Integrand size = 22, antiderivative size = 119 \[ \int \frac {1+x^{10}}{\sqrt {1+x^4} \left (-1+x^{10}\right )} \, dx=-\frac {1}{5} \sqrt {2 \left (-1+\sqrt {5}\right )} \arctan \left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt {1+x^4}}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{5 \sqrt {2}}-\frac {1}{5} \sqrt {2 \left (1+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt {1+x^4}}\right ) \]
-1/5*(-2+2*5^(1/2))^(1/2)*arctan(1/2*(2+2*5^(1/2))^(1/2)*x/(x^4+1)^(1/2))- 1/10*arctanh(2^(1/2)*x/(x^4+1)^(1/2))*2^(1/2)-1/5*(2+2*5^(1/2))^(1/2)*arct anh(1/2*(-2+2*5^(1/2))^(1/2)*x/(x^4+1)^(1/2))
Time = 1.64 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.90 \[ \int \frac {1+x^{10}}{\sqrt {1+x^4} \left (-1+x^{10}\right )} \, dx=-\frac {2 \sqrt {-1+\sqrt {5}} \arctan \left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} x}{\sqrt {1+x^4}}\right )+\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )+2 \sqrt {1+\sqrt {5}} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} x}{\sqrt {1+x^4}}\right )}{5 \sqrt {2}} \]
-1/5*(2*Sqrt[-1 + Sqrt[5]]*ArcTan[(Sqrt[(1 + Sqrt[5])/2]*x)/Sqrt[1 + x^4]] + ArcTanh[(Sqrt[2]*x)/Sqrt[1 + x^4]] + 2*Sqrt[1 + Sqrt[5]]*ArcTanh[(Sqrt[ (-1 + Sqrt[5])/2]*x)/Sqrt[1 + x^4]])/Sqrt[2]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 2.90 (sec) , antiderivative size = 871, normalized size of antiderivative = 7.32, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {7276, 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.
| \(\displaystyle \int \frac {x^{10}+1}{\sqrt {x^4+1} \left (x^{10}-1\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {1}{\sqrt {x^4+1}}+\frac {2}{\left (x^{10}-1\right ) \sqrt {x^4+1}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(-1)^{4/5} \left (1+\sqrt [5]{-1}\right ) \left (1+(-1)^{4/5}\right ) \arctan \left (\frac {\sqrt {\sqrt [5]{-1}-(-1)^{4/5}} x}{\sqrt {x^4+1}}\right )}{5 \left (1+(-1)^{3/5}\right ) \sqrt {\sqrt [5]{-1}-(-1)^{4/5}}}-\frac {\arctan \left (\frac {\sqrt {\sqrt [5]{-1}-(-1)^{4/5}} x}{\sqrt {x^4+1}}\right )}{5 \sqrt {\sqrt [5]{-1}-(-1)^{4/5}}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{5 \sqrt {2}}-\frac {\left (1-\sqrt [5]{-1}+(-1)^{2/5}\right ) \left (1-(-1)^{3/5}\right ) \text {arctanh}\left (\frac {\sqrt {(-1)^{2/5}-(-1)^{3/5}} x}{\sqrt {x^4+1}}\right )}{5 \sqrt {(-1)^{2/5}-(-1)^{3/5}}}-\frac {\text {arctanh}\left (\frac {\sqrt {(-1)^{2/5}-(-1)^{3/5}} x}{\sqrt {x^4+1}}\right )}{5 \sqrt {(-1)^{2/5}-(-1)^{3/5}}}-\frac {\left (1-(-1)^{4/5}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{5 \left (1+(-1)^{3/5}\right ) \sqrt {x^4+1}}-\frac {\left (1-\sqrt [5]{-1}+(-1)^{2/5}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{5 \sqrt {x^4+1}}-\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{5 \left (1+(-1)^{2/5}\right ) \sqrt {x^4+1}}-\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{5 \left (1-\sqrt [5]{-1}\right ) \sqrt {x^4+1}}+\frac {2 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{5 \sqrt {x^4+1}}+\frac {\sqrt [5]{-1} \left (1+\sqrt [5]{-1}\right ) \left (1+(-1)^{2/5}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4} (-1)^{4/5} \left (1-\sqrt [5]{-1}\right )^2,2 \arctan (x),\frac {1}{2}\right )}{10 \sqrt {x^4+1}}+\frac {\left (1-(-1)^{2/5}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (-\frac {1}{4} (-1)^{3/5} \left (1+(-1)^{2/5}\right )^2,2 \arctan (x),\frac {1}{2}\right )}{10 \left (1+(-1)^{2/5}\right ) \sqrt {x^4+1}}+\frac {(-1)^{3/5} \left (1-\sqrt [5]{-1}\right ) \left (1+(-1)^{3/5}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4} (-1)^{2/5} \left (1-(-1)^{3/5}\right )^2,2 \arctan (x),\frac {1}{2}\right )}{10 \sqrt {x^4+1}}+\frac {\left (1-(-1)^{4/5}\right )^2 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (-\frac {1}{4} \sqrt [5]{-1} \left (1+(-1)^{4/5}\right )^2,2 \arctan (x),\frac {1}{2}\right )}{10 \left (1+(-1)^{3/5}\right ) \sqrt {x^4+1}}\) |
-1/5*ArcTan[(Sqrt[(-1)^(1/5) - (-1)^(4/5)]*x)/Sqrt[1 + x^4]]/Sqrt[(-1)^(1/ 5) - (-1)^(4/5)] + ((-1)^(4/5)*(1 + (-1)^(1/5))*(1 + (-1)^(4/5))*ArcTan[(S qrt[(-1)^(1/5) - (-1)^(4/5)]*x)/Sqrt[1 + x^4]])/(5*(1 + (-1)^(3/5))*Sqrt[( -1)^(1/5) - (-1)^(4/5)]) - ArcTanh[(Sqrt[2]*x)/Sqrt[1 + x^4]]/(5*Sqrt[2]) - ArcTanh[(Sqrt[(-1)^(2/5) - (-1)^(3/5)]*x)/Sqrt[1 + x^4]]/(5*Sqrt[(-1)^(2 /5) - (-1)^(3/5)]) - ((1 - (-1)^(1/5) + (-1)^(2/5))*(1 - (-1)^(3/5))*ArcTa nh[(Sqrt[(-1)^(2/5) - (-1)^(3/5)]*x)/Sqrt[1 + x^4]])/(5*Sqrt[(-1)^(2/5) - (-1)^(3/5)]) + (2*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan [x], 1/2])/(5*Sqrt[1 + x^4]) - ((1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*Elli pticF[2*ArcTan[x], 1/2])/(5*(1 - (-1)^(1/5))*Sqrt[1 + x^4]) - ((1 + x^2)*S qrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(5*(1 + (-1)^(2/5) )*Sqrt[1 + x^4]) - ((1 - (-1)^(1/5) + (-1)^(2/5))*(1 + x^2)*Sqrt[(1 + x^4) /(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(5*Sqrt[1 + x^4]) - ((1 - (-1)^ (4/5))*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/ (5*(1 + (-1)^(3/5))*Sqrt[1 + x^4]) + ((-1)^(1/5)*(1 + (-1)^(1/5))*(1 + (-1 )^(2/5))*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticPi[((-1)^(4/5)*(1 - (-1)^(1/5))^2)/4, 2*ArcTan[x], 1/2])/(10*Sqrt[1 + x^4]) + ((1 - (-1)^(2/5 ))*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticPi[-1/4*((-1)^(3/5)*(1 + (-1)^(2/5))^2), 2*ArcTan[x], 1/2])/(10*(1 + (-1)^(2/5))*Sqrt[1 + x^4]) + ( (-1)^(3/5)*(1 - (-1)^(1/5))*(1 + (-1)^(3/5))*(1 + x^2)*Sqrt[(1 + x^4)/(...
3.18.71.3.1 Defintions of rubi rules used
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 5.25 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.92
| method | result | size |
| elliptic | \(\frac {\left (-\frac {\ln \left (1+\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right )}{10}+\frac {\ln \left (-1+\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right )}{10}-\frac {4 \,\operatorname {arctanh}\left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{x \sqrt {\sqrt {5}-1}}\right )}{5 \sqrt {\sqrt {5}-1}}+\frac {4 \arctan \left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{x \sqrt {\sqrt {5}+1}}\right )}{5 \sqrt {\sqrt {5}+1}}\right ) \sqrt {2}}{2}\) | \(109\) |
| default | \(\frac {\left (2 \arctan \left (\frac {\sqrt {5}\, x^{2}-x^{2}+\sqrt {5}-4 x -1}{\sqrt {2+2 \sqrt {5}}\, \sqrt {x^{4}+1}}\right )-2 \arctan \left (\frac {\sqrt {5}\, x^{2}-x^{2}+\sqrt {5}+4 x -1}{\sqrt {2+2 \sqrt {5}}\, \sqrt {x^{4}+1}}\right )\right ) \sqrt {-2+2 \sqrt {5}}}{20}+\frac {\sqrt {2}\, \left (\operatorname {arctanh}\left (\frac {\left (x^{2}+x +1\right ) \sqrt {2}}{\sqrt {x^{4}+1}}\right )-\operatorname {arctanh}\left (\frac {\left (x^{2}-x +1\right ) \sqrt {2}}{\sqrt {x^{4}+1}}\right )\right )}{20}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {5}\, x^{2}+x^{2}+\sqrt {5}-4 x +1}{\sqrt {-2+2 \sqrt {5}}\, \sqrt {x^{4}+1}}\right ) \sqrt {2+2 \sqrt {5}}}{10}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {5}\, x^{2}+x^{2}+\sqrt {5}+4 x +1}{\sqrt {-2+2 \sqrt {5}}\, \sqrt {x^{4}+1}}\right ) \sqrt {2+2 \sqrt {5}}}{10}\) | \(234\) |
| pseudoelliptic | \(\frac {\left (2 \arctan \left (\frac {\sqrt {5}\, x^{2}-x^{2}+\sqrt {5}-4 x -1}{\sqrt {2+2 \sqrt {5}}\, \sqrt {x^{4}+1}}\right )-2 \arctan \left (\frac {\sqrt {5}\, x^{2}-x^{2}+\sqrt {5}+4 x -1}{\sqrt {2+2 \sqrt {5}}\, \sqrt {x^{4}+1}}\right )\right ) \sqrt {-2+2 \sqrt {5}}}{20}+\frac {\sqrt {2}\, \left (\operatorname {arctanh}\left (\frac {\left (x^{2}+x +1\right ) \sqrt {2}}{\sqrt {x^{4}+1}}\right )-\operatorname {arctanh}\left (\frac {\left (x^{2}-x +1\right ) \sqrt {2}}{\sqrt {x^{4}+1}}\right )\right )}{20}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {5}\, x^{2}+x^{2}+\sqrt {5}-4 x +1}{\sqrt {-2+2 \sqrt {5}}\, \sqrt {x^{4}+1}}\right ) \sqrt {2+2 \sqrt {5}}}{10}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {5}\, x^{2}+x^{2}+\sqrt {5}+4 x +1}{\sqrt {-2+2 \sqrt {5}}\, \sqrt {x^{4}+1}}\right ) \sqrt {2+2 \sqrt {5}}}{10}\) | \(234\) |
| trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +\sqrt {x^{4}+1}}{\left (1+x \right ) \left (-1+x \right )}\right )}{10}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right )^{2}-1\right ) \ln \left (\frac {25 \operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right )^{2}-1\right ) \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right )^{2} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right )^{2}-1\right ) x^{4}+2 \sqrt {x^{4}+1}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right )^{2}-1\right )}{\left (25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right )^{2} x +x^{2}-x +1\right ) \left (25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right )^{2} x -x^{2}-x -1\right )}\right )}{5}+\operatorname {RootOf}\left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right ) \ln \left (\frac {-125 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right )^{3} x^{2}-5 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right ) x^{4}+5 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right ) x^{2}+2 \sqrt {x^{4}+1}\, x -5 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right )}{\left (25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right )^{2} x +x^{2}+1\right ) \left (25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right )^{2} x -x^{2}-1\right )}\right )\) | \(381\) |
1/2*(-1/10*ln(1+1/2*2^(1/2)/x*(x^4+1)^(1/2))+1/10*ln(-1+1/2*2^(1/2)/x*(x^4 +1)^(1/2))-4/5/(5^(1/2)-1)^(1/2)*arctanh(2^(1/2)/x*(x^4+1)^(1/2)/(5^(1/2)- 1)^(1/2))+4/5/(5^(1/2)+1)^(1/2)*arctan(2^(1/2)/x*(x^4+1)^(1/2)/(5^(1/2)+1) ^(1/2)))*2^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 545 vs. \(2 (81) = 162\).
Time = 0.32 (sec) , antiderivative size = 545, normalized size of antiderivative = 4.58 \[ \int \frac {1+x^{10}}{\sqrt {1+x^4} \left (-1+x^{10}\right )} \, dx=\frac {1}{20} \, \sqrt {2} \log \left (\frac {x^{4} - 2 \, \sqrt {2} \sqrt {x^{4} + 1} x + 2 \, x^{2} + 1}{x^{4} - 2 \, x^{2} + 1}\right ) - \frac {1}{20} \, \sqrt {-2 \, \sqrt {5} + 2} \log \left (-\frac {4 \, {\left (3 \, x^{5} - x^{3} + \sqrt {5} {\left (x^{5} - x^{3} + x\right )} + 3 \, x\right )} \sqrt {x^{4} + 1} + {\left (3 \, x^{8} - 5 \, x^{6} + 9 \, x^{4} - 5 \, x^{2} + \sqrt {5} {\left (x^{8} - 3 \, x^{6} + 3 \, x^{4} - 3 \, x^{2} + 1\right )} + 3\right )} \sqrt {-2 \, \sqrt {5} + 2}}{x^{8} + x^{6} + x^{4} + x^{2} + 1}\right ) + \frac {1}{20} \, \sqrt {-2 \, \sqrt {5} + 2} \log \left (-\frac {4 \, {\left (3 \, x^{5} - x^{3} + \sqrt {5} {\left (x^{5} - x^{3} + x\right )} + 3 \, x\right )} \sqrt {x^{4} + 1} - {\left (3 \, x^{8} - 5 \, x^{6} + 9 \, x^{4} - 5 \, x^{2} + \sqrt {5} {\left (x^{8} - 3 \, x^{6} + 3 \, x^{4} - 3 \, x^{2} + 1\right )} + 3\right )} \sqrt {-2 \, \sqrt {5} + 2}}{x^{8} + x^{6} + x^{4} + x^{2} + 1}\right ) - \frac {1}{20} \, \sqrt {2 \, \sqrt {5} + 2} \log \left (-\frac {4 \, {\left (3 \, x^{5} - x^{3} - \sqrt {5} {\left (x^{5} - x^{3} + x\right )} + 3 \, x\right )} \sqrt {x^{4} + 1} + {\left (3 \, x^{8} - 5 \, x^{6} + 9 \, x^{4} - 5 \, x^{2} - \sqrt {5} {\left (x^{8} - 3 \, x^{6} + 3 \, x^{4} - 3 \, x^{2} + 1\right )} + 3\right )} \sqrt {2 \, \sqrt {5} + 2}}{x^{8} + x^{6} + x^{4} + x^{2} + 1}\right ) + \frac {1}{20} \, \sqrt {2 \, \sqrt {5} + 2} \log \left (-\frac {4 \, {\left (3 \, x^{5} - x^{3} - \sqrt {5} {\left (x^{5} - x^{3} + x\right )} + 3 \, x\right )} \sqrt {x^{4} + 1} - {\left (3 \, x^{8} - 5 \, x^{6} + 9 \, x^{4} - 5 \, x^{2} - \sqrt {5} {\left (x^{8} - 3 \, x^{6} + 3 \, x^{4} - 3 \, x^{2} + 1\right )} + 3\right )} \sqrt {2 \, \sqrt {5} + 2}}{x^{8} + x^{6} + x^{4} + x^{2} + 1}\right ) \]
1/20*sqrt(2)*log((x^4 - 2*sqrt(2)*sqrt(x^4 + 1)*x + 2*x^2 + 1)/(x^4 - 2*x^ 2 + 1)) - 1/20*sqrt(-2*sqrt(5) + 2)*log(-(4*(3*x^5 - x^3 + sqrt(5)*(x^5 - x^3 + x) + 3*x)*sqrt(x^4 + 1) + (3*x^8 - 5*x^6 + 9*x^4 - 5*x^2 + sqrt(5)*( x^8 - 3*x^6 + 3*x^4 - 3*x^2 + 1) + 3)*sqrt(-2*sqrt(5) + 2))/(x^8 + x^6 + x ^4 + x^2 + 1)) + 1/20*sqrt(-2*sqrt(5) + 2)*log(-(4*(3*x^5 - x^3 + sqrt(5)* (x^5 - x^3 + x) + 3*x)*sqrt(x^4 + 1) - (3*x^8 - 5*x^6 + 9*x^4 - 5*x^2 + sq rt(5)*(x^8 - 3*x^6 + 3*x^4 - 3*x^2 + 1) + 3)*sqrt(-2*sqrt(5) + 2))/(x^8 + x^6 + x^4 + x^2 + 1)) - 1/20*sqrt(2*sqrt(5) + 2)*log(-(4*(3*x^5 - x^3 - sq rt(5)*(x^5 - x^3 + x) + 3*x)*sqrt(x^4 + 1) + (3*x^8 - 5*x^6 + 9*x^4 - 5*x^ 2 - sqrt(5)*(x^8 - 3*x^6 + 3*x^4 - 3*x^2 + 1) + 3)*sqrt(2*sqrt(5) + 2))/(x ^8 + x^6 + x^4 + x^2 + 1)) + 1/20*sqrt(2*sqrt(5) + 2)*log(-(4*(3*x^5 - x^3 - sqrt(5)*(x^5 - x^3 + x) + 3*x)*sqrt(x^4 + 1) - (3*x^8 - 5*x^6 + 9*x^4 - 5*x^2 - sqrt(5)*(x^8 - 3*x^6 + 3*x^4 - 3*x^2 + 1) + 3)*sqrt(2*sqrt(5) + 2 ))/(x^8 + x^6 + x^4 + x^2 + 1))
\[ \int \frac {1+x^{10}}{\sqrt {1+x^4} \left (-1+x^{10}\right )} \, dx=\int \frac {\left (x^{2} + 1\right ) \left (x^{8} - x^{6} + x^{4} - x^{2} + 1\right )}{\left (x - 1\right ) \left (x + 1\right ) \sqrt {x^{4} + 1} \left (x^{4} - x^{3} + x^{2} - x + 1\right ) \left (x^{4} + x^{3} + x^{2} + x + 1\right )}\, dx \]
Integral((x**2 + 1)*(x**8 - x**6 + x**4 - x**2 + 1)/((x - 1)*(x + 1)*sqrt( x**4 + 1)*(x**4 - x**3 + x**2 - x + 1)*(x**4 + x**3 + x**2 + x + 1)), x)
\[ \int \frac {1+x^{10}}{\sqrt {1+x^4} \left (-1+x^{10}\right )} \, dx=\int { \frac {x^{10} + 1}{{\left (x^{10} - 1\right )} \sqrt {x^{4} + 1}} \,d x } \]
\[ \int \frac {1+x^{10}}{\sqrt {1+x^4} \left (-1+x^{10}\right )} \, dx=\int { \frac {x^{10} + 1}{{\left (x^{10} - 1\right )} \sqrt {x^{4} + 1}} \,d x } \]
Timed out. \[ \int \frac {1+x^{10}}{\sqrt {1+x^4} \left (-1+x^{10}\right )} \, dx=\int \frac {x^{10}+1}{\sqrt {x^4+1}\,\left (x^{10}-1\right )} \,d x \]