Integrand size = 14, antiderivative size = 238 \[ \int \frac {1}{\left (1+x+x^2+x^3+x^4\right )^2} \, dx=-\frac {2}{5 \left (2+\left (1+\sqrt {5}\right ) x+2 x^2\right )}+\frac {4 \left (5+\sqrt {5}+\left (5-\sqrt {5}\right ) x\right )}{5 \left (5+\sqrt {5}\right ) \left (2+\left (1-\sqrt {5}\right ) x+2 x^2\right ) \left (2+\left (1+\sqrt {5}\right ) x+2 x^2\right )}-\frac {2}{25} \sqrt {2 \left (25-11 \sqrt {5}\right )} \arctan \left (\frac {1-\sqrt {5}+4 x}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )+\frac {2}{25} \sqrt {2 \left (25+11 \sqrt {5}\right )} \arctan \left (\frac {1}{2} \sqrt {\frac {1}{10} \left (5+\sqrt {5}\right )} \left (1+\sqrt {5}+4 x\right )\right )-\frac {2 \log \left (2+x-\sqrt {5} x+2 x^2\right )}{5 \sqrt {5}}+\frac {2 \log \left (2+x+\sqrt {5} x+2 x^2\right )}{5 \sqrt {5}} \] Output:
-2/(10+5*(5^(1/2)+1)*x+10*x^2)+4/5*(5+5^(1/2)+(5-5^(1/2))*x)/(5+5^(1/2))/( 2+x*(-5^(1/2)+1)+2*x^2)/(2+(5^(1/2)+1)*x+2*x^2)-2/25*(50-22*5^(1/2))^(1/2) *arctan((1-5^(1/2)+4*x)/(10+2*5^(1/2))^(1/2))+2/25*(50+22*5^(1/2))^(1/2)*a rctan(1/20*(50+10*5^(1/2))^(1/2)*(1+5^(1/2)+4*x))-2/25*ln(2+x-x*5^(1/2)+2* x^2)*5^(1/2)+2/25*ln(2+x+x*5^(1/2)+2*x^2)*5^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.37 \[ \int \frac {1}{\left (1+x+x^2+x^3+x^4\right )^2} \, dx=\frac {1}{5} \left (-\frac {(-1+x) x}{1+x+x^2+x^3+x^4}-2 \text {RootSum}\left [1+\text {$\#$1}+\text {$\#$1}^2+\text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-2 \log (x-\text {$\#$1})+\log (x-\text {$\#$1}) \text {$\#$1}}{1+2 \text {$\#$1}+3 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ]\right ) \] Input:
Integrate[(1 + x + x^2 + x^3 + x^4)^(-2),x]
Output:
(-(((-1 + x)*x)/(1 + x + x^2 + x^3 + x^4)) - 2*RootSum[1 + #1 + #1^2 + #1^ 3 + #1^4 & , (-2*Log[x - #1] + Log[x - #1]*#1)/(1 + 2*#1 + 3*#1^2 + 4*#1^3 ) & ])/5
Time = 0.73 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.50, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, 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.
| \(\displaystyle \int \frac {1}{\left (x^4+x^3+x^2+x+1\right )^2} \, dx\) |
| \(\Big \downarrow \) 2492 |
| \(\displaystyle \int \left (\frac {2 \left (1-\sqrt {5}\right ) (x+1)}{5 \left (2 x^2+\left (1-\sqrt {5}\right ) x+2\right )^2}+\frac {2 \left (1+\sqrt {5}\right ) (x+1)}{5 \left (2 x^2+\left (1+\sqrt {5}\right ) x+2\right )^2}-\frac {2 \left (4 x-2 \sqrt {5}+3\right )}{5 \sqrt {5} \left (2 x^2+\left (1-\sqrt {5}\right ) x+2\right )}+\frac {2 \left (4 x+2 \sqrt {5}+3\right )}{5 \sqrt {5} \left (2 x^2+\left (1+\sqrt {5}\right ) x+2\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{25} \sqrt {2 \left (5-\sqrt {5}\right )} \arctan \left (\frac {4 x-\sqrt {5}+1}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )+\frac {1}{25} \sqrt {2 \left (65-29 \sqrt {5}\right )} \arctan \left (\frac {4 x-\sqrt {5}+1}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )+\frac {1}{25} \sqrt {2 \left (65+29 \sqrt {5}\right )} \arctan \left (\frac {4 x+\sqrt {5}+1}{\sqrt {2 \left (5-\sqrt {5}\right )}}\right )+\frac {1}{25} \sqrt {2 \left (5+\sqrt {5}\right )} \arctan \left (\frac {4 x+\sqrt {5}+1}{\sqrt {2 \left (5-\sqrt {5}\right )}}\right )-\frac {\left (1+\sqrt {5}\right ) \left (-\left (\left (3-\sqrt {5}\right ) x\right )-\sqrt {5}+3\right )}{5 \left (5-\sqrt {5}\right ) \left (2 x^2+\left (1+\sqrt {5}\right ) x+2\right )}-\frac {\left (1-\sqrt {5}\right ) \left (-\left (\left (3+\sqrt {5}\right ) x\right )+\sqrt {5}+3\right )}{5 \left (5+\sqrt {5}\right ) \left (2 x^2+\left (1-\sqrt {5}\right ) x+2\right )}-\frac {2 \log \left (2 x^2+\left (1-\sqrt {5}\right ) x+2\right )}{5 \sqrt {5}}+\frac {2 \log \left (2 x^2+\left (1+\sqrt {5}\right ) x+2\right )}{5 \sqrt {5}}\) |
Input:
Int[(1 + x + x^2 + x^3 + x^4)^(-2),x]
Output:
-1/5*((1 - Sqrt[5])*(3 + Sqrt[5] - (3 + Sqrt[5])*x))/((5 + Sqrt[5])*(2 + ( 1 - Sqrt[5])*x + 2*x^2)) - ((1 + Sqrt[5])*(3 - Sqrt[5] - (3 - Sqrt[5])*x)) /(5*(5 - Sqrt[5])*(2 + (1 + Sqrt[5])*x + 2*x^2)) + (Sqrt[2*(65 - 29*Sqrt[5 ])]*ArcTan[(1 - Sqrt[5] + 4*x)/Sqrt[2*(5 + Sqrt[5])]])/25 - (Sqrt[2*(5 - S qrt[5])]*ArcTan[(1 - Sqrt[5] + 4*x)/Sqrt[2*(5 + Sqrt[5])]])/25 + (Sqrt[2*( 5 + Sqrt[5])]*ArcTan[(1 + Sqrt[5] + 4*x)/Sqrt[2*(5 - Sqrt[5])]])/25 + (Sqr t[2*(65 + 29*Sqrt[5])]*ArcTan[(1 + Sqrt[5] + 4*x)/Sqrt[2*(5 - Sqrt[5])]])/ 25 - (2*Log[2 + (1 - Sqrt[5])*x + 2*x^2])/(5*Sqrt[5]) + (2*Log[2 + (1 + Sq rt[5])*x + 2*x^2])/(5*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 = 0.16 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.31
| method | result | size |
| risch | \(\frac {-\frac {1}{5} x^{2}+\frac {1}{5} x}{x^{4}+x^{3}+x^{2}+x +1}+\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+\textit {\_Z}^{3}+\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{\sum }\frac {\left (2-\textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{4 \textit {\_R}^{3}+3 \textit {\_R}^{2}+2 \textit {\_R} +1}\right )}{5}\) | \(73\) |
| default | \(-\frac {2 \left (\frac {\sqrt {5}\, x}{2}-\frac {\sqrt {5}}{2}\right )}{25 \left (x^{2}+\frac {x}{2}-\frac {\sqrt {5}\, x}{2}+1\right )}-\frac {2 \ln \left (2+x -\sqrt {5}\, x +2 x^{2}\right ) \sqrt {5}}{25}+\frac {8 \left (\frac {\sqrt {5}\, \left (-\sqrt {5}+1\right )}{2}+5-2 \sqrt {5}\right ) \arctan \left (\frac {1-\sqrt {5}+4 x}{\sqrt {10+2 \sqrt {5}}}\right )}{25 \sqrt {10+2 \sqrt {5}}}+\frac {\frac {\sqrt {5}\, x}{25}-\frac {\sqrt {5}}{25}}{x^{2}+\frac {x}{2}+\frac {\sqrt {5}\, x}{2}+1}+\frac {2 \ln \left (2+x +\sqrt {5}\, x +2 x^{2}\right ) \sqrt {5}}{25}+\frac {8 \left (-\frac {\sqrt {5}\, \left (\sqrt {5}+1\right )}{2}+5+2 \sqrt {5}\right ) \arctan \left (\frac {1+\sqrt {5}+4 x}{\sqrt {10-2 \sqrt {5}}}\right )}{25 \sqrt {10-2 \sqrt {5}}}\) | \(199\) |
Input:
int(1/(x^4+x^3+x^2+x+1)^2,x,method=_RETURNVERBOSE)
Output:
(-1/5*x^2+1/5*x)/(x^4+x^3+x^2+x+1)+2/5*sum((2-_R)/(4*_R^3+3*_R^2+2*_R+1)*l n(x-_R),_R=RootOf(_Z^4+_Z^3+_Z^2+_Z+1))
Time = 0.11 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\left (1+x+x^2+x^3+x^4\right )^2} \, dx=\frac {2 \, {\left (x^{4} + x^{3} + x^{2} + x + 1\right )} \sqrt {22 \, \sqrt {5} + 50} \arctan \left (\frac {1}{20} \, {\left (\sqrt {5} {\left (6 \, x - 1\right )} - 10 \, x + 5\right )} \sqrt {22 \, \sqrt {5} + 50}\right ) - 2 \, {\left (x^{4} + x^{3} + x^{2} + x + 1\right )} \sqrt {-22 \, \sqrt {5} + 50} \arctan \left (\frac {1}{20} \, {\left (\sqrt {5} {\left (6 \, x - 1\right )} + 10 \, x - 5\right )} \sqrt {-22 \, \sqrt {5} + 50}\right ) + 2 \, \sqrt {5} {\left (x^{4} + x^{3} + x^{2} + x + 1\right )} \log \left (2 \, x^{2} + \sqrt {5} x + x + 2\right ) - 2 \, \sqrt {5} {\left (x^{4} + x^{3} + x^{2} + x + 1\right )} \log \left (2 \, x^{2} - \sqrt {5} x + x + 2\right ) - 5 \, x^{2} + 5 \, x}{25 \, {\left (x^{4} + x^{3} + x^{2} + x + 1\right )}} \] Input:
integrate(1/(x^4+x^3+x^2+x+1)^2,x, algorithm="fricas")
Output:
1/25*(2*(x^4 + x^3 + x^2 + x + 1)*sqrt(22*sqrt(5) + 50)*arctan(1/20*(sqrt( 5)*(6*x - 1) - 10*x + 5)*sqrt(22*sqrt(5) + 50)) - 2*(x^4 + x^3 + x^2 + x + 1)*sqrt(-22*sqrt(5) + 50)*arctan(1/20*(sqrt(5)*(6*x - 1) + 10*x - 5)*sqrt (-22*sqrt(5) + 50)) + 2*sqrt(5)*(x^4 + x^3 + x^2 + x + 1)*log(2*x^2 + sqrt (5)*x + x + 2) - 2*sqrt(5)*(x^4 + x^3 + x^2 + x + 1)*log(2*x^2 - sqrt(5)*x + x + 2) - 5*x^2 + 5*x)/(x^4 + x^3 + x^2 + x + 1)
Leaf count of result is larger than twice the leaf count of optimal. 1360 vs. \(2 (207) = 414\).
Time = 0.65 (sec) , antiderivative size = 1360, normalized size of antiderivative = 5.71 \[ \int \frac {1}{\left (1+x+x^2+x^3+x^4\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(1/(x**4+x**3+x**2+x+1)**2,x)
Output:
(-x**2 + x)/(5*x**4 + 5*x**3 + 5*x**2 + 5*x + 5) + 2*sqrt(5)*log(x**2 + x* (-8229/802 - 85*sqrt(5)*sqrt(201 - 88*sqrt(5))/802 + 315*sqrt(201 - 88*sqr t(5))/401 + 1928*sqrt(5)/401) - 30944507*sqrt(5)/643204 - 3418085*sqrt(201 - 88*sqrt(5))/643204 + 1570197*sqrt(5)*sqrt(201 - 88*sqrt(5))/643204 + 69 646079/643204)/25 - 2*sqrt(5)*log(x**2 + x*(-1928*sqrt(5)/401 - 8229/802 + 85*sqrt(5)*sqrt(88*sqrt(5) + 201)/802 + 315*sqrt(88*sqrt(5) + 201)/401) - 1570197*sqrt(5)*sqrt(88*sqrt(5) + 201)/643204 - 3418085*sqrt(88*sqrt(5) + 201)/643204 + 30944507*sqrt(5)/643204 + 69646079/643204)/25 - 2*sqrt(-2*s qrt(5)*sqrt(88*sqrt(5) + 201)/625 + 18/125)*atan(802*sqrt(2)*x/(170*sqrt(5 )*sqrt(-sqrt(5)*sqrt(88*sqrt(5) + 201) + 45) + 382*sqrt(-sqrt(5)*sqrt(88*s qrt(5) + 201) + 45) + 21*sqrt(5)*sqrt(88*sqrt(5) + 201)*sqrt(-sqrt(5)*sqrt (88*sqrt(5) + 201) + 45)) - 3856*sqrt(10)/(340*sqrt(5)*sqrt(-sqrt(5)*sqrt( 88*sqrt(5) + 201) + 45) + 764*sqrt(-sqrt(5)*sqrt(88*sqrt(5) + 201) + 45) + 42*sqrt(5)*sqrt(88*sqrt(5) + 201)*sqrt(-sqrt(5)*sqrt(88*sqrt(5) + 201) + 45)) - 8229*sqrt(2)/(340*sqrt(5)*sqrt(-sqrt(5)*sqrt(88*sqrt(5) + 201) + 45 ) + 764*sqrt(-sqrt(5)*sqrt(88*sqrt(5) + 201) + 45) + 42*sqrt(5)*sqrt(88*sq rt(5) + 201)*sqrt(-sqrt(5)*sqrt(88*sqrt(5) + 201) + 45)) + 85*sqrt(10)*sqr t(88*sqrt(5) + 201)/(340*sqrt(5)*sqrt(-sqrt(5)*sqrt(88*sqrt(5) + 201) + 45 ) + 764*sqrt(-sqrt(5)*sqrt(88*sqrt(5) + 201) + 45) + 42*sqrt(5)*sqrt(88*sq rt(5) + 201)*sqrt(-sqrt(5)*sqrt(88*sqrt(5) + 201) + 45)) + 630*sqrt(2)*...
\[ \int \frac {1}{\left (1+x+x^2+x^3+x^4\right )^2} \, dx=\int { \frac {1}{{\left (x^{4} + x^{3} + x^{2} + x + 1\right )}^{2}} \,d x } \] Input:
integrate(1/(x^4+x^3+x^2+x+1)^2,x, algorithm="maxima")
Output:
-1/5*(x^2 - x)/(x^4 + x^3 + x^2 + x + 1) - 2/5*integrate((x - 2)/(x^4 + x^ 3 + x^2 + x + 1), x)
Time = 0.14 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.52 \[ \int \frac {1}{\left (1+x+x^2+x^3+x^4\right )^2} \, dx=-\frac {2}{25} \, \sqrt {-22 \, \sqrt {5} + 50} \arctan \left (\frac {4 \, x - \sqrt {5} + 1}{\sqrt {2 \, \sqrt {5} + 10}}\right ) + \frac {2}{25} \, \sqrt {22 \, \sqrt {5} + 50} \arctan \left (\frac {4 \, x + \sqrt {5} + 1}{\sqrt {-2 \, \sqrt {5} + 10}}\right ) + \frac {2}{25} \, \sqrt {5} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {5} + 1\right )} + 1\right ) - \frac {2}{25} \, \sqrt {5} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {5} - 1\right )} + 1\right ) - \frac {x^{2} - x}{5 \, {\left (x^{4} + x^{3} + x^{2} + x + 1\right )}} \] Input:
integrate(1/(x^4+x^3+x^2+x+1)^2,x, algorithm="giac")
Output:
-2/25*sqrt(-22*sqrt(5) + 50)*arctan((4*x - sqrt(5) + 1)/sqrt(2*sqrt(5) + 1 0)) + 2/25*sqrt(22*sqrt(5) + 50)*arctan((4*x + sqrt(5) + 1)/sqrt(-2*sqrt(5 ) + 10)) + 2/25*sqrt(5)*log(x^2 + 1/2*x*(sqrt(5) + 1) + 1) - 2/25*sqrt(5)* log(x^2 - 1/2*x*(sqrt(5) - 1) + 1) - 1/5*(x^2 - x)/(x^4 + x^3 + x^2 + x + 1)
Time = 21.28 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.48 \[ \int \frac {1}{\left (1+x+x^2+x^3+x^4\right )^2} \, dx=\left (\sum _{k=1}^4\ln \left (-\frac {8\,x}{125}-\mathrm {root}\left (z^4+\frac {12\,z^2}{125}+\frac {176\,z}{3125}+\frac {496}{78125},z,k\right )\,\left (\frac {76\,x}{25}+\mathrm {root}\left (z^4+\frac {12\,z^2}{125}+\frac {176\,z}{3125}+\frac {496}{78125},z,k\right )\,\left (25\,\mathrm {root}\left (z^4+\frac {12\,z^2}{125}+\frac {176\,z}{3125}+\frac {496}{78125},z,k\right )+18\,x+14\right )+\frac {44}{25}\right )+\frac {16}{125}\right )\,\mathrm {root}\left (z^4+\frac {12\,z^2}{125}+\frac {176\,z}{3125}+\frac {496}{78125},z,k\right )\right )+\frac {\frac {x}{5}-\frac {x^2}{5}}{x^4+x^3+x^2+x+1} \] Input:
int(1/(x + x^2 + x^3 + x^4 + 1)^2,x)
Output:
symsum(log(16/125 - root(z^4 + (12*z^2)/125 + (176*z)/3125 + 496/78125, z, k)*((76*x)/25 + root(z^4 + (12*z^2)/125 + (176*z)/3125 + 496/78125, z, k) *(25*root(z^4 + (12*z^2)/125 + (176*z)/3125 + 496/78125, z, k) + 18*x + 14 ) + 44/25) - (8*x)/125)*root(z^4 + (12*z^2)/125 + (176*z)/3125 + 496/78125 , z, k), k, 1, 4) + (x/5 - x^2/5)/(x + x^2 + x^3 + x^4 + 1)
\[ \int \frac {1}{\left (1+x+x^2+x^3+x^4\right )^2} \, dx=\frac {-4 \left (\int \frac {x^{3}}{x^{8}+2 x^{7}+3 x^{6}+4 x^{5}+5 x^{4}+4 x^{3}+3 x^{2}+2 x +1}d x \right ) x^{4}-4 \left (\int \frac {x^{3}}{x^{8}+2 x^{7}+3 x^{6}+4 x^{5}+5 x^{4}+4 x^{3}+3 x^{2}+2 x +1}d x \right ) x^{3}-4 \left (\int \frac {x^{3}}{x^{8}+2 x^{7}+3 x^{6}+4 x^{5}+5 x^{4}+4 x^{3}+3 x^{2}+2 x +1}d x \right ) x^{2}-4 \left (\int \frac {x^{3}}{x^{8}+2 x^{7}+3 x^{6}+4 x^{5}+5 x^{4}+4 x^{3}+3 x^{2}+2 x +1}d x \right ) x -4 \left (\int \frac {x^{3}}{x^{8}+2 x^{7}+3 x^{6}+4 x^{5}+5 x^{4}+4 x^{3}+3 x^{2}+2 x +1}d x \right )-3 \left (\int \frac {x^{2}}{x^{8}+2 x^{7}+3 x^{6}+4 x^{5}+5 x^{4}+4 x^{3}+3 x^{2}+2 x +1}d x \right ) x^{4}-3 \left (\int \frac {x^{2}}{x^{8}+2 x^{7}+3 x^{6}+4 x^{5}+5 x^{4}+4 x^{3}+3 x^{2}+2 x +1}d x \right ) x^{3}-3 \left (\int \frac {x^{2}}{x^{8}+2 x^{7}+3 x^{6}+4 x^{5}+5 x^{4}+4 x^{3}+3 x^{2}+2 x +1}d x \right ) x^{2}-3 \left (\int \frac {x^{2}}{x^{8}+2 x^{7}+3 x^{6}+4 x^{5}+5 x^{4}+4 x^{3}+3 x^{2}+2 x +1}d x \right ) x -3 \left (\int \frac {x^{2}}{x^{8}+2 x^{7}+3 x^{6}+4 x^{5}+5 x^{4}+4 x^{3}+3 x^{2}+2 x +1}d x \right )-2 \left (\int \frac {x}{x^{8}+2 x^{7}+3 x^{6}+4 x^{5}+5 x^{4}+4 x^{3}+3 x^{2}+2 x +1}d x \right ) x^{4}-2 \left (\int \frac {x}{x^{8}+2 x^{7}+3 x^{6}+4 x^{5}+5 x^{4}+4 x^{3}+3 x^{2}+2 x +1}d x \right ) x^{3}-2 \left (\int \frac {x}{x^{8}+2 x^{7}+3 x^{6}+4 x^{5}+5 x^{4}+4 x^{3}+3 x^{2}+2 x +1}d x \right ) x^{2}-2 \left (\int \frac {x}{x^{8}+2 x^{7}+3 x^{6}+4 x^{5}+5 x^{4}+4 x^{3}+3 x^{2}+2 x +1}d x \right ) x -2 \left (\int \frac {x}{x^{8}+2 x^{7}+3 x^{6}+4 x^{5}+5 x^{4}+4 x^{3}+3 x^{2}+2 x +1}d x \right )-1}{x^{4}+x^{3}+x^{2}+x +1} \] Input:
int(1/(x^4+x^3+x^2+x+1)^2,x)
Output:
( - 4*int(x**3/(x**8 + 2*x**7 + 3*x**6 + 4*x**5 + 5*x**4 + 4*x**3 + 3*x**2 + 2*x + 1),x)*x**4 - 4*int(x**3/(x**8 + 2*x**7 + 3*x**6 + 4*x**5 + 5*x**4 + 4*x**3 + 3*x**2 + 2*x + 1),x)*x**3 - 4*int(x**3/(x**8 + 2*x**7 + 3*x**6 + 4*x**5 + 5*x**4 + 4*x**3 + 3*x**2 + 2*x + 1),x)*x**2 - 4*int(x**3/(x**8 + 2*x**7 + 3*x**6 + 4*x**5 + 5*x**4 + 4*x**3 + 3*x**2 + 2*x + 1),x)*x - 4 *int(x**3/(x**8 + 2*x**7 + 3*x**6 + 4*x**5 + 5*x**4 + 4*x**3 + 3*x**2 + 2* x + 1),x) - 3*int(x**2/(x**8 + 2*x**7 + 3*x**6 + 4*x**5 + 5*x**4 + 4*x**3 + 3*x**2 + 2*x + 1),x)*x**4 - 3*int(x**2/(x**8 + 2*x**7 + 3*x**6 + 4*x**5 + 5*x**4 + 4*x**3 + 3*x**2 + 2*x + 1),x)*x**3 - 3*int(x**2/(x**8 + 2*x**7 + 3*x**6 + 4*x**5 + 5*x**4 + 4*x**3 + 3*x**2 + 2*x + 1),x)*x**2 - 3*int(x* *2/(x**8 + 2*x**7 + 3*x**6 + 4*x**5 + 5*x**4 + 4*x**3 + 3*x**2 + 2*x + 1), x)*x - 3*int(x**2/(x**8 + 2*x**7 + 3*x**6 + 4*x**5 + 5*x**4 + 4*x**3 + 3*x **2 + 2*x + 1),x) - 2*int(x/(x**8 + 2*x**7 + 3*x**6 + 4*x**5 + 5*x**4 + 4* x**3 + 3*x**2 + 2*x + 1),x)*x**4 - 2*int(x/(x**8 + 2*x**7 + 3*x**6 + 4*x** 5 + 5*x**4 + 4*x**3 + 3*x**2 + 2*x + 1),x)*x**3 - 2*int(x/(x**8 + 2*x**7 + 3*x**6 + 4*x**5 + 5*x**4 + 4*x**3 + 3*x**2 + 2*x + 1),x)*x**2 - 2*int(x/( x**8 + 2*x**7 + 3*x**6 + 4*x**5 + 5*x**4 + 4*x**3 + 3*x**2 + 2*x + 1),x)*x - 2*int(x/(x**8 + 2*x**7 + 3*x**6 + 4*x**5 + 5*x**4 + 4*x**3 + 3*x**2 + 2 *x + 1),x) - 1)/(x**4 + x**3 + x**2 + x + 1)