Integrand size = 54, antiderivative size = 327 \[ \int \frac {-21504 x^3-3072 x^{11}-63744 x^{19}-3840 x^{27}}{-1024-192 x^8-8688 x^{16}-1632 x^{24}+72 x^{32}-12 x^{40}+x^{48}} \, dx=-2 \arctan \left (1-x^2\right )-2 \arctan \left (1+x^2\right )+\frac {\arctan \left (1-\sqrt {2}-\sqrt {2} x^2\right )}{\sqrt {2}}-\frac {\arctan \left (1+\sqrt {2}-\sqrt {2} x^2\right )}{\sqrt {2}}-\frac {\arctan \left (1+\sqrt {2}-2 x^2-\sqrt {2} x^2\right )}{\sqrt {2}}+\frac {\arctan \left (1-\sqrt {2}+2 x^2-\sqrt {2} x^2\right )}{\sqrt {2}}+\frac {\arctan \left (1-\sqrt {2}+\sqrt {2} x^2\right )}{\sqrt {2}}-\frac {\arctan \left (1+\sqrt {2}+\sqrt {2} x^2\right )}{\sqrt {2}}+\frac {\arctan \left (1-\sqrt {2}-2 x^2+\sqrt {2} x^2\right )}{\sqrt {2}}-\frac {\arctan \left (1+\sqrt {2}+2 x^2+\sqrt {2} x^2\right )}{\sqrt {2}}+\text {arctanh}\left (\frac {x^4}{4}\right )-\frac {\log \left (4-8 \sqrt {2} x^4+16 x^8-4 \sqrt {2} x^{12}+x^{16}\right )}{2 \sqrt {2}}+\frac {\log \left (4+8 \sqrt {2} x^4+16 x^8+4 \sqrt {2} x^{12}+x^{16}\right )}{2 \sqrt {2}} \] Output:
2*arctan(x^2-1)-2*arctan(x^2+1)-1/2*arctan(-1+2^(1/2)+x^2*2^(1/2))*2^(1/2) +1/2*arctan(-1-2^(1/2)+x^2*2^(1/2))*2^(1/2)+1/2*arctan(x^2*2^(1/2)+2*x^2-2 ^(1/2)-1)*2^(1/2)-1/2*arctan(x^2*2^(1/2)-2*x^2+2^(1/2)-1)*2^(1/2)+1/2*arct an(1-2^(1/2)+x^2*2^(1/2))*2^(1/2)-1/2*arctan(1+2^(1/2)+x^2*2^(1/2))*2^(1/2 )+1/2*arctan(1-2^(1/2)-2*x^2+x^2*2^(1/2))*2^(1/2)-1/2*arctan(1+2^(1/2)+2*x ^2+x^2*2^(1/2))*2^(1/2)+arctanh(1/4*x^4)-1/4*ln(4-8*2^(1/2)*x^4+16*x^8-4*2 ^(1/2)*x^12+x^16)*2^(1/2)+1/4*ln(4+8*2^(1/2)*x^4+16*x^8+4*2^(1/2)*x^12+x^1 6)*2^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.07 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.87 \[ \int \frac {-21504 x^3-3072 x^{11}-63744 x^{19}-3840 x^{27}}{-1024-192 x^8-8688 x^{16}-1632 x^{24}+72 x^{32}-12 x^{40}+x^{48}} \, dx=\frac {1}{2} \left (-4 \arctan \left (1-x^2\right )-4 \arctan \left (1+x^2\right )-\log \left (2-x^2\right )-\log \left (2+x^2\right )+\log \left (2-2 x+x^2\right )+\log \left (2+2 x+x^2\right )-\text {RootSum}\left [2-4 \text {$\#$1}^2+2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x-\text {$\#$1})+\log (x-\text {$\#$1}) \text {$\#$1}^4}{-1+\text {$\#$1}^2+\text {$\#$1}^6}\&\right ]+\text {RootSum}\left [2+4 \text {$\#$1}^2+2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x-\text {$\#$1})+\log (x-\text {$\#$1}) \text {$\#$1}^4}{1+\text {$\#$1}^2+\text {$\#$1}^6}\&\right ]+\text {RootSum}\left [2-4 \text {$\#$1}^2+6 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log (x-\text {$\#$1})}{-1+3 \text {$\#$1}^2-3 \text {$\#$1}^4+\text {$\#$1}^6}\&\right ]-\text {RootSum}\left [2+4 \text {$\#$1}^2+6 \text {$\#$1}^4+4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log (x-\text {$\#$1})}{1+3 \text {$\#$1}^2+3 \text {$\#$1}^4+\text {$\#$1}^6}\&\right ]\right ) \] Input:
Integrate[(-21504*x^3 - 3072*x^11 - 63744*x^19 - 3840*x^27)/(-1024 - 192*x ^8 - 8688*x^16 - 1632*x^24 + 72*x^32 - 12*x^40 + x^48),x]
Output:
(-4*ArcTan[1 - x^2] - 4*ArcTan[1 + x^2] - Log[2 - x^2] - Log[2 + x^2] + Lo g[2 - 2*x + x^2] + Log[2 + 2*x + x^2] - RootSum[2 - 4*#1^2 + 2*#1^4 + #1^8 & , (-Log[x - #1] + Log[x - #1]*#1^4)/(-1 + #1^2 + #1^6) & ] + RootSum[2 + 4*#1^2 + 2*#1^4 + #1^8 & , (-Log[x - #1] + Log[x - #1]*#1^4)/(1 + #1^2 + #1^6) & ] + RootSum[2 - 4*#1^2 + 6*#1^4 - 4*#1^6 + #1^8 & , Log[x - #1]/( -1 + 3*#1^2 - 3*#1^4 + #1^6) & ] - RootSum[2 + 4*#1^2 + 6*#1^4 + 4*#1^6 + #1^8 & , Log[x - #1]/(1 + 3*#1^2 + 3*#1^4 + #1^6) & ])/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.
| \(\displaystyle \int \frac {-3840 x^{27}-63744 x^{19}-3072 x^{11}-21504 x^3}{x^{48}-12 x^{40}+72 x^{32}-1632 x^{24}-8688 x^{16}-192 x^8-1024} \, dx\) |
| \(\Big \downarrow \) 2029 |
| \(\displaystyle \int \frac {x^3 \left (-3840 x^{24}-63744 x^{16}-3072 x^8-21504\right )}{x^{48}-12 x^{40}+72 x^{32}-1632 x^{24}-8688 x^{16}-192 x^8-1024}dx\) |
| \(\Big \downarrow \) 2460 |
| \(\displaystyle \int \left (\frac {x-1}{x^2-2 x+2}-\frac {x}{x^2-2}-\frac {x}{x^2+2}+\frac {x+1}{x^2+2 x+2}+\frac {4 x}{x^4-2 x^2+2}-\frac {4 x}{x^4+2 x^2+2}-\frac {4 x \left (x^4-1\right )}{x^8+2 x^4-4 x^2+2}+\frac {4 x \left (x^4-1\right )}{x^8+2 x^4+4 x^2+2}+\frac {4 x}{x^8-4 x^6+6 x^4-4 x^2+2}-\frac {4 x}{x^8+4 x^6+6 x^4+4 x^2+2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \text {Subst}\left (\int \frac {x^2}{x^4+2 x^2-4 x+2}dx,x,x^2\right )+2 \text {Subst}\left (\int \frac {x^2}{x^4+2 x^2+4 x+2}dx,x,x^2\right )+\frac {2}{3} \arctan \left (-\frac {2}{x^2}-\sqrt {2}+1\right )+\frac {2}{3} \arctan \left (-\frac {2}{x^2}+\sqrt {2}+1\right )+\frac {2}{3} \arctan \left (\frac {2}{x^2}-\sqrt {2}+1\right )+\frac {2}{3} \arctan \left (\frac {2}{x^2}+\sqrt {2}+1\right )-2 \arctan \left (1-x^2\right )-2 \arctan \left (x^2+1\right )+\frac {\arctan \left (1-\sqrt {2} \left (1-x^2\right )\right )}{\sqrt {2}}-\frac {\arctan \left (\sqrt {2} \left (1-x^2\right )+1\right )}{\sqrt {2}}+\frac {\arctan \left (1-\sqrt {2} \left (x^2+1\right )\right )}{\sqrt {2}}-\frac {\arctan \left (\sqrt {2} \left (x^2+1\right )+1\right )}{\sqrt {2}}-\frac {\arctan \left (\frac {\sqrt {2} \left (1-x^2\right )}{x^4}\right )}{\sqrt {2}}-\frac {\arctan \left (\frac {\sqrt {2} \left (x^2+1\right )}{x^4}\right )}{\sqrt {2}}-\frac {1}{2} \log \left (2-x^2\right )-\frac {1}{2} \log \left (x^2+2\right )+\frac {1}{2} \log \left (x^2-2 x+2\right )+\frac {1}{2} \log \left (x^2+2 x+2\right )+\frac {\log \left (\left (x^2-1\right )^2-\sqrt {2} \left (1-x^2\right )+1\right )}{2 \sqrt {2}}-\frac {\log \left (\left (x^2-1\right )^2+\sqrt {2} \left (1-x^2\right )+1\right )}{2 \sqrt {2}}+\frac {\log \left (\left (x^2+1\right )^2-\sqrt {2} \left (x^2+1\right )+1\right )}{2 \sqrt {2}}-\frac {\log \left (\left (x^2+1\right )^2+\sqrt {2} \left (x^2+1\right )+1\right )}{2 \sqrt {2}}+\frac {\log \left (\frac {-\sqrt {2} x^4+2 x^4-2 \sqrt {2} x^2+2 x^2+2}{x^4}\right )}{6 \sqrt {2}}+\frac {\log \left (\frac {-\sqrt {2} x^4+2 x^4+2 \sqrt {2} x^2-2 x^2+2}{x^4}\right )}{6 \sqrt {2}}-\frac {\log \left (\frac {\left (2+\sqrt {2}\right ) x^4-2 \left (1+\sqrt {2}\right ) x^2+2}{x^4}\right )}{6 \sqrt {2}}-\frac {\log \left (\frac {\left (2+\sqrt {2}\right ) x^4+2 \left (1+\sqrt {2}\right ) x^2+2}{x^4}\right )}{6 \sqrt {2}}\) |
Input:
Int[(-21504*x^3 - 3072*x^11 - 63744*x^19 - 3840*x^27)/(-1024 - 192*x^8 - 8 688*x^16 - 1632*x^24 + 72*x^32 - 12*x^40 + x^48),x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.96 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.17
| method | result | size |
| risch | \(-\frac {\ln \left (x^{4}-4\right )}{2}+2 \arctan \left (\frac {x^{4}}{2}\right )+\frac {\ln \left (x^{4}+4\right )}{2}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (x^{8}+4 \textit {\_R} \,x^{4}+2 \textit {\_R}^{2}\right )\right )}{2}\) | \(55\) |
| default | \(2 \arctan \left (\frac {x^{4}}{2}\right )+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}+8 \textit {\_Z} +4\right )}{\sum }\frac {\left (1+\textit {\_R} \right ) \ln \left (x^{4}-\textit {\_R} \right )}{\textit {\_R}^{3}-3 \textit {\_R}^{2}+4 \textit {\_R} +2}\right )-\frac {\ln \left (x^{4}-4\right )}{2}+\frac {\ln \left (x^{4}+4\right )}{2}+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+4 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}-8 \textit {\_Z} +4\right )}{\sum }\frac {\left (1-\textit {\_R} \right ) \ln \left (x^{4}-\textit {\_R} \right )}{\textit {\_R}^{3}+3 \textit {\_R}^{2}+4 \textit {\_R} -2}\right )\) | \(130\) |
Input:
int((-3840*x^27-63744*x^19-3072*x^11-21504*x^3)/(x^48-12*x^40+72*x^32-1632 *x^24-8688*x^16-192*x^8-1024),x,method=_RETURNVERBOSE)
Output:
-1/2*ln(x^4-4)+2*arctan(1/2*x^4)+1/2*ln(x^4+4)+1/2*sum(_R*ln(x^8+4*_R*x^4+ 2*_R^2),_R=RootOf(_Z^4+1))
Time = 0.13 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.52 \[ \int \frac {-21504 x^3-3072 x^{11}-63744 x^{19}-3840 x^{27}}{-1024-192 x^8-8688 x^{16}-1632 x^{24}+72 x^{32}-12 x^{40}+x^{48}} \, dx=\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {7}{6} \, x^{8} + \frac {1}{6} \, \sqrt {2} {\left (x^{12} + 14 \, x^{4}\right )} + \frac {4}{3}\right ) + \frac {1}{2} \, \sqrt {2} \arctan \left (-\frac {7}{6} \, x^{8} + \frac {1}{6} \, \sqrt {2} {\left (x^{12} + 14 \, x^{4}\right )} - \frac {4}{3}\right ) + \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{4} \, \sqrt {2} x^{4} + \frac {3}{4}\right ) + \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{4} \, \sqrt {2} x^{4} - \frac {3}{4}\right ) + \frac {1}{4} \, \sqrt {2} \log \left (x^{16} + 16 \, x^{8} + 4 \, \sqrt {2} {\left (x^{12} + 2 \, x^{4}\right )} + 4\right ) - \frac {1}{4} \, \sqrt {2} \log \left (x^{16} + 16 \, x^{8} - 4 \, \sqrt {2} {\left (x^{12} + 2 \, x^{4}\right )} + 4\right ) + 2 \, \arctan \left (\frac {1}{2} \, x^{4}\right ) + \frac {1}{2} \, \log \left (x^{4} + 4\right ) - \frac {1}{2} \, \log \left (x^{4} - 4\right ) \] Input:
integrate((-3840*x^27-63744*x^19-3072*x^11-21504*x^3)/(x^48-12*x^40+72*x^3 2-1632*x^24-8688*x^16-192*x^8-1024),x, algorithm="fricas")
Output:
1/2*sqrt(2)*arctan(7/6*x^8 + 1/6*sqrt(2)*(x^12 + 14*x^4) + 4/3) + 1/2*sqrt (2)*arctan(-7/6*x^8 + 1/6*sqrt(2)*(x^12 + 14*x^4) - 4/3) + 1/2*sqrt(2)*arc tan(1/4*sqrt(2)*x^4 + 3/4) + 1/2*sqrt(2)*arctan(1/4*sqrt(2)*x^4 - 3/4) + 1 /4*sqrt(2)*log(x^16 + 16*x^8 + 4*sqrt(2)*(x^12 + 2*x^4) + 4) - 1/4*sqrt(2) *log(x^16 + 16*x^8 - 4*sqrt(2)*(x^12 + 2*x^4) + 4) + 2*arctan(1/2*x^4) + 1 /2*log(x^4 + 4) - 1/2*log(x^4 - 4)
Time = 0.28 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.69 \[ \int \frac {-21504 x^3-3072 x^{11}-63744 x^{19}-3840 x^{27}}{-1024-192 x^8-8688 x^{16}-1632 x^{24}+72 x^{32}-12 x^{40}+x^{48}} \, dx=- \frac {\sqrt {2} \left (- 2 \operatorname {atan}{\left (\frac {\sqrt {2} x^{4}}{4} - \frac {3}{4} \right )} - 2 \operatorname {atan}{\left (\frac {\sqrt {2} x^{12}}{6} - \frac {7 x^{8}}{6} + \frac {7 \sqrt {2} x^{4}}{3} - \frac {4}{3} \right )}\right )}{4} - \frac {\sqrt {2} \left (- 2 \operatorname {atan}{\left (\frac {\sqrt {2} x^{4}}{4} + \frac {3}{4} \right )} - 2 \operatorname {atan}{\left (\frac {\sqrt {2} x^{12}}{6} + \frac {7 x^{8}}{6} + \frac {7 \sqrt {2} x^{4}}{3} + \frac {4}{3} \right )}\right )}{4} - \frac {\log {\left (x^{4} - 4 \right )}}{2} + \frac {\log {\left (x^{4} + 4 \right )}}{2} - \frac {\sqrt {2} \log {\left (x^{16} - 4 \sqrt {2} x^{12} + 16 x^{8} - 8 \sqrt {2} x^{4} + 4 \right )}}{4} + \frac {\sqrt {2} \log {\left (x^{16} + 4 \sqrt {2} x^{12} + 16 x^{8} + 8 \sqrt {2} x^{4} + 4 \right )}}{4} + 2 \operatorname {atan}{\left (\frac {x^{4}}{2} \right )} \] Input:
integrate((-3840*x**27-63744*x**19-3072*x**11-21504*x**3)/(x**48-12*x**40+ 72*x**32-1632*x**24-8688*x**16-192*x**8-1024),x)
Output:
-sqrt(2)*(-2*atan(sqrt(2)*x**4/4 - 3/4) - 2*atan(sqrt(2)*x**12/6 - 7*x**8/ 6 + 7*sqrt(2)*x**4/3 - 4/3))/4 - sqrt(2)*(-2*atan(sqrt(2)*x**4/4 + 3/4) - 2*atan(sqrt(2)*x**12/6 + 7*x**8/6 + 7*sqrt(2)*x**4/3 + 4/3))/4 - log(x**4 - 4)/2 + log(x**4 + 4)/2 - sqrt(2)*log(x**16 - 4*sqrt(2)*x**12 + 16*x**8 - 8*sqrt(2)*x**4 + 4)/4 + sqrt(2)*log(x**16 + 4*sqrt(2)*x**12 + 16*x**8 + 8 *sqrt(2)*x**4 + 4)/4 + 2*atan(x**4/2)
\[ \int \frac {-21504 x^3-3072 x^{11}-63744 x^{19}-3840 x^{27}}{-1024-192 x^8-8688 x^{16}-1632 x^{24}+72 x^{32}-12 x^{40}+x^{48}} \, dx=\int { -\frac {768 \, {\left (5 \, x^{27} + 83 \, x^{19} + 4 \, x^{11} + 28 \, x^{3}\right )}}{x^{48} - 12 \, x^{40} + 72 \, x^{32} - 1632 \, x^{24} - 8688 \, x^{16} - 192 \, x^{8} - 1024} \,d x } \] Input:
integrate((-3840*x^27-63744*x^19-3072*x^11-21504*x^3)/(x^48-12*x^40+72*x^3 2-1632*x^24-8688*x^16-192*x^8-1024),x, algorithm="maxima")
Output:
4*integrate((x^5 - x)/(x^8 + 2*x^4 + 4*x^2 + 2), x) - 4*integrate((x^5 - x )/(x^8 + 2*x^4 - 4*x^2 + 2), x) - 4*integrate(x/(x^8 + 4*x^6 + 6*x^4 + 4*x ^2 + 2), x) + 4*integrate(x/(x^8 - 4*x^6 + 6*x^4 - 4*x^2 + 2), x) - 4*inte grate(x/(x^4 + 2*x^2 + 2), x) + 4*integrate(x/(x^4 - 2*x^2 + 2), x) + 1/2* log(x^2 + 2*x + 2) + 1/2*log(x^2 - 2*x + 2) - 1/2*log(x^2 + 2) - 1/2*log(x ^2 - 2)
Timed out. \[ \int \frac {-21504 x^3-3072 x^{11}-63744 x^{19}-3840 x^{27}}{-1024-192 x^8-8688 x^{16}-1632 x^{24}+72 x^{32}-12 x^{40}+x^{48}} \, dx=\text {Timed out} \] Input:
integrate((-3840*x^27-63744*x^19-3072*x^11-21504*x^3)/(x^48-12*x^40+72*x^3 2-1632*x^24-8688*x^16-192*x^8-1024),x, algorithm="giac")
Output:
Timed out
Time = 10.43 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.25 \[ \int \frac {-21504 x^3-3072 x^{11}-63744 x^{19}-3840 x^{27}}{-1024-192 x^8-8688 x^{16}-1632 x^{24}+72 x^{32}-12 x^{40}+x^{48}} \, dx=\text {Too large to display} \] Input:
int((21504*x^3 + 3072*x^11 + 63744*x^19 + 3840*x^27)/(192*x^8 + 8688*x^16 + 1632*x^24 - 72*x^32 + 12*x^40 - x^48 + 1024),x)
Output:
2*atan(x^4/2) - atan((x^4*1i)/4)*1i - 2^(1/2)*atan((2^(1/2)*x^4*(764886904 45726928928043971784974348792916843261154049524815686201788737984099646078 28394633776665441993199104289579518086625046676785965701785495403705254975 311367263415128406680534025128252141595525120000000000000000 + 40898633798 79310808240510565117378645620131517107279988428287019008178399655663770556 23104992286116205196899694450685882227368471498014546349028880705248228162 1118053336022968167868837660179578098810880000000000000000i))/(x^8*(293468 31061130009252612269359037033812279539608056712352274639097967630495164070 94596156420924881651011292025308521594227224682847941527791318946052689434 314233121329187787843712100715697107929923584000000000000000000 - 88975141 61733455211409716533450140584178882022520312410133204002926738496885752093 18040895963451069985307526836268164816087840490451455059573801649163193173 422562302519776359628166296867018140874178560000000000000000i) - (17795028 32346691042281943306690028116835776404504062482026640800585347699377150418 63608179192690213997061505367253632963217568098090291011914760329832638634 6845124605039552719256332593734036281748357120000000000000000 + 5869366212 22600185052245387180740676245590792161134247045492781959352609903281418919 23128418497633020225840506170431884544493656958830555826378921053788686284 66242658375575687424201431394215859847168000000000000000000i)))*(1/2 + 1i/ 2) - 2^(1/2)*atan((2^(1/2)*x^4*(764886904457269289280439717849743487929...
\[ \int \frac {-21504 x^3-3072 x^{11}-63744 x^{19}-3840 x^{27}}{-1024-192 x^8-8688 x^{16}-1632 x^{24}+72 x^{32}-12 x^{40}+x^{48}} \, dx=\int \frac {-3840 x^{27}-63744 x^{19}-3072 x^{11}-21504 x^{3}}{x^{48}-12 x^{40}+72 x^{32}-1632 x^{24}-8688 x^{16}-192 x^{8}-1024}d x \] Input:
int((-3840*x^27-63744*x^19-3072*x^11-21504*x^3)/(x^48-12*x^40+72*x^32-1632 *x^24-8688*x^16-192*x^8-1024),x)
Output:
int((-3840*x^27-63744*x^19-3072*x^11-21504*x^3)/(x^48-12*x^40+72*x^32-1632 *x^24-8688*x^16-192*x^8-1024),x)