Integrand size = 18, antiderivative size = 175 \[ \int \frac {1}{\left (1-x+x^2-x^3+x^4\right )^2} \, dx=\frac {x (1+x)}{5 \left (1-x+x^2-x^3+x^4\right )}+\frac {2}{25} \sqrt {2 \left (25-11 \sqrt {5}\right )} \arctan \left (\frac {1-\sqrt {5}-4 x}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )-\frac {2}{25} \sqrt {2 \left (25+11 \sqrt {5}\right )} \arctan \left (\frac {1+\sqrt {5}-4 x}{\sqrt {2 \left (5-\sqrt {5}\right )}}\right )+\frac {2 \log \left (2-\left (1-\sqrt {5}\right ) x+2 x^2\right )}{5 \sqrt {5}}-\frac {2 \log \left (2-\left (1+\sqrt {5}\right ) x+2 x^2\right )}{5 \sqrt {5}} \] Output:
x*(1+x)/(5*x^4-5*x^3+5*x^2-5*x+5)+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)*arctan((1+5^(1/ 2)-4*x)/(10-2*5^(1/2))^(1/2))+2/25*ln(2-x*(-5^(1/2)+1)+2*x^2)*5^(1/2)-2/25 *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 = 0.02 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.55 \[ \int \frac {1}{\left (1-x+x^2-x^3+x^4\right )^2} \, dx=\frac {x+x^2}{5 \left (1-x+x^2-x^3+x^4\right )}+\frac {2}{5} \text {RootSum}\left [1-\text {$\#$1}+\text {$\#$1}^2-\text {$\#$1}^3+\text {$\#$1}^4\&,\frac {2 \log (x-\text {$\#$1})+\log (x-\text {$\#$1}) \text {$\#$1}}{-1+2 \text {$\#$1}-3 \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ] \] Input:
Integrate[(1 - x + x^2 - x^3 + x^4)^(-2),x]
Output:
(x + x^2)/(5*(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
Leaf count is larger than twice the leaf count of optimal. \(359\) vs. \(2(175)=350\).
Time = 0.71 (sec) , antiderivative size = 359, normalized size of antiderivative = 2.05, 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.
| \(\displaystyle \int \frac {1}{\left (x^4-x^3+x^2-x+1\right )^2} \, dx\) |
| \(\Big \downarrow \) 2492 |
| \(\displaystyle \int \left (-\frac {2 \left (-4 x-2 \sqrt {5}+3\right )}{5 \sqrt {5} \left (2 x^2-\left (1-\sqrt {5}\right ) x+2\right )}+\frac {2 \left (-4 x+2 \sqrt {5}+3\right )}{5 \sqrt {5} \left (2 x^2-\left (1+\sqrt {5}\right ) x+2\right )}+\frac {2 \left (1-\sqrt {5}\right ) (1-x)}{5 \left (2 x^2-\left (1-\sqrt {5}\right ) x+2\right )^2}+\frac {2 \left (1+\sqrt {5}\right ) (1-x)}{5 \left (2 x^2-\left (1+\sqrt {5}\right ) x+2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{25} \sqrt {2 \left (5-\sqrt {5}\right )} \arctan \left (\frac {-4 x-\sqrt {5}+1}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )-\frac {1}{25} \sqrt {2 \left (65-29 \sqrt {5}\right )} \arctan \left (\frac {-4 x-\sqrt {5}+1}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )-\frac {1}{25} \sqrt {2 \left (65+29 \sqrt {5}\right )} \arctan \left (\frac {-4 x+\sqrt {5}+1}{\sqrt {2 \left (5-\sqrt {5}\right )}}\right )-\frac {1}{25} \sqrt {2 \left (5+\sqrt {5}\right )} \arctan \left (\frac {-4 x+\sqrt {5}+1}{\sqrt {2 \left (5-\sqrt {5}\right )}}\right )+\frac {\left (1+\sqrt {5}\right ) \left (\left (3-\sqrt {5}\right ) x-\sqrt {5}+3\right )}{5 \left (5-\sqrt {5}\right ) \left (2 x^2-\left (1+\sqrt {5}\right ) x+2\right )}+\frac {\left (1-\sqrt {5}\right ) \left (\left (3+\sqrt {5}\right ) x+\sqrt {5}+3\right )}{5 \left (5+\sqrt {5}\right ) \left (2 x^2-\left (1-\sqrt {5}\right ) x+2\right )}+\frac {2 \log \left (2 x^2-\left (1-\sqrt {5}\right ) x+2\right )}{5 \sqrt {5}}-\frac {2 \log \left (2 x^2-\left (1+\sqrt {5}\right ) x+2\right )}{5 \sqrt {5}}\) |
Input:
Int[(1 - x + x^2 - x^3 + x^4)^(-2),x]
Output:
((1 - Sqrt[5])*(3 + Sqrt[5] + (3 + Sqrt[5])*x))/(5*(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 - Sqrt [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 - (Sqrt[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 + Sqrt[ 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 = 0.15 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.45
| method | result | size |
| risch | \(\frac {\frac {1}{5} x^{2}+\frac {1}{5} x}{x^{4}-x^{3}+x^{2}-x +1}+\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-\textit {\_Z}^{3}+\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{\sum }\frac {\left (2+\textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{4 \textit {\_R}^{3}-3 \textit {\_R}^{2}+2 \textit {\_R} -1}\right )}{5}\) | \(79\) |
| default | \(\frac {-\frac {\sqrt {5}\, x}{25}-\frac {\sqrt {5}}{25}}{x^{2}-\frac {x}{2}+\frac {\sqrt {5}\, x}{2}+1}+\frac {2 \ln \left (2-x +\sqrt {5}\, x +2 x^{2}\right ) \sqrt {5}}{25}+\frac {8 \left (-\frac {\sqrt {5}\, \left (\sqrt {5}-1\right )}{2}+5-2 \sqrt {5}\right ) \arctan \left (\frac {-1+\sqrt {5}+4 x}{\sqrt {10+2 \sqrt {5}}}\right )}{25 \sqrt {10+2 \sqrt {5}}}-\frac {2 \left (-\frac {\sqrt {5}\, x}{2}-\frac {\sqrt {5}}{2}\right )}{25 \left (x^{2}-\frac {x}{2}-\frac {\sqrt {5}\, x}{2}+1\right )}-\frac {2 \ln \left (2-x -\sqrt {5}\, x +2 x^{2}\right ) \sqrt {5}}{25}+\frac {8 \left (\frac {\sqrt {5}\, \left (-\sqrt {5}-1\right )}{2}+5+2 \sqrt {5}\right ) \arctan \left (\frac {-1-\sqrt {5}+4 x}{\sqrt {10-2 \sqrt {5}}}\right )}{25 \sqrt {10-2 \sqrt {5}}}\) | \(203\) |
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)*ln (x-_R),_R=RootOf(_Z^4-_Z^3+_Z^2-_Z+1))
Time = 0.08 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\left (1-x+x^2-x^3+x^4\right )^2} \, dx=\frac {2 \, {\left (x^{4} - x^{3} + x^{2} - x + 1\right )} \sqrt {22 \, \sqrt {5} + 50} \arctan \left (\frac {1}{20} \, {\left (\sqrt {5} {\left (6 \, x + 1\right )} - 10 \, x - 5\right )} \sqrt {22 \, \sqrt {5} + 50}\right ) - 2 \, {\left (x^{4} - x^{3} + x^{2} - x + 1\right )} \sqrt {-22 \, \sqrt {5} + 50} \arctan \left (\frac {1}{20} \, {\left (\sqrt {5} {\left (6 \, x + 1\right )} + 10 \, x + 5\right )} \sqrt {-22 \, \sqrt {5} + 50}\right ) + 2 \, \sqrt {5} {\left (x^{4} - x^{3} + x^{2} - x + 1\right )} \log \left (2 \, x^{2} + \sqrt {5} x - x + 2\right ) - 2 \, \sqrt {5} {\left (x^{4} - x^{3} + x^{2} - x + 1\right )} \log \left (2 \, x^{2} - \sqrt {5} x - x + 2\right ) + 5 \, x^{2} + 5 \, x}{25 \, {\left (x^{4} - x^{3} + x^{2} - x + 1\right )}} \] 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. \(2 (153) = 306\).
Time = 0.62 (sec) , antiderivative size = 1360, normalized size of antiderivative = 7.77 \[ \int \frac {1}{\left (1-x+x^2-x^3+x^4\right )^2} \, dx=\text {Too large to display} \] 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*( -1928*sqrt(5)/401 - 315*sqrt(201 - 88*sqrt(5))/401 + 85*sqrt(5)*sqrt(201 - 88*sqrt(5))/802 + 8229/802) - 30944507*sqrt(5)/643204 - 3418085*sqrt(201 - 88*sqrt(5))/643204 + 1570197*sqrt(5)*sqrt(201 - 88*sqrt(5))/643204 + 696 46079/643204)/25 + 2*sqrt(5)*log(x**2 + x*(-315*sqrt(88*sqrt(5) + 201)/401 - 85*sqrt(5)*sqrt(88*sqrt(5) + 201)/802 + 8229/802 + 1928*sqrt(5)/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*sq rt(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*sq rt(5) + 201) + 45) + 21*sqrt(5)*sqrt(88*sqrt(5) + 201)*sqrt(-sqrt(5)*sqrt( 88*sqrt(5) + 201) + 45)) - 630*sqrt(2)*sqrt(88*sqrt(5) + 201)/(340*sqrt(5) *sqrt(-sqrt(5)*sqrt(88*sqrt(5) + 201) + 45) + 764*sqrt(-sqrt(5)*sqrt(88*sq rt(5) + 201) + 45) + 42*sqrt(5)*sqrt(88*sqrt(5) + 201)*sqrt(-sqrt(5)*sqrt( 88*sqrt(5) + 201) + 45)) - 85*sqrt(10)*sqrt(88*sqrt(5) + 201)/(340*sqrt(5) *sqrt(-sqrt(5)*sqrt(88*sqrt(5) + 201) + 45) + 764*sqrt(-sqrt(5)*sqrt(88*sq rt(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) + 4 2*sqrt(5)*sqrt(88*sqrt(5) + 201)*sqrt(-sqrt(5)*sqrt(88*sqrt(5) + 201) +...
\[ \int \frac {1}{\left (1-x+x^2-x^3+x^4\right )^2} \, dx=\int { \frac {1}{{\left (x^{4} - x^{3} + x^{2} - x + 1\right )}^{2}} \,d x } \] 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 = 0.12 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.72 \[ \int \frac {1}{\left (1-x+x^2-x^3+x^4\right )^2} \, dx=-\frac {2}{25} \, \sqrt {-22 \, \sqrt {5} + 50} \arctan \left (\frac {4 \, x + \sqrt {5} - 1}{\sqrt {2 \, \sqrt {5} + 10}}\right ) + \frac {2}{25} \, \sqrt {22 \, \sqrt {5} + 50} \arctan \left (\frac {4 \, x - \sqrt {5} - 1}{\sqrt {-2 \, \sqrt {5} + 10}}\right ) - \frac {2}{25} \, \sqrt {5} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {5} + 1\right )} + 1\right ) + \frac {2}{25} \, \sqrt {5} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {5} - 1\right )} + 1\right ) + \frac {x^{2} + x}{5 \, {\left (x^{4} - x^{3} + x^{2} - x + 1\right )}} \] 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 = 21.23 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\left (1-x+x^2-x^3+x^4\right )^2} \, dx=\left (\sum _{k=1}^4\ln \left (\frac {8\,x}{125}+\mathrm {root}\left (z^4+\frac {12\,z^2}{125}-\frac {176\,z}{3125}+\frac {496}{78125},z,k\right )\,\left (-\frac {76\,x}{25}+\mathrm {root}\left (z^4+\frac {12\,z^2}{125}-\frac {176\,z}{3125}+\frac {496}{78125},z,k\right )\,\left (25\,\mathrm {root}\left (z^4+\frac {12\,z^2}{125}-\frac {176\,z}{3125}+\frac {496}{78125},z,k\right )+18\,x-14\right )+\frac {44}{25}\right )+\frac {16}{125}\right )\,\mathrm {root}\left (z^4+\frac {12\,z^2}{125}-\frac {176\,z}{3125}+\frac {496}{78125},z,k\right )\right )+\frac {\frac {x^2}{5}+\frac {x}{5}}{x^4-x^3+x^2-x+1} \] Input:
int(1/(x^2 - x - x^3 + x^4 + 1)^2,x)
Output:
symsum(log((8*x)/125 + root(z^4 + (12*z^2)/125 - (176*z)/3125 + 496/78125, z, k)*(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) - (76*x )/25 + 44/25) + 16/125)*root(z^4 + (12*z^2)/125 - (176*z)/3125 + 496/78125 , z, k), k, 1, 4) + (x/5 + x^2/5)/(x^2 - x - 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*in t(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)