\(\int \frac {-1+x^{10}}{\sqrt {1+x^4} (1+x^{10})} \, dx\) [1770]

Optimal result

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

[Out]

-1/10*arctan(2^(1/2)*x/(x^4+1)^(1/2))*2^(1/2)-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))-1/5*(-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*ArcTan[(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]) - (8*Defer[Int][1/(Sqrt[1 + x^4]*(1 - x^2 + x^4 - x^6 + x^8)), x])/5 + (6*Defer[Int
][x^2/(Sqrt[1 + x^4]*(1 - x^2 + x^4 - x^6 + x^8)), x])/5 - (4*Defer[Int][x^4/(Sqrt[1 + x^4]*(1 - x^2 + x^4 - x
^6 + x^8)), x])/5 + (2*Defer[Int][x^6/(Sqrt[1 + x^4]*(1 - x^2 + x^4 - x^6 + x^8)), 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 \\ & = -\left (2 \int \frac {1}{\sqrt {1+x^4} \left (1+x^{10}\right )} \, dx\right )+\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+2 x^4-x^6}{5 \sqrt {1+x^4} \left (1-x^2+x^4-x^6+x^8\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 {2}{5} \int \frac {1}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx-\frac {2}{5} \int \frac {4-3 x^2+2 x^4-x^6}{\sqrt {1+x^4} \left (1-x^2+x^4-x^6+x^8\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 {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 {2}{5} \int \left (\frac {4}{\sqrt {1+x^4} \left (1-x^2+x^4-x^6+x^8\right )}-\frac {3 x^2}{\sqrt {1+x^4} \left (1-x^2+x^4-x^6+x^8\right )}+\frac {2 x^4}{\sqrt {1+x^4} \left (1-x^2+x^4-x^6+x^8\right )}-\frac {x^6}{\sqrt {1+x^4} \left (1-x^2+x^4-x^6+x^8\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} \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^6}{\sqrt {1+x^4} \left (1-x^2+x^4-x^6+x^8\right )} \, dx-\frac {4}{5} \int \frac {x^4}{\sqrt {1+x^4} \left (1-x^2+x^4-x^6+x^8\right )} \, dx+\frac {6}{5} \int \frac {x^2}{\sqrt {1+x^4} \left (1-x^2+x^4-x^6+x^8\right )} \, dx-\frac {8}{5} \int \frac {1}{\sqrt {1+x^4} \left (1-x^2+x^4-x^6+x^8\right )} \, dx \\ & = -\frac {\arctan \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 {2}{5} \int \frac {x^6}{\sqrt {1+x^4} \left (1-x^2+x^4-x^6+x^8\right )} \, dx-\frac {4}{5} \int \frac {x^4}{\sqrt {1+x^4} \left (1-x^2+x^4-x^6+x^8\right )} \, dx+\frac {6}{5} \int \frac {x^2}{\sqrt {1+x^4} \left (1-x^2+x^4-x^6+x^8\right )} \, dx-\frac {8}{5} \int \frac {1}{\sqrt {1+x^4} \left (1-x^2+x^4-x^6+x^8\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.71 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.90 \[ \int \frac {-1+x^{10}}{\sqrt {1+x^4} \left (1+x^{10}\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )+2 \sqrt {1+\sqrt {5}} \arctan \left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} x}{\sqrt {1+x^4}}\right )+2 \sqrt {-1+\sqrt {5}} \text {arctanh}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} x}{\sqrt {1+x^4}}\right )}{5 \sqrt {2}} \]

[In]

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

[Out]

-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]

Maple [A] (verified)

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	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x +\sqrt {x^{4}+1}}{x^{2}+1}\right )}{10}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}+25 \textit {\_Z}^{2}-1\right )^{2}+1\right ) \ln \left (\frac {25 \operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}+25 \textit {\_Z}^{2}-1\right )^{2}+1\right ) \operatorname {RootOf}\left (625 \textit {\_Z}^{4}+25 \textit {\_Z}^{2}-1\right )^{2} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}+25 \textit {\_Z}^{2}-1\right )^{2}+1\right ) x^{4}+2 \sqrt {x^{4}+1}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}+25 \textit {\_Z}^{2}-1\right )^{2}+1\right )}{25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}+25 \textit {\_Z}^{2}-1\right )^{2} x^{2}+x^{4}+1}\right )}{5}+\operatorname {RootOf}\left (625 \textit {\_Z}^{4}+25 \textit {\_Z}^{2}-1\right ) \ln \left (-\frac {-125 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}+25 \textit {\_Z}^{2}-1\right )^{3} x^{2}-5 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}+25 \textit {\_Z}^{2}-1\right ) x^{4}-5 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}+25 \textit {\_Z}^{2}-1\right ) x^{2}+2 \sqrt {x^{4}+1}\, x -5 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}+25 \textit {\_Z}^{2}-1\right )}{25 \operatorname {RootOf}\left (625 \textit {\_Z}^{4}+25 \textit {\_Z}^{2}-1\right )^{2} x^{2}-x^{4}+x^{2}-1}\right )\)	\(327\)

[In]

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

[Out]

-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)

Fricas [B] (verification not implemented)

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

[In]

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

[Out]

-1/10*sqrt(2)*arctan(sqrt(2)*x/sqrt(x^4 + 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*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))

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

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)