\(\int \frac {\sqrt {1+x^2-2 x^6} (1+4 x^6)}{(-1-4 x^2+2 x^6) (-1-2 x^2+2 x^6)} \, dx\) [886]

Optimal result

Integrand size = 50, antiderivative size = 67 \[ \int \frac {\sqrt {1+x^2-2 x^6} \left (1+4 x^6\right )}{\left (-1-4 x^2+2 x^6\right ) \left (-1-2 x^2+2 x^6\right )} \, dx=-\frac {1}{2} \arctan \left (\frac {x}{\sqrt {1+x^2-2 x^6}}\right )-\frac {1}{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x \sqrt {1+x^2-2 x^6}}{-1-x^2+2 x^6}\right ) \]

[Out]

-1/2*arctan(x/(-2*x^6+x^2+1)^(1/2))-1/2*3^(1/2)*arctan(3^(1/2)*x*(-2*x^6+x^2+1)^(1/2)/(2*x^6-x^2-1))

Rubi [F]

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

[In]

Int[(Sqrt[1 + x^2 - 2*x^6]*(1 + 4*x^6))/((-1 - 4*x^2 + 2*x^6)*(-1 - 2*x^2 + 2*x^6)),x]

[Out]

-2*Defer[Int][Sqrt[1 + x^2 - 2*x^6]/(-1 - 4*x^2 + 2*x^6), x] + 3*Defer[Int][(x^4*Sqrt[1 + x^2 - 2*x^6])/(-1 -
4*x^2 + 2*x^6), x] + Defer[Int][Sqrt[1 + x^2 - 2*x^6]/(-1 - 2*x^2 + 2*x^6), x] - 3*Defer[Int][(x^4*Sqrt[1 + x^
2 - 2*x^6])/(-1 - 2*x^2 + 2*x^6), x]

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

Mathematica [A] (verified)

Time = 1.46 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.78 \[ \int \frac {\sqrt {1+x^2-2 x^6} \left (1+4 x^6\right )}{\left (-1-4 x^2+2 x^6\right ) \left (-1-2 x^2+2 x^6\right )} \, dx=\frac {1}{2} \left (-\arctan \left (\frac {x}{\sqrt {1+x^2-2 x^6}}\right )+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{\sqrt {1+x^2-2 x^6}}\right )\right ) \]

[In]

Integrate[(Sqrt[1 + x^2 - 2*x^6]*(1 + 4*x^6))/((-1 - 4*x^2 + 2*x^6)*(-1 - 2*x^2 + 2*x^6)),x]

[Out]

(-ArcTan[x/Sqrt[1 + x^2 - 2*x^6]] + Sqrt[3]*ArcTan[(Sqrt[3]*x)/Sqrt[1 + x^2 - 2*x^6]])/2

Maple [A] (verified)

Time = 4.54 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.70

method	result	size
pseudoelliptic	\(\frac {\arctan \left (\frac {\sqrt {-2 x^{6}+x^{2}+1}}{x}\right )}{2}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \sqrt {-2 x^{6}+x^{2}+1}}{3 x}\right )}{2}\)	\(47\)
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{6}+2 \sqrt {-2 x^{6}+x^{2}+1}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{2 x^{6}-2 x^{2}-1}\right )}{4}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{6}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{2}+6 \sqrt {-2 x^{6}+x^{2}+1}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right )}{2 x^{6}-4 x^{2}-1}\right )}{4}\)	\(133\)

[In]

int((-2*x^6+x^2+1)^(1/2)*(4*x^6+1)/(2*x^6-4*x^2-1)/(2*x^6-2*x^2-1),x,method=_RETURNVERBOSE)

[Out]

1/2*arctan(1/x*(-2*x^6+x^2+1)^(1/2))-1/2*3^(1/2)*arctan(1/3*3^(1/2)/x*(-2*x^6+x^2+1)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt {1+x^2-2 x^6} \left (1+4 x^6\right )}{\left (-1-4 x^2+2 x^6\right ) \left (-1-2 x^2+2 x^6\right )} \, dx=-\frac {1}{4} \, \sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} \sqrt {-2 \, x^{6} + x^{2} + 1} x}{2 \, x^{6} + 2 \, x^{2} - 1}\right ) + \frac {1}{4} \, \arctan \left (\frac {2 \, \sqrt {-2 \, x^{6} + x^{2} + 1} x}{2 \, x^{6} - 1}\right ) \]

[In]

integrate((-2*x^6+x^2+1)^(1/2)*(4*x^6+1)/(2*x^6-4*x^2-1)/(2*x^6-2*x^2-1),x, algorithm="fricas")

[Out]

-1/4*sqrt(3)*arctan(2*sqrt(3)*sqrt(-2*x^6 + x^2 + 1)*x/(2*x^6 + 2*x^2 - 1)) + 1/4*arctan(2*sqrt(-2*x^6 + x^2 +
1)*x/(2*x^6 - 1))

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

integrate((-2*x^6+x^2+1)^(1/2)*(4*x^6+1)/(2*x^6-4*x^2-1)/(2*x^6-2*x^2-1),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate((-2*x^6+x^2+1)^(1/2)*(4*x^6+1)/(2*x^6-4*x^2-1)/(2*x^6-2*x^2-1),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

int(((4*x^6 + 1)*(x^2 - 2*x^6 + 1)^(1/2))/((2*x^2 - 2*x^6 + 1)*(4*x^2 - 2*x^6 + 1)),x)

[Out]

int(((4*x^6 + 1)*(x^2 - 2*x^6 + 1)^(1/2))/((2*x^2 - 2*x^6 + 1)*(4*x^2 - 2*x^6 + 1)), x)