\(\int \frac {-1+2 x^6}{(1+x^6) \sqrt {1-2 x^2+x^6}} \, dx\) [354]

Optimal result

Integrand size = 29, antiderivative size = 29 \[ \int \frac {-1+2 x^6}{\left (1+x^6\right ) \sqrt {1-2 x^2+x^6}} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt {1-2 x^2+x^6}}\right )}{\sqrt {2}} \]

[Out]

-1/2*arctan(2^(1/2)*x/(x^6-2*x^2+1)^(1/2))*2^(1/2)

Rubi [F]

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

[In]

Int[(-1 + 2*x^6)/((1 + x^6)*Sqrt[1 - 2*x^2 + x^6]),x]

[Out]

2*Defer[Int][1/Sqrt[1 - 2*x^2 + x^6], x] - (I/2)*Defer[Int][1/((I - x)*Sqrt[1 - 2*x^2 + x^6]), x] - (I/2)*Defe
r[Int][1/((I + x)*Sqrt[1 - 2*x^2 + x^6]), x] - (Sqrt[1 - I*Sqrt[3]]*Defer[Int][1/((Sqrt[1 - I*Sqrt[3]] - Sqrt[
2]*x)*Sqrt[1 - 2*x^2 + x^6]), x])/2 - (Sqrt[1 + I*Sqrt[3]]*Defer[Int][1/((Sqrt[1 + I*Sqrt[3]] - Sqrt[2]*x)*Sqr
t[1 - 2*x^2 + x^6]), x])/2 - (Sqrt[1 - I*Sqrt[3]]*Defer[Int][1/((Sqrt[1 - I*Sqrt[3]] + Sqrt[2]*x)*Sqrt[1 - 2*x
^2 + x^6]), x])/2 - (Sqrt[1 + I*Sqrt[3]]*Defer[Int][1/((Sqrt[1 + I*Sqrt[3]] + Sqrt[2]*x)*Sqrt[1 - 2*x^2 + x^6]
), x])/2

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2}{\sqrt {1-2 x^2+x^6}}-\frac {3}{\left (1+x^6\right ) \sqrt {1-2 x^2+x^6}}\right ) \, dx \\ & = 2 \int \frac {1}{\sqrt {1-2 x^2+x^6}} \, dx-3 \int \frac {1}{\left (1+x^6\right ) \sqrt {1-2 x^2+x^6}} \, dx \\ & = 2 \int \frac {1}{\sqrt {1-2 x^2+x^6}} \, dx-3 \int \left (\frac {1}{3 \left (1+x^2\right ) \sqrt {1-2 x^2+x^6}}+\frac {2-x^2}{3 \left (1-x^2+x^4\right ) \sqrt {1-2 x^2+x^6}}\right ) \, dx \\ & = 2 \int \frac {1}{\sqrt {1-2 x^2+x^6}} \, dx-\int \frac {1}{\left (1+x^2\right ) \sqrt {1-2 x^2+x^6}} \, dx-\int \frac {2-x^2}{\left (1-x^2+x^4\right ) \sqrt {1-2 x^2+x^6}} \, dx \\ & = 2 \int \frac {1}{\sqrt {1-2 x^2+x^6}} \, dx-\int \left (\frac {i}{2 (i-x) \sqrt {1-2 x^2+x^6}}+\frac {i}{2 (i+x) \sqrt {1-2 x^2+x^6}}\right ) \, dx-\int \left (\frac {-1-i \sqrt {3}}{\left (-1-i \sqrt {3}+2 x^2\right ) \sqrt {1-2 x^2+x^6}}+\frac {-1+i \sqrt {3}}{\left (-1+i \sqrt {3}+2 x^2\right ) \sqrt {1-2 x^2+x^6}}\right ) \, dx \\ & = -\left (\frac {1}{2} i \int \frac {1}{(i-x) \sqrt {1-2 x^2+x^6}} \, dx\right )-\frac {1}{2} i \int \frac {1}{(i+x) \sqrt {1-2 x^2+x^6}} \, dx+2 \int \frac {1}{\sqrt {1-2 x^2+x^6}} \, dx-\left (-1-i \sqrt {3}\right ) \int \frac {1}{\left (-1-i \sqrt {3}+2 x^2\right ) \sqrt {1-2 x^2+x^6}} \, dx-\left (-1+i \sqrt {3}\right ) \int \frac {1}{\left (-1+i \sqrt {3}+2 x^2\right ) \sqrt {1-2 x^2+x^6}} \, dx \\ & = -\left (\frac {1}{2} i \int \frac {1}{(i-x) \sqrt {1-2 x^2+x^6}} \, dx\right )-\frac {1}{2} i \int \frac {1}{(i+x) \sqrt {1-2 x^2+x^6}} \, dx+2 \int \frac {1}{\sqrt {1-2 x^2+x^6}} \, dx-\left (-1-i \sqrt {3}\right ) \int \left (\frac {\sqrt {1+i \sqrt {3}}}{2 \left (-1-i \sqrt {3}\right ) \left (\sqrt {1+i \sqrt {3}}-\sqrt {2} x\right ) \sqrt {1-2 x^2+x^6}}+\frac {\sqrt {1+i \sqrt {3}}}{2 \left (-1-i \sqrt {3}\right ) \left (\sqrt {1+i \sqrt {3}}+\sqrt {2} x\right ) \sqrt {1-2 x^2+x^6}}\right ) \, dx-\left (-1+i \sqrt {3}\right ) \int \left (\frac {\sqrt {1-i \sqrt {3}}}{2 \left (-1+i \sqrt {3}\right ) \left (\sqrt {1-i \sqrt {3}}-\sqrt {2} x\right ) \sqrt {1-2 x^2+x^6}}+\frac {\sqrt {1-i \sqrt {3}}}{2 \left (-1+i \sqrt {3}\right ) \left (\sqrt {1-i \sqrt {3}}+\sqrt {2} x\right ) \sqrt {1-2 x^2+x^6}}\right ) \, dx \\ & = -\left (\frac {1}{2} i \int \frac {1}{(i-x) \sqrt {1-2 x^2+x^6}} \, dx\right )-\frac {1}{2} i \int \frac {1}{(i+x) \sqrt {1-2 x^2+x^6}} \, dx+2 \int \frac {1}{\sqrt {1-2 x^2+x^6}} \, dx-\frac {1}{2} \sqrt {1-i \sqrt {3}} \int \frac {1}{\left (\sqrt {1-i \sqrt {3}}-\sqrt {2} x\right ) \sqrt {1-2 x^2+x^6}} \, dx-\frac {1}{2} \sqrt {1-i \sqrt {3}} \int \frac {1}{\left (\sqrt {1-i \sqrt {3}}+\sqrt {2} x\right ) \sqrt {1-2 x^2+x^6}} \, dx-\frac {1}{2} \sqrt {1+i \sqrt {3}} \int \frac {1}{\left (\sqrt {1+i \sqrt {3}}-\sqrt {2} x\right ) \sqrt {1-2 x^2+x^6}} \, dx-\frac {1}{2} \sqrt {1+i \sqrt {3}} \int \frac {1}{\left (\sqrt {1+i \sqrt {3}}+\sqrt {2} x\right ) \sqrt {1-2 x^2+x^6}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.73 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {-1+2 x^6}{\left (1+x^6\right ) \sqrt {1-2 x^2+x^6}} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt {1-2 x^2+x^6}}\right )}{\sqrt {2}} \]

[In]

Integrate[(-1 + 2*x^6)/((1 + x^6)*Sqrt[1 - 2*x^2 + x^6]),x]

[Out]

-(ArcTan[(Sqrt[2]*x)/Sqrt[1 - 2*x^2 + x^6]]/Sqrt[2])

Maple [A] (verified)

Time = 2.90 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93

method	result	size
pseudoelliptic	\(\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x^{6}-2 x^{2}+1}}{2 x}\right )}{2}\)	\(27\)
trager	\(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{6}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{2}+4 \sqrt {x^{6}-2 x^{2}+1}\, x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )}{\left (x^{2}+1\right ) \left (x^{4}-x^{2}+1\right )}\right )}{4}\)	\(74\)

[In]

int((2*x^6-1)/(x^6+1)/(x^6-2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2*2^(1/2)*arctan(1/2*2^(1/2)/x*(x^6-2*x^2+1)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.24 \[ \int \frac {-1+2 x^6}{\left (1+x^6\right ) \sqrt {1-2 x^2+x^6}} \, dx=-\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {x^{6} - 2 \, x^{2} + 1} x}{x^{6} - 4 \, x^{2} + 1}\right ) \]

[In]

integrate((2*x^6-1)/(x^6+1)/(x^6-2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

-1/4*sqrt(2)*arctan(2*sqrt(2)*sqrt(x^6 - 2*x^2 + 1)*x/(x^6 - 4*x^2 + 1))

Sympy [F]

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

[In]

integrate((2*x**6-1)/(x**6+1)/(x**6-2*x**2+1)**(1/2),x)

[Out]

Integral((2*x**6 - 1)/(sqrt((x - 1)*(x + 1)*(x**4 + x**2 - 1))*(x**2 + 1)*(x**4 - x**2 + 1)), x)

Maxima [F]

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

[In]

integrate((2*x^6-1)/(x^6+1)/(x^6-2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate((2*x^6 - 1)/(sqrt(x^6 - 2*x^2 + 1)*(x^6 + 1)), x)

Giac [F]

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

[In]

integrate((2*x^6-1)/(x^6+1)/(x^6-2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate((2*x^6 - 1)/(sqrt(x^6 - 2*x^2 + 1)*(x^6 + 1)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {-1+2 x^6}{\left (1+x^6\right ) \sqrt {1-2 x^2+x^6}} \, dx=\int \frac {2\,x^6-1}{\left (x^6+1\right )\,\sqrt {x^6-2\,x^2+1}} \,d x \]

[In]

int((2*x^6 - 1)/((x^6 + 1)*(x^6 - 2*x^2 + 1)^(1/2)),x)

[Out]

int((2*x^6 - 1)/((x^6 + 1)*(x^6 - 2*x^2 + 1)^(1/2)), x)