**4 + x**2 *(-64*_t**6 + 8*_t**2) + x*(-256*_t**7 + 32*_t**3) + 1))) - RootSum(16*_t* *4 + 4*_t**2 - 1, Lambda(_t, _t*log(-16*_t**4 + x**4 + x**2*(-64*_t**6 + 8 *_t**2) + x*(-256*_t**7 + 32*_t**3) + 1)))
\[ \int \frac {-8+24 x^4-272 x^8-252 x^{12}+244 x^{16}-296 x^{20}-16 x^{24}-12 x^{28}}{1-2 x^4+69 x^8-236 x^{12}-34 x^{16}-114 x^{20}+4 x^{24}-8 x^{28}+x^{32}} \, dx=\int { -\frac {4 \, {\left (3 \, x^{28} + 4 \, x^{24} + 74 \, x^{20} - 61 \, x^{16} + 63 \, x^{12} + 68 \, x^{8} - 6 \, x^{4} + 2\right )}}{x^{32} - 8 \, x^{28} + 4 \, x^{24} - 114 \, x^{20} - 34 \, x^{16} - 236 \, x^{12} + 69 \, x^{8} - 2 \, x^{4} + 1} \,d x } \] Input:
integrate((-12*x^28-16*x^24-296*x^20+244*x^16-252*x^12-272*x^8+24*x^4-8)/( x^32-8*x^28+4*x^24-114*x^20-34*x^16-236*x^12+69*x^8-2*x^4+1),x, algorithm= "maxima")
Output:
-4*integrate((3*x^28 + 4*x^24 + 74*x^20 - 61*x^16 + 63*x^12 + 68*x^8 - 6*x ^4 + 2)/(x^32 - 8*x^28 + 4*x^24 - 114*x^20 - 34*x^16 - 236*x^12 + 69*x^8 - 2*x^4 + 1), x)
Time = 0.26 (sec) , antiderivative size = 353, normalized size of antiderivative = 0.73 \[ \int \frac {-8+24 x^4-272 x^8-252 x^{12}+244 x^{16}-296 x^{20}-16 x^{24}-12 x^{28}}{1-2 x^4+69 x^8-236 x^{12}-34 x^{16}-114 x^{20}+4 x^{24}-8 x^{28}+x^{32}} \, dx=-\sqrt {\frac {1}{2}} \sqrt {\sqrt {5} + 1} {\left (\arctan \left (\frac {x^{3} + x}{\sqrt {2 \, \sqrt {5} + 2}}\right ) - \arctan \left (-\frac {2 \, x^{7} + 4 \, x^{5} - x^{3} {\left (\sqrt {5} - 1\right )} + 3 \, x {\left (\sqrt {5} + 1\right )}}{4 \, \sqrt {\sqrt {5} + 2}}\right )\right )} - \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 1} \arctan \left (\frac {2 \, {\left (x^{2} + x\right )}}{\sqrt {2 \, \sqrt {5} - 2}}\right ) - \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 1} \arctan \left (-\frac {2 \, {\left (x^{2} - x\right )}}{\sqrt {2 \, \sqrt {5} - 2}}\right ) - \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 1} \log \left ({\left | 2344 \, x^{4} + 2344 \, x^{2} + 2344 \, x \sqrt {2 \, \sqrt {5} - 2} + 1172 \, \sqrt {5} - 1172 \right |}\right ) + \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} - 1} \log \left ({\left | 2344 \, x^{4} + 2344 \, x^{2} - 2344 \, x \sqrt {2 \, \sqrt {5} - 2} + 1172 \, \sqrt {5} - 1172 \right |}\right ) - \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} + 1} \log \left ({\left | 104 \, x^{2} + 104 \, x + 52 \, \sqrt {2 \, \sqrt {5} + 2} \right |}\right ) + \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} + 1} \log \left ({\left | 104 \, x^{2} + 104 \, x - 52 \, \sqrt {2 \, \sqrt {5} + 2} \right |}\right ) + \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} + 1} \log \left ({\left | 104 \, x^{2} - 104 \, x + 52 \, \sqrt {2 \, \sqrt {5} + 2} \right |}\right ) - \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} + 1} \log \left ({\left | 104 \, x^{2} - 104 \, x - 52 \, \sqrt {2 \, \sqrt {5} + 2} \right |}\right ) \] Input:
integrate((-12*x^28-16*x^24-296*x^20+244*x^16-252*x^12-272*x^8+24*x^4-8)/( x^32-8*x^28+4*x^24-114*x^20-34*x^16-236*x^12+69*x^8-2*x^4+1),x, algorithm= "giac")
Output:
-sqrt(1/2)*sqrt(sqrt(5) + 1)*(arctan((x^3 + x)/sqrt(2*sqrt(5) + 2)) - arct an(-1/4*(2*x^7 + 4*x^5 - x^3*(sqrt(5) - 1) + 3*x*(sqrt(5) + 1))/sqrt(sqrt( 5) + 2))) - sqrt(1/2)*sqrt(sqrt(5) - 1)*arctan(2*(x^2 + x)/sqrt(2*sqrt(5) - 2)) - sqrt(1/2)*sqrt(sqrt(5) - 1)*arctan(-2*(x^2 - x)/sqrt(2*sqrt(5) - 2 )) - 1/2*sqrt(1/2)*sqrt(sqrt(5) - 1)*log(abs(2344*x^4 + 2344*x^2 + 2344*x* sqrt(2*sqrt(5) - 2) + 1172*sqrt(5) - 1172)) + 1/2*sqrt(1/2)*sqrt(sqrt(5) - 1)*log(abs(2344*x^4 + 2344*x^2 - 2344*x*sqrt(2*sqrt(5) - 2) + 1172*sqrt(5 ) - 1172)) - 1/2*sqrt(1/2)*sqrt(sqrt(5) + 1)*log(abs(104*x^2 + 104*x + 52* sqrt(2*sqrt(5) + 2))) + 1/2*sqrt(1/2)*sqrt(sqrt(5) + 1)*log(abs(104*x^2 + 104*x - 52*sqrt(2*sqrt(5) + 2))) + 1/2*sqrt(1/2)*sqrt(sqrt(5) + 1)*log(abs (104*x^2 - 104*x + 52*sqrt(2*sqrt(5) + 2))) - 1/2*sqrt(1/2)*sqrt(sqrt(5) + 1)*log(abs(104*x^2 - 104*x - 52*sqrt(2*sqrt(5) + 2)))
Time = 10.62 (sec) , antiderivative size = 477, normalized size of antiderivative = 0.99 \[ \int \frac {-8+24 x^4-272 x^8-252 x^{12}+244 x^{16}-296 x^{20}-16 x^{24}-12 x^{28}}{1-2 x^4+69 x^8-236 x^{12}-34 x^{16}-114 x^{20}+4 x^{24}-8 x^{28}+x^{32}} \, dx=\text {Too large to display} \] Input:
int((272*x^8 - 24*x^4 + 252*x^12 - 244*x^16 + 296*x^20 + 16*x^24 + 12*x^28 + 8)/(2*x^4 - 69*x^8 + 236*x^12 + 34*x^16 + 114*x^20 - 4*x^24 + 8*x^28 - x^32 - 1),x)
Output:
(2^(1/2)*atan((2^(1/2)*x*(5^(1/2) - 1)^(1/2)*16115474033830642766388091296 183347357627609590253855994778226853596903348633600000000000000i)/(4454173 68797007436283527902825556339892841344322745233246803542196855368581120000 0000000000*5^(1/2) + 72071266578904940407175332396722205597707827037989513 29842156009659795977011200000000000000*5^(1/2)*x^2 + 720712665789049404071 7533239672220559770782703798951329842156009659795977011200000000000000*5^( 1/2)*x^4 + 161154740338306427663880912961833473576276095902538559947782268 53596903348633600000000000000*x^2 + 16115474033830642766388091296183347357 627609590253855994778226853596903348633600000000000000*x^4 + 9960079627810 91371859978745108887772061315196437045032721627659735103826821120000000000 0000) + (2^(1/2)*5^(1/2)*x*(5^(1/2) - 1)^(1/2)*720712665789049404071753323 9672220559770782703798951329842156009659795977011200000000000000i)/(445417 36879700743628352790282555633989284134432274523324680354219685536858112000 00000000000*5^(1/2) + 7207126657890494040717533239672220559770782703798951 329842156009659795977011200000000000000*5^(1/2)*x^2 + 72071266578904940407 17533239672220559770782703798951329842156009659795977011200000000000000*5^ (1/2)*x^4 + 16115474033830642766388091296183347357627609590253855994778226 853596903348633600000000000000*x^2 + 1611547403383064276638809129618334735 7627609590253855994778226853596903348633600000000000000*x^4 + 996007962781 09137185997874510888777206131519643704503272162765973510382682112000000...
\[ \int \frac {-8+24 x^4-272 x^8-252 x^{12}+244 x^{16}-296 x^{20}-16 x^{24}-12 x^{28}}{1-2 x^4+69 x^8-236 x^{12}-34 x^{16}-114 x^{20}+4 x^{24}-8 x^{28}+x^{32}} \, dx=-12 \left (\int \frac {x^{28}}{x^{32}-8 x^{28}+4 x^{24}-114 x^{20}-34 x^{16}-236 x^{12}+69 x^{8}-2 x^{4}+1}d x \right )-16 \left (\int \frac {x^{24}}{x^{32}-8 x^{28}+4 x^{24}-114 x^{20}-34 x^{16}-236 x^{12}+69 x^{8}-2 x^{4}+1}d x \right )-296 \left (\int \frac {x^{20}}{x^{32}-8 x^{28}+4 x^{24}-114 x^{20}-34 x^{16}-236 x^{12}+69 x^{8}-2 x^{4}+1}d x \right )+244 \left (\int \frac {x^{16}}{x^{32}-8 x^{28}+4 x^{24}-114 x^{20}-34 x^{16}-236 x^{12}+69 x^{8}-2 x^{4}+1}d x \right )-252 \left (\int \frac {x^{12}}{x^{32}-8 x^{28}+4 x^{24}-114 x^{20}-34 x^{16}-236 x^{12}+69 x^{8}-2 x^{4}+1}d x \right )-272 \left (\int \frac {x^{8}}{x^{32}-8 x^{28}+4 x^{24}-114 x^{20}-34 x^{16}-236 x^{12}+69 x^{8}-2 x^{4}+1}d x \right )+24 \left (\int \frac {x^{4}}{x^{32}-8 x^{28}+4 x^{24}-114 x^{20}-34 x^{16}-236 x^{12}+69 x^{8}-2 x^{4}+1}d x \right )-8 \left (\int \frac {1}{x^{32}-8 x^{28}+4 x^{24}-114 x^{20}-34 x^{16}-236 x^{12}+69 x^{8}-2 x^{4}+1}d x \right ) \] Input:
int((-12*x^28-16*x^24-296*x^20+244*x^16-252*x^12-272*x^8+24*x^4-8)/(x^32-8 *x^28+4*x^24-114*x^20-34*x^16-236*x^12+69*x^8-2*x^4+1),x)
Output:
4*( - 3*int(x**28/(x**32 - 8*x**28 + 4*x**24 - 114*x**20 - 34*x**16 - 236* x**12 + 69*x**8 - 2*x**4 + 1),x) - 4*int(x**24/(x**32 - 8*x**28 + 4*x**24 - 114*x**20 - 34*x**16 - 236*x**12 + 69*x**8 - 2*x**4 + 1),x) - 74*int(x** 20/(x**32 - 8*x**28 + 4*x**24 - 114*x**20 - 34*x**16 - 236*x**12 + 69*x**8 - 2*x**4 + 1),x) + 61*int(x**16/(x**32 - 8*x**28 + 4*x**24 - 114*x**20 - 34*x**16 - 236*x**12 + 69*x**8 - 2*x**4 + 1),x) - 63*int(x**12/(x**32 - 8* x**28 + 4*x**24 - 114*x**20 - 34*x**16 - 236*x**12 + 69*x**8 - 2*x**4 + 1) ,x) - 68*int(x**8/(x**32 - 8*x**28 + 4*x**24 - 114*x**20 - 34*x**16 - 236* x**12 + 69*x**8 - 2*x**4 + 1),x) + 6*int(x**4/(x**32 - 8*x**28 + 4*x**24 - 114*x**20 - 34*x**16 - 236*x**12 + 69*x**8 - 2*x**4 + 1),x) - 2*int(1/(x* *32 - 8*x**28 + 4*x**24 - 114*x**20 - 34*x**16 - 236*x**12 + 69*x**8 - 2*x **4 + 1),x))