Integrand size = 9, antiderivative size = 163 \[ \int \frac {1}{1-x^{10}} \, dx=\frac {1}{20} \sqrt {10-2 \sqrt {5}} \arctan \left (\frac {\sqrt {10-2 \sqrt {5}} x}{2 \left (1-x^2\right )}\right )+\frac {1}{20} \sqrt {10+2 \sqrt {5}} \arctan \left (\frac {\sqrt {10+2 \sqrt {5}} x}{2 \left (1-x^2\right )}\right )+\frac {\text {arctanh}(x)}{5}+\frac {1}{20} \left (1-\sqrt {5}\right ) \text {arctanh}\left (\frac {\left (1-\sqrt {5}\right ) x}{2 \left (1+x^2\right )}\right )+\frac {1}{20} \left (1+\sqrt {5}\right ) \text {arctanh}\left (\frac {\left (1+\sqrt {5}\right ) x}{2 \left (1+x^2\right )}\right ) \] Output:
1/20*(10-2*5^(1/2))^(1/2)*arctan((10-2*5^(1/2))^(1/2)*x/(-2*x^2+2))+1/20*( 10+2*5^(1/2))^(1/2)*arctan((10+2*5^(1/2))^(1/2)*x/(-2*x^2+2))+1/5*arctanh( x)+1/20*(-5^(1/2)+1)*arctanh((-5^(1/2)+1)*x/(2*x^2+2))+1/20*(5^(1/2)+1)*ar ctanh((5^(1/2)+1)*x/(2*x^2+2))
Time = 0.14 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.77 \[ \int \frac {1}{1-x^{10}} \, dx=\frac {1}{40} \left (-2 \sqrt {10-2 \sqrt {5}} \arctan \left (\frac {1+\sqrt {5}-4 x}{\sqrt {10-2 \sqrt {5}}}\right )+2 \sqrt {2 \left (5+\sqrt {5}\right )} \arctan \left (\frac {1-\sqrt {5}+4 x}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )+2 \sqrt {2 \left (5+\sqrt {5}\right )} \arctan \left (\frac {-1+\sqrt {5}+4 x}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )+2 \sqrt {10-2 \sqrt {5}} \arctan \left (\frac {1+\sqrt {5}+4 x}{\sqrt {10-2 \sqrt {5}}}\right )-4 \log (1-x)+4 \log (1+x)-\left (-1+\sqrt {5}\right ) \log \left (1-\frac {1}{2} \left (-1+\sqrt {5}\right ) x+x^2\right )+\left (-1+\sqrt {5}\right ) \log \left (1+\frac {1}{2} \left (-1+\sqrt {5}\right ) x+x^2\right )-\left (1+\sqrt {5}\right ) \log \left (1-\frac {1}{2} \left (1+\sqrt {5}\right ) x+x^2\right )+\left (1+\sqrt {5}\right ) \log \left (\frac {1}{2} \left (2+x+\sqrt {5} x+2 x^2\right )\right )\right ) \] Input:
Integrate[(1 - x^10)^(-1),x]
Output:
(-2*Sqrt[10 - 2*Sqrt[5]]*ArcTan[(1 + Sqrt[5] - 4*x)/Sqrt[10 - 2*Sqrt[5]]] + 2*Sqrt[2*(5 + Sqrt[5])]*ArcTan[(1 - Sqrt[5] + 4*x)/Sqrt[2*(5 + Sqrt[5])] ] + 2*Sqrt[2*(5 + Sqrt[5])]*ArcTan[(-1 + Sqrt[5] + 4*x)/Sqrt[2*(5 + Sqrt[5 ])]] + 2*Sqrt[10 - 2*Sqrt[5]]*ArcTan[(1 + Sqrt[5] + 4*x)/Sqrt[10 - 2*Sqrt[ 5]]] - 4*Log[1 - x] + 4*Log[1 + x] - (-1 + Sqrt[5])*Log[1 - ((-1 + Sqrt[5] )*x)/2 + x^2] + (-1 + Sqrt[5])*Log[1 + ((-1 + Sqrt[5])*x)/2 + x^2] - (1 + Sqrt[5])*Log[1 - ((1 + Sqrt[5])*x)/2 + x^2] + (1 + Sqrt[5])*Log[(2 + x + S qrt[5]*x + 2*x^2)/2])/40
Time = 0.84 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.99, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.889, Rules used = {754, 27, 219, 1142, 25, 1083, 217, 1103}
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.
| \(\displaystyle \int \frac {1}{1-x^{10}} \, dx\) |
| \(\Big \downarrow \) 754 |
| \(\displaystyle \frac {1}{5} \int \frac {1}{1-x^2}dx+\frac {1}{5} \int \frac {4-\left (1-\sqrt {5}\right ) x}{2 \left (2 x^2-\left (1-\sqrt {5}\right ) x+2\right )}dx+\frac {1}{5} \int \frac {\left (1-\sqrt {5}\right ) x+4}{2 \left (2 x^2+\left (1-\sqrt {5}\right ) x+2\right )}dx+\frac {1}{5} \int \frac {4-\left (1+\sqrt {5}\right ) x}{2 \left (2 x^2-\left (1+\sqrt {5}\right ) x+2\right )}dx+\frac {1}{5} \int \frac {\left (1+\sqrt {5}\right ) x+4}{2 \left (2 x^2+\left (1+\sqrt {5}\right ) x+2\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \int \frac {1}{1-x^2}dx+\frac {1}{10} \int \frac {4-\left (1-\sqrt {5}\right ) x}{2 x^2-\left (1-\sqrt {5}\right ) x+2}dx+\frac {1}{10} \int \frac {\left (1-\sqrt {5}\right ) x+4}{2 x^2+\left (1-\sqrt {5}\right ) x+2}dx+\frac {1}{10} \int \frac {4-\left (1+\sqrt {5}\right ) x}{2 x^2-\left (1+\sqrt {5}\right ) x+2}dx+\frac {1}{10} \int \frac {\left (1+\sqrt {5}\right ) x+4}{2 x^2+\left (1+\sqrt {5}\right ) x+2}dx\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{10} \int \frac {4-\left (1-\sqrt {5}\right ) x}{2 x^2-\left (1-\sqrt {5}\right ) x+2}dx+\frac {1}{10} \int \frac {\left (1-\sqrt {5}\right ) x+4}{2 x^2+\left (1-\sqrt {5}\right ) x+2}dx+\frac {1}{10} \int \frac {4-\left (1+\sqrt {5}\right ) x}{2 x^2-\left (1+\sqrt {5}\right ) x+2}dx+\frac {1}{10} \int \frac {\left (1+\sqrt {5}\right ) x+4}{2 x^2+\left (1+\sqrt {5}\right ) x+2}dx+\frac {\text {arctanh}(x)}{5}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{2} \left (5+\sqrt {5}\right ) \int \frac {1}{2 x^2-\left (1-\sqrt {5}\right ) x+2}dx-\frac {1}{4} \left (1-\sqrt {5}\right ) \int -\frac {-4 x-\sqrt {5}+1}{2 x^2-\left (1-\sqrt {5}\right ) x+2}dx\right )+\frac {1}{10} \left (\frac {1}{2} \left (5+\sqrt {5}\right ) \int \frac {1}{2 x^2+\left (1-\sqrt {5}\right ) x+2}dx+\frac {1}{4} \left (1-\sqrt {5}\right ) \int \frac {4 x-\sqrt {5}+1}{2 x^2+\left (1-\sqrt {5}\right ) x+2}dx\right )+\frac {1}{10} \left (\frac {1}{2} \left (5-\sqrt {5}\right ) \int \frac {1}{2 x^2-\left (1+\sqrt {5}\right ) x+2}dx-\frac {1}{4} \left (1+\sqrt {5}\right ) \int -\frac {-4 x+\sqrt {5}+1}{2 x^2-\left (1+\sqrt {5}\right ) x+2}dx\right )+\frac {1}{10} \left (\frac {1}{2} \left (5-\sqrt {5}\right ) \int \frac {1}{2 x^2+\left (1+\sqrt {5}\right ) x+2}dx+\frac {1}{4} \left (1+\sqrt {5}\right ) \int \frac {4 x+\sqrt {5}+1}{2 x^2+\left (1+\sqrt {5}\right ) x+2}dx\right )+\frac {\text {arctanh}(x)}{5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{2} \left (5+\sqrt {5}\right ) \int \frac {1}{2 x^2-\left (1-\sqrt {5}\right ) x+2}dx+\frac {1}{4} \left (1-\sqrt {5}\right ) \int \frac {-4 x-\sqrt {5}+1}{2 x^2-\left (1-\sqrt {5}\right ) x+2}dx\right )+\frac {1}{10} \left (\frac {1}{2} \left (5+\sqrt {5}\right ) \int \frac {1}{2 x^2+\left (1-\sqrt {5}\right ) x+2}dx+\frac {1}{4} \left (1-\sqrt {5}\right ) \int \frac {4 x-\sqrt {5}+1}{2 x^2+\left (1-\sqrt {5}\right ) x+2}dx\right )+\frac {1}{10} \left (\frac {1}{2} \left (5-\sqrt {5}\right ) \int \frac {1}{2 x^2-\left (1+\sqrt {5}\right ) x+2}dx+\frac {1}{4} \left (1+\sqrt {5}\right ) \int \frac {-4 x+\sqrt {5}+1}{2 x^2-\left (1+\sqrt {5}\right ) x+2}dx\right )+\frac {1}{10} \left (\frac {1}{2} \left (5-\sqrt {5}\right ) \int \frac {1}{2 x^2+\left (1+\sqrt {5}\right ) x+2}dx+\frac {1}{4} \left (1+\sqrt {5}\right ) \int \frac {4 x+\sqrt {5}+1}{2 x^2+\left (1+\sqrt {5}\right ) x+2}dx\right )+\frac {\text {arctanh}(x)}{5}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{4} \left (1+\sqrt {5}\right ) \int \frac {-4 x+\sqrt {5}+1}{2 x^2-\left (1+\sqrt {5}\right ) x+2}dx-\left (5-\sqrt {5}\right ) \int \frac {1}{-\left (4 x-\sqrt {5}-1\right )^2-2 \left (5-\sqrt {5}\right )}d\left (4 x-\sqrt {5}-1\right )\right )+\frac {1}{10} \left (\frac {1}{4} \left (1-\sqrt {5}\right ) \int \frac {4 x-\sqrt {5}+1}{2 x^2+\left (1-\sqrt {5}\right ) x+2}dx-\left (5+\sqrt {5}\right ) \int \frac {1}{-\left (4 x-\sqrt {5}+1\right )^2-2 \left (5+\sqrt {5}\right )}d\left (4 x-\sqrt {5}+1\right )\right )+\frac {1}{10} \left (\frac {1}{4} \left (1-\sqrt {5}\right ) \int \frac {-4 x-\sqrt {5}+1}{2 x^2-\left (1-\sqrt {5}\right ) x+2}dx-\left (5+\sqrt {5}\right ) \int \frac {1}{-\left (4 x+\sqrt {5}-1\right )^2-2 \left (5+\sqrt {5}\right )}d\left (4 x+\sqrt {5}-1\right )\right )+\frac {1}{10} \left (\frac {1}{4} \left (1+\sqrt {5}\right ) \int \frac {4 x+\sqrt {5}+1}{2 x^2+\left (1+\sqrt {5}\right ) x+2}dx-\left (5-\sqrt {5}\right ) \int \frac {1}{-\left (4 x+\sqrt {5}+1\right )^2-2 \left (5-\sqrt {5}\right )}d\left (4 x+\sqrt {5}+1\right )\right )+\frac {\text {arctanh}(x)}{5}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{10} \left (\frac {1}{4} \left (1-\sqrt {5}\right ) \int \frac {-4 x-\sqrt {5}+1}{2 x^2-\left (1-\sqrt {5}\right ) x+2}dx+\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \arctan \left (\frac {4 x+\sqrt {5}-1}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )\right )+\frac {1}{10} \left (\frac {1}{4} \left (1-\sqrt {5}\right ) \int \frac {4 x-\sqrt {5}+1}{2 x^2+\left (1-\sqrt {5}\right ) x+2}dx+\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \arctan \left (\frac {4 x-\sqrt {5}+1}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )\right )+\frac {1}{10} \left (\frac {1}{4} \left (1+\sqrt {5}\right ) \int \frac {-4 x+\sqrt {5}+1}{2 x^2-\left (1+\sqrt {5}\right ) x+2}dx+\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \arctan \left (\frac {4 x-\sqrt {5}-1}{\sqrt {2 \left (5-\sqrt {5}\right )}}\right )\right )+\frac {1}{10} \left (\frac {1}{4} \left (1+\sqrt {5}\right ) \int \frac {4 x+\sqrt {5}+1}{2 x^2+\left (1+\sqrt {5}\right ) x+2}dx+\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \arctan \left (\frac {4 x+\sqrt {5}+1}{\sqrt {2 \left (5-\sqrt {5}\right )}}\right )\right )+\frac {\text {arctanh}(x)}{5}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{10} \left (\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \arctan \left (\frac {4 x+\sqrt {5}-1}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )-\frac {1}{4} \left (1-\sqrt {5}\right ) \log \left (2 x^2-\left (1-\sqrt {5}\right ) x+2\right )\right )+\frac {1}{10} \left (\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \arctan \left (\frac {4 x-\sqrt {5}+1}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )+\frac {1}{4} \left (1-\sqrt {5}\right ) \log \left (2 x^2+\left (1-\sqrt {5}\right ) x+2\right )\right )+\frac {1}{10} \left (\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \arctan \left (\frac {4 x-\sqrt {5}-1}{\sqrt {2 \left (5-\sqrt {5}\right )}}\right )-\frac {1}{4} \left (1+\sqrt {5}\right ) \log \left (2 x^2-\left (1+\sqrt {5}\right ) x+2\right )\right )+\frac {1}{10} \left (\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \arctan \left (\frac {4 x+\sqrt {5}+1}{\sqrt {2 \left (5-\sqrt {5}\right )}}\right )+\frac {1}{4} \left (1+\sqrt {5}\right ) \log \left (2 x^2+\left (1+\sqrt {5}\right ) x+2\right )\right )+\frac {\text {arctanh}(x)}{5}\) |
Input:
Int[(1 - x^10)^(-1),x]
Output:
ArcTanh[x]/5 + (Sqrt[(5 + Sqrt[5])/2]*ArcTan[(-1 + Sqrt[5] + 4*x)/Sqrt[2*( 5 + Sqrt[5])]] - ((1 - Sqrt[5])*Log[2 - (1 - Sqrt[5])*x + 2*x^2])/4)/10 + (Sqrt[(5 + Sqrt[5])/2]*ArcTan[(1 - Sqrt[5] + 4*x)/Sqrt[2*(5 + Sqrt[5])]] + ((1 - Sqrt[5])*Log[2 + (1 - Sqrt[5])*x + 2*x^2])/4)/10 + (Sqrt[(5 - Sqrt[ 5])/2]*ArcTan[(-1 - Sqrt[5] + 4*x)/Sqrt[2*(5 - Sqrt[5])]] - ((1 + Sqrt[5]) *Log[2 - (1 + Sqrt[5])*x + 2*x^2])/4)/10 + (Sqrt[(5 - Sqrt[5])/2]*ArcTan[( 1 + Sqrt[5] + 4*x)/Sqrt[2*(5 - Sqrt[5])]] + ((1 + Sqrt[5])*Log[2 + (1 + Sq rt[5])*x + 2*x^2])/4)/10
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[-a /b, n]], s = Denominator[Rt[-a/b, n]], k, u}, Simp[u = Int[(r - s*Cos[(2*k* Pi)/n]*x)/(r^2 - 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x] + Int[(r + s*Cos[(2 *k*Pi)/n]*x)/(r^2 + 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x]; 2*(r^2/(a*n)) Int[1/(r^2 - s^2*x^2), x] + 2*(r/(a*n)) Sum[u, {k, 1, (n - 2)/4}], x]] / ; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && NegQ[a/b]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.41 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.40
| method | result | size |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+\textit {\_Z}^{3}+\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{\sum }\textit {\_R} \ln \left (\textit {\_R} +x \right )\right )}{10}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-\textit {\_Z}^{3}+\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{\sum }\textit {\_R} \ln \left (\textit {\_R} +x \right )\right )}{10}+\frac {\ln \left (1+x \right )}{10}-\frac {\ln \left (-1+x \right )}{10}\) | \(66\) |
| meijerg | \(-\frac {x \left (\ln \left (1-\left (x^{10}\right )^{\frac {1}{10}}\right )-\ln \left (1+\left (x^{10}\right )^{\frac {1}{10}}\right )+\cos \left (\frac {\pi }{5}\right ) \ln \left (1-2 \cos \left (\frac {\pi }{5}\right ) \left (x^{10}\right )^{\frac {1}{10}}+\left (x^{10}\right )^{\frac {1}{5}}\right )-2 \sin \left (\frac {\pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{5}\right ) \left (x^{10}\right )^{\frac {1}{10}}}{1-\cos \left (\frac {\pi }{5}\right ) \left (x^{10}\right )^{\frac {1}{10}}}\right )+\cos \left (\frac {2 \pi }{5}\right ) \ln \left (1-2 \cos \left (\frac {2 \pi }{5}\right ) \left (x^{10}\right )^{\frac {1}{10}}+\left (x^{10}\right )^{\frac {1}{5}}\right )-2 \sin \left (\frac {2 \pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {2 \pi }{5}\right ) \left (x^{10}\right )^{\frac {1}{10}}}{1-\cos \left (\frac {2 \pi }{5}\right ) \left (x^{10}\right )^{\frac {1}{10}}}\right )-\cos \left (\frac {2 \pi }{5}\right ) \ln \left (1+2 \cos \left (\frac {2 \pi }{5}\right ) \left (x^{10}\right )^{\frac {1}{10}}+\left (x^{10}\right )^{\frac {1}{5}}\right )-2 \sin \left (\frac {2 \pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {2 \pi }{5}\right ) \left (x^{10}\right )^{\frac {1}{10}}}{1+\cos \left (\frac {2 \pi }{5}\right ) \left (x^{10}\right )^{\frac {1}{10}}}\right )-\cos \left (\frac {\pi }{5}\right ) \ln \left (1+2 \cos \left (\frac {\pi }{5}\right ) \left (x^{10}\right )^{\frac {1}{10}}+\left (x^{10}\right )^{\frac {1}{5}}\right )-2 \sin \left (\frac {\pi }{5}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{5}\right ) \left (x^{10}\right )^{\frac {1}{10}}}{1+\cos \left (\frac {\pi }{5}\right ) \left (x^{10}\right )^{\frac {1}{10}}}\right )\right )}{10 \left (x^{10}\right )^{\frac {1}{10}}}\) | \(254\) |
| default | \(-\frac {\ln \left (-1+x \right )}{10}-\frac {\left (-\sqrt {5}-1\right ) \ln \left (x \sqrt {5}+2 x^{2}+x +2\right )}{40}-\frac {\left (-4-\frac {\left (\sqrt {5}+1\right ) \left (-\sqrt {5}-1\right )}{4}\right ) \arctan \left (\frac {\sqrt {5}+4 x +1}{\sqrt {10-2 \sqrt {5}}}\right )}{5 \sqrt {10-2 \sqrt {5}}}+\frac {\left (-\sqrt {5}+1\right ) \ln \left (-x \sqrt {5}+2 x^{2}+x +2\right )}{40}+\frac {\left (4-\frac {\left (-\sqrt {5}+1\right )^{2}}{4}\right ) \arctan \left (\frac {-\sqrt {5}+4 x +1}{\sqrt {10+2 \sqrt {5}}}\right )}{5 \sqrt {10+2 \sqrt {5}}}-\frac {\left (\sqrt {5}+1\right ) \ln \left (-x \sqrt {5}+2 x^{2}-x +2\right )}{40}-\frac {\left (-4-\frac {\left (\sqrt {5}+1\right ) \left (-\sqrt {5}-1\right )}{4}\right ) \arctan \left (\frac {-\sqrt {5}+4 x -1}{\sqrt {10-2 \sqrt {5}}}\right )}{5 \sqrt {10-2 \sqrt {5}}}+\frac {\left (\sqrt {5}-1\right ) \ln \left (x \sqrt {5}+2 x^{2}-x +2\right )}{40}+\frac {\left (4-\frac {\left (\sqrt {5}-1\right )^{2}}{4}\right ) \arctan \left (\frac {\sqrt {5}+4 x -1}{\sqrt {10+2 \sqrt {5}}}\right )}{5 \sqrt {10+2 \sqrt {5}}}+\frac {\ln \left (1+x \right )}{10}\) | \(288\) |
Input:
int(1/(-x^10+1),x,method=_RETURNVERBOSE)
Output:
1/10*sum(_R*ln(_R+x),_R=RootOf(_Z^4+_Z^3+_Z^2+_Z+1))+1/10*sum(_R*ln(_R+x), _R=RootOf(_Z^4-_Z^3+_Z^2-_Z+1))+1/10*ln(1+x)-1/10*ln(-1+x)
Leaf count of result is larger than twice the leaf count of optimal. 251 vs. \(2 (113) = 226\).
Time = 0.08 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.54 \[ \int \frac {1}{1-x^{10}} \, dx=\frac {1}{40} \, {\left (\sqrt {5} + 1\right )} \log \left (2 \, x^{2} + \sqrt {5} x + x + 2\right ) + \frac {1}{40} \, {\left (\sqrt {5} - 1\right )} \log \left (2 \, x^{2} + \sqrt {5} x - x + 2\right ) - \frac {1}{40} \, {\left (\sqrt {5} - 1\right )} \log \left (2 \, x^{2} - \sqrt {5} x + x + 2\right ) - \frac {1}{40} \, {\left (\sqrt {5} + 1\right )} \log \left (2 \, x^{2} - \sqrt {5} x - x + 2\right ) - \frac {1}{10} \, \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} \arctan \left (\frac {1}{10} \, {\left (\sqrt {5} {\left (2 \, x + 3\right )} - 10 \, x - 5\right )} \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {5}{2}}\right ) - \frac {1}{10} \, \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} \arctan \left (\frac {1}{10} \, {\left (\sqrt {5} {\left (2 \, x - 3\right )} - 10 \, x + 5\right )} \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {5}{2}}\right ) + \frac {1}{10} \, \sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} \arctan \left (\frac {1}{10} \, {\left (\sqrt {5} {\left (2 \, x + 3\right )} + 10 \, x + 5\right )} \sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {5}{2}}\right ) + \frac {1}{10} \, \sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {5}{2}} \arctan \left (\frac {1}{10} \, {\left (\sqrt {5} {\left (2 \, x - 3\right )} + 10 \, x - 5\right )} \sqrt {-\frac {1}{2} \, \sqrt {5} + \frac {5}{2}}\right ) + \frac {1}{10} \, \log \left (x + 1\right ) - \frac {1}{10} \, \log \left (x - 1\right ) \] Input:
integrate(1/(-x^10+1),x, algorithm="fricas")
Output:
1/40*(sqrt(5) + 1)*log(2*x^2 + sqrt(5)*x + x + 2) + 1/40*(sqrt(5) - 1)*log (2*x^2 + sqrt(5)*x - x + 2) - 1/40*(sqrt(5) - 1)*log(2*x^2 - sqrt(5)*x + x + 2) - 1/40*(sqrt(5) + 1)*log(2*x^2 - sqrt(5)*x - x + 2) - 1/10*sqrt(1/2* sqrt(5) + 5/2)*arctan(1/10*(sqrt(5)*(2*x + 3) - 10*x - 5)*sqrt(1/2*sqrt(5) + 5/2)) - 1/10*sqrt(1/2*sqrt(5) + 5/2)*arctan(1/10*(sqrt(5)*(2*x - 3) - 1 0*x + 5)*sqrt(1/2*sqrt(5) + 5/2)) + 1/10*sqrt(-1/2*sqrt(5) + 5/2)*arctan(1 /10*(sqrt(5)*(2*x + 3) + 10*x + 5)*sqrt(-1/2*sqrt(5) + 5/2)) + 1/10*sqrt(- 1/2*sqrt(5) + 5/2)*arctan(1/10*(sqrt(5)*(2*x - 3) + 10*x - 5)*sqrt(-1/2*sq rt(5) + 5/2)) + 1/10*log(x + 1) - 1/10*log(x - 1)
Time = 3.67 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.43 \[ \int \frac {1}{1-x^{10}} \, dx=- \frac {\log {\left (x - 1 \right )}}{10} + \frac {\log {\left (x + 1 \right )}}{10} - \operatorname {RootSum} {\left (10000 t^{4} - 1000 t^{3} + 100 t^{2} - 10 t + 1, \left ( t \mapsto t \log {\left (- 10 t + x \right )} \right )\right )} - \operatorname {RootSum} {\left (10000 t^{4} + 1000 t^{3} + 100 t^{2} + 10 t + 1, \left ( t \mapsto t \log {\left (- 10 t + x \right )} \right )\right )} \] Input:
integrate(1/(-x**10+1),x)
Output:
-log(x - 1)/10 + log(x + 1)/10 - RootSum(10000*_t**4 - 1000*_t**3 + 100*_t **2 - 10*_t + 1, Lambda(_t, _t*log(-10*_t + x))) - RootSum(10000*_t**4 + 1 000*_t**3 + 100*_t**2 + 10*_t + 1, Lambda(_t, _t*log(-10*_t + x)))
\[ \int \frac {1}{1-x^{10}} \, dx=\int { -\frac {1}{x^{10} - 1} \,d x } \] Input:
integrate(1/(-x^10+1),x, algorithm="maxima")
Output:
1/10*integrate((x^3 + 2*x^2 + 3*x + 4)/(x^4 + x^3 + x^2 + x + 1), x) - 1/1 0*integrate((x^3 - 2*x^2 + 3*x - 4)/(x^4 - x^3 + x^2 - x + 1), x) + 1/10*l og(x + 1) - 1/10*log(x - 1)
Time = 0.13 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.37 \[ \int \frac {1}{1-x^{10}} \, dx=\frac {1}{40} \, {\left (\sqrt {5} + 1\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {5} + 1\right )} + 1\right ) - \frac {1}{40} \, {\left (\sqrt {5} + 1\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {5} + 1\right )} + 1\right ) + \frac {1}{40} \, {\left (\sqrt {5} - 1\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {5} - 1\right )} + 1\right ) - \frac {1}{40} \, {\left (\sqrt {5} - 1\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {5} - 1\right )} + 1\right ) + \frac {1}{20} \, \sqrt {2 \, \sqrt {5} + 10} \arctan \left (\frac {4 \, x + \sqrt {5} - 1}{\sqrt {2 \, \sqrt {5} + 10}}\right ) + \frac {1}{20} \, \sqrt {2 \, \sqrt {5} + 10} \arctan \left (\frac {4 \, x - \sqrt {5} + 1}{\sqrt {2 \, \sqrt {5} + 10}}\right ) + \frac {1}{20} \, \sqrt {-2 \, \sqrt {5} + 10} \arctan \left (\frac {4 \, x + \sqrt {5} + 1}{\sqrt {-2 \, \sqrt {5} + 10}}\right ) + \frac {1}{20} \, \sqrt {-2 \, \sqrt {5} + 10} \arctan \left (\frac {4 \, x - \sqrt {5} - 1}{\sqrt {-2 \, \sqrt {5} + 10}}\right ) + \frac {1}{10} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{10} \, \log \left ({\left | x - 1 \right |}\right ) \] Input:
integrate(1/(-x^10+1),x, algorithm="giac")
Output:
1/40*(sqrt(5) + 1)*log(x^2 + 1/2*x*(sqrt(5) + 1) + 1) - 1/40*(sqrt(5) + 1) *log(x^2 - 1/2*x*(sqrt(5) + 1) + 1) + 1/40*(sqrt(5) - 1)*log(x^2 + 1/2*x*( sqrt(5) - 1) + 1) - 1/40*(sqrt(5) - 1)*log(x^2 - 1/2*x*(sqrt(5) - 1) + 1) + 1/20*sqrt(2*sqrt(5) + 10)*arctan((4*x + sqrt(5) - 1)/sqrt(2*sqrt(5) + 10 )) + 1/20*sqrt(2*sqrt(5) + 10)*arctan((4*x - sqrt(5) + 1)/sqrt(2*sqrt(5) + 10)) + 1/20*sqrt(-2*sqrt(5) + 10)*arctan((4*x + sqrt(5) + 1)/sqrt(-2*sqrt (5) + 10)) + 1/20*sqrt(-2*sqrt(5) + 10)*arctan((4*x - sqrt(5) - 1)/sqrt(-2 *sqrt(5) + 10)) + 1/10*log(abs(x + 1)) - 1/10*log(abs(x - 1))
Time = 0.33 (sec) , antiderivative size = 852, normalized size of antiderivative = 5.23 \[ \int \frac {1}{1-x^{10}} \, dx=\text {Too large to display} \] Input:
int(-1/(x^10 - 1),x)
Output:
atan((((2^(1/2)*(- 5^(1/2) - 5)^(1/2))/40 - 5^(1/2)/40 + 1/40)*(10*x - (5* 2^(1/2)*(- 5^(1/2) - 5)^(1/2))/2 + (5*5^(1/2))/2 - 5/2)*1i + ((2^(1/2)*(- 5^(1/2) - 5)^(1/2))/40 - 5^(1/2)/40 + 1/40)*(10*x + (5*2^(1/2)*(- 5^(1/2) - 5)^(1/2))/2 - (5*5^(1/2))/2 + 5/2)*1i)/(((2^(1/2)*(- 5^(1/2) - 5)^(1/2)) /40 - 5^(1/2)/40 + 1/40)*(10*x - (5*2^(1/2)*(- 5^(1/2) - 5)^(1/2))/2 + (5* 5^(1/2))/2 - 5/2) - ((2^(1/2)*(- 5^(1/2) - 5)^(1/2))/40 - 5^(1/2)/40 + 1/4 0)*(10*x + (5*2^(1/2)*(- 5^(1/2) - 5)^(1/2))/2 - (5*5^(1/2))/2 + 5/2)))*(( 2^(1/2)*(- 5^(1/2) - 5)^(1/2)*1i)/20 - (5^(1/2)*1i)/20 + 1i/20) - (atan(x* 1i)*1i)/5 + atan((((2^(1/2)*(- 5^(1/2) - 5)^(1/2))/40 + 5^(1/2)/40 - 1/40) *(10*x - (5*2^(1/2)*(- 5^(1/2) - 5)^(1/2))/2 - (5*5^(1/2))/2 + 5/2)*1i + ( (2^(1/2)*(- 5^(1/2) - 5)^(1/2))/40 + 5^(1/2)/40 - 1/40)*(10*x + (5*2^(1/2) *(- 5^(1/2) - 5)^(1/2))/2 + (5*5^(1/2))/2 - 5/2)*1i)/(((2^(1/2)*(- 5^(1/2) - 5)^(1/2))/40 + 5^(1/2)/40 - 1/40)*(10*x - (5*2^(1/2)*(- 5^(1/2) - 5)^(1 /2))/2 - (5*5^(1/2))/2 + 5/2) - ((2^(1/2)*(- 5^(1/2) - 5)^(1/2))/40 + 5^(1 /2)/40 - 1/40)*(10*x + (5*2^(1/2)*(- 5^(1/2) - 5)^(1/2))/2 + (5*5^(1/2))/2 - 5/2)))*((2^(1/2)*(- 5^(1/2) - 5)^(1/2)*1i)/20 + (5^(1/2)*1i)/20 - 1i/20 ) + atan(((5^(1/2)/40 + (2^(1/2)*(5^(1/2) - 5)^(1/2))/40 + 1/40)*((5*5^(1/ 2))/2 - 10*x + (5*2^(1/2)*(5^(1/2) - 5)^(1/2))/2 + 5/2)*1i - (5^(1/2)/40 + (2^(1/2)*(5^(1/2) - 5)^(1/2))/40 + 1/40)*(10*x + (5*5^(1/2))/2 + (5*2^(1/ 2)*(5^(1/2) - 5)^(1/2))/2 + 5/2)*1i)/((5^(1/2)/40 + (2^(1/2)*(5^(1/2) -...
\[ \int \frac {1}{1-x^{10}} \, dx=-\left (\int \frac {1}{x^{10}-1}d x \right ) \] Input:
int(1/(-x^10+1),x)
Output:
- int(1/(x**10 - 1),x)