\(\int \frac {\sqrt {-1+x^6} (1+2 x^6)}{x^2 (-1+x^2+x^6)} \, dx\) [293]

Optimal result

Integrand size = 30, antiderivative size = 26 \[ \int \frac {\sqrt {-1+x^6} \left (1+2 x^6\right )}{x^2 \left (-1+x^2+x^6\right )} \, dx=\frac {\sqrt {-1+x^6}}{x}+\arctan \left (\frac {x}{\sqrt {-1+x^6}}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

Sqrt[-1 + x^6]/x + (3*(1 + Sqrt[3])*x*Sqrt[-1 + x^6])/(2*(1 - (1 + Sqrt[3])*x^2)) + (3*3^(1/4)*x*(1 - x^2)*Sqr
t[(1 + x^2 + x^4)/(1 - (1 + Sqrt[3])*x^2)^2]*EllipticE[ArcCos[(1 - (1 - Sqrt[3])*x^2)/(1 - (1 + Sqrt[3])*x^2)]
, (2 + Sqrt[3])/4])/(2*Sqrt[-((x^2*(1 - x^2))/(1 - (1 + Sqrt[3])*x^2)^2)]*Sqrt[-1 + x^6]) + (3^(3/4)*(1 - Sqrt
[3])*x*(1 - x^2)*Sqrt[(1 + x^2 + x^4)/(1 - (1 + Sqrt[3])*x^2)^2]*EllipticF[ArcCos[(1 - (1 - Sqrt[3])*x^2)/(1 -
(1 + Sqrt[3])*x^2)], (2 + Sqrt[3])/4])/(4*Sqrt[-((x^2*(1 - x^2))/(1 - (1 + Sqrt[3])*x^2)^2)]*Sqrt[-1 + x^6])
+ Defer[Int][Sqrt[-1 + x^6]/(-1 + x^2 + x^6), x] + 3*Defer[Int][(x^4*Sqrt[-1 + x^6])/(-1 + x^2 + x^6), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {\sqrt {-1+x^6}}{x^2}+\frac {\left (1+3 x^4\right ) \sqrt {-1+x^6}}{-1+x^2+x^6}\right ) \, dx \\ & = -\int \frac {\sqrt {-1+x^6}}{x^2} \, dx+\int \frac {\left (1+3 x^4\right ) \sqrt {-1+x^6}}{-1+x^2+x^6} \, dx \\ & = \frac {\sqrt {-1+x^6}}{x}-3 \int \frac {x^4}{\sqrt {-1+x^6}} \, dx+\int \left (\frac {\sqrt {-1+x^6}}{-1+x^2+x^6}+\frac {3 x^4 \sqrt {-1+x^6}}{-1+x^2+x^6}\right ) \, dx \\ & = \frac {\sqrt {-1+x^6}}{x}+\frac {3}{2} \int \frac {-1+\sqrt {3}-2 x^4}{\sqrt {-1+x^6}} \, dx+3 \int \frac {x^4 \sqrt {-1+x^6}}{-1+x^2+x^6} \, dx+\frac {1}{2} \left (3 \left (1-\sqrt {3}\right )\right ) \int \frac {1}{\sqrt {-1+x^6}} \, dx+\int \frac {\sqrt {-1+x^6}}{-1+x^2+x^6} \, dx \\ & = \frac {\sqrt {-1+x^6}}{x}+\frac {3 \left (1+\sqrt {3}\right ) x \sqrt {-1+x^6}}{2 \left (1-\left (1+\sqrt {3}\right ) x^2\right )}+\frac {3 \sqrt [4]{3} x \left (1-x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1-\left (1+\sqrt {3}\right ) x^2\right )^2}} E\left (\arccos \left (\frac {1-\left (1-\sqrt {3}\right ) x^2}{1-\left (1+\sqrt {3}\right ) x^2}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2 \sqrt {-\frac {x^2 \left (1-x^2\right )}{\left (1-\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {-1+x^6}}+\frac {3^{3/4} \left (1-\sqrt {3}\right ) x \left (1-x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1-\left (1+\sqrt {3}\right ) x^2\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {1-\left (1-\sqrt {3}\right ) x^2}{1-\left (1+\sqrt {3}\right ) x^2}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt {-\frac {x^2 \left (1-x^2\right )}{\left (1-\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {-1+x^6}}+3 \int \frac {x^4 \sqrt {-1+x^6}}{-1+x^2+x^6} \, dx+\int \frac {\sqrt {-1+x^6}}{-1+x^2+x^6} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 2.99 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-1+x^6} \left (1+2 x^6\right )}{x^2 \left (-1+x^2+x^6\right )} \, dx=\frac {\sqrt {-1+x^6}}{x}+\arctan \left (\frac {x}{\sqrt {-1+x^6}}\right ) \]

[In]

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

[Out]

Sqrt[-1 + x^6]/x + ArcTan[x/Sqrt[-1 + x^6]]

Maple [A] (verified)

Time = 2.09 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08

method	result	size
pseudoelliptic	\(\frac {-\arctan \left (\frac {\sqrt {x^{6}-1}}{x}\right ) x +\sqrt {x^{6}-1}}{x}\)	\(28\)
trager	\(\frac {\sqrt {x^{6}-1}}{x}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{6}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 x \sqrt {x^{6}-1}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{6}+x^{2}-1}\right )}{2}\)	\(73\)
risch	\(\frac {\sqrt {x^{6}-1}}{x}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{6}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}-2 x \sqrt {x^{6}-1}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{6}+x^{2}-1}\right )}{2}\)	\(73\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.50 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.54 \[ \int \frac {\sqrt {-1+x^6} \left (1+2 x^6\right )}{x^2 \left (-1+x^2+x^6\right )} \, dx=\frac {x \arctan \left (\frac {2 \, \sqrt {x^{6} - 1} x}{x^{6} - x^{2} - 1}\right ) + 2 \, \sqrt {x^{6} - 1}}{2 \, x} \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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