Integrand size = 22, antiderivative size = 119 \[ \int \frac {-1+x^{10}}{\sqrt {1+x^4} \left (1+x^{10}\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{5 \sqrt {2}}-\frac {1}{5} \sqrt {2 \left (1+\sqrt {5}\right )} \arctan \left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt {1+x^4}}\right )-\frac {1}{5} \sqrt {2 \left (-1+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt {1+x^4}}\right ) \]
-1/10*arctan(2^(1/2)*x/(x^4+1)^(1/2))*2^(1/2)-1/5*(2+2*5^(1/2))^(1/2)*arct an(1/2*(-2+2*5^(1/2))^(1/2)*x/(x^4+1)^(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 = 1.71 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.90 \[ \int \frac {-1+x^{10}}{\sqrt {1+x^4} \left (1+x^{10}\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )+2 \sqrt {1+\sqrt {5}} \arctan \left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} x}{\sqrt {1+x^4}}\right )+2 \sqrt {-1+\sqrt {5}} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} x}{\sqrt {1+x^4}}\right )}{5 \sqrt {2}} \]
-1/5*(ArcTan[(Sqrt[2]*x)/Sqrt[1 + x^4]] + 2*Sqrt[1 + Sqrt[5]]*ArcTan[(Sqrt [(-1 + Sqrt[5])/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 = 4.02 (sec) , antiderivative size = 1494, normalized size of antiderivative = 12.55, 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 |
| \(\displaystyle \int \left (\frac {1}{\sqrt {x^4+1}}-\frac {2}{\sqrt {x^4+1} \left (x^{10}+1\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{5 \sqrt {2}}+\frac {1}{5} (-1)^{4/5} \left (1+(-1)^{2/5}\right ) \sqrt {(-1)^{2/5}-(-1)^{3/5}} \arctan \left (\frac {\sqrt {(-1)^{2/5}-(-1)^{3/5}} x}{\sqrt {x^4+1}}\right )-\frac {\left (1-\sqrt [5]{-1}+(-1)^{2/5}\right ) \left (1-(-1)^{3/5}\right ) \arctan \left (\frac {\sqrt {(-1)^{2/5}-(-1)^{3/5}} x}{\sqrt {x^4+1}}\right )}{5 \sqrt {(-1)^{2/5}-(-1)^{3/5}}}-\frac {\sqrt [10]{-1} \arctan \left (\frac {x^2+\sqrt [5]{-1}}{\sqrt {-1-(-1)^{2/5}} \sqrt {x^4+1}}\right )}{10 \sqrt {-1-(-1)^{2/5}}}+\frac {\sqrt [10]{-1} \arctan \left (\frac {(-1)^{4/5} \left (x^2+(-1)^{3/5}\right )}{\sqrt {(-1)^{3/5}-(-1)^{4/5}} \sqrt {x^4+1}}\right )}{10 \sqrt {(-1)^{3/5}-(-1)^{4/5}}}-\frac {(-1)^{7/10} \arctan \left (\frac {(-1)^{3/5} \left (\sqrt [5]{-1} x^2+1\right )}{\sqrt {\sqrt [5]{-1}+(-1)^{3/5}} \sqrt {x^4+1}}\right )}{10 \sqrt {\sqrt [5]{-1}+(-1)^{3/5}}}-\frac {(-1)^{9/10} \arctan \left (\frac {(-1)^{3/5} \left ((-1)^{3/5} x^2+1\right )}{\sqrt {\sqrt [5]{-1}-(-1)^{2/5}} \sqrt {x^4+1}}\right )}{10 \sqrt {\sqrt [5]{-1}-(-1)^{2/5}}}+\frac {1}{10} (-1)^{3/5} \sqrt {\sqrt [5]{-1}-(-1)^{4/5}} \left (1+(-1)^{4/5}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt [5]{-1}-(-1)^{4/5}} x}{\sqrt {x^4+1}}\right )-\frac {\left (1-(-1)^{4/5}\right ) \left (1+(-1)^{4/5}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt [5]{-1}-(-1)^{4/5}} x}{\sqrt {x^4+1}}\right )}{10 \left (1+(-1)^{3/5}\right ) \sqrt {\sqrt [5]{-1}-(-1)^{4/5}}}-\frac {\text {arctanh}\left (\frac {\sqrt {\sqrt [5]{-1}-(-1)^{4/5}} x}{\sqrt {x^4+1}}\right )}{5 \sqrt {\sqrt [5]{-1}-(-1)^{4/5}}}-\frac {(-1)^{9/10} \text {arctanh}\left (\frac {(-1)^{4/5} \left (x^2+\sqrt [5]{-1}\right )}{\sqrt {1-(-1)^{3/5}} \sqrt {x^4+1}}\right )}{10 \sqrt {1-(-1)^{3/5}}}-\frac {i \text {arctanh}\left (\frac {\sqrt [5]{-1} \left (x^2+(-1)^{3/5}\right )}{\sqrt {(-1)^{2/5}-(-1)^{3/5}} \sqrt {x^4+1}}\right )}{10 \sqrt {(-1)^{2/5}-(-1)^{3/5}}}-\frac {\sqrt [10]{-1} \text {arctanh}\left (\frac {\sqrt [5]{-1} x^2+1}{\sqrt {1+(-1)^{2/5}} \sqrt {x^4+1}}\right )}{10 \sqrt {1+(-1)^{2/5}}}-\frac {i \text {arctanh}\left (\frac {\sqrt [5]{-1} \left ((-1)^{3/5} x^2+1\right )}{\sqrt {(-1)^{2/5}-(-1)^{3/5}} \sqrt {x^4+1}}\right )}{10 \sqrt {(-1)^{2/5}-(-1)^{3/5}}}+\frac {(-1)^{4/5} \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 )}{10 \left (1+(-1)^{3/5}\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 )}{10 \left (1+(-1)^{3/5}\right ) \sqrt {x^4+1}}-\frac {(-1)^{3/5} \left (1-(-1)^{3/5}\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+\sqrt [5]{-1}\right ) \sqrt {x^4+1}}-\frac {\left (1-(-1)^{3/5}\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 )}{10 \left (1+\sqrt [5]{-1}\right ) \sqrt {x^4+1}}-\frac {\sqrt [5]{-1} \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 {\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 )}{10 \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 {\left (1-\sqrt [5]{-1}\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 \left (1+\sqrt [5]{-1}\right ) \sqrt {x^4+1}}-\frac {\left (1-(-1)^{3/5}+2 (-1)^{4/5}\right ) \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 )}{20 \left (1+(-1)^{3/5}\right ) \sqrt {x^4+1}}+\frac {\left (1-\sqrt [5]{-1}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{4} \left (2+\sqrt [5]{-1}-(-1)^{4/5}\right ),2 \arctan (x),\frac {1}{2}\right )}{20 \left (1+\sqrt [5]{-1}\right ) \sqrt {x^4+1}}\) |
-1/5*ArcTan[(Sqrt[2]*x)/Sqrt[1 + x^4]]/Sqrt[2] - ((1 - (-1)^(1/5) + (-1)^( 2/5))*(1 - (-1)^(3/5))*ArcTan[(Sqrt[(-1)^(2/5) - (-1)^(3/5)]*x)/Sqrt[1 + x ^4]])/(5*Sqrt[(-1)^(2/5) - (-1)^(3/5)]) + ((-1)^(4/5)*(1 + (-1)^(2/5))*Sqr t[(-1)^(2/5) - (-1)^(3/5)]*ArcTan[(Sqrt[(-1)^(2/5) - (-1)^(3/5)]*x)/Sqrt[1 + x^4]])/5 - ((-1)^(1/10)*ArcTan[((-1)^(1/5) + x^2)/(Sqrt[-1 - (-1)^(2/5) ]*Sqrt[1 + x^4])])/(10*Sqrt[-1 - (-1)^(2/5)]) + ((-1)^(1/10)*ArcTan[((-1)^ (4/5)*((-1)^(3/5) + x^2))/(Sqrt[(-1)^(3/5) - (-1)^(4/5)]*Sqrt[1 + x^4])])/ (10*Sqrt[(-1)^(3/5) - (-1)^(4/5)]) - ((-1)^(7/10)*ArcTan[((-1)^(3/5)*(1 + (-1)^(1/5)*x^2))/(Sqrt[(-1)^(1/5) + (-1)^(3/5)]*Sqrt[1 + x^4])])/(10*Sqrt[ (-1)^(1/5) + (-1)^(3/5)]) - ((-1)^(9/10)*ArcTan[((-1)^(3/5)*(1 + (-1)^(3/5 )*x^2))/(Sqrt[(-1)^(1/5) - (-1)^(2/5)]*Sqrt[1 + x^4])])/(10*Sqrt[(-1)^(1/5 ) - (-1)^(2/5)]) - ArcTanh[(Sqrt[(-1)^(1/5) - (-1)^(4/5)]*x)/Sqrt[1 + x^4] ]/(5*Sqrt[(-1)^(1/5) - (-1)^(4/5)]) - ((1 - (-1)^(4/5))*(1 + (-1)^(4/5))*A rcTanh[(Sqrt[(-1)^(1/5) - (-1)^(4/5)]*x)/Sqrt[1 + x^4]])/(10*(1 + (-1)^(3/ 5))*Sqrt[(-1)^(1/5) - (-1)^(4/5)]) + ((-1)^(3/5)*Sqrt[(-1)^(1/5) - (-1)^(4 /5)]*(1 + (-1)^(4/5))*ArcTanh[(Sqrt[(-1)^(1/5) - (-1)^(4/5)]*x)/Sqrt[1 + x ^4]])/10 - ((-1)^(9/10)*ArcTanh[((-1)^(4/5)*((-1)^(1/5) + x^2))/(Sqrt[1 - (-1)^(3/5)]*Sqrt[1 + x^4])])/(10*Sqrt[1 - (-1)^(3/5)]) - ((I/10)*ArcTanh[( (-1)^(1/5)*((-1)^(3/5) + x^2))/(Sqrt[(-1)^(2/5) - (-1)^(3/5)]*Sqrt[1 + x^4 ])])/Sqrt[(-1)^(2/5) - (-1)^(3/5)] - ((-1)^(1/10)*ArcTanh[(1 + (-1)^(1/...
3.18.70.3.1 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 = 4.59 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.72
| method | result | size |
| default | \(-\frac {\arctan \left (\frac {\sqrt {2}\, x}{\sqrt {x^{4}+1}}\right ) \sqrt {2}}{10}+\frac {\arctan \left (\frac {2 \sqrt {x^{4}+1}}{\sqrt {-2+2 \sqrt {5}}\, x}\right ) \sqrt {2+2 \sqrt {5}}}{5}-\frac {\operatorname {arctanh}\left (\frac {2 \sqrt {x^{4}+1}}{\sqrt {2+2 \sqrt {5}}\, x}\right ) \sqrt {-2+2 \sqrt {5}}}{5}\) | \(86\) |
| pseudoelliptic | \(-\frac {\arctan \left (\frac {\sqrt {2}\, x}{\sqrt {x^{4}+1}}\right ) \sqrt {2}}{10}+\frac {\arctan \left (\frac {2 \sqrt {x^{4}+1}}{\sqrt {-2+2 \sqrt {5}}\, x}\right ) \sqrt {2+2 \sqrt {5}}}{5}-\frac {\operatorname {arctanh}\left (\frac {2 \sqrt {x^{4}+1}}{\sqrt {2+2 \sqrt {5}}\, x}\right ) \sqrt {-2+2 \sqrt {5}}}{5}\) | \(86\) |
| elliptic | \(\frac {\left (\frac {4 \arctan \left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{x \sqrt {\sqrt {5}-1}}\right )}{5 \sqrt {\sqrt {5}-1}}-\frac {4 \,\operatorname {arctanh}\left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{x \sqrt {\sqrt {5}+1}}\right )}{5 \sqrt {\sqrt {5}+1}}+\frac {\arctan \left (\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right )}{5}\right ) \sqrt {2}}{2}\) | \(87\) |
| trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x +\sqrt {x^{4}+1}}{x^{2}+1}\right )}{10}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}+25 \textit {\_Z}^{2}-1\right )^{2}+1\right ) \ln \left (\frac {25 \operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}+25 \textit {\_Z}^{2}-1\right )^{2}+1\right ) \operatorname {RootOf}\left (625 \textit {\_Z}^{4}+25 \textit {\_Z}^{2}-1\right )^{2} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}+25 \textit {\_Z}^{2}-1\right )^{2}+1\right ) x^{4}+2 \sqrt {x^{4}+1}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}+25 \textit {\_Z}^{2}-1\right )^{2}+1\right )}{25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}+25 \textit {\_Z}^{2}-1\right )^{2} x^{2}+x^{4}+1}\right )}{5}+\operatorname {RootOf}\left (625 \textit {\_Z}^{4}+25 \textit {\_Z}^{2}-1\right ) \ln \left (-\frac {-125 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}+25 \textit {\_Z}^{2}-1\right )^{3} x^{2}-5 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}+25 \textit {\_Z}^{2}-1\right ) x^{4}-5 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}+25 \textit {\_Z}^{2}-1\right ) x^{2}+2 \sqrt {x^{4}+1}\, x -5 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}+25 \textit {\_Z}^{2}-1\right )}{25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}+25 \textit {\_Z}^{2}-1\right )^{2} x^{2}-x^{4}+x^{2}-1}\right )\) | \(327\) |
-1/10*arctan(2^(1/2)*x/(x^4+1)^(1/2))*2^(1/2)+1/5*arctan(2/(-2+2*5^(1/2))^ (1/2)/x*(x^4+1)^(1/2))*(2+2*5^(1/2))^(1/2)-1/5*arctanh(2/(2+2*5^(1/2))^(1/ 2)/x*(x^4+1)^(1/2))*(-2+2*5^(1/2))^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 521 vs. \(2 (81) = 162\).
Time = 0.33 (sec) , antiderivative size = 521, normalized size of antiderivative = 4.38 \[ \int \frac {-1+x^{10}}{\sqrt {1+x^4} \left (1+x^{10}\right )} \, dx=-\frac {1}{10} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} x}{\sqrt {x^{4} + 1}}\right ) - \frac {1}{20} \, \sqrt {2 \, \sqrt {5} - 2} \log \left (-\frac {4 \, {\left (3 \, x^{5} + x^{3} + \sqrt {5} {\left (x^{5} + x^{3} + x\right )} + 3 \, x\right )} \sqrt {x^{4} + 1} + {\left (3 \, x^{8} + 5 \, x^{6} + 9 \, x^{4} + 5 \, x^{2} + \sqrt {5} {\left (x^{8} + 3 \, x^{6} + 3 \, x^{4} + 3 \, x^{2} + 1\right )} + 3\right )} \sqrt {2 \, \sqrt {5} - 2}}{x^{8} - x^{6} + x^{4} - x^{2} + 1}\right ) + \frac {1}{20} \, \sqrt {2 \, \sqrt {5} - 2} \log \left (-\frac {4 \, {\left (3 \, x^{5} + x^{3} + \sqrt {5} {\left (x^{5} + x^{3} + x\right )} + 3 \, x\right )} \sqrt {x^{4} + 1} - {\left (3 \, x^{8} + 5 \, x^{6} + 9 \, x^{4} + 5 \, x^{2} + \sqrt {5} {\left (x^{8} + 3 \, x^{6} + 3 \, x^{4} + 3 \, x^{2} + 1\right )} + 3\right )} \sqrt {2 \, \sqrt {5} - 2}}{x^{8} - x^{6} + x^{4} - x^{2} + 1}\right ) - \frac {1}{20} \, \sqrt {-2 \, \sqrt {5} - 2} \log \left (-\frac {4 \, {\left (3 \, x^{5} + x^{3} - \sqrt {5} {\left (x^{5} + x^{3} + x\right )} + 3 \, x\right )} \sqrt {x^{4} + 1} + {\left (3 \, x^{8} + 5 \, x^{6} + 9 \, x^{4} + 5 \, x^{2} - \sqrt {5} {\left (x^{8} + 3 \, x^{6} + 3 \, x^{4} + 3 \, x^{2} + 1\right )} + 3\right )} \sqrt {-2 \, \sqrt {5} - 2}}{x^{8} - x^{6} + x^{4} - x^{2} + 1}\right ) + \frac {1}{20} \, \sqrt {-2 \, \sqrt {5} - 2} \log \left (-\frac {4 \, {\left (3 \, x^{5} + x^{3} - \sqrt {5} {\left (x^{5} + x^{3} + x\right )} + 3 \, x\right )} \sqrt {x^{4} + 1} - {\left (3 \, x^{8} + 5 \, x^{6} + 9 \, x^{4} + 5 \, x^{2} - \sqrt {5} {\left (x^{8} + 3 \, x^{6} + 3 \, x^{4} + 3 \, x^{2} + 1\right )} + 3\right )} \sqrt {-2 \, \sqrt {5} - 2}}{x^{8} - x^{6} + x^{4} - x^{2} + 1}\right ) \]
-1/10*sqrt(2)*arctan(sqrt(2)*x/sqrt(x^4 + 1)) - 1/20*sqrt(2*sqrt(5) - 2)*l og(-(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 + 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*sqr t(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)*sqr t(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))
Timed out. \[ \int \frac {-1+x^{10}}{\sqrt {1+x^4} \left (1+x^{10}\right )} \, dx=\text {Timed out} \]
\[ \int \frac {-1+x^{10}}{\sqrt {1+x^4} \left (1+x^{10}\right )} \, dx=\int { \frac {x^{10} - 1}{{\left (x^{10} + 1\right )} \sqrt {x^{4} + 1}} \,d x } \]
\[ \int \frac {-1+x^{10}}{\sqrt {1+x^4} \left (1+x^{10}\right )} \, dx=\int { \frac {x^{10} - 1}{{\left (x^{10} + 1\right )} \sqrt {x^{4} + 1}} \,d x } \]
Timed out. \[ \int \frac {-1+x^{10}}{\sqrt {1+x^4} \left (1+x^{10}\right )} \, dx=\int \frac {x^{10}-1}{\sqrt {x^4+1}\,\left (x^{10}+1\right )} \,d x \]