\(\int \frac {1+2 x^6}{\sqrt {-1+x^6} (-1+x^2+x^6)} \, dx\) [64]

Optimal result

Integrand size = 27, antiderivative size = 14 \[ \int \frac {1+2 x^6}{\sqrt {-1+x^6} \left (-1+x^2+x^6\right )} \, dx=-\arctan \left (\frac {x}{\sqrt {-1+x^6}}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 3.10 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1+2 x^6}{\sqrt {-1+x^6} \left (-1+x^2+x^6\right )} \, dx=-\arctan \left (\frac {x}{\sqrt {-1+x^6}}\right ) \]

[In]

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

[Out]

-ArcTan[x/Sqrt[-1 + x^6]]

Maple [A] (verified)

Time = 2.70 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93

method	result	size
pseudoelliptic	\(\arctan \left (\frac {\sqrt {x^{6}-1}}{x}\right )\)	\(13\)
trager	\(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{6}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 x \sqrt {x^{6}-1}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{6}+x^{2}-1}\right )}{2}\)	\(62\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (12) = 24\).

Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.79 \[ \int \frac {1+2 x^6}{\sqrt {-1+x^6} \left (-1+x^2+x^6\right )} \, dx=-\frac {1}{2} \, \arctan \left (\frac {2 \, \sqrt {x^{6} - 1} x}{x^{6} - x^{2} - 1}\right ) \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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