\(\int \frac {1+x^{10}}{\sqrt {1+x^4} (-1+x^{10})} \, dx\) [1771]

Optimal result

Integrand size = 22, antiderivative size = 119 \[ \int \frac {1+x^{10}}{\sqrt {1+x^4} \left (-1+x^{10}\right )} \, dx=-\frac {1}{5} \sqrt {2 \left (-1+\sqrt {5}\right )} \arctan \left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt {1+x^4}}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{5 \sqrt {2}}-\frac {1}{5} \sqrt {2 \left (1+\sqrt {5}\right )} \text {arctanh}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt {1+x^4}}\right ) \]

[Out]

-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)*arctanh(1/2*(-2+2*5^(1/2))^(1/2)*x/(x^4+1)^(1/2))

Rubi [F]

\[ \int \frac {1+x^{10}}{\sqrt {1+x^4} \left (-1+x^{10}\right )} \, dx=\int \frac {1+x^{10}}{\sqrt {1+x^4} \left (-1+x^{10}\right )} \, dx \]

[In]

Int[(1 + x^10)/(Sqrt[1 + x^4]*(-1 + x^10)),x]

[Out]

-1/5*ArcTanh[(Sqrt[2]*x)/Sqrt[1 + x^4]]/Sqrt[2] + (2*(1 + x^2)*Sqrt[(1 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[
x], 1/2])/(5*Sqrt[1 + x^4]) - (4*Defer[Int][1/(Sqrt[1 + x^4]*(1 - x + x^2 - x^3 + x^4)), x])/5 + (3*Defer[Int]
[x/(Sqrt[1 + x^4]*(1 - x + x^2 - x^3 + x^4)), x])/5 - (2*Defer[Int][x^2/(Sqrt[1 + x^4]*(1 - x + x^2 - x^3 + x^
4)), x])/5 + Defer[Int][x^3/(Sqrt[1 + x^4]*(1 - x + x^2 - x^3 + x^4)), x]/5 - (4*Defer[Int][1/(Sqrt[1 + x^4]*(
1 + x + x^2 + x^3 + x^4)), x])/5 - (3*Defer[Int][x/(Sqrt[1 + x^4]*(1 + x + x^2 + x^3 + x^4)), x])/5 - (2*Defer
[Int][x^2/(Sqrt[1 + x^4]*(1 + x + x^2 + x^3 + x^4)), x])/5 - Defer[Int][x^3/(Sqrt[1 + x^4]*(1 + x + x^2 + x^3
+ x^4)), x]/5

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{\sqrt {1+x^4}}+\frac {2}{\sqrt {1+x^4} \left (-1+x^{10}\right )}\right ) \, dx \\ & = 2 \int \frac {1}{\sqrt {1+x^4} \left (-1+x^{10}\right )} \, dx+\int \frac {1}{\sqrt {1+x^4}} \, dx \\ & = \frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{2 \sqrt {1+x^4}}+2 \int \left (\frac {1}{5 \left (-1+x^2\right ) \sqrt {1+x^4}}+\frac {-4+3 x-2 x^2+x^3}{10 \sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )}+\frac {-4-3 x-2 x^2-x^3}{10 \sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )}\right ) \, dx \\ & = \frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{2 \sqrt {1+x^4}}+\frac {1}{5} \int \frac {-4+3 x-2 x^2+x^3}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )} \, dx+\frac {1}{5} \int \frac {-4-3 x-2 x^2-x^3}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx+\frac {2}{5} \int \frac {1}{\left (-1+x^2\right ) \sqrt {1+x^4}} \, dx \\ & = \frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{2 \sqrt {1+x^4}}-\frac {1}{5} \int \frac {1}{\sqrt {1+x^4}} \, dx-\frac {1}{5} \int \frac {-1-x^2}{\left (-1+x^2\right ) \sqrt {1+x^4}} \, dx+\frac {1}{5} \int \left (-\frac {4}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )}+\frac {3 x}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )}-\frac {2 x^2}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )}+\frac {x^3}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )}\right ) \, dx+\frac {1}{5} \int \left (-\frac {4}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )}-\frac {3 x}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )}-\frac {2 x^2}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )}-\frac {x^3}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )}\right ) \, dx \\ & = \frac {2 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{5 \sqrt {1+x^4}}+\frac {1}{5} \int \frac {x^3}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )} \, dx-\frac {1}{5} \int \frac {x^3}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx+\frac {1}{5} \text {Subst}\left (\int \frac {1}{-1+2 x^2} \, dx,x,\frac {x}{\sqrt {1+x^4}}\right )-\frac {2}{5} \int \frac {x^2}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )} \, dx-\frac {2}{5} \int \frac {x^2}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx+\frac {3}{5} \int \frac {x}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )} \, dx-\frac {3}{5} \int \frac {x}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx-\frac {4}{5} \int \frac {1}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )} \, dx-\frac {4}{5} \int \frac {1}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx \\ & = -\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{5 \sqrt {2}}+\frac {2 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{5 \sqrt {1+x^4}}+\frac {1}{5} \int \frac {x^3}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )} \, dx-\frac {1}{5} \int \frac {x^3}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx-\frac {2}{5} \int \frac {x^2}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )} \, dx-\frac {2}{5} \int \frac {x^2}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx+\frac {3}{5} \int \frac {x}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )} \, dx-\frac {3}{5} \int \frac {x}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx-\frac {4}{5} \int \frac {1}{\sqrt {1+x^4} \left (1-x+x^2-x^3+x^4\right )} \, dx-\frac {4}{5} \int \frac {1}{\sqrt {1+x^4} \left (1+x+x^2+x^3+x^4\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.64 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.90 \[ \int \frac {1+x^{10}}{\sqrt {1+x^4} \left (-1+x^{10}\right )} \, dx=-\frac {2 \sqrt {-1+\sqrt {5}} \arctan \left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} x}{\sqrt {1+x^4}}\right )+\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )+2 \sqrt {1+\sqrt {5}} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} x}{\sqrt {1+x^4}}\right )}{5 \sqrt {2}} \]

[In]

Integrate[(1 + x^10)/(Sqrt[1 + x^4]*(-1 + x^10)),x]

[Out]

-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]

Maple [A] (verified)

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	\(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +\sqrt {x^{4}+1}}{\left (1+x \right ) \left (-1+x \right )}\right )}{10}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right )^{2}-1\right ) \ln \left (\frac {25 \operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right )^{2}-1\right ) \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right )^{2} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right )^{2}-1\right ) x^{4}+2 \sqrt {x^{4}+1}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right )^{2}-1\right )}{\left (25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right )^{2} x +x^{2}-x +1\right ) \left (25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right )^{2} x -x^{2}-x -1\right )}\right )}{5}+\operatorname {RootOf}\left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right ) \ln \left (\frac {-125 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right )^{3} x^{2}-5 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right ) x^{4}+5 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right ) x^{2}+2 \sqrt {x^{4}+1}\, x -5 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right )}{\left (25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right )^{2} x +x^{2}+1\right ) \left (25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}-25 \textit {\_Z}^{2}-1\right )^{2} x -x^{2}-1\right )}\right )\)	\(381\)

[In]

int((x^10+1)/(x^4+1)^(1/2)/(x^10-1),x,method=_RETURNVERBOSE)

[Out]

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)*arc
tanh(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)

Fricas [B] (verification not implemented)

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 \[ \int \frac {1+x^{10}}{\sqrt {1+x^4} \left (-1+x^{10}\right )} \, dx=\frac {1}{20} \, \sqrt {2} \log \left (\frac {x^{4} - 2 \, \sqrt {2} \sqrt {x^{4} + 1} x + 2 \, x^{2} + 1}{x^{4} - 2 \, x^{2} + 1}\right ) - \frac {1}{20} \, \sqrt {-2 \, \sqrt {5} + 2} \log \left (-\frac {4 \, {\left (3 \, x^{5} - x^{3} + \sqrt {5} {\left (x^{5} - x^{3} + x\right )} + 3 \, x\right )} \sqrt {x^{4} + 1} + {\left (3 \, x^{8} - 5 \, x^{6} + 9 \, x^{4} - 5 \, x^{2} + \sqrt {5} {\left (x^{8} - 3 \, x^{6} + 3 \, x^{4} - 3 \, x^{2} + 1\right )} + 3\right )} \sqrt {-2 \, \sqrt {5} + 2}}{x^{8} + x^{6} + x^{4} + x^{2} + 1}\right ) + \frac {1}{20} \, \sqrt {-2 \, \sqrt {5} + 2} \log \left (-\frac {4 \, {\left (3 \, x^{5} - x^{3} + \sqrt {5} {\left (x^{5} - x^{3} + x\right )} + 3 \, x\right )} \sqrt {x^{4} + 1} - {\left (3 \, x^{8} - 5 \, x^{6} + 9 \, x^{4} - 5 \, x^{2} + \sqrt {5} {\left (x^{8} - 3 \, x^{6} + 3 \, x^{4} - 3 \, x^{2} + 1\right )} + 3\right )} \sqrt {-2 \, \sqrt {5} + 2}}{x^{8} + x^{6} + x^{4} + x^{2} + 1}\right ) - \frac {1}{20} \, \sqrt {2 \, \sqrt {5} + 2} \log \left (-\frac {4 \, {\left (3 \, x^{5} - x^{3} - \sqrt {5} {\left (x^{5} - x^{3} + x\right )} + 3 \, x\right )} \sqrt {x^{4} + 1} + {\left (3 \, x^{8} - 5 \, x^{6} + 9 \, x^{4} - 5 \, x^{2} - \sqrt {5} {\left (x^{8} - 3 \, x^{6} + 3 \, x^{4} - 3 \, x^{2} + 1\right )} + 3\right )} \sqrt {2 \, \sqrt {5} + 2}}{x^{8} + x^{6} + x^{4} + x^{2} + 1}\right ) + \frac {1}{20} \, \sqrt {2 \, \sqrt {5} + 2} \log \left (-\frac {4 \, {\left (3 \, x^{5} - x^{3} - \sqrt {5} {\left (x^{5} - x^{3} + x\right )} + 3 \, x\right )} \sqrt {x^{4} + 1} - {\left (3 \, x^{8} - 5 \, x^{6} + 9 \, x^{4} - 5 \, x^{2} - \sqrt {5} {\left (x^{8} - 3 \, x^{6} + 3 \, x^{4} - 3 \, x^{2} + 1\right )} + 3\right )} \sqrt {2 \, \sqrt {5} + 2}}{x^{8} + x^{6} + x^{4} + x^{2} + 1}\right ) \]

[In]

integrate((x^10+1)/(x^4+1)^(1/2)/(x^10-1),x, algorithm="fricas")

[Out]

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
+ 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*sqr
t(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 - 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))

Sympy [F]

\[ \int \frac {1+x^{10}}{\sqrt {1+x^4} \left (-1+x^{10}\right )} \, dx=\int \frac {\left (x^{2} + 1\right ) \left (x^{8} - x^{6} + x^{4} - x^{2} + 1\right )}{\left (x - 1\right ) \left (x + 1\right ) \sqrt {x^{4} + 1} \left (x^{4} - x^{3} + x^{2} - x + 1\right ) \left (x^{4} + x^{3} + x^{2} + x + 1\right )}\, dx \]

[In]

integrate((x**10+1)/(x**4+1)**(1/2)/(x**10-1),x)

[Out]

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)

Maxima [F]

\[ \int \frac {1+x^{10}}{\sqrt {1+x^4} \left (-1+x^{10}\right )} \, dx=\int { \frac {x^{10} + 1}{{\left (x^{10} - 1\right )} \sqrt {x^{4} + 1}} \,d x } \]

[In]

integrate((x^10+1)/(x^4+1)^(1/2)/(x^10-1),x, algorithm="maxima")

[Out]

integrate((x^10 + 1)/((x^10 - 1)*sqrt(x^4 + 1)), x)

Giac [F]

\[ \int \frac {1+x^{10}}{\sqrt {1+x^4} \left (-1+x^{10}\right )} \, dx=\int { \frac {x^{10} + 1}{{\left (x^{10} - 1\right )} \sqrt {x^{4} + 1}} \,d x } \]

[In]

integrate((x^10+1)/(x^4+1)^(1/2)/(x^10-1),x, algorithm="giac")

[Out]

integrate((x^10 + 1)/((x^10 - 1)*sqrt(x^4 + 1)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1+x^{10}}{\sqrt {1+x^4} \left (-1+x^{10}\right )} \, dx=\int \frac {x^{10}+1}{\sqrt {x^4+1}\,\left (x^{10}-1\right )} \,d x \]

[In]

int((x^10 + 1)/((x^4 + 1)^(1/2)*(x^10 - 1)),x)

[Out]

int((x^10 + 1)/((x^4 + 1)^(1/2)*(x^10 - 1)), x)