Integrand size = 18, antiderivative size = 389 \[ \int \frac {1}{\left (1-x+x^2-x^3+x^4\right )^3} \, dx=\frac {6 \left (2 \left (15+8 \sqrt {5}\right )+\left (15+11 \sqrt {5}\right ) x\right )}{25 \left (5-\sqrt {5}\right ) \left (2-\left (1-\sqrt {5}\right ) x+2 x^2\right )^2}+\frac {3 \left (6+\sqrt {5}+2 x\right )}{25 \left (2-\left (1-\sqrt {5}\right ) x+2 x^2\right )}-\frac {8 \left (5-\sqrt {5}-\left (5+\sqrt {5}\right ) x\right )}{5 \left (5-\sqrt {5}\right ) \left (2-\left (1-\sqrt {5}\right ) x+2 x^2\right )^2 \left (2-\left (1+\sqrt {5}\right ) x+2 x^2\right )^2}-\frac {8 \left (45+11 \sqrt {5}-6 \left (15-\sqrt {5}\right ) x\right )}{5 \left (5-\sqrt {5}\right )^2 \left (2-\left (1-\sqrt {5}\right ) x+2 x^2\right )^2 \left (2-\left (1+\sqrt {5}\right ) x+2 x^2\right )}+\frac {3}{125} \sqrt {1025-422 \sqrt {5}} \arctan \left (\frac {1-\sqrt {5}-4 x}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )-\frac {3}{125} \sqrt {1025+422 \sqrt {5}} \arctan \left (\frac {1+\sqrt {5}-4 x}{\sqrt {2 \left (5-\sqrt {5}\right )}}\right )+\frac {3 \log \left (2-\left (1-\sqrt {5}\right ) x+2 x^2\right )}{10 \sqrt {5}}-\frac {3 \log \left (2-\left (1+\sqrt {5}\right ) x+2 x^2\right )}{10 \sqrt {5}} \] Output:
6/25*(30+16*5^(1/2)+(15+11*5^(1/2))*x)/(5-5^(1/2))/(2-x*(-5^(1/2)+1)+2*x^2 )^2+3*(6+5^(1/2)+2*x)/(50-25*x*(-5^(1/2)+1)+50*x^2)-8/5*(5-5^(1/2)-(5+5^(1 /2))*x)/(5-5^(1/2))/(2-x*(-5^(1/2)+1)+2*x^2)^2/(2-(5^(1/2)+1)*x+2*x^2)^2-8 /5*(45+11*5^(1/2)-6*(15-5^(1/2))*x)/(5-5^(1/2))^2/(2-x*(-5^(1/2)+1)+2*x^2) ^2/(2-(5^(1/2)+1)*x+2*x^2)+3/125*(1025-422*5^(1/2))^(1/2)*arctan((1-5^(1/2 )-4*x)/(10+2*5^(1/2))^(1/2))-3/125*(1025+422*5^(1/2))^(1/2)*arctan((1+5^(1 /2)-4*x)/(10-2*5^(1/2))^(1/2))+3/50*ln(2-x*(-5^(1/2)+1)+2*x^2)*5^(1/2)-3/5 0*ln(2-(5^(1/2)+1)*x+2*x^2)*5^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.31 \[ \int \frac {1}{\left (1-x+x^2-x^3+x^4\right )^3} \, dx=\frac {1}{50} \left (\frac {x \left (14+11 x+9 x^5+6 x^6\right )}{\left (1-x+x^2-x^3+x^4\right )^2}+6 \text {RootSum}\left [1-\text {$\#$1}+\text {$\#$1}^2-\text {$\#$1}^3+\text {$\#$1}^4\&,\frac {6 \log (x-\text {$\#$1})+6 \log (x-\text {$\#$1}) \text {$\#$1}+\log (x-\text {$\#$1}) \text {$\#$1}^2}{-1+2 \text {$\#$1}-3 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ]\right ) \] Input:
Integrate[(1 - x + x^2 - x^3 + x^4)^(-3),x]
Output:
((x*(14 + 11*x + 9*x^5 + 6*x^6))/(1 - x + x^2 - x^3 + x^4)^2 + 6*RootSum[1 - #1 + #1^2 - #1^3 + #1^4 & , (6*Log[x - #1] + 6*Log[x - #1]*#1 + Log[x - #1]*#1^2)/(-1 + 2*#1 - 3*#1^2 + 4*#1^3) & ])/50
Time = 1.17 (sec) , antiderivative size = 637, normalized size of antiderivative = 1.64, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2492, 2009}
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.
| \(\displaystyle \int \frac {1}{\left (x^4-x^3+x^2-x+1\right )^3} \, dx\) |
| \(\Big \downarrow \) 2492 |
| \(\displaystyle \int \left (-\frac {3 \left (-10 x-5 \sqrt {5}+11\right )}{25 \sqrt {5} \left (2 x^2-\left (1-\sqrt {5}\right ) x+2\right )}+\frac {3 \left (-10 x+5 \sqrt {5}+11\right )}{25 \sqrt {5} \left (2 x^2-\left (1+\sqrt {5}\right ) x+2\right )}+\frac {2 \left (2 \left (5-4 \sqrt {5}\right )-\left (9-5 \sqrt {5}\right ) x\right )}{25 \left (2 x^2-\left (1-\sqrt {5}\right ) x+2\right )^2}+\frac {2 \left (2 \left (5+4 \sqrt {5}\right )-\left (9+5 \sqrt {5}\right ) x\right )}{25 \left (2 x^2-\left (1+\sqrt {5}\right ) x+2\right )^2}-\frac {4 \left (2-\left (1-\sqrt {5}\right ) x\right )}{5 \sqrt {5} \left (2 x^2-\left (1-\sqrt {5}\right ) x+2\right )^3}+\frac {4 \left (2-\left (1+\sqrt {5}\right ) x\right )}{5 \sqrt {5} \left (2 x^2-\left (1+\sqrt {5}\right ) x+2\right )^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {6}{125} \sqrt {5-2 \sqrt {5}} \arctan \left (\frac {-4 x-\sqrt {5}+1}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )+\frac {3}{125} \sqrt {2 \left (145-61 \sqrt {5}\right )} \arctan \left (\frac {-4 x-\sqrt {5}+1}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )+\frac {3}{125} \sqrt {365-158 \sqrt {5}} \arctan \left (\frac {-4 x-\sqrt {5}+1}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )-\frac {3}{125} \sqrt {365+158 \sqrt {5}} \arctan \left (\frac {-4 x+\sqrt {5}+1}{\sqrt {2 \left (5-\sqrt {5}\right )}}\right )-\frac {3}{125} \sqrt {2 \left (145+61 \sqrt {5}\right )} \arctan \left (\frac {-4 x+\sqrt {5}+1}{\sqrt {2 \left (5-\sqrt {5}\right )}}\right )+\frac {6}{125} \sqrt {5+2 \sqrt {5}} \arctan \left (\frac {-4 x+\sqrt {5}+1}{\sqrt {2 \left (5-\sqrt {5}\right )}}\right )+\frac {3 \left (-4 x-\sqrt {5}+1\right )}{25 \left (5+\sqrt {5}\right ) \left (2 x^2-\left (1-\sqrt {5}\right ) x+2\right )}-\frac {2 \left (-3 \left (1-3 \sqrt {5}\right ) x+\sqrt {5}+7\right )}{25 \left (5+\sqrt {5}\right ) \left (2 x^2-\left (1-\sqrt {5}\right ) x+2\right )}+\frac {3 \left (-4 x+\sqrt {5}+1\right )}{25 \left (5-\sqrt {5}\right ) \left (2 x^2-\left (1+\sqrt {5}\right ) x+2\right )}-\frac {2 \left (-3 \left (1+3 \sqrt {5}\right ) x-\sqrt {5}+7\right )}{25 \left (5-\sqrt {5}\right ) \left (2 x^2-\left (1+\sqrt {5}\right ) x+2\right )}-\frac {2 \left (\left (1+\sqrt {5}\right ) x-\sqrt {5}+1\right )}{5 \sqrt {5} \left (5+\sqrt {5}\right ) \left (2 x^2-\left (1-\sqrt {5}\right ) x+2\right )^2}+\frac {2 \left (\left (1-\sqrt {5}\right ) x+\sqrt {5}+1\right )}{5 \sqrt {5} \left (5-\sqrt {5}\right ) \left (2 x^2-\left (1+\sqrt {5}\right ) x+2\right )^2}+\frac {3 \log \left (2 x^2-\left (1-\sqrt {5}\right ) x+2\right )}{10 \sqrt {5}}-\frac {3 \log \left (2 x^2-\left (1+\sqrt {5}\right ) x+2\right )}{10 \sqrt {5}}\) |
Input:
Int[(1 - x + x^2 - x^3 + x^4)^(-3),x]
Output:
(-2*(1 - Sqrt[5] + (1 + Sqrt[5])*x))/(5*Sqrt[5]*(5 + Sqrt[5])*(2 - (1 - Sq rt[5])*x + 2*x^2)^2) + (3*(1 - Sqrt[5] - 4*x))/(25*(5 + Sqrt[5])*(2 - (1 - Sqrt[5])*x + 2*x^2)) - (2*(7 + Sqrt[5] - 3*(1 - 3*Sqrt[5])*x))/(25*(5 + S qrt[5])*(2 - (1 - Sqrt[5])*x + 2*x^2)) + (2*(1 + Sqrt[5] + (1 - Sqrt[5])*x ))/(5*Sqrt[5]*(5 - Sqrt[5])*(2 - (1 + Sqrt[5])*x + 2*x^2)^2) + (3*(1 + Sqr t[5] - 4*x))/(25*(5 - Sqrt[5])*(2 - (1 + Sqrt[5])*x + 2*x^2)) - (2*(7 - Sq rt[5] - 3*(1 + 3*Sqrt[5])*x))/(25*(5 - Sqrt[5])*(2 - (1 + Sqrt[5])*x + 2*x ^2)) + (3*Sqrt[365 - 158*Sqrt[5]]*ArcTan[(1 - Sqrt[5] - 4*x)/Sqrt[2*(5 + S qrt[5])]])/125 + (3*Sqrt[2*(145 - 61*Sqrt[5])]*ArcTan[(1 - Sqrt[5] - 4*x)/ Sqrt[2*(5 + Sqrt[5])]])/125 + (6*Sqrt[5 - 2*Sqrt[5]]*ArcTan[(1 - Sqrt[5] - 4*x)/Sqrt[2*(5 + Sqrt[5])]])/125 + (6*Sqrt[5 + 2*Sqrt[5]]*ArcTan[(1 + Sqr t[5] - 4*x)/Sqrt[2*(5 - Sqrt[5])]])/125 - (3*Sqrt[2*(145 + 61*Sqrt[5])]*Ar cTan[(1 + Sqrt[5] - 4*x)/Sqrt[2*(5 - Sqrt[5])]])/125 - (3*Sqrt[365 + 158*S qrt[5]]*ArcTan[(1 + Sqrt[5] - 4*x)/Sqrt[2*(5 - Sqrt[5])]])/125 + (3*Log[2 - (1 - Sqrt[5])*x + 2*x^2])/(10*Sqrt[5]) - (3*Log[2 - (1 + Sqrt[5])*x + 2* x^2])/(10*Sqrt[5])
Int[(Px_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2 + (d_.)*(x_)^3 + (e_.)*(x_)^4) ^(p_), x_Symbol] :> Simp[e^p Int[ExpandIntegrand[Px*(b/d + ((d + Sqrt[e*( (b^2 - 4*a*c)/a) + 8*a*d*(e/b)])/(2*e))*x + x^2)^p*(b/d + ((d - Sqrt[e*((b^ 2 - 4*a*c)/a) + 8*a*d*(e/b)])/(2*e))*x + x^2)^p, x], x], x] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Px, x] && ILtQ[p, 0] && EqQ[a*d^2 - b^2*e, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.22 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.24
| method | result | size |
| risch | \(\frac {\frac {3}{25} x^{7}+\frac {9}{50} x^{6}+\frac {11}{50} x^{2}+\frac {7}{25} x}{\left (x^{4}-x^{3}+x^{2}-x +1\right )^{2}}+\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-\textit {\_Z}^{3}+\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{\sum }\frac {\left (\textit {\_R}^{2}+6 \textit {\_R} +6\right ) \ln \left (x -\textit {\_R} \right )}{4 \textit {\_R}^{3}-3 \textit {\_R}^{2}+2 \textit {\_R} -1}\right )}{25}\) | \(94\) |
| default | \(\frac {\frac {4 \left (\frac {15}{8}-\frac {21 \sqrt {5}}{8}\right ) x^{3}}{125}+\frac {4 \left (\frac {29 \sqrt {5}}{16}-\frac {135}{16}\right ) x^{2}}{125}+\frac {4 \left (-\frac {23 \sqrt {5}}{8}+\frac {5}{8}\right ) x}{125}-\frac {3 \sqrt {5}}{125}}{\left (x^{2}-\frac {x}{2}+\frac {\sqrt {5}\, x}{2}+1\right )^{2}}+\frac {3 \ln \left (2-x +\sqrt {5}\, x +2 x^{2}\right ) \sqrt {5}}{50}+\frac {12 \left (-\frac {5 \sqrt {5}\, \left (\sqrt {5}-1\right )}{4}-9 \sqrt {5}+15\right ) \arctan \left (\frac {-1+\sqrt {5}+4 x}{\sqrt {10+2 \sqrt {5}}}\right )}{125 \sqrt {10+2 \sqrt {5}}}-\frac {4 \left (\left (-\frac {15}{8}-\frac {21 \sqrt {5}}{8}\right ) x^{3}+\left (\frac {29 \sqrt {5}}{16}+\frac {135}{16}\right ) x^{2}+\left (-\frac {23 \sqrt {5}}{8}-\frac {5}{8}\right ) x -\frac {3 \sqrt {5}}{4}\right )}{125 \left (x^{2}-\frac {x}{2}-\frac {\sqrt {5}\, x}{2}+1\right )^{2}}-\frac {3 \ln \left (2-x -\sqrt {5}\, x +2 x^{2}\right ) \sqrt {5}}{50}+\frac {12 \left (\frac {5 \sqrt {5}\, \left (-\sqrt {5}-1\right )}{4}+9 \sqrt {5}+15\right ) \arctan \left (\frac {-1-\sqrt {5}+4 x}{\sqrt {10-2 \sqrt {5}}}\right )}{125 \sqrt {10-2 \sqrt {5}}}\) | \(253\) |
Input:
int(1/(x^4-x^3+x^2-x+1)^3,x,method=_RETURNVERBOSE)
Output:
(3/25*x^7+9/50*x^6+11/50*x^2+7/25*x)/(x^4-x^3+x^2-x+1)^2+3/25*sum((_R^2+6* _R+6)/(4*_R^3-3*_R^2+2*_R-1)*ln(x-_R),_R=RootOf(_Z^4-_Z^3+_Z^2-_Z+1))
Time = 0.08 (sec) , antiderivative size = 330, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\left (1-x+x^2-x^3+x^4\right )^3} \, dx=\frac {30 \, x^{7} + 45 \, x^{6} + 6 \, {\left (x^{8} - 2 \, x^{7} + 3 \, x^{6} - 4 \, x^{5} + 5 \, x^{4} - 4 \, x^{3} + 3 \, x^{2} - 2 \, x + 1\right )} \sqrt {422 \, \sqrt {5} + 1025} \arctan \left (\frac {1}{895} \, {\left (\sqrt {5} {\left (31 \, x + 1\right )} - 35 \, x - 30\right )} \sqrt {422 \, \sqrt {5} + 1025}\right ) - 2 \, {\left (x^{8} - 2 \, x^{7} + 3 \, x^{6} - 4 \, x^{5} + 5 \, x^{4} - 4 \, x^{3} + 3 \, x^{2} - 2 \, x + 1\right )} \sqrt {-3798 \, \sqrt {5} + 9225} \arctan \left (\frac {1}{2685} \, {\left (\sqrt {5} {\left (31 \, x + 1\right )} + 35 \, x + 30\right )} \sqrt {-3798 \, \sqrt {5} + 9225}\right ) + 15 \, \sqrt {5} {\left (x^{8} - 2 \, x^{7} + 3 \, x^{6} - 4 \, x^{5} + 5 \, x^{4} - 4 \, x^{3} + 3 \, x^{2} - 2 \, x + 1\right )} \log \left (2 \, x^{2} + \sqrt {5} x - x + 2\right ) - 15 \, \sqrt {5} {\left (x^{8} - 2 \, x^{7} + 3 \, x^{6} - 4 \, x^{5} + 5 \, x^{4} - 4 \, x^{3} + 3 \, x^{2} - 2 \, x + 1\right )} \log \left (2 \, x^{2} - \sqrt {5} x - x + 2\right ) + 55 \, x^{2} + 70 \, x}{250 \, {\left (x^{8} - 2 \, x^{7} + 3 \, x^{6} - 4 \, x^{5} + 5 \, x^{4} - 4 \, x^{3} + 3 \, x^{2} - 2 \, x + 1\right )}} \] Input:
integrate(1/(x^4-x^3+x^2-x+1)^3,x, algorithm="fricas")
Output:
1/250*(30*x^7 + 45*x^6 + 6*(x^8 - 2*x^7 + 3*x^6 - 4*x^5 + 5*x^4 - 4*x^3 + 3*x^2 - 2*x + 1)*sqrt(422*sqrt(5) + 1025)*arctan(1/895*(sqrt(5)*(31*x + 1) - 35*x - 30)*sqrt(422*sqrt(5) + 1025)) - 2*(x^8 - 2*x^7 + 3*x^6 - 4*x^5 + 5*x^4 - 4*x^3 + 3*x^2 - 2*x + 1)*sqrt(-3798*sqrt(5) + 9225)*arctan(1/2685 *(sqrt(5)*(31*x + 1) + 35*x + 30)*sqrt(-3798*sqrt(5) + 9225)) + 15*sqrt(5) *(x^8 - 2*x^7 + 3*x^6 - 4*x^5 + 5*x^4 - 4*x^3 + 3*x^2 - 2*x + 1)*log(2*x^2 + sqrt(5)*x - x + 2) - 15*sqrt(5)*(x^8 - 2*x^7 + 3*x^6 - 4*x^5 + 5*x^4 - 4*x^3 + 3*x^2 - 2*x + 1)*log(2*x^2 - sqrt(5)*x - x + 2) + 55*x^2 + 70*x)/( x^8 - 2*x^7 + 3*x^6 - 4*x^5 + 5*x^4 - 4*x^3 + 3*x^2 - 2*x + 1)
Leaf count of result is larger than twice the leaf count of optimal. 1445 vs. \(2 (333) = 666\).
Time = 0.65 (sec) , antiderivative size = 1445, normalized size of antiderivative = 3.71 \[ \int \frac {1}{\left (1-x+x^2-x^3+x^4\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(1/(x**4-x**3+x**2-x+1)**3,x)
Output:
(6*x**7 + 9*x**6 + 11*x**2 + 14*x)/(50*x**8 - 100*x**7 + 150*x**6 - 200*x* *5 + 250*x**4 - 200*x**3 + 150*x**2 - 100*x + 50) - 3*sqrt(5)*log(x**2 + x *(-15796955*sqrt(5)/13115509 - 429675*sqrt(2)*sqrt(23823 - 5275*sqrt(5))/2 6231018 + 18333*sqrt(10)*sqrt(23823 - 5275*sqrt(5))/13115509 + 41064583/13 115509) - 540905909434020*sqrt(5)/172016576329081 - 23080955957875*sqrt(2) *sqrt(23823 - 5275*sqrt(5))/688066305316324 + 7519430041655*sqrt(10)*sqrt( 23823 - 5275*sqrt(5))/688066305316324 + 1624519191908706/172016576329081)/ 50 + 3*sqrt(5)*log(x**2 + x*(-429675*sqrt(2)*sqrt(5275*sqrt(5) + 23823)/26 231018 - 18333*sqrt(10)*sqrt(5275*sqrt(5) + 23823)/13115509 + 15796955*sqr t(5)/13115509 + 41064583/13115509) - 23080955957875*sqrt(2)*sqrt(5275*sqrt (5) + 23823)/688066305316324 - 7519430041655*sqrt(10)*sqrt(5275*sqrt(5) + 23823)/688066305316324 + 540905909434020*sqrt(5)/172016576329081 + 1624519 191908706/172016576329081)/50 - 2*sqrt(-9*sqrt(10)*sqrt(5275*sqrt(5) + 238 23)/31250 + 459/2500)*atan(52462036*x/(183330*sqrt(5)*sqrt(-2*sqrt(10)*sqr t(5275*sqrt(5) + 23823) + 1275) + 1701818*sqrt(-2*sqrt(10)*sqrt(5275*sqrt( 5) + 23823) + 1275) + 5729*sqrt(10)*sqrt(5275*sqrt(5) + 23823)*sqrt(-2*sqr t(10)*sqrt(5275*sqrt(5) + 23823) + 1275)) - 429675*sqrt(2)*sqrt(5275*sqrt( 5) + 23823)/(183330*sqrt(5)*sqrt(-2*sqrt(10)*sqrt(5275*sqrt(5) + 23823) + 1275) + 1701818*sqrt(-2*sqrt(10)*sqrt(5275*sqrt(5) + 23823) + 1275) + 5729 *sqrt(10)*sqrt(5275*sqrt(5) + 23823)*sqrt(-2*sqrt(10)*sqrt(5275*sqrt(5)...
\[ \int \frac {1}{\left (1-x+x^2-x^3+x^4\right )^3} \, dx=\int { \frac {1}{{\left (x^{4} - x^{3} + x^{2} - x + 1\right )}^{3}} \,d x } \] Input:
integrate(1/(x^4-x^3+x^2-x+1)^3,x, algorithm="maxima")
Output:
1/50*(6*x^7 + 9*x^6 + 11*x^2 + 14*x)/(x^8 - 2*x^7 + 3*x^6 - 4*x^5 + 5*x^4 - 4*x^3 + 3*x^2 - 2*x + 1) + 3/25*integrate((x^2 + 6*x + 6)/(x^4 - x^3 + x ^2 - x + 1), x)
Time = 0.13 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.36 \[ \int \frac {1}{\left (1-x+x^2-x^3+x^4\right )^3} \, dx=-\frac {3}{125} \, \sqrt {-422 \, \sqrt {5} + 1025} \arctan \left (\frac {4 \, x + \sqrt {5} - 1}{\sqrt {2 \, \sqrt {5} + 10}}\right ) + \frac {3}{125} \, \sqrt {422 \, \sqrt {5} + 1025} \arctan \left (\frac {4 \, x - \sqrt {5} - 1}{\sqrt {-2 \, \sqrt {5} + 10}}\right ) - \frac {3}{50} \, \sqrt {5} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {5} + 1\right )} + 1\right ) + \frac {3}{50} \, \sqrt {5} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {5} - 1\right )} + 1\right ) + \frac {6 \, x^{7} + 9 \, x^{6} + 11 \, x^{2} + 14 \, x}{50 \, {\left (x^{4} - x^{3} + x^{2} - x + 1\right )}^{2}} \] Input:
integrate(1/(x^4-x^3+x^2-x+1)^3,x, algorithm="giac")
Output:
-3/125*sqrt(-422*sqrt(5) + 1025)*arctan((4*x + sqrt(5) - 1)/sqrt(2*sqrt(5) + 10)) + 3/125*sqrt(422*sqrt(5) + 1025)*arctan((4*x - sqrt(5) - 1)/sqrt(- 2*sqrt(5) + 10)) - 3/50*sqrt(5)*log(x^2 - 1/2*x*(sqrt(5) + 1) + 1) + 3/50* sqrt(5)*log(x^2 + 1/2*x*(sqrt(5) - 1) + 1) + 1/50*(6*x^7 + 9*x^6 + 11*x^2 + 14*x)/(x^4 - x^3 + x^2 - x + 1)^2
Time = 21.36 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.39 \[ \int \frac {1}{\left (1-x+x^2-x^3+x^4\right )^3} \, dx=\left (\sum _{k=1}^4\ln \left (\frac {4887\,x}{15625}+\mathrm {root}\left (z^4+\frac {162\,z^2}{625}-\frac {5697\,z}{78125}+\frac {437481}{48828125},z,k\right )\,\left (-\frac {1026\,x}{625}+\mathrm {root}\left (z^4+\frac {162\,z^2}{625}-\frac {5697\,z}{78125}+\frac {437481}{48828125},z,k\right )\,\left (25\,\mathrm {root}\left (z^4+\frac {162\,z^2}{625}-\frac {5697\,z}{78125}+\frac {437481}{48828125},z,k\right )+\frac {111\,x}{5}-\frac {72}{5}\right )+\frac {1314}{625}\right )+\frac {5967}{15625}\right )\,\mathrm {root}\left (z^4+\frac {162\,z^2}{625}-\frac {5697\,z}{78125}+\frac {437481}{48828125},z,k\right )\right )+\frac {\frac {3\,x^7}{25}+\frac {9\,x^6}{50}+\frac {11\,x^2}{50}+\frac {7\,x}{25}}{x^8-2\,x^7+3\,x^6-4\,x^5+5\,x^4-4\,x^3+3\,x^2-2\,x+1} \] Input:
int(1/(x^2 - x - x^3 + x^4 + 1)^3,x)
Output:
symsum(log((4887*x)/15625 + root(z^4 + (162*z^2)/625 - (5697*z)/78125 + 43 7481/48828125, z, k)*(root(z^4 + (162*z^2)/625 - (5697*z)/78125 + 437481/4 8828125, z, k)*(25*root(z^4 + (162*z^2)/625 - (5697*z)/78125 + 437481/4882 8125, z, k) + (111*x)/5 - 72/5) - (1026*x)/625 + 1314/625) + 5967/15625)*r oot(z^4 + (162*z^2)/625 - (5697*z)/78125 + 437481/48828125, z, k), k, 1, 4 ) + ((7*x)/25 + (11*x^2)/50 + (9*x^6)/50 + (3*x^7)/25)/(3*x^2 - 2*x - 4*x^ 3 + 5*x^4 - 4*x^5 + 3*x^6 - 2*x^7 + x^8 + 1)
\[ \int \frac {1}{\left (1-x+x^2-x^3+x^4\right )^3} \, dx=\text {too large to display} \] Input:
int(1/(x^4-x^3+x^2-x+1)^3,x)
Output:
( - 24*int(x**5/(x**12 - 3*x**11 + 6*x**10 - 10*x**9 + 15*x**8 - 18*x**7 + 19*x**6 - 18*x**5 + 15*x**4 - 10*x**3 + 6*x**2 - 3*x + 1),x)*x**8 + 48*in t(x**5/(x**12 - 3*x**11 + 6*x**10 - 10*x**9 + 15*x**8 - 18*x**7 + 19*x**6 - 18*x**5 + 15*x**4 - 10*x**3 + 6*x**2 - 3*x + 1),x)*x**7 - 72*int(x**5/(x **12 - 3*x**11 + 6*x**10 - 10*x**9 + 15*x**8 - 18*x**7 + 19*x**6 - 18*x**5 + 15*x**4 - 10*x**3 + 6*x**2 - 3*x + 1),x)*x**6 + 96*int(x**5/(x**12 - 3* x**11 + 6*x**10 - 10*x**9 + 15*x**8 - 18*x**7 + 19*x**6 - 18*x**5 + 15*x** 4 - 10*x**3 + 6*x**2 - 3*x + 1),x)*x**5 - 120*int(x**5/(x**12 - 3*x**11 + 6*x**10 - 10*x**9 + 15*x**8 - 18*x**7 + 19*x**6 - 18*x**5 + 15*x**4 - 10*x **3 + 6*x**2 - 3*x + 1),x)*x**4 + 96*int(x**5/(x**12 - 3*x**11 + 6*x**10 - 10*x**9 + 15*x**8 - 18*x**7 + 19*x**6 - 18*x**5 + 15*x**4 - 10*x**3 + 6*x **2 - 3*x + 1),x)*x**3 - 72*int(x**5/(x**12 - 3*x**11 + 6*x**10 - 10*x**9 + 15*x**8 - 18*x**7 + 19*x**6 - 18*x**5 + 15*x**4 - 10*x**3 + 6*x**2 - 3*x + 1),x)*x**2 + 48*int(x**5/(x**12 - 3*x**11 + 6*x**10 - 10*x**9 + 15*x**8 - 18*x**7 + 19*x**6 - 18*x**5 + 15*x**4 - 10*x**3 + 6*x**2 - 3*x + 1),x)* x - 24*int(x**5/(x**12 - 3*x**11 + 6*x**10 - 10*x**9 + 15*x**8 - 18*x**7 + 19*x**6 - 18*x**5 + 15*x**4 - 10*x**3 + 6*x**2 - 3*x + 1),x) + 44*int(x** 4/(x**12 - 3*x**11 + 6*x**10 - 10*x**9 + 15*x**8 - 18*x**7 + 19*x**6 - 18* x**5 + 15*x**4 - 10*x**3 + 6*x**2 - 3*x + 1),x)*x**8 - 88*int(x**4/(x**12 - 3*x**11 + 6*x**10 - 10*x**9 + 15*x**8 - 18*x**7 + 19*x**6 - 18*x**5 +...