\(\int \frac {x (2+x^6)}{\sqrt {-1+x^6} (-1-x^4+x^6)} \, dx\) [113]

Optimal result

Integrand size = 28, antiderivative size = 16 \[ \int \frac {x \left (2+x^6\right )}{\sqrt {-1+x^6} \left (-1-x^4+x^6\right )} \, dx=-\text {arctanh}\left (\frac {x^2}{\sqrt {-1+x^6}}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

-((Sqrt[2 - Sqrt[3]]*(1 - x^2)*Sqrt[(1 + x^2 + x^4)/(1 - Sqrt[3] - x^2)^2]*EllipticF[ArcSin[(1 + Sqrt[3] - x^2
)/(1 - Sqrt[3] - x^2)], -7 + 4*Sqrt[3]])/(3^(1/4)*Sqrt[-((1 - x^2)/(1 - Sqrt[3] - x^2)^2)]*Sqrt[-1 + x^6])) +
(3*Defer[Subst][Defer[Int][1/(Sqrt[-1 + x^3]*(-1 - x^2 + x^3)), x], x, x^2])/2 + Defer[Subst][Defer[Int][x^2/(
Sqrt[-1 + x^3]*(-1 - x^2 + x^3)), x], x, x^2]/2

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {2+x^3}{\sqrt {-1+x^3} \left (-1-x^2+x^3\right )} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{\sqrt {-1+x^3}}+\frac {3+x^2}{\sqrt {-1+x^3} \left (-1-x^2+x^3\right )}\right ) \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^3}} \, dx,x,x^2\right )+\frac {1}{2} \text {Subst}\left (\int \frac {3+x^2}{\sqrt {-1+x^3} \left (-1-x^2+x^3\right )} \, dx,x,x^2\right ) \\ & = -\frac {\sqrt {2-\sqrt {3}} \left (1-x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1-\sqrt {3}-x^2\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x^2}{1-\sqrt {3}-x^2}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x^2}{\left (1-\sqrt {3}-x^2\right )^2}} \sqrt {-1+x^6}}+\frac {1}{2} \text {Subst}\left (\int \left (\frac {3}{\sqrt {-1+x^3} \left (-1-x^2+x^3\right )}+\frac {x^2}{\sqrt {-1+x^3} \left (-1-x^2+x^3\right )}\right ) \, dx,x,x^2\right ) \\ & = -\frac {\sqrt {2-\sqrt {3}} \left (1-x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1-\sqrt {3}-x^2\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x^2}{1-\sqrt {3}-x^2}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x^2}{\left (1-\sqrt {3}-x^2\right )^2}} \sqrt {-1+x^6}}+\frac {1}{2} \text {Subst}\left (\int \frac {x^2}{\sqrt {-1+x^3} \left (-1-x^2+x^3\right )} \, dx,x,x^2\right )+\frac {3}{2} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^3} \left (-1-x^2+x^3\right )} \, dx,x,x^2\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 10.80 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {x \left (2+x^6\right )}{\sqrt {-1+x^6} \left (-1-x^4+x^6\right )} \, dx=-\text {arctanh}\left (\frac {x^2}{\sqrt {-1+x^6}}\right ) \]

[In]

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

[Out]

-ArcTanh[x^2/Sqrt[-1 + x^6]]

Maple [A] (verified)

Time = 1.46 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94

method	result	size
pseudoelliptic	\(-\operatorname {arctanh}\left (\frac {\sqrt {x^{6}-1}}{x^{2}}\right )\)	\(15\)
trager	\(\frac {\ln \left (-\frac {-x^{6}-x^{4}+2 x^{2} \sqrt {x^{6}-1}+1}{x^{6}-x^{4}-1}\right )}{2}\)	\(42\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (14) = 28\).

Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 2.25 \[ \int \frac {x \left (2+x^6\right )}{\sqrt {-1+x^6} \left (-1-x^4+x^6\right )} \, dx=\frac {1}{2} \, \log \left (\frac {x^{6} + x^{4} - 2 \, \sqrt {x^{6} - 1} x^{2} - 1}{x^{6} - x^{4} - 1}\right ) \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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